% !TEX program = pdflatex %===================================================================== % landmark_IV.tex % % Single-file manuscript for pdfLaTeX. % Compile twice: % pdflatex -interaction=nonstopmode -halt-on-error landmark_IV.tex % pdflatex -interaction=nonstopmode -halt-on-error landmark_IV.tex % % No external figures, bibliography database, shell escape, or companion % auxiliary files are required. % % Verification status: % This source has not been compiled in this interface. % No checked page count, mechanized proof, or independent review is claimed. % % Mathematical scope: % Three separate represented models are treated. % No equality of their Hilbert spaces, response maps, or generators is % inferred from their common binary-instrument pattern. % % Corrections incorporated: % - Model 3 measures a sourced curvature pairing, not a first-order % inhomogeneous scalar correction. % - The two classical Newton--wave prescriptions are compared explicitly. % Their difference is a residual pure gauge; they are not identical maps. % - Model 2's spectral parameter removes gravitational IR and UV masks % and a scalar UV mask. % - The number-preserving Model 2 kernel need not be Hilbert--Schmidt. % - A densely defined symmetric operator is closable; closability is not % listed as an unresolved problem. % - The quadratic metric form in Model 3 is distinguished from an % unregulated metric operator field and from a cubic scalar correction. % - All measured Model 2 and Model 3 dynamics retain a number cutoff. %===================================================================== \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{tikz} \usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds,matrix} % Blue gravitational, green scalar, purple operators and instruments, % orange errors and restrictions, gray support and scope. \definecolor{mblue}{HTML}{1D4E89} \definecolor{sgreen}{HTML}{1F7A4D} \definecolor{rpurple}{HTML}{5B4B8A} \definecolor{corange}{HTML}{C05621} \definecolor{sgray}{HTML}{6B6560} \tikzset{ mbox/.style={draw=mblue,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=mblue!8}, sbox/.style={draw=sgreen,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=sgreen!10}, pbox/.style={draw=rpurple,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=rpurple!8}, cbox/.style={draw=corange,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=corange!10}, gbox/.style={draw=sgray,dashed,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=sgray!8}, ibox/.style={draw,rounded corners=2pt,align=center,font=\footnotesize,inner sep=3.5pt}, arr/.style={-{Stealth[length=2mm]},semithick}, arrg/.style={-{Stealth[length=2mm]},semithick,draw=sgray,dashed}, arro/.style={-{Stealth[length=2mm]},semithick,draw=corange} } \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Id}{I} \newcommand{\dd}{\,\mathrm d} \newcommand{\HH}{\mathcal H} \newcommand{\hh}{\mathfrak h} \newcommand{\DD}{\mathcal D} \newcommand{\QQ}{\mathscr Q} \newcommand{\Sch}{\mathcal S} \newcommand{\NN}{\mathsf N} \newcommand{\TT}{\mathrm{TT}} \newcommand{\HS}{\mathrm{HS}} \newcommand{\NW}{\mathrm{NW}} \newcommand{\op}{\mathrm{op}} \newcommand{\ret}{\mathrm{ret}} \newcommand{\Coh}{\operatorname{Coh}} \newcommand{\dG}{\mathrm d\Gamma} \newcommand{\Dom}{\operatorname{Dom}} \newcommand{\Ran}{\operatorname{Ran}} \newcommand{\Span}{\operatorname{span}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\Rea}{\operatorname{Re}} \newcommand{\Ima}{\operatorname{Im}} \newcommand{\cB}{\mathcal B} \newcommand{\cR}{\mathcal R} \newcommand{\cT}{\mathcal T} \newcommand{\cJ}{\mathcal J} \newcommand{\cI}{\mathcal I} \newcommand{\Lie}{\mathcal L} \newcommand{\flip}{\mathsf F} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\ip}[2]{\left\langle #1,#2\right\rangle} \newcommand{\br}[1]{\langle #1\rangle} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\proj}[1]{\ket{#1}\bra{#1}} \newcommand{\scopebox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \title{\textbf{Represented records for three named scalar-field models}\\[3pt] \large Finite-plane corrections, dense-TT corrections, and sourced curvature pairings} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We construct represented binary measurements in three separate models. On a fixed nondegenerate transverse-traceless plane, a first-order inhomogeneous scalar correction becomes a quadratic oscillator--field operator. It is essentially self-adjoint, a smooth initial-time-edge regulator can be removed, and two gravitational coherent preparations with the same scalar factor yield a positive Ramsey contrast. On an infinite-dimensional vacuum transverse-traceless completion, the analogous bilinear functional has explicit four-leg kernels. Their spectral masks can be removed in operator norm after a fixed particle-number compression, with separate Hilbert--Schmidt and operator-norm coefficient estimates and a controlled record remainder. Independently, a specified Newton--wave response to normal-ordered massive scalar stress defines a metric-valued form on scalar Fock space. A compact exterior curvature pairing gives a positive vacuum--coherent record at fixed scalar momentum cutoff and sufficiently large finite number cutoff. Every record theorem includes an exact zero-interpolation control. The three representations, sources, and cutoffs are not identified. The measured dense-TT and sourced models retain their number cutoffs, and all apparatuses are global controlled-unitary instruments. \end{abstract} \begin{center} \scriptsize \renewcommand{\arraystretch}{1.2} \begin{tabular}{@{}p{.18\linewidth}p{.20\linewidth} p{.45\linewidth}p{.08\linewidth}@{}} \toprule Model & Operator class & Cutoffs in the theorem & \(\lambda=0\)\\ \midrule Finite TT plane & ESA quadratic & Edge \(\varepsilon\) removable; plane fixed & Exact\\ Dense TT & Bounded \(A_{R,M}\) & \(R,M\) named; \(R\to\infty\) at fixed \(M\) & Exact\\ \(k=0\) stress form & Bounded \(A_{\Lambda,M}\) & \(\Lambda,M\) named & Exact\\ \bottomrule \end{tabular} \end{center} \noindent Classical sourced \(j\) exists. It is not an argument of \(A^{(1)}\) or \(A^{(2)}\), and it is not the cubic \(S_{t_0}(j)\phi\). \tableofcontents %===================================================================== \section{Problem, three models, and cutoff grammar} \label{sec:problem} %===================================================================== The question is whether a specified classical functional admits a complete chain \[ \text{classical functional} \longrightarrow \text{form or operator} \longrightarrow \text{instrument} \longrightarrow \text{record}. \] Each arrow requires its own definition and proof. A classical gauge-invariant expression does not automatically define an operator. A symmetric form cannot automatically be exponentiated. A unitary system channel does not by itself specify its controlled realization. We answer this question in three named models. They are three models, not three approximations to one operator. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.19\linewidth}XX@{}} \toprule Model & Measured quantity & Representation\\ \midrule 1: finite vacuum TT plane & A first-order inhomogeneous scalar correction & \(L^2(\R)\otimes\Gamma_s(L^2(\R^3))\).\\ \addlinespace 2: dense vacuum TT & The same type of bilinear correction on an infinite-dimensional TT space & \(\Gamma_s(\hh_g)\otimes\Gamma_s(\hh_\phi)\).\\ \addlinespace 3: sourced stress at \(k=0\) & A curvature pairing of a prescribed linear response to scalar stress & \(\Gamma_s(L^2(\R^3))\), without a graviton factor.\\ \bottomrule \end{tabularx} \end{center} Figure~\ref{fig:three-models} separates the three classical quantities, representations, generators, and record endpoints. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries,align=center] (title) {Common measurement pattern; different models.}; \matrix (M) [matrix of nodes, below=2.5mm of title, nodes={text width=2.72cm, align=center, anchor=center, font=\scriptsize}, column sep=1.4mm, row sep=1.5mm] { |[gbox]| {lane} & |[gbox]| {classical quantity} & |[gbox]| {representation} & |[gbox]| {generator} & |[gbox]| {record} \\ |[ibox]| {Model 1} & |[mbox]| {$\ell_{0,f}+q\ell_{e,f}+\pi\ell_{r,f}$, formal coefficient bookkeeping} & |[pbox]| {$\HH_1=L^2(\R)\otimes\Gamma_s(L^2(\R^3))$} & |[pbox]| {essentially self-adjoint numerical quadratic polynomial} & |[pbox]| {$c_1-e_\varepsilon$} \\ |[ibox]| {Model 2} & |[mbox]| {vacuum-TT bilinear $\cB_f(k,\phi)$} & |[pbox]| {$\HH_2=\Gamma_s(\hh_g)\otimes\Gamma_s(\hh_\phi)$} & |[pbox]| {bounded number-compressed generator} & |[pbox]| {$c_2-e_M-e_R(M)$} \\ |[ibox]| {Model 3} & |[sbox]| {$\int J\,\cR^{\NW}T(\phi)$} & |[sbox]| {scalar Fock space only} & |[pbox]| {bounded compression of a quadratic scalar response} & |[pbox]| {$c_3-e_M^{(3)}$} \\ }; \node[gbox, below=3mm of M, text width=14.2cm] {Model 3 measures sourced curvature, not the first-order scalar correction.}; \end{tikzpicture} \caption{Three distinct represented models. Models 1 and 2 represent vacuum-gravity scalar corrections, whereas Model 3 represents a curvature pairing of a prescribed scalar-stress response. The common binary-instrument construction does not identify their Hilbert spaces, generators, or classical sources.} \label{fig:three-models} \end{figure} The third quantity is not the first-order scalar correction of the first two models. It is a quadratic scalar composite obtained by reconstructing a sourced metric coefficient and smearing its linearized curvature. Its independent inclusion is the reason for the broader title. The numerical parameter \(\lambda\in\R\) belongs to each specified measurement generator. It interpolates between that generator and its declared control. It is not identified with Newton's constant, nor with evaluation of a nilpotent formal parameter at a real number. The distinction between an identified measurement semantics and a represented implementation is also present in the Enter calculus \cite{EnterCalculus}. We use no measurement-uniqueness theorem here. In particular, the infinite-dimensional operators below are not asserted to be primitives of a finite-qubit language. \subsection{Cutoff grammar} The symbols have different meanings: \begin{description}[style=nextline,leftmargin=1.7em] \item[\(\varepsilon\), Model 1.] A smooth replacement of a sharp initial-time edge in a scalar smearing. It is removed at the unitary and record levels. \item[\(R\), Model 2.] A simultaneous gravitational infrared and ultraviolet mask and a scalar ultraviolet mask. It is removed at fixed number cutoff \(M\). \item[\(M\), Models 2 and 3.] A particle-number projection on the Hilbert space of that model. It remains finite in every measured evolution considered here. Statements uniform for sufficiently large \(M\) do not assert an instrument limit as \(M\to\infty\). \item[\(\Lambda\), Model 3.] A scalar momentum cutoff. It is retained in its measurement theorem. A classical or coherent expectation limit is used only to select one sufficiently large finite value. \item[\(N\), Model 2.] A finite-rank projection on gravitational one-particle kernel legs. It is not a particle-number cutoff. \end{description} \scopebox{Scope}{% No theorem constructs a full reduced Einstein--scalar phase space, a sourced graviton representation, an unregulated cubic scalar correction, or a compact system--probe field coupling. Models 1 and 2 remain vacuum gravitational models. Model 3 has no independent quantum metric degree of freedom. The controlled-unitary apparatus is admitted globally in each model.} Figure~\ref{fig:cutoff-grammar} records which regulators are removed and which remain in the measurement theorems. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries] (title) {Cutoff ledger}; \matrix (M) [matrix of nodes, below=2mm of title, nodes={text width=3.55cm, align=center, anchor=center, font=\scriptsize}, column sep=1.4mm, row sep=1.3mm] { |[gbox]| {parameter} & |[gbox]| {meaning} & |[gbox]| {status} \\ |[cbox]| {Model 1: $\varepsilon$} & |[sbox]| {initial-time-edge smoothing} & |[pbox]| {removable} \\ |[mbox]| {Model 1: fixed TT plane} & |[mbox]| {part of the model} & |[gbox]| {not a regulator} \\ |[cbox]| {Model 2: $R$} & |[mbox]| {gravitational IR and UV masks, plus a scalar UV mask} & |[pbox]| {removed at fixed finite $M$} \\ |[cbox]| {Model 2: $M$} & |[pbox]| {total-number compression of the mixed quadratic} & |[cbox]| {retained} \\ |[gbox]| {Model 2: $N$} & |[mbox]| {finite-rank gravitational one-particle projection} & |[gbox]| {not particle-number compression} \\ |[cbox]| {Model 3: $\Lambda$} & |[sbox]| {retained scalar momentum cutoff} & |[cbox]| {expectation limit only} \\ |[cbox]| {Model 3: $M$} & |[pbox]| {retained finite number cutoff} & |[cbox]| {remains in the instrument} \\ }; \node[pbox, below=3mm of M.west, anchor=west, text width=6.4cm] (rlim) {$R\to\infty$ at fixed finite $M$, and joint $R,N$ approximation at fixed $M$}; \node[cbox, right=3mm of rlim, text width=5.6cm] (stop) {STOP. Measured $M\to\infty$: not asserted.}; \node[sbox, below=2mm of rlim, text width=6.4cm] (lam) {$\Lambda\to\infty$ selects one finite $\Lambda$. Expectation only, not an instrument limit.}; \node[gbox, below=2mm of stop, text width=5.6cm] {$\lambda$ is numerical interpolation. $\kappa^2=0$ is formal bookkeeping.}; \end{tikzpicture} \caption{Cutoff grammar. The parameters regulate different objects and have different limit statements. Model 1 removes the edge regulator. Model 2 removes spectral masks at fixed number cutoff. Model 3 retains momentum and number cutoffs; its momentum-limit argument concerns coherent expectations only. Finite-rank projection is not particle-number compression.} \label{fig:cutoff-grammar} \end{figure} \subsection{The three principal endpoints} Theorem~\ref{thm:model1} gives the Model 1 record, including removal of \(\varepsilon\) and exact equality at \(\lambda=0\). Theorem~\ref{thm:model2} gives the Model 2 fixed-\(M\) spectral limit and uniformly calibrated contrast. Theorem~\ref{thm:model3} gives the Model 3 record at fixed \(\Lambda\) and finite \(M\), with both interpolation-off and source-off controls. The classical constructions needed to identify the functionals are included in their stated domains. No result about a larger equation space is imported. %===================================================================== \section{Shared conventions and elementary measurement facts} \label{sec:shared} %===================================================================== We use \(\hbar=1\), signature \((-1,1,1,1)\), and \[ \Box=-\partial_t^2+\Delta,\qquad P=\Box-m^2,\qquad \Omega_m=\sqrt{-\Delta+m^2},\qquad m>0. \] A fixed reference time is denoted by \(t_0\). The spatial Fourier transform is \[ \widehat u(p)=(2\pi)^{-3/2} \int e^{-ip\cdot x}u(x)\dd x. \] Inner products are antilinear in their first entry. Tensor contraction is included in inner products of tensor-valued functions. The scalar Fock convention is \[ [a(u),a^\dagger(v)]=\ip uv\,I,\qquad \Phi(u)=a(u)+a^\dagger(u), \] so \(\Phi\) is real linear and \[ [\Phi(u),\Phi(v)]=2i\Ima\ip uv\,I. \] The normalized coherent vector is \[ \Coh(z)=e^{-\norm z^2/2} \bigoplus_{n=0}^{\infty}\frac{z^{\otimes n}}{\sqrt{n!}}. \] Its number distribution is Poisson with mean \(\norm z^2\). \subsection{The fixed-time scalar map} For a smooth symmetric tensor \(h\), let \[ D_\nu(h)=\partial^\mu h_{\mu\nu} -\frac12\partial_\nu h,\qquad h=\eta^{\mu\nu}h_{\mu\nu}, \] and define \begin{equation} L_h\phi =-h^{\mu\nu}\partial_\mu\partial_\nu\phi -D^\nu(h)\partial_\nu\phi. \label{eq:L} \end{equation} For \((K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu\) and \(X_\xi=\xi^\mu\partial_\mu\), \begin{equation} D(K\xi)=\Box\xi^\flat,\qquad L_{K\xi}=-[P,X_\xi]. \label{eq:gauge-identities} \end{equation} These identities follow by differentiating the displayed formulas; the second also follows by differentiating scalar covariance under a diffeomorphism. For a free scalar \(P\phi=0\), set \begin{equation} S_{t_0}(h)\phi(t) =\int_{t_0}^{t} \frac{\sin((t-s)\Omega_m)}{\Omega_m}L_h\phi(s)\dd s. \label{eq:S} \end{equation} It solves \[ PS_{t_0}(h)\phi=-L_h\phi,\qquad (S_{t_0}(h)\phi,\partial_tS_{t_0}(h)\phi)|_{t_0}=(0,0). \] \begin{lemma}[Finite-slab Cauchy estimate] \label{lem:wave} For \(b\in L^1(I;H^s(\R^3))\), \(s\ge0\), and a compact time interval \(I\) containing \(t_0\), the zero-data solution of \((\partial_t^2-\Delta+m^2)u=b\) is the sine integral in \eqref{eq:S} with \(b\) in place of \(L_h\phi\), and \[ \sup_{t\in I}\left( \norm{u(t)}_{H^{s+1}}+\norm{\partial_tu(t)}_{H^s}\right) \le C_m\int_I\norm{b(s)}_{H^s}\dd s. \] It is unique in this class. Smooth data and source give a smooth solution. Propagation has speed at most one. \end{lemma} \begin{proof} The Fourier multipliers are \(\sin((t-s)\omega_p)/\omega_p\) and \(\cos((t-s)\omega_p)\), with \(\omega_p=\sqrt{|p|^2+m^2}\). Their weighted bounds give the estimate and differentiation proves the equation. Uniqueness follows from the homogeneous energy identity. For finite propagation integrate \[ \partial_t\frac{|\partial_tu|^2+|\nabla u|^2+m^2|u|^2}{2} -\nabla\cdot(\partial_tu\,\nabla u)=b\,\partial_tu \] over a truncated causal cone. In a source-free cone the outward characteristic flux is nonnegative; zero initial energy implies zero solution there. Smoothness follows by applying the estimates at all orders. \end{proof} If \(\phi\) has compact smooth data, its support is spatially bounded on every finite slab. Thus \eqref{eq:S} is well defined for every smooth \(h\), even one with a Coulomb tail: \(L_h\) does not enlarge the support of \(\phi\). \subsection{One Ramsey pattern, different generators} For any self-adjoint \(A\), real \(\tau,\theta\), put \begin{equation} U=e^{i\tau A},\qquad K_\pm=\frac12(I\pm e^{-i\theta}U). \label{eq:Kraus} \end{equation} The associated Schr\"odinger operations are \[ \cI_\pm(\rho)=K_\pm\rho K_\pm^*. \] \begin{lemma}[Instrument, calibration, and remainder] \label{lem:Ramsey} The operations in \eqref{eq:Kraus} form a normal binary CP instrument, and \begin{equation} p_+(\Psi) =\frac12+\frac12\Rea\left( e^{-i\theta}\ip{\Psi}{e^{i\tau A}\Psi}\right). \label{eq:prob} \end{equation} For normalized \(\Psi\in\Dom A\), \begin{equation} \left|\ip{\Psi}{e^{i\tau A}\Psi} -1-i\tau\ip{\Psi}{A\Psi}\right| \le\frac{\tau^2}{2}\norm{A\Psi}^2. \label{eq:Taylor} \end{equation} For bounded self-adjoint \(A,B\), \[ \norm{e^{i\tau A}-e^{i\tau B}} \le|\tau|\norm{A-B}, \] and the change in a single probability at fixed \(\theta\) is at most half this norm. \end{lemma} \begin{proof} Expansion gives \(\sum_\pm K_\pm^*K_\pm=I\), proving normalization. Bounded Kraus maps are normal and CP. The same expansion proves \eqref{eq:prob}. Integrating the scalar inequality \(|e^{ix}-1-ix|\le x^2/2\) against the spectral measure gives \eqref{eq:Taylor}. The operator bound follows by differentiating \(e^{i(\tau-s)A}e^{isB}\) and integrating. Formula \eqref{eq:prob} gives the factor \(1/2\). \end{proof} The realization is a qubit initially in \(\ket+\), a controlled \(U\), and an equatorial qubit readout. This proves the claimed global dilation. It does not prove spacetime localization of the control. Figure~\ref{fig:Ramsey-instrument} gives the controlled-unitary realization of the common binary instrument. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[gbox, text width=12.4cm] at (3.1,2.35) {global admitted apparatus; no localization theorem}; \node[pbox] (plus) at (0,1.2) {$\ket+$}; \node[sbox] (rho) at (0,0) {$\rho$}; \draw (plus.east) -- (6.5,1.2); \draw (rho.east) -- (6.5,0); \node[pbox, minimum width=2.3cm] (U) at (2.6,0) {$U=e^{i\tau A}$}; \fill[rpurple] (2.6,1.2) circle (1.6pt); \draw[rpurple] (2.6,1.2) -- (U.north); \node[font=\scriptsize, above] at (2.6,1.2) {control}; \node[pbox] (meas) at (5.5,1.2) {$\ket{\pm_\theta}$}; \node[cbox, text width=3.3cm] at (5.5,-0.05) {$\theta=\pi/2$: imaginary-part readout}; \node[align=center, font=\footnotesize] at (2.5,-1.15) {$C_U=\ket0\bra0\otimes I+\ket1\bra1\otimes U$}; \node[pbox] at (2.5,-1.9) {$K_\pm=\tfrac12\bigl(I\pm e^{-i\theta}U\bigr)$}; \node[align=center, font=\footnotesize, text width=12cm] at (3.0,-2.75) {$p_+(\Psi)=\tfrac12+\tfrac12\Rea\!\left(e^{-i\theta}\ip{\Psi}{U\Psi}\right)$}; \end{tikzpicture} \caption{Global binary Ramsey instrument. An ancilla prepared in \(\ket+\), controlled \(U=e^{i\tau A}\), and equatorial readout produce the Kraus operators \(K_\pm=(I\pm e^{-i\theta}U)/2\). The construction supplies a normal CP instrument and global dilation, not a compact relativistic system--probe interaction.} \label{fig:Ramsey-instrument} \end{figure} \subsection{A self-contained quadratic-domain lemma} Let \(\mathcal F=\Gamma_s(\hh)\), let \(\NN\) be its number operator, and let \(\DD_{\mathrm{fin}}\) be the union of its finite number sectors. This includes arbitrary vectors in each sector. \begin{lemma}[Uniform finite-band estimate] \label{lem:analytic} Suppose a symmetric operator \(A\) preserves \(\DD_{\mathrm{fin}}\), changes particle number by at most two, and satisfies \[ \norm{A\Psi}\le C(n+2)\norm\Psi \quad\text{on }\Ran I_{\{\NN\le n\}}. \] Then \(A\) is essentially self-adjoint on \(\DD_{\mathrm{fin}}\). If also \(\norm{A\Psi}\le C\norm{(\NN+2)\Psi}\), its closure is defined on \(\Dom\NN\) and obeys that estimate there. \end{lemma} \begin{proof} Iteration gives \[ \norm{A^j\Psi} \le C^j\prod_{\ell=1}^j(n+2\ell)\norm\Psi \le(2C)^j\frac{(n+j)!}{n!}\norm\Psi. \label{eq:analytic-est} \] Thus the power series \(U(t)\Psi=\sum_j(it)^jA^j\Psi/j!\) converges for \(2C|t|<1\), with a radius independent of \(n\). We give the analytic-vector argument explicitly. In a smaller interval, the differentiated double series for \(\ip{U(t)\Psi}{U(t)\Chi}\) is absolutely convergent. Symmetry cancels its derivative term by term, so \(U(t)\) is an isometry on the dense finite-particle space and extends to an isometry on \(\mathcal F\). For \(2C(|s|+|t|)<1\), the double series for \(U(s)U(t)\Psi\) converges absolutely by the preceding bound and the binomial identity. It equals \(U(s+t)\Psi\). Hence \(U(t)\) is unitary for sufficiently small \(t\), with inverse \(U(-t)\). If \(A^*h=ih\), then symmetry and the series give \[ \ip h{U(t)\Psi}=e^t\ip h\Psi. \] Density implies \(U(t)^*h=e^th\), impossible for a nonzero \(h\) and a nonzero small real \(t\). The same argument excludes \(A^*h=-ih\). Therefore both deficiency spaces vanish. Indeed, the ranges of the closure of \(A\pm i\) are closed by \(\norm{(A\pm i)\Psi}^2=\norm{A\Psi}^2+\norm\Psi^2\); their orthogonal complements are those deficiency spaces. Surjectivity of both ranges proves self-adjointness of the closure. Finally, number truncations converge in the number graph norm. The assumed graph bound makes their \(A\)-images Cauchy. Closedness proves the last assertion. \end{proof} This lemma will be used to specify the uncompressed generator of Model 1. For Model 2 the measured dynamics will instead be defined only by its bounded compressions. Figure~\ref{fig:finite-band-domain} summarizes the finite-band analytic-vector argument used for the Model 1 generator. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries, align=center] at (4.6,3.55) {Model 1 domain argument}; \node[ibox, minimum width=1.15cm] (n0) at (0,2.7) {$0$}; \node[ibox, minimum width=1.15cm] (n1) at (0,2.05) {$1$}; \node[ibox, minimum width=1.15cm] (n2) at (0,1.4) {$2$}; \node[ibox, minimum width=1.15cm] (nn) at (0,0.75) {$n$}; \node[ibox, minimum width=1.15cm] (np) at (0,0.1) {$n+1$}; \node[ibox, minimum width=1.15cm] (npp) at (0,-0.55) {$n+2$}; \draw[arr, corange] (n0.east) -- ++(0.55,0) |- (n2.east); \node[cbox, font=\scriptsize, right=2mm of n1] {$|\Delta N|\le 2$}; \node[pbox, text width=7.6cm, anchor=west] at (3.3,2.55) {$\norm{A\Psi_{\le n}}\le C(n+2)\norm{\Psi_{\le n}}$}; \node[pbox, text width=7.6cm, anchor=west] at (3.3,1.55) {$\norm{A^j\Psi_{\le n}}\le(2C)^j\dfrac{(n+j)!}{n!}\norm{\Psi_{\le n}}$}; \node[sbox, anchor=west] (av) at (3.3,0.55) {analytic-vector series, $2C|t|<1$}; \node[pbox, right=2mm of av] (loc) {local unitary group}; \node[pbox, below=2mm of av] (def) {vanishing deficiency spaces}; \node[pbox, below=2mm of loc] (esa) {essential self-adjointness}; \draw[arr] (av)--(loc); \draw[arr] (av)--(def); \draw[arr] (loc)--(esa); \draw[arr] (def)--(esa); \node[gbox, text width=10.6cm, below=8mm of def] {$\norm{A\Psi}\le C\norm{(\NN+2)\Psi} \Longrightarrow \Dom\NN\subset\Dom\overline A$}; \node[cbox, text width=10.6cm, below=2mm of def, yshift=-11mm] {Not an uncompressed measured dynamics for Models 2 or 3.}; \end{tikzpicture} \caption{Finite-band domain argument. Linear growth of the number-sector bound and number changes of at most two yield analytic finite-particle vectors with a common convergence radius. The resulting local unitary construction eliminates the deficiency spaces. The number graph estimate also gives the stated domain inclusion for the closure.} \label{fig:finite-band-domain} \end{figure} %===================================================================== \section{Model 1: the fixed vacuum TT plane} \label{sec:model1} %===================================================================== \subsection{A nondegenerate plane and its admitted section} For \(p\ne0\), define \[ P_{ij}(p)=\delta_{ij}-\frac{p_ip_j}{|p|^2}, \] \begin{equation} \Pi_{ij,lm}(p) =\frac12\left( P_{il}P_{jm}+P_{im}P_{jl}-P_{ij}P_{lm}\right). \label{eq:TT} \end{equation} It is the orthogonal projector onto symmetric transverse traceless spatial tensors. Let \(p_*=(0,0,1)\), \(H=\operatorname{diag}(1,-1,0)\), and choose a nonnegative even smooth bump \(\chi\) supported sufficiently near \(p_*\) and \(-p_*\), away from zero. Put \[ \widehat A_{ij}(p)=\chi(p)(\Pi(p)H)_{ij}. \] Choose \(\chi\) nonzero where \(\Pi(p)H\ne0\). Then \(A\) is real Schwartz and TT, and \[ H^{ij}A_{ij}(0) =(2\pi)^{-3/2}\int\chi(p)|\Pi(p)H|^2\dd p>0. \] In particular \(A(0)\ne0\). Let \(D_0=\sqrt{-\Delta}\), \(M_A=\int A_{ij}A_{ij}\dd x\), and set \[ e_{ij}(t)=\cos((t-t_0)D_0)A_{ij},\qquad r_{ij}(t)=\frac1{M_A} \frac{\sin((t-t_0)D_0)}{D_0}A_{ij}, \qquad e_{0\mu}=r_{0\mu}=0. \] These waves are Schwartz on finite time slabs. \begin{proposition}[Plane and section] \label{prop:plane} With \[ \sigma_{\mathrm{plane}}(a,b) =\int(a_{ij}\partial_tb_{ij} -\partial_ta_{ij}b_{ij})\dd x, \] one has \(\sigma_{\mathrm{plane}}(e,r)=1\). Every \(k=qe+\pi r\) is a linearized vacuum solution, and no nonzero such \(k\) is pure gauge. For \[ \mathcal G=C_c^\infty(\R^{1,3};T\R^{1,3}),\qquad \mathcal E_h=\Span_\R\{e,r\}+K\mathcal G, \] the decomposition \(h=k+K\xi\) is unique, and the retarded harmonic map \[ w(h)=\bigl(G_{\ret}^{\Box}D(h)\bigr)^\sharp \] satisfies \(w(h)=\xi\) and \(h-Kw(h)=k\). \end{proposition} \begin{proof} The Cauchy data are \((e,\dot e)=(A,0)\) and \((r,\dot r)=(0,A/M_A)\), giving the symplectic pairing. TT wave evolution gives \(D(k)=0\), zero trace, and \(\Box k=0\); substitution into the linearized Einstein tensor gives vacuum. Linearized curvature is unchanged by \(K\xi\). For a temporal-gauge TT field, \(R^{(1)}_{0i0j}(k)=-\tfrac12\partial_t^2k_{ij}\). A pure gauge member of the plane would therefore satisfy \(\Delta k=0\). A harmonic Schwartz function vanishes, so \(k=0\). For the section, \eqref{eq:gauge-identities} gives \(D(h)=\Box\xi^\flat\). Since \(\xi\) is compactly supported, the retarded zero-past solution of that equation is \(\xi^\flat\). A retarded solution can be constructed using zero Cauchy data before the compact source; Lemma~\ref{lem:wave} with \(m=0\) gives existence and uniqueness. Uniqueness of \(k\) follows from the curvature argument, and a compactly supported Minkowski Killing field is zero. Thus the decomposition and the section are unique. \end{proof} The displayed symplectic normalization belongs to this plane model. It will not be silently substituted for the distinct normalization of the dense-TT model. Figure~\ref{fig:fixed-TT-plane} displays the canonical plane and the unique section on its admitted compact-gauge extension. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries] at (2.1,3.5) {(a) fixed canonical plane}; \draw[->, mblue] (0,0.4) -- (3.1,0.4) node[right] {$q$}; \draw[->, mblue] (0.3,0) -- (0.3,2.7) node[above] {$\pi$}; \node[mbox] at (2.15,0.95) {$e$}; \node[mbox] at (0.95,2.05) {$r$}; \node[align=center, font=\scriptsize] at (2.0,-0.35) {$k=qe+\pi r$,\quad $\sigma_{\mathrm{plane}}(e,r)=1$}; \node[mbox, text width=4.5cm, font=\scriptsize] at (2.0,-1.25) {$(e,\dot e)|_{t_0}=(A,0)$\\ $(r,\dot r)|_{t_0}=(0,A/M_A)$}; \node[gbox, text width=4.6cm, font=\scriptsize] at (2.0,-2.25) {fixed nondegenerate vacuum TT plane}; \node[font=\small\bfseries] at (8.6,3.5) {(b) admitted decomposition}; \node[mbox] (h) at (6.7,2.2) {$h=k+K\xi$}; \node[pbox] (w) at (8.8,2.2) {$w(h)=\xi$}; \node[mbox] (k) at (10.9,2.2) {$h-Kw(h)=k$}; \draw[arr] (h)--(w); \draw[arr] (w)--(k); \node[gbox, text width=5.2cm, font=\scriptsize] at (8.8,1.15) {$\xi\in C_c^\infty$,\ $D(k)=0$, unique on the declared domain}; \node[mbox, text width=5.2cm, font=\scriptsize] at (8.8,0.2) {$R^{(1)}_{0i0j}(k)=-\tfrac12\ddot k_{ij}$}; \node[gbox, text width=5.2cm, font=\scriptsize] at (8.8,-0.85) {A pure-gauge plane member would be a harmonic Schwartz field, hence zero.}; \node[cbox, text width=5.2cm, font=\scriptsize] at (8.8,-1.9) {Model choice, not a gravitational-sector cutoff.}; \end{tikzpicture} \caption{Fixed vacuum TT plane. The normalized waves \(e,r\) define one canonical gravitational degree of freedom. On \(\operatorname{Span}\{e,r\}+K\mathcal G\), the retarded harmonic map extracts the compact gauge parameter and returns the unique plane representative. The plane is a model choice, not a gravitational-sector cutoff.} \label{fig:fixed-TT-plane} \end{figure} \subsection{Scalar identification and formal coefficients} Let \(\phi_0\) be a free scalar with smooth compact initial data. Lemma~\ref{lem:wave} defines \(S_{t_0}(h)\phi_0\), with support in the causal evolution of those data on each side of \(t_0\). If \(H_{t_0}(u)\) denotes the homogeneous scalar solution with the Cauchy data of \(u\) at \(t_0\), then \begin{equation} S_{t_0}(K\xi)\phi_0 =X_\xi\phi_0-H_{t_0}(X_\xi\phi_0). \label{eq:Sgauge} \end{equation} Both sides have zero data and solve \(P\psi=P(X_\xi\phi_0)\), by \eqref{eq:gauge-identities}; uniqueness proves the equality. \begin{lemma}[No invisible plane direction] \label{lem:injective} If \(S_{t_0}(k)\phi_0=0\) for every compact-data free scalar and \(k\in\Span\{e,r\}\), then \(k=0\). \end{lemma} \begin{proof} The field equation gives \(k^{ij}\partial_i\partial_j\phi_0=0\) for every such solution. At any chosen time and point, arbitrary spatial Hessians can be prescribed using compact smooth position data there. Propagating those data to \(t_0\) preserves compactness. Therefore \(k\) pairs to zero with every symmetric matrix at every point. \end{proof} For completeness, the admitted first-order family consists of \[ (h,\phi_0,\phi_1) = \left(k+K\xi,\phi_0,S_{t_0}(k)\phi_0+X_\xi\phi_0\right). \] The complete coefficient expression \[ \int f\left[\phi_0+\kappa \bigl(\phi_1-X_{w(h)}\phi_0\bigr)\right]\dd^4x, \qquad \kappa^2=0, \] is unchanged by adding a compact gauge parameter, since both \(w(h)\) and \(\phi_1\) change by that parameter's prescribed action. On the selected section it is \begin{equation} F_f^{[1]} =\ell_{0,f}+\kappa(q\ell_{e,f}+\pi\ell_{r,f}), \quad \ell_{0,f}=\int f\phi_0,\quad \ell_{a,f}=\int fS_{t_0}(a)\phi_0. \label{eq:Fplane} \end{equation} No sourced second metric coefficient occurs here. \subsection{Three scalar one-particle vectors} Take real \(f\in C_c^\infty((t_0,T)\times\R^3)\), and define \begin{equation} v_f(s,x)=\int_s^T \left(\frac{\sin((t-s)\Omega_m)}{\Omega_m}f(t,\cdot)\right)(x) \dd t. \label{eq:vf} \end{equation} This sign convention is fixed throughout the paper: \((\partial_s^2+\Omega_m^2)v_f=f\), with zero terminal data. It differs by a minus sign from the advanced inverse of \(P\). Exchanging the finite-time integrals gives, for TT \(a\), \[ \ell_{a,f} =-\int_{t_0}^T v_fa^{ij}\partial_i\partial_j\phi_0\dd^4x. \] Consequently set \begin{equation} b_{a,f}=-\partial_i\partial_j(a^{ij}v_f),\qquad g_{a,f}=I_{\{t\ge t_0\}}b_{a,f},\qquad g_{0,f}=f. \label{eq:gplane} \end{equation} Then \(\ell_{a,f}=\int g_{a,f}\phi_0\). The spatial support on the finite slab is compact, all spatial derivatives are bounded, and the only possible spacetime nonsmoothness is the initial-time edge. For such a real \(g\), define \[ \alpha_g=\int_{t_0}^T \cos((t-t_0)\Omega_m)g(t)\dd t,\qquad \beta_g=\int_{t_0}^T \frac{\sin((t-t_0)\Omega_m)}{\Omega_m}g(t)\dd t. \] Spatial smoothness and finite propagation imply \(\alpha_g,\beta_g\in C_c^\infty(\R^3)\). If \((\varphi,\varpi)\) are the scalar data, \[ \ell_g=\int(\alpha_g\varphi+\beta_g\varpi)\dd x. \] Define \begin{equation} u_g(p)=\frac1{\sqrt{2\omega_p}} \int_{t_0}^T e^{i\omega_p(t-t_0)}\widehat g(t,p)\dd t =\frac{\widehat\alpha_g}{\sqrt{2\omega}} +i\sqrt{\frac{\omega}{2}}\widehat\beta_g. \label{eq:ug} \end{equation} Since \(m>0\), these vectors are Schwartz. Write \(u_{a,f}=u_{g_{a,f}}\). \begin{lemma}[Scalar symplectic correspondence] \label{lem:scalarCCR} For real \(g,h\), \[ 2\Ima\ip{u_g}{u_h} =\int(\alpha_g\beta_h-\beta_g\alpha_h)\dd x. \] Thus \(\ell_g\mapsto\Phi(u_g)\) has the canonical commutator for scalar initial linear functionals. \end{lemma} \begin{proof} Expand \eqref{eq:ug}. The two diagonal terms have real inner products for real data. Plancherel's identity identifies the imaginary cross terms with half the displayed expression. \end{proof} Declare the plane Poisson bracket by \(\{q,\pi\}=1\), the scalar bracket by Lemma~\ref{lem:scalarCCR}, and mixed brackets zero. This is a specified initial-data bracket, not the reduced bracket of a coupled Einstein--scalar theory. The coefficient quantization \[ \widehat F_f^{[1]} =\Phi(u_{0,f}) +\kappa\left(Q\Phi(u_{e,f})+\Pi_0\Phi(u_{r,f})\right) \] satisfies \begin{equation} [\widehat F_f^{[1]},\widehat F_h^{[1]}] =i\,\operatorname{Op}\{F_f^{[1]},F_h^{[1]}\} \pmod{\kappa^2}. \label{eq:formalCCR} \end{equation} Indeed, the surviving terms contain only one scalar CCR; brackets between two correction terms carry \(\kappa^2\). \subsection{The numerical self-adjoint generator} On \[ \HH_1=L^2(\R)\otimes\Gamma_s(L^2(\R^3)), \qquad Q\psi(q)=q\psi(q),\quad \Pi_0\psi(q)=-i\partial_q\psi(q), \] define initially on the total finite-particle domain \begin{equation} A_f^{(1)}(\lambda) =I\otimes\Phi(u_{0,f}) +\lambda\left( Q\otimes\Phi(u_{e,f})+\Pi_0\otimes\Phi(u_{r,f})\right). \label{eq:A1} \end{equation} The oscillator is identified with a single Fock mode: \(Q=(b+b^\dagger)/\sqrt2\), \(\Pi_0=-i(b-b^\dagger)/\sqrt2\). \begin{proposition}[Model 1 operator] \label{prop:A1} For every real \(\lambda\), \eqref{eq:A1} is essentially self-adjoint on the total finite-particle domain. Its closure is defined on \(\Dom\NN\). It acts only on the gravitational oscillator and \[ V_f=\Span_\C\{u_{0,f},u_{e,f},u_{r,f}\} \] in the scalar one-particle space, and is the identity on the orthogonal Fock factor. \end{proposition} \begin{proof} The standard sector estimates are \[ \norm{a(u)\Psi}\le\norm u\norm{\NN^{1/2}\Psi},\qquad \norm{a^\dagger(u)\Psi} \le\norm u\norm{(\NN+1)^{1/2}\Psi}. \] Products of two such operators obey a constant multiple of \(\norm{(\NN+2)\Psi}\). Expanding the two mixed terms in \eqref{eq:A1} gives a symmetric operator, changing number by at most two, with the estimates of Lemma~\ref{lem:analytic}. Its factors commute because the species differ. The lemma proves essential self-adjointness and the domain inclusion. Fock factorization over \(V_f\oplus V_f^\perp\) gives the active factor assertion first on finite-particle vectors and then for the unique self-adjoint closure. \end{proof} There is no evaluation homomorphism sending the nilpotent \(\kappa\) in \eqref{eq:Fplane} to a nonzero real \(\lambda\). Equation \eqref{eq:A1} is a separately defined numerical polynomial using the same coefficients. Its spectral functions are predictions of this named model, not higher-order predictions of a formal gravitational expansion. Figure~\ref{fig:scalar-to-generator} traces the coefficient construction while separating formal correspondence from numerical spectral calculus. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[sbox, text width=8.5cm] (ell) {$\ell_{a,f}=\int fS_{t_0}(a)\phi_0$}; \node[sbox, below=2.2mm of ell, text width=8.5cm] (vf) {$v_f(s)=\displaystyle\int_s^T\frac{\sin((t-s)\Omega_m)}{\Omega_m}f(t)\,dt$}; \node[sbox, below=2.2mm of vf, text width=8.5cm] (bg) {$b_{a,f}=-\partial_i\partial_j(a^{ij}v_f)$,\quad $g_{a,f}=I_{\{t\ge t_0\}}b_{a,f}$}; \node[sbox, below=2.2mm of bg] (ab) {Cauchy coefficients $(\alpha_g,\beta_g)$}; \node[sbox, below=2.2mm of ab, text width=8.5cm] (ug) {$u_g=\dfrac{\widehat\alpha_g}{\sqrt{2\omega}}+i\sqrt{\omega/2}\,\widehat\beta_g$}; \node[gbox, below=2.2mm of ug] (vfspan) {$V_f=\Span_\C\{u_0,u_e,u_r\}$}; \node[gbox, dashed, below left=4mm and 2mm of vfspan, text width=5.3cm] (formal) {Formal:\\ $\widehat F^{[1]}=\Phi(u_0)+\kappa[Q\Phi(u_e)+\Pi_0\Phi(u_r)]$,\\ $\kappa^2=0$}; \node[pbox, below right=4mm and 2mm of vfspan, text width=5.5cm] (num) {Numerical:\\ $A^{(1)}(\lambda)=\Phi(u_0)+\lambda[Q\Phi(u_e)+\Pi_0\Phi(u_r)]$}; \draw[arrg] (vfspan.south) -- (formal.north); \draw[arr] (vfspan.south) -- (num.north); \node[cbox, text width=8.2cm, below=12mm of vfspan] {same coefficients; no evaluation homomorphism $\kappa\mapsto\lambda\neq 0$}; \draw[arr] (ell)--(vf); \draw[arr] (vf)--(bg); \draw[arr] (bg)--(ab); \draw[arr] (ab)--(ug); \draw[arr] (ug)--(vfspan); \end{tikzpicture} \caption{Coefficient construction and numerical representation. The fixed-time response determines effective scalar smearings and three one-particle vectors. These coefficients enter both a first-order formal expression and a separately defined numerical self-adjoint polynomial. Numerical interpolation does not evaluate the nilpotent formal parameter.} \label{fig:scalar-to-generator} \end{figure} \subsection{Removing the initial-time edge} Choose \(0\le\vartheta\le1\), smooth, with \(\vartheta(s)=0\) for \(s\le0\) and \(1\) for \(s\ge1\). Set \[ g_{a,f,\varepsilon}(t,x) =\vartheta((t-t_0)/\varepsilon)b_{a,f}(t,x),\qquad a=e,r, \] and leave \(g_{0,f}\) unchanged. For \(0<\varepsilon\le1\), these are genuine compact smooth spacetime smearings. Put \[ B_{a,f}=\frac1{\sqrt{2m}} \sup_{t_0\le t\le t_0+1}\norm{b_{a,f}(t)}_2. \] Equation \eqref{eq:ug} and Minkowski's integral inequality give \begin{equation} \norm{u_{a,f,\varepsilon}-u_{a,f}}\le B_{a,f}\varepsilon. \label{eq:edge} \end{equation} Let \(A^{(1)}_{f,\varepsilon}(\lambda)\) use these regulated vectors. All these operators satisfy the preceding proof with a common constant \[ \overline C= 16\left[1+\norm{u_{0,f}} +|\lambda|(\norm{u_{e,f}}+\norm{u_{r,f}}+B_{e,f}+B_{r,f})\right] \] in the number estimates. Their difference has the same estimates with \[ D_\varepsilon= 16|\lambda|\sum_{a=e,r} \norm{u_{a,f,\varepsilon}-u_{a,f}} \le16|\lambda|(B_{e,f}+B_{r,f})\varepsilon. \] \begin{proposition}[Model 1 unitary edge limit] \label{prop:edge-limit} For every fixed real \(\tau,\lambda\), \[ e^{i\tau A^{(1)}_{f,\varepsilon}(\lambda)} \longrightarrow e^{i\tau A^{(1)}_f(\lambda)} \quad\text{strongly}. \] If \(2\overline C|\tau|<1\), then on particle number at most \(n\), \begin{equation} \norm{(U_\varepsilon-U)\Psi} \le \frac{2D_\varepsilon|\tau|(n+1)} {(1-2\overline C|\tau|)^{n+2}}\norm\Psi. \label{eq:edge-unitary} \end{equation} For a total coherent vector of mean number \(\mu\), an upper bound is \begin{equation} \mathcal E_\varepsilon(\mu;\tau) = \inf_{n\ge0}\left[ 2\left(e^{-\mu}\sum_{k>n}\frac{\mu^k}{k!}\right)^{1/2} +\frac{2D_\varepsilon|\tau|(n+1)} {(1-2\overline C|\tau|)^{n+2}}\right], \label{eq:Ecoh} \end{equation} which tends to zero. \end{proposition} \begin{proof} On finite-particle vectors use the convergent exponential series from Lemma~\ref{lem:analytic}. Telescoping the \(j\)-th powers gives the bound \[ jD_\varepsilon\overline C^{j-1} \prod_{\ell=1}^j(n+2\ell)\norm\Psi. \] Summing with coefficients \(|\tau|^j/j!\), and using \[ \sum_{j\ge1}j\binom{n+j}{j}x^{j-1} =\frac{n+1}{(1-x)^{n+2}}, \] proves \eqref{eq:edge-unitary}. Density and the norm bound two on a difference of unitaries give strong convergence for small times. For arbitrary \(\tau\), divide it into finitely many sufficiently small steps and use the group law and strong convergence of uniformly bounded products. For a coherent vector, truncate at number \(n\), use \eqref{eq:edge-unitary} on the truncation, and bound the two tail images by twice the tail norm. This gives \eqref{eq:Ecoh}. First choose \(n\) and then let \(\varepsilon\to0\). \end{proof} Figure~\ref{fig:edge-removal} shows the coefficient estimate and the truncation argument behind strong unitary edge removal. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries] at (1.8,3.3) {(a) temporal smoothing}; \draw[->, sgray] (0,0.2) -- (3.4,0.2) node[right, font=\scriptsize] {$t$}; \draw[sgray, dashed, thick] (0.3,0.5) -- (1.15,0.5) -- (1.15,1.7) -- (3.0,1.7); \draw[sgreen, thick] (0.3,0.5) .. controls (1.0,0.5) and (1.25,1.7) .. (1.9,1.7) -- (3.0,1.7); \node[font=\scriptsize, sgray] at (0.7,0.85) {$I_{\{t\ge t_0\}}$}; \node[font=\scriptsize, sgreen] at (2.35,1.35) {$\vartheta((t-t_0)/\varepsilon)$}; \node[gbox, font=\scriptsize, text width=3.3cm] at (1.7,-0.55) {transition $[t_0,t_0+\varepsilon]$\\ schematic multiplier profiles}; \node[font=\small\bfseries] at (7.2,3.3) {(b) estimates}; \node[sbox, text width=4.8cm, font=\scriptsize] at (7.2,2.35) {$\norm{u_{a,f,\varepsilon}-u_{a,f}}\le B_{a,f}\varepsilon$}; \node[pbox, text width=5.2cm, font=\scriptsize] at (7.2,1.15) {$\norm{(U_\varepsilon-U)\Psi_{\le n}} \le\dfrac{2D_\varepsilon|\tau|(n+1)}{(1-2\overline C|\tau|)^{n+2}}\norm{\Psi_{\le n}}$}; \node[cbox, font=\scriptsize] at (7.2,0.15) {condition $2\overline C|\tau|<1$}; \node[font=\small\bfseries] at (7.2,-0.7) {(c) coherent truncation}; \node[cbox, font=\scriptsize] at (5.5,-1.4) {Poisson tail}; \node[cbox, font=\scriptsize] at (8.8,-1.4) {finite-sector edge error}; \node[gbox, font=\scriptsize, text width=5.4cm] at (7.2,-2.25) {choose $n$, then remove $\varepsilon$\\ arbitrary fixed time: finite subdivision}; \node[pbox, font=\scriptsize] at (7.2,-3.15) {convergence is strong, not operator norm}; \node[gbox, font=\scriptsize, text width=5.4cm] at (7.2,-4.0) {No gravitational-sector limit is taken.}; \end{tikzpicture} \caption{Initial-time-edge removal in Model 1. Smoothing changes the scalar coefficients by \(O(\varepsilon)\). Finite-sector analytic estimates and coherent Poisson tails yield strong unitary convergence. The displayed estimate is a small-time bound; arbitrary fixed times are handled by subdivision. No gravitational-sector limit is taken.} \label{fig:edge-removal} \end{figure} %===================================================================== \section{Model 1: a fixed detector and its contrast} \label{sec:model1-record} %===================================================================== \subsection{An explicit positive detector coefficient} Let \(A_0=A(0)\), \(c_0=|A_0|^2>0\). Choose a real compact bump \(\eta_0=1\) near zero and take scalar data \[ \varphi_*(x)=-\frac12(A_0)_{ij}x^ix^j\eta_0(x), \qquad \varpi_*=0. \] Let \(\phi_*\) be the free solution and \(\psi_*=S_{t_0}(e)\phi_*\). Its zero correction data imply \[ \partial_t^2\psi_*(t_0,0)=L_e\phi_*(t_0,0)=c_0. \] Define finite constants on \(0\le s\le1,\ |x|\le1\): \[ M_t=\sup|\partial_t^3\psi_*(t_0+s,x)|,\qquad M_x=\sup|\nabla_x\partial_t^2\psi_*(t_0+s,x)|. \] Set \[ \delta=\min\left\{\frac12,\frac{c_0}{4(1+M_t)}\right\}, \qquad r_0=\min\left\{\frac12,\frac{c_0}{4(1+M_x)}\right\}. \] Then \[ \partial_t^2\psi_*(t_0+s,x)\ge c_0/2,\qquad \psi_*(t_0+s,x)\ge c_0s^2/4 \] for \(0\le s\le\delta,\ |x|\le r_0\). Choose nonnegative unit-integral bumps \(\beta_t\in C_c^\infty((1/2,1))\) and \(\beta_x\in C_c^\infty(B(0,1))\), and define \[ f_*(t,x)=\frac1{\delta r_0^3} \beta_t((t-t_0)/\delta)\beta_x(x/r_0). \] Consequently, \begin{equation} d_*=\ell_{e,f_*}(\phi_*) =\int f_*\psi_*\dd^4x \ge d_0:=\frac{c_0\delta^2}{16}>0. \label{eq:d1} \end{equation} The scalar coherent label for real data is \[ z_{\varphi,\varpi} =\sqrt{\frac{\omega}{2}}\widehat\varphi +\frac{i}{\sqrt{2\omega}}\widehat\varpi. \] Direct expansion of \eqref{eq:ug} gives \[ \ip{\Coh(z_{\varphi,\varpi})} {\Phi(u_g)\Coh(z_{\varphi,\varpi})} =2\Rea\ip{u_g}{z_{\varphi,\varpi}}=\ell_g(\varphi,\varpi). \] Put \(z_*=z_{\varphi_*,0}\), and choose \[ \Psi_0=\Coh(0)\otimes\Coh(z_*),\qquad \Psi_*=\Coh(1/\sqrt2)\otimes\Coh(z_*). \] Their gravitational covariances agree; their \(Q\)-means differ by one and their \(\Pi_0\)-means by zero. Hence \begin{equation} \ip{\Psi_*}{A^{(1)}_{f_*}(\lambda)\Psi_*} -\ip{\Psi_0}{A^{(1)}_{f_*}(\lambda)\Psi_0} =\lambda d_*. \label{eq:mean1} \end{equation} \begin{theorem}[Model 1 record: fixed plane and removable edge] \label{thm:model1} The fixed plane, detector, and states just constructed have the following property. For each fixed \(\lambda\ne0\), choose \[ \mu_0=\norm{z_*}^2,\qquad \mu_*=\mu_0+\frac12, \] \[ S_1=\overline C^{\,2} \sum_{s\in\{0,*\}}(\mu_s^2+5\mu_s+4), \] and fix \[ \theta=\pi/2,\qquad 0<\tau\le \min\left\{\frac1{8\overline C}, \frac{|\lambda|d_0}{S_1}\right\}. \] For the instruments of \(A^{(1)}_{f_*,\varepsilon}(\lambda)\), \[ \left|p_{+,\varepsilon}(\Psi_*) -p_{+,\varepsilon}(\Psi_0)\right| \ge c_1-e_\varepsilon, \qquad c_1=\frac{\tau|\lambda|d_0}{4}>0, \] where \[ e_\varepsilon= \frac12\left[ \mathcal E_\varepsilon(\mu_*;\tau) +\mathcal E_\varepsilon(\mu_0;\tau)\right]\longrightarrow0. \] At \(\varepsilon=0\) the contrast is at least \(c_1\). The plane is fixed; no gravitational-sector limit is involved. For \(\lambda=0\), the two probabilities agree exactly for every \(\varepsilon,\tau,\theta\). \end{theorem} \begin{proof} The graph bound and the Poisson moments give \[ \sum_{s\in\{0,*\}}\norm{A^{(1)}_{f_*}(\lambda)\Psi_s}^2 \le S_1. \] For \(\theta=\pi/2\), formula \eqref{eq:prob} reads the imaginary part of the unitary expectation. Equations \eqref{eq:Taylor} and \eqref{eq:mean1} give \[ |\Delta p|\ge\frac{\tau|\lambda|d_0}{2} -\frac{\tau^2S_1}{4}\ge c_1. \] Proposition~\ref{prop:edge-limit} and the probability norm bound in Lemma~\ref{lem:Ramsey} give the stated \(e_\varepsilon\). At \(\lambda=0\), both the regulated and unregulated generator act only on the scalar factor, which is identical in the two preparations. This proves exact equality. \end{proof} Figure~\ref{fig:model1-record} displays the fixed witness, preparation change, and exact interpolation-off control. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries] (wlab) {Witness}; \node[sbox, below=2mm of wlab, text width=4.6cm, font=\scriptsize] (phi) {$\varphi_*=-\tfrac12(A_0)_{ij}x^ix^j\eta_0$,\ $\varpi_*=0$}; \node[mbox, below=2mm of phi, text width=4.6cm, font=\scriptsize] (c0) {$\ddot\psi_*(t_0,0)=c_0=|A_0|^2>0$}; \node[gbox, below=2mm of c0, text width=4.6cm, font=\scriptsize] (d) {small future support of $f_*$\\ $d_*=\int f_*\psi_*\ge d_0>0$}; \node[font=\small\bfseries, right=8mm of wlab] (plab) {Preparation}; \node[pbox, below=2mm of plab, text width=5.2cm, font=\scriptsize] (psi) {$\Psi_0=\Coh(0)\otimes\Coh(z_*)$\\ $\Psi_*=\Coh(1/\sqrt2)\otimes\Coh(z_*)$}; \node[sbox, below=2mm of psi, text width=5.2cm, font=\scriptsize] {same scalar factor; equal oscillator covariances; $\Delta Q=1$, $\Delta\Pi_0=0$}; \node[font=\small\bfseries, right=8mm of plab] (rlab) {Record}; \node[pbox, below=2mm of rlab, text width=4.8cm, font=\scriptsize] (rec) {$\Delta\langle A^{(1)}\rangle=\lambda d_*$\\ $|\Delta p_\varepsilon|\ge c_1-e_\varepsilon$}; \node[sbox, below=2mm of rec, text width=4.8cm, font=\scriptsize] (off) {$\lambda=0\Rightarrow U=I_g\otimes U_\phi\Rightarrow\Delta p=0$}; \node[gbox, below=3mm of d, text width=14.8cm, font=\scriptsize] {This contrast does not exclude every classical stochastic model.}; \end{tikzpicture} \caption{Model 1 record certificate. A local Hessian choice produces a positive detector coefficient. Gravitational coherent displacement at fixed scalar preparation changes the generator mean by \(\lambda d_*\), yielding a calibrated positive Ramsey contrast after edge removal. At \(\lambda=0\), the two records agree exactly.} \label{fig:model1-record} \end{figure} \subsection{What is and is not detected} The contrast of Model 1 changes gravitational preparation at fixed scalar preparation. It vanishes when the mixed term is removed. It is not, by itself, a test excluding every classical stochastic description of that preparation. It is not a gravitational quadrature alone. By \eqref{eq:d1}, \(u_{e,f_*}\ne0\), and on the finite-particle domain \[ [A^{(1)}_{f_*}(\lambda),I\otimes\Phi(iu_{e,f_*})] =2i\Ima\ip{u_{0,f_*}}{iu_{e,f_*}}I +2i\lambda\norm{u_{e,f_*}}^2Q +2i\lambda\Ima\ip{u_{r,f_*}}{iu_{e,f_*}}\Pi_0. \] For \(\lambda\ne0\), this is nonzero because \(I,Q,\Pi_0\) are linearly independent. A gravitational-only operator would commute with every scalar quadrature. \subsection{Compensator and homogeneous-data dependence} Let \(C,C'\) be right inverses of \(K\) on the admitted compact gauge class, and put \(A_C=C'-C\). Then \(A_CK=0\), and, on the same underlying coefficient triple, \[ F_f^{C'}-F_f^C =-\kappa\int fX_{A_Ch}\phi_0\dd^4x. \] If \(A_Ce=\eta_e\) and \(A_Cr=\eta_r\) are smooth near \(\supp f\), integration by parts gives \[ F_f^{C'}-F_f^C =\kappa(q\ell_{d_e}+\pi\ell_{d_r}), \qquad d_a=\partial_\mu(f\eta_a^\mu). \] These compact smooth smearings have one-particle vectors and change the represented polynomial by \[ \kappa\left(Q\Phi(u_{d_e})+\Pi_0\Phi(u_{d_r})\right). \] They need not vanish. For example, \(C'(qe+\pi r+K\xi)=\xi+q\eta\) is another right inverse for any fixed smooth \(\eta\). Changing zero correction data by a homogeneous solution instead adds \(\kappa\int fH\phi_0\), a distinct change. No identification of these prescriptions follows from their right-inverse property. %===================================================================== \section{Model 2: dense vacuum TT form and the actual cutoffs} \label{sec:model2} %===================================================================== \subsection{One-particle spaces and normalization} Define \[ \hh_g= \left\{z\in L^2(\R^3,\dd p;\Sym^2\C^3): p_iz_{ij}=0,\ z_{ii}=0\ \text{a.e.}\right\}, \qquad \hh_\phi=L^2(\R^3,\dd q). \] The measure on \(\hh_g\) is \(\dd p\), not \(\dd p/(2|p|)\). Frequency factors are in the modes \[ u_p(t,x)= \frac{e^{-i|p|(t-t_0)+ip\cdot x}} {(2\pi)^{3/2}\sqrt{2|p|}},\qquad v_q(t,x)= \frac{e^{-iE_q(t-t_0)+iq\cdot x}} {(2\pi)^{3/2}\sqrt{2E_q}},\quad E_q=\sqrt{|q|^2+m^2}. \] The TT field and scalar field are represented by \[ \widehat k_{ij} =2\int(u_pa_{g,ij}(p)+\bar u_pa_{g,ij}^\dagger(p))\dd p, \qquad \widehat\phi =\int(v_qa_\phi(q)+\bar v_qa_\phi^\dagger(q))\dd q \] on \[ \HH_2=\Gamma_s(\hh_g)\otimes\Gamma_s(\hh_\phi). \] The tensor CCR contain \(\Pi_{ij,lm}(p)\). The factor two in the gravitational field fixes the classical symplectic convention \[ \sigma_g(k,k') =\frac14\int(k_{ij}\dot k'_{ij}-\dot k_{ij}k'_{ij})\dd x. \] For real TT data \((q,p)\), the coherent label is \[ z(\ell)=\frac12\left( \sqrt{\frac{|\ell|}{2}}\widehat q(\ell) +\frac{i}{\sqrt{2|\ell|}}\widehat p(\ell)\right), \] and Fourier inversion gives \[ \sigma_g(k_z,k_{z'})=2\Ima\ip z{z'}. \] This fixes Model 2's normalization independently of Model 1. A dense test space is \[ \mathfrak d_g= \{z\in C_c^\infty(\R^3\setminus\{0\};\Sym^2\C^3):\Pi z=z\}, \qquad \mathfrak d_\phi=C_c^\infty(\R^3). \] Density follows by cutting off away from zero, smoothing on annuli, and applying the smooth projection \(\Pi\). Let \(\DD_2\) be the algebraic finite-particle space generated by these labels. \subsection{The bilinear and its four legs} For TT waves and free scalars with data in \(H^2\times H^1\), \[ \cB_f(k,\phi)=\int fS_{t_0}(k)\phi =-\int_{t_0}^T v_fk_{ij}\partial_i\partial_j\phi\,\dd^4x. \label{eq:Bclassical} \] The exchange of integrations follows from the finite-slab estimates. Moreover, \begin{equation} |\cB_f(k,\phi)| \le C_{f,I,m} (\norm{k(t_0)}_{H^2}+\norm{\dot k(t_0)}_{H^1}) (\norm{\phi(t_0)}_{H^2}+\norm{\dot\phi(t_0)}_{H^1}). \label{eq:Bcontinuity} \end{equation} Indeed, \(H^2(\R^3)\subset L^\infty\), so \(\norm{k_{ij}\partial_i\partial_j\phi}_2 \le C\norm k_{H^2}\norm\phi_{H^2}\). Free evolution and Lemma~\ref{lem:wave} prove the estimate. Use signs \(u_p^+=u_p,\ u_p^-=\bar u_p\), and similarly for \(v_q^\delta\). The four kernels are \begin{equation} H_{ij}^{\epsilon\delta}(p,q) =2\Pi_{ij,lm}(p)q_lq_m \int_{t_0}^T\int v_f(t,x) u_p^\epsilon(t,x)v_q^\delta(t,x)\dd x\dd t. \label{eq:Hfour} \end{equation} Here \(a^+=a,\ a^-=a^\dagger\), so the quadratic expression is \[ B=\sum_{\epsilon,\delta}\iint H_{ij}^{\epsilon\delta}(p,q) a_{g,ij}^\epsilon(p)a_\phi^\delta(q)\dd p\dd q. \] For real \(f\), \[ H^{-\epsilon,-\delta} =\overline{H^{\epsilon\delta}}. \label{eq:Hreal} \] On \(\DD_2\), these formulas initially define a symmetric sesquilinear form: every nonzero matrix element is a finite sum of kernel integrals over compact external supports away from gravitational zero momentum. The local bound \[ |H^{\epsilon\delta}(p,q)| \le C_f|p|^{-1/2}\br q^{3/2} \] makes them finite. \subsection{The exact mask family} For \(R\ge2\), take smooth real even masks satisfying \begin{equation} \begin{gathered} 0\le\alpha_R,\beta_R\le1,\qquad \alpha_R=1\quad(R^{-1}\le|p|\le R),\\ \supp\alpha_R\subset\{(2R)^{-1}<|p|<2R\},\\ \beta_R=1\quad(|q|\le R),\qquad \supp\beta_R\subset\{|q|<2R\},\\ H_R^{\epsilon\delta} =\alpha_R(p)\beta_R(q)H^{\epsilon\delta}. \end{gathered} \label{eq:masks} \end{equation} No derivative bounds or monotonicity in \(R\) are assumed. These are gravitational IR and UV masks and a scalar UV mask. Put \(L_R=H_R^{-+}:\hh_\phi\to\hh_g\) and \(P_R=H_R^{--}\in\hh_g\otimes\hh_\phi\). On \(\hh=\hh_g\oplus\hh_\phi\), set \[ T_R=\begin{pmatrix}0&L_R\\L_R^*&0\end{pmatrix}, \qquad K_R=(i_g\otimes i_\phi)P_R+ \flip(i_g\otimes i_\phi)P_R. \] For a symmetric pair tensor \(K\), use \[ a^\dagger(K)\Psi_n =\sqrt{(n+1)(n+2)}\,S_{n+2}(K\otimes\Psi_n). \] \begin{lemma}[Exact quadratic reduction] \label{lem:quadratic2} At finite \(R\), \begin{equation} B_R=\dG(T_R)+\frac12(a^\dagger(K_R)+a(K_R)), \label{eq:quadratic2} \end{equation} with no constant term, and \[ \norm{T_R}=\norm{L_R},\qquad \norm{K_R}=\sqrt2\norm{P_R}_{\HS}. \] For bounded \(T,T'\) and pair tensors \(K,K'\), \begin{align} \norm{(B-B')\Psi} &\le(\norm{T-T'}+\norm{K-K'}) \norm{(\NN+2)\Psi},\label{eq:graph2}\\ \norm{\Pi_M(B-B')\Pi_M} &\le M\norm{T-T'} +\sqrt{M(M-1)}\norm{K-K'}.\label{eq:compress2} \end{align} \end{lemma} \begin{proof} The \((-+)\) and \((+-)\) terms are adjoint cross-species number-preserving terms. They give \(\dG(T_R)\); moving the species past each other produces no scalar. For an elementary \(P_R=g\otimes v\), \(\tfrac12a^\dagger(K_R)=a_g^\dagger(g)a_\phi^\dagger(v)\). Linearity and Hilbert tensor approximation prove the identity. The two species slots are orthogonal, giving the pair norm. On the \(n\)-particle sector the number term is bounded by \(n\norm T\); pair creation and annihilation have bounds \(\sqrt{(n+1)(n+2)}\norm K\) and \(\sqrt{n(n-1)}\norm K\). The triangle inequality gives \eqref{eq:graph2}. After compression, each of creation and annihilation has norm at most \(\sqrt{M(M-1)}\norm K\); the half-sum has the same upper bound. This gives \eqref{eq:compress2}. \end{proof} Figure~\ref{fig:four-leg-quadratic} organizes the four kernels into number-preserving and pair coefficients. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\scriptsize, mblue] at (-0.2,2.15) {$a_g$}; \node[font=\scriptsize, mblue] at (-0.2,0.55) {$a_g^\dagger$}; \node[font=\scriptsize, sgreen] at (1.8,3.15) {$a_\phi$}; \node[font=\scriptsize, sgreen] at (5.2,3.15) {$a_\phi^\dagger$}; \node[mbox, text width=2.8cm] (pp) at (1.8,2.15) {$H^{++}a_ga_\phi$\\ pair annihilation}; \node[pbox, text width=2.8cm] (pm) at (5.2,2.15) {$H^{+-}a_ga_\phi^\dagger$\\ number-preserving}; \node[pbox, text width=2.8cm] (mp) at (1.8,0.55) {$H^{-+}a_g^\dagger a_\phi$\\ number-preserving}; \node[mbox, text width=2.8cm] (mm) at (5.2,0.55) {$H^{--}a_g^\dagger a_\phi^\dagger$\\ pair creation}; \node[pbox, anchor=west, text width=6.4cm, font=\scriptsize] at (7.4,2.35) {$L_R=H_R^{-+}$,\quad $P_R=H_R^{--}$}; \node[pbox, anchor=west, font=\scriptsize] at (7.4,1.35) {$T_R=\bigl(\begin{smallmatrix}0 & L_R \\ L_R^* & 0\end{smallmatrix}\bigr)$}; \node[pbox, anchor=west, text width=6.4cm, font=\scriptsize] at (7.4,0.35) {symmetrized cross-species pair tensor $K_R$}; \node[pbox, text width=12.6cm] at (6.2,-1.15) {$B_R=\dG(T_R)+\tfrac12\bigl[a^\dagger(K_R)+a(K_R)\bigr]$}; \node[ibox, font=\scriptsize] at (3.2,-2.15) {$\norm{T_R}=\norm{L_R}$,\quad $\norm{K_R}=\sqrt2\norm{P_R}_{\HS}$}; \node[gbox, font=\scriptsize, text width=6.8cm] at (9.2,-2.15) {no constant term: different species commute}; \end{tikzpicture} \caption{Four-leg reduction of the dense-TT bilinear. The mixed legs form the off-diagonal number coefficient, and the pair legs form a symmetric cross-species tensor. Their quadratic reduction has no constant term because reordering the distinct species produces no contraction.} \label{fig:four-leg-quadratic} \end{figure} Let \(\Pi_M=I_{\{\NN_g+\NN_\phi\le M\}}\) and \(Q_{\phi,M}=I_{\{\NN_\phi\le M\}}\). Writing \[ F_{0,M}=Q_{\phi,M}\Phi(u_{0,f})Q_{\phi,M}, \] define \begin{equation} A^{(2)}_{R,M}(\lambda) =I_g\otimes F_{0,M}+\lambda\Pi_MB_R\Pi_M. \label{eq:A2} \end{equation} It is bounded self-adjoint for finite \(R,M\). The scalar free term has its own number cutoff. Replacing it by a total-number compression would introduce a false gravitational dependence at \(\lambda=0\). %===================================================================== \section{Model 2: spectral removal at fixed number cutoff} \label{sec:model2-limit} %===================================================================== \subsection{Transfer variables and the time factor} Set \(\ell=T-t_0\), \(\gamma=(2\pi)^{-3}\), and define \[ F(s,k)=\int v_f(t_0+s,x)e^{ik\cdot x}\dd x,\qquad J(k,\nu)=\int_0^\ell F(s,k)e^{-i\nu s}\dd s. \] The four uncut legs are \begin{align*} H^{++}&=G(p,q)J(p+q,|p|+E_q),\\ H^{+-}&=G(p,q)J(p-q,|p|-E_q),\\ H^{-+}&=G(p,q)J(-p+q,-|p|+E_q),\\ H^{--}&=G(p,q)J(-p-q,-|p|-E_q), \end{align*} where \[ G(p,q)=\gamma\frac{\Pi(p)(q\otimes q)}{\sqrt{|p|E_q}}. \] \begin{lemma}[TT and finite-time gains] \label{lem:gains} For every integer \(n\ge0\), \[ |J(k,\nu)|\le C_n\br k^{-n}(1+|\nu|)^{-1}. \] For \(k=\epsilon p+\delta q\), \[ \Pi(p)(q\otimes q)=\Pi(p)(k\otimes k),\qquad |\Pi(p)(k\otimes k)|^2 =\frac12\left(|k|^2-|\widehat p\cdot k|^2\right)^2. \] Consequently, for every \(n\), \begin{equation} |H^{\epsilon\delta}(p,q)| \le \frac{C_n\br{\epsilon p+\delta q}^{-n}} {\sqrt{|p|E_q}\,[1+|\epsilon|p|+\delta E_q|]}. \label{eq:master2} \end{equation} \end{lemma} \begin{proof} Spatial integrations by parts give arbitrary decay of \(F\) and \(\partial_sF\), uniformly on the finite interval. Since \(F\) vanishes near \(\ell\), integration by parts in time for \(|\nu|>1\) gives \[ J(k,\nu)=\frac{F(0,k)}{i\nu} +\frac1{i\nu}\int_0^\ell\partial_sF(s,k)e^{-i\nu s}\dd s. \] The initial endpoint has not been omitted. For \(|\nu|\le1\) use the original integral. The TT projection annihilates each term containing a factor \(p\). Substituting \(\delta q=k-\epsilon p\) gives the identity. The norm is the squared norm of the trace-free part of \(k_\perp\otimes k_\perp\) on a two-dimensional transverse plane. Absorb the factor \(|k|^2\) into the arbitrary spatial decay. \end{proof} \subsection{Pair and number estimates} Let \(P=H^{--}\) and \(L=H^{-+}\). \begin{proposition}[Evaluated spectral remainders] \label{prop:rates} The pair kernel \(P\) is Hilbert--Schmidt, and \(L\) defines a bounded compact operator \(\hh_\phi\to\hh_g\). There are finite constants \(C_P,C_L\), independent of the choices of masks satisfying \eqref{eq:masks}, such that \begin{equation} \norm{P_R-P}_{\HS}\le C_PR^{-1/2},\qquad \norm{L_R-L}_{\op}\le C_LR^{-1}. \label{eq:rates} \end{equation} For \(f\ne0\), \(L\) is not Hilbert--Schmidt; more precisely, \[ cR\le\norm{L_R}_{\HS}^2\le CR \] for all sufficiently large \(R\), with \(c,C>0\). \end{proposition} \begin{proof} For pair signs the denominator in \eqref{eq:master2} contains \(1+|p|+E_q\). Put \(k=p+q\), and use \(E_q^{-1}\le C_m\br k\br p^{-1}\). With arbitrarily large \(n\), \begin{equation} |P(p,q)|^2 \le C_n\br{p+q}^{-n}\frac1{|p|\br p^3}. \label{eq:Pmajor} \end{equation} The radial integral of the last factor is finite: \[ 4\pi\int_0^\infty r(1+r^2)^{-3/2}\dd r<\infty. \] Its integral for \(|p|R\) is \(O(R^{-1})\). For the scalar UV tail split \(|p|<1\) and \(|p|\ge1\). On the latter, \eqref{eq:Pmajor} and \(\br p^{-4}\le C\br{p+q}^4\br q^{-4}\) give a bound whose integral over \(|q|>R\) is \(O(R^{-1})\). On the former, spatial transfer decay gives \(C_n|p|^{-1}\br q^{-n}\), with arbitrarily rapid integrated decay. Since \[ |1-\alpha_R\beta_R| \le I_{\{|p|R\}}+I_{\{|q|>R\}}, \] the pair estimate follows. For the mixed signs, discard the bounded inverse frequency factor, but retain transfer decay. For \(|p|\ge1\), Peetre's inequality gives both bounds \begin{equation} |L(p,q)|\le C_n\br{p-q}^{-n}\br p^{-1}, \qquad |L(p,q)|\le C_n\br{p-q}^{-n}\br q^{-1}. \label{eq:Schur2} \end{equation} For \(|p|<1\), \[ |L(p,q)|\le C_n|p|^{-1/2}\br q^{-n}. \] The low-momentum part is Hilbert--Schmidt and the high part is bounded by the Schur test. The low gravitational tail has squared integral \(O(R^{-2})\). For \(|p|>R\) or \(|q|>R\), the appropriate bound in \eqref{eq:Schur2} has both Schur integrals \(O(R^{-1})\); the remaining low-\(p\), high-\(q\) part decays faster. This proves the operator-norm rate. Each \(L_R\) is Hilbert--Schmidt, so its norm limit \(L\) is compact. For the last assertion, the squared mixed estimate gives \[ |L(p,q)|^2\le C_n\br{p-q}^{-n}\frac1{|p|\br p}. \] Integration over \(|p|<2R\) proves the upper bound \(CR\). For a lower bound write \(p=rn,\ q=rn+k\), with \(n\in S^2\) and \(k\) in a fixed compact set. Uniformly there, \[ rL(rn,rn+k) \longrightarrow \gamma\Pi(n)(k\otimes k)J(k,n\cdot k). \label{eq:mixedasymptotic} \] The function \(J\) is the Fourier transform of the compactly supported integrable function \(I_{[0,\ell]}(s)v_f(t_0+s,x)\); hence it is real analytic. It is nonzero if \(f\ne0\), because \((\partial_s^2+\Omega_m^2)v_f=f\). It therefore cannot vanish on the open set \(|\nu|<|k|\). Choose a nonzero value there and an \(n\) with \(n\cdot k=\nu\). The TT norm is strictly positive. Continuity gives open sets of positive measure in \(n,k\) on which the limit in \eqref{eq:mixedasymptotic} is bounded below. For \(r_0\le r\le R/2\), both masks equal one when \(R\) is large. The Jacobian is \(r^2\dd r\dd n\dd k\), so integrating the lower bound \(c_0^2r^{-2}\) gives a quantity bounded below by \(cR\). The same region without radial truncation proves that \(L\) is not Hilbert--Schmidt. \end{proof} This is a distinction of operator ideals, not an obstruction to the number coefficient required by the quadratic estimate. Figure~\ref{fig:model2-kernel-ideals} contrasts the pair and number-leg estimates and their different operator ideals. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[pbox, text width=12.8cm, font=\scriptsize] (ker) {$H^{\epsilon\delta}(p,q)= \gamma\dfrac{\Pi(p)(q\otimes q)}{\sqrt{|p|E_q}} J(\epsilon p+\delta q,\ \epsilon|p|+\delta E_q)$}; \node[mbox, below=3mm of ker, text width=6cm, font=\scriptsize] (g1) {$\Pi(p)(q\otimes q)=\Pi(p)(k\otimes k)$}; \node[sbox, right=3mm of g1, text width=6cm, font=\scriptsize] (g2) {$|J(k,\nu)|\le C_n\langle k\rangle^{-n}(1+|\nu|)^{-1}$}; \node[gbox, below=2mm of g1, text width=12.6cm] (end) {initial endpoint retained}; \node[mbox, below=4mm of end, text width=6cm, font=\scriptsize] (pair) {Pair branch: $\nu=-(|p|+E_q)$\\ $P\in\mathrm{HS}$,\ $\norm{P_R-P}_{\HS}\le C_PR^{-1/2}$}; \node[pbox, right=3mm of pair, text width=6.2cm, font=\scriptsize] (mix) {Mixed branch: transfer near $q=p$\\ possible frequency cancellation\\ $L$ compact and bounded,\ $\norm{L_R-L}_{\op}\le C_LR^{-1}$}; \draw[arr] (end.south) -- (pair.north); \draw[arr] (end.south) -- (mix.north); \node[cbox, below=3mm of mix, text width=6.2cm, font=\scriptsize] {$f\neq 0$:\ $\norm{L_R}_{\HS}^2\asymp R$,\quad $L\notin\mathrm{HS}$}; \node[gbox, below=2mm of pair, text width=6cm, font=\scriptsize] {asymptotic labels only; no numerical convergence curve}; \end{tikzpicture} \caption{Different coefficient topologies in Model 2. TT projection and finite-time integration provide transfer gains. The pair coefficient converges in Hilbert--Schmidt norm. The mixed coefficient converges in operator norm to a compact operator, but for nonzero detector smearing its Hilbert--Schmidt norm diverges with the spectral cutoff.} \label{fig:model2-kernel-ideals} \end{figure} \subsection{The fixed-\(M\) limit} Define \[ T=\begin{pmatrix}0&L\\L^*&0\end{pmatrix}, \qquad K=(i_g\otimes i_\phi)P+\flip(i_g\otimes i_\phi)P. \] The quadratic with these coefficients obeys \eqref{eq:graph2} and defines a symmetric operator \(B_\infty\) on \(\Dom\NN\). Its form agrees with the original dense pairing on \(\DD_2\): for those test vectors, the masks eventually equal one on all active external supports. \begin{corollary}[Model 2 generator limit at fixed \(M\)] \label{cor:fixedM} For every fixed finite \(M\ge2\), \[ B_{\infty,M}=\Pi_MB_\infty\Pi_M \] is bounded self-adjoint, and \begin{equation} \delta_M(R):=\norm{B_{R,M}-B_{\infty,M}} \le \frac{MC_L}{R} +\frac{\sqrt{2M(M-1)}C_P}{\sqrt R} =:\overline\delta_M(R). \label{eq:deltaM} \end{equation} Also, on \(\Dom\NN\), \[ \norm{(B_R-B_\infty)\Psi} \le(C_LR^{-1}+\sqrt2C_PR^{-1/2}) \norm{(\NN+2)\Psi}. \] \end{corollary} \begin{proof} The off-diagonal block norms give \(\norm{T_R-T}=\norm{L_R-L}\), and the pair tensor difference has norm \(\sqrt2\norm{P_R-P}_{\HS}\). Apply Lemma~\ref{lem:quadratic2}. \end{proof} The symmetric operator \(B_\infty\) is closable. No claim that closability remains undecided is made. An uncompressed measured evolution is simply not used in the following theorem. \subsection{Witness, calibration, and record} Choose a nonzero real TT packet with compact Fourier support away from zero and initial velocity zero, as in the construction of Section~\ref{sec:model1}. Its Model 2 label \(z_*\) belongs to \(\mathfrak d_g\). The Hessian argument of Lemma~\ref{lem:injective}, applied to this wave, gives a compact-data free scalar \(\phi_*\) such that \(\psi_*=S_{t_0}(k_{z_*})\phi_*\) is nonzero at a future point. Choose a nonnegative bump \(b\) there and put \[ f_*=b\psi_*,\qquad d_*=\cB_{f_*}(k_{z_*},\phi_*)=\int b\psi_*^2\dd^4x>0. \] The scalar label \(v_*\) is Schwartz by its compact smooth data. The coherent contraction of the four kernels is \begin{equation} \cB_f(k_z,\phi_v) =2\Rea\left(\ip z{Lv}+\ip{z\otimes v}{P}\right). \label{eq:coherent2} \end{equation} It follows first by inserting the one-point functions in \eqref{eq:Bclassical}; boundedness and the Hilbert tensor norm then extend the pairing continuously in the labels. Let \[ d_R=\cB_{f_*}(C_{g,R}k_{z_*},C_{\phi,R}\phi_*). \] If \(z_*\) is supported in \(r_*\le|p|\le R_*\), then for \(R\ge\max\{2,r_*^{-1},R_*\}\), \(\alpha_Rz_*=z_*\). For every integer \(j\ge0\), \begin{equation} |d_R-d_*| \le2(\norm L+\norm P_{\HS})\norm{z_*} R^{-j}\norm{|q|^jv_*}. \label{eq:dR} \end{equation} This uses \(\norm{(1-\beta_R)v_*}\le R^{-j}\norm{|q|^jv_*}\). Exact eventual equality is not asserted, because the scalar label need not have compact momentum support. Fix \(a\ne0\), and use \[ \Psi_0=\Omega_g\otimes\Coh(v_*),\qquad \Psi_a=\Coh(az_*)\otimes\Coh(v_*). \] Put \[ C_B=\norm L+\sqrt2\norm P_{\HS},\qquad \mu_0=\norm{v_*}^2,\quad \mu_a=\mu_0+a^2\norm{z_*}^2. \] The cutoff masks are contractions, so \(\norm{B_R\Psi}\le C_B\norm{(\NN+2)\Psi}\), uniformly in \(R\), including \(R=\infty\). Finite number bandwidth gives \[ \norm{(B_{R,M}-B_R)\Psi} \le2C_B\norm{(\NN+2)I_{\{\NN>M-2\}}\Psi}. \] Therefore define \begin{equation} E_M=2C_B\sum_{s\in\{0,a\}} \left[ e^{-\mu_s}\sum_{n>M-2}(n+2)^2\frac{\mu_s^n}{n!} \right]^{1/2}\longrightarrow0. \label{eq:EM2} \end{equation} The mean contrast obeys \[ \ip{\Psi_a}{B_{R,M}\Psi_a} -\ip{\Psi_0}{B_{R,M}\Psi_0} =a\,d_R+\epsilon_{R,M},\qquad |\epsilon_{R,M}|\le E_M. \] This error cannot be suppressed: particle-number projections do not commute with coherent displacement. \begin{theorem}[Model 2 record: spectral removal at fixed \(M\)] \label{thm:model2} Use the preceding detector and states, and fix \(a,\lambda\ne0\). Set \[ d=|\lambda a|d_*, \] \[ Q_s=2\norm{u_{0,f_*}}\sqrt{\norm{v_*}^2+1} +|\lambda|C_B\sqrt{\mu_s^2+5\mu_s+4}, \qquad S_2=Q_0^2+Q_a^2. \] Choose, independently of \(R,M\), \[ \theta=\pi/2,\qquad 0<\tau\le d/S_2,\qquad c_2=\tau d/4. \] For every fixed finite \(M\ge2\), the Model 2 instruments satisfy \[ \Delta_{R,M} :=|p^{(2)}_{+,R,M}(\Psi_a)-p^{(2)}_{+,R,M}(\Psi_0)| \ge c_2-e_M-e_R(M), \] where \[ e_M=\frac{\tau|\lambda|}{2}E_M,\qquad e_R(M)=|\tau\lambda|\,\overline\delta_M(R). \] At that fixed \(M\), the generators, unitaries, and binary probabilities converge in operator norm, operator norm, and uniformly over states, respectively, as \(R\to\infty\). For all sufficiently large finite \(M\), the limiting contrast is at least \(c_2/2\). At \(\lambda=0\), the two probabilities agree exactly for every \(R,M,\tau,\theta\), including the fixed-\(M\) spectral limit. No number-cutoff instrument limit is asserted. \end{theorem} \begin{proof} At \(R=\infty\), the absolute generator mean difference is at least \(d-|\lambda|E_M\). The free scalar expectations cancel. The number bounds and Poisson moments give \[ \norm{A^{(2)}_{R,M}(\lambda)\Psi_s}\le Q_s \] uniformly in \(R,M\). Lemma~\ref{lem:Ramsey} yields \[ \Delta_{\infty,M} \ge\frac{\tau}{2}(d-|\lambda|E_M)-\frac{\tau^2S_2}{4} \ge c_2-e_M. \] The generator norm difference is \(|\lambda|\delta_M(R)\). The unitary difference is at most \(|\tau\lambda|\delta_M(R)\); each probability costs half this amount, so the contrast costs at most \(e_R(M)\). This proves all convergence and contrast statements. Since \(E_M\to0\), one may fix a finite \(M\) with \(e_M\le c_2/2\). At \(\lambda=0\), \[ e^{i\tau A^{(2)}_{R,M}(0)} =I_g\otimes e^{i\tau F_{0,M}}. \] The scalar preparations coincide, proving the exact control. \end{proof} Figure~\ref{fig:model2-record} separates spectral and number-tail record errors and shows why the free term uses scalar-only compression. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[pbox, text width=5.4cm, font=\scriptsize] (p0) {$\Psi_0=\Omega_g\otimes\Coh(v_*)$}; \node[pbox, right=4mm of p0, text width=5.4cm, font=\scriptsize] (pa) {$\Psi_a=\Coh(az_*)\otimes\Coh(v_*)$}; \node[sbox, below=2mm of p0, text width=11.4cm] {scalar factor identical}; \node[sbox, below=2mm of p0, yshift=-9mm, text width=11.4cm, font=\scriptsize] (wit) {$f_*=b\psi_*$,\qquad $d_*=\int b\psi_*^2>0$}; \node[pbox, below=2mm of wit, text width=11.4cm, font=\scriptsize] (gen) {$A^{(2)}_{R,M}=I_g\otimes F_{0,M}+\lambda\Pi_MB_R\Pi_M$}; \node[pbox, below=2mm of gen, font=\scriptsize] {$\Delta_{R,M}\ge c_2-e_M-e_R(M)$}; \node[cbox, below left=4mm and 0mm of gen, text width=5.4cm, font=\scriptsize] (em) {$e_M$: coherent number tails}; \node[cbox, below right=4mm and 0mm of gen, text width=5.4cm, font=\scriptsize] (er) {$e_R(M)$: spectral generator convergence}; \node[sbox, below=16mm of gen, text width=11.4cm, font=\scriptsize] (good) {$\lambda=0$:\ $U=I_g\otimes e^{i\tau F_{0,M}}$,\ $\Delta p=0$\\ scalar-only compression of the free term}; \node[cbox, below=2mm of good, text width=11.4cm, font=\scriptsize] (bad) {not used: total-number compression of the free scalar term\\ would introduce gravitational-number dependence}; \draw[corange] (bad.north west) -- (bad.south east); \node[gbox, below=2mm of bad, text width=11.4cm, font=\scriptsize] {Coherent displacement is not claimed to commute with $\Pi_M$.}; \end{tikzpicture} \caption{Model 2 record and exact control. A gravitational displacement at fixed scalar preparation gives a positive bilinear witness. The calibrated record includes separate spectral and coherent number-tail errors. Scalar-only compression of the free term preserves exact equality at \(\lambda=0\); total-number compression would not preserve that factorization.} \label{fig:model2-record} \end{figure} \subsection{Finite-rank gravitational kernels} Let \(\mathsf P_N\) be finite-rank orthogonal projections on \(\hh_g\), increasing strongly to \(I\). Project the gravitational legs: \[ L_{R,N}=\mathsf P_NL_R,\qquad P_{R,N}=(\mathsf P_N\otimes I)P_R, \] using adjoint or conjugate projections on the other two legs. The induced \(B_{R,N,M}\) and \(A^{(2)}_{R,N,M}\) use the same number projections and the same free scalar term. At fixed \(R,M\), Hilbert--Schmidt approximation of the cutoff kernels and Lemma~\ref{lem:quadratic2} give operator-norm convergence as \(N\to\infty\). There is also a joint estimate at fixed \(M\): \[ \norm{B_{R,N,M}-B_{\infty,M}} \le\overline\delta_M(R) +M\rho_N+\sqrt{2M(M-1)}\,\sigma_N, \] where \[ \rho_N=\norm{(\mathsf P_N-I)L}\to0,\qquad \sigma_N=\norm{((\mathsf P_N-I)\otimes I)P}\to0. \] The first convergence uses compactness of \(L\), and the second uses Hilbert tensor approximation. Thus any claimed interchange of \(R,N\) at fixed \(M\) is supported by an explicit bound. No interchange with measured \(M\to\infty\) dynamics is claimed. Figure~\ref{fig:fixed-M-convergence} summarizes the spectral and finite-rank limits supported by the fixed-\(M\) estimates. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[pbox, font=\small\bfseries] (M) {fixed finite $M$}; \node[mbox] (br) at (-2.2,-1.3) {$B_{R,M}$}; \node[pbox] (bi) at (2.2,-1.3) {$B_{\infty,M}$}; \draw[arr] (br)--(bi); \node[cbox, font=\scriptsize, text width=5.4cm] at (0,-2.25) {$\overline\delta_M(R)=MC_LR^{-1}+\sqrt{2M(M-1)}C_PR^{-1/2}$}; \node[pbox, font=\scriptsize] (A) at (0,-3.15) {$A^{(2)}_{R,M}\to A^{(2)}_{\infty,M}$}; \node[pbox, font=\scriptsize] (U) at (0,-3.9) {$U_{R,M}\to U_{\infty,M}$}; \node[pbox, font=\scriptsize] (pr) at (0,-4.65) {probabilities, uniformly over states}; \draw[arr] (bi.south) -- (A.north); \draw[arr] (A)--(U); \draw[arr] (U)--(pr); \node[mbox, font=\scriptsize] (brn) at (-2.2,-5.7) {$B_{R,N,M}$}; \node[gbox, font=\scriptsize, text width=6.6cm] at (2.4,-5.7) {remainder $\overline\delta_M(R)+M\rho_N+\sqrt{2M(M-1)}\sigma_N$}; \draw[arrg] (brn.east) -- (0.4,-5.7); \node[font=\scriptsize, align=center] at (0,-6.55) {$\rho_N\to 0$ by compactness of $L$\\ $\sigma_N\to 0$ by Hilbert-tensor approximation}; \node[cbox, font=\scriptsize] at (0,-7.55) {STOP. Measured $M\to\infty$: not asserted.}; \node[gbox, font=\scriptsize] at (0,-8.35) {$\Pi_M$ may have infinite rank.}; \end{tikzpicture} \caption{Model 2 convergence at fixed number cutoff. Separate operator-norm and Hilbert--Schmidt coefficient estimates give bounded-generator convergence, hence unitary and uniform probability convergence. Finite-rank gravitational projection has an additional explicit remainder. These results do not supply a measured number-cutoff removal theorem.} \label{fig:fixed-M-convergence} \end{figure} \subsection{A covariant-algebra non-identification} \begin{proposition}[Raw functional obstruction] \label{prop:raw} The unsectioned functional \[ h\longmapsto\int fS_{t_0}(h)\phi_0 \] at a fixed free scalar is not, in general, invariant under compact metric gauge changes. In particular, some such functionals cannot be written for all admitted representatives as \(\int h_{\mu\nu}J^{\mu\nu}\) with compact smooth symmetric \(\partial_\mu J^{\mu\nu}=0\). The retarded harmonic section above is not a left inverse on every spacelike-compact pure gauge. \end{proposition} \begin{proof} Choose \(\xi\) compactly supported strictly after \(t_0\). Equation \eqref{eq:Sgauge} then gives \[ \int fS_{t_0}(K\xi)\phi_0 =\int f\,\xi^\mu\partial_\mu\phi_0, \] which is nonzero for suitable data and a test supported where a scalar derivative is nonzero. A compact conserved tensor current would instead annihilate \(K\xi\) by integration by parts. For the last statement choose a nonzero vector wave \(\xi\) with compact Cauchy data and \(K\xi\ne0\). It is spacelike compact and \(\Box\xi=0\), so \(G_{\ret}^{\Box}D(K\xi)=0\), not \(\xi\). \end{proof} Accordingly, this TT completion and its represented operators are not identified here with a full covariant vacuum-gravity observable algebra. No compact-Cauchy nondegeneracy theorem is used on Minkowski space. %===================================================================== \section{Model 3: the classical sourced response} \label{sec:classical3} %===================================================================== The role of a sourced metric coefficient is different from the vacuum perturbations in Models 1 and 2. The formal coefficient equations motivating it are \[ G^{(1)}(k)=0,\qquad G^{(1)}(j)+G^{(2)}[k,k]=T(\eta,\phi_0),\qquad P\phi_0=0,\qquad P\phi_1=-L_k\phi_0. \] Only the \(k=0\) source is represented in Model 3. In particular, \(j\) is not substituted for \(k\) in a first-order scalar correction. \subsection{Two explicit response prescriptions} For a symmetric tensor \(h\), write \[ (\mathsf r h)_{\mu\nu} =h_{\mu\nu}-\frac12\eta_{\mu\nu} \eta^{\alpha\beta}h_{\alpha\beta}, \qquad \bar h=\mathsf r h. \] In four dimensions \(\mathsf r^2=I\), and \[ G^{(1)}_{\mu\nu}(h) =-\frac12\Box\bar h_{\mu\nu} +\partial_{(\mu}D_{\nu)}(h) -\frac12\eta_{\mu\nu}\partial^\alpha D_\alpha(h). \label{eq:G1} \] Let \(\QQ\) consist of smooth conserved symmetric sources \(Q\in C^\infty(\R;\Sch(\R^3))\), with complexification allowed. Schwartz seminorms and all time derivatives are locally uniformly bounded. Conservation means \(\partial^\mu Q_{\mu\nu}=0\). Fix \[ (\Delta^{-1}b)(x) =-\frac1{4\pi}\int\frac{b(y)}{|x-y|}\dd y. \] For \(b\in\Sch\), this is smooth, and \[ |\partial^\alpha\Delta^{-1}b(x)| \le C_\alpha\br x^{-1-|\alpha|},\qquad \Delta^{-1}b(x) =-\frac{\int b}{4\pi|x|}+O(|x|^{-2}). \label{eq:Newtonestimate} \] To verify these statements, split the integral into \(|y|<|x|/2\) and its complement. In the first region expand or differentiate the kernel; in the second, use rapid decay and move derivatives onto \(b\) near the integrable singularity. The resulting bounds use finitely many Schwartz seminorms. In particular \(\nabla\Delta^{-1}b\in L^2\), although \(\Delta^{-1}b\) need not be in \(L^2\). Put \[ \rho=Q_{00}(t_0),\quad J_i=Q_{0i}(t_0),\quad a=-2\Delta^{-1}\rho,\quad V_i=2\Delta^{-1}J_i, \] \[ B_{ij}=\partial_iV_j+\partial_jV_i -\delta_{ij}\partial_\ell V_\ell. \] Then \(\Delta a=-2\rho\) and \(\partial_jB_{ji}=2J_i\). Define two trace-reversed data prescriptions: \begin{center} \small \begin{tabular}{@{}c|ccc|ccc@{}} \toprule & \(\bar j_{00}\) & \(\bar j_{0i}\) & \(\bar j_{ij}\) & \(\dot{\bar j}_{00}\) & \(\dot{\bar j}_{0i}\) & \(\dot{\bar j}_{ij}\)\\ \midrule \(\cR^{\mathrm C}_{t_0}\) & \(a\) & \(0\) & \(0\) & \(0\) & \(0\) & \(B_{ij}\)\\ \(\cR^{\NW}_{t_0}\) & \(a\) & \(-V_i\) & \(0\) & \(-\partial_iV_i\) & \(0\) & \(0\)\\ \bottomrule \end{tabular} \end{center} For either row, with data \((a_{\mu\nu},b_{\mu\nu})\), evolve \begin{equation} \bar j(t)=\cos((t-t_0)D_0)a +\frac{\sin((t-t_0)D_0)}{D_0}b +2\int_{t_0}^t \frac{\sin((t-s)D_0)}{D_0}Q(s)\dd s, \qquad j=\mathsf r\bar j. \label{eq:response} \end{equation} Model 3 uses \(\cR^{\NW}_{t_0}\). \begin{proposition}[Classical response and constraint propagation] \label{prop:response3} Both prescriptions are linear maps on \(\QQ\), give smooth solutions, and satisfy \[ \Box\bar j=-2Q,\qquad D(j)=0,\qquad G^{(1)}(j)=Q. \] On every finite time interval, \[ \nabla j,\ \partial_tj\in C(I;L^2), \] and the response is continuous from finitely many source Schwartz seminorms to every prescribed compact \(C^s\) seminorm. The metric has Coulomb-order spatial tails, not a general \(L^2\) bound. \end{proposition} \begin{proof} The Newton estimates give finite derivative energy of the position data and finite \(L^2\) velocity data. The massless wave energy estimate, proved as in Lemma~\ref{lem:wave}, applies to \eqref{eq:response}. Smoothness follows by differentiation or the Fourier formula; the potentially singular initial multipliers \(|p|^{-2}\) and \(|p|^{-1}\) are locally integrable in three dimensions. Set \[ d_\nu=-\partial_t\bar j_{0\nu}+\partial_i\bar j_{i\nu}. \] Both data rows have \(d(t_0)=0\). For the first row, \[ \dot d_0=-\Delta a-2\rho=0,\qquad \dot d_i=-2J_i+\partial_jB_{ji}=0. \] For the second row, \[ \dot d_0=-\Delta a-2\rho=0,\qquad \dot d_i=-\Delta(-V_i)-2J_i=0. \] Conservation gives \(\Box d=0\), so uniqueness gives \(d=0\). Equation \eqref{eq:G1} then proves the sourced equation. For continuity, finitely many source seminorms control arbitrary prescribed finite Fourier decay of the source and its time derivatives. The initial multipliers are locally integrable and have polynomial growth after differentiation. Fourier inversion on a compact time interval gives every stated compact \(C^s\) bound. The Newton bounds, propagated by the finite-time Kirchhoff formula, give Coulomb-order tails. They do not make a nonzero \(1/|x|\) term square integrable. \end{proof} \subsection{The initial-data comparison is executed} \begin{proposition}[The two maps differ by a specified residual gauge] \label{prop:mapcomparison} For every \(Q\in\QQ\), define a covector wave by \[ \xi_0=0,\qquad \xi_i(t)=-\frac{\sin((t-t_0)D_0)}{D_0}V_i. \] Then \begin{equation} \cR^{\NW}_{t_0}Q-\cR^{\mathrm C}_{t_0}Q=K\xi. \label{eq:mapdifference} \end{equation} Thus the maps are not generally identical, but their pairings with compact conserved symmetric test tensors agree. Their linearized curvature tensors also agree. The field \(\xi\) is smooth, but is not generally compactly supported. This comparison does not identify it with a gauge parameter in the smaller compact-gauge classes used above. \end{proposition} \begin{proof} Both responses solve the same reduced wave equation, so their trace-reversed difference is homogeneous. For \(\Box\xi=0\), \[ \overline{K\xi}_{\mu\nu} =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu -\eta_{\mu\nu}\partial^\alpha\xi_\alpha. \] At \(t_0\), its only nonzero components are \(\overline{K\xi}_{0i}=-V_i\). Its velocity components are \[ \partial_t\overline{K\xi}_{00}=-\partial_iV_i,\qquad \partial_t\overline{K\xi}_{0i}=0,\qquad \partial_t\overline{K\xi}_{ij}=-B_{ij}. \] These are exactly the difference of the two table rows. Wave uniqueness proves \eqref{eq:mapdifference}. A compact conserved test annihilates \(K\xi\) by integration by parts, even when \(\xi\) is noncompact. Linearized curvature annihilates every smooth pure gauge. \end{proof} This proves more than a comparison criterion but less than equality of the response maps. Arbitrary metric smearings remain prescription dependent. Figure~\ref{fig:response-comparison} displays the executed initial-data comparison of the two classical response maps. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[sbox] (Q) {$Q$ conserved}; \node[mbox, above right=2mm and 6mm of Q, text width=5.6cm, font=\scriptsize] (C) {$\cR^{\mathrm C}_{t_0}$\\ $\bar j=(a,0,0)$,\quad $\dot{\bar j}=(0,0,B_{ij})$}; \node[mbox, below right=2mm and 6mm of Q, text width=5.6cm, font=\scriptsize] (N) {$\cR^{\NW}_{t_0}$\\ $\bar j=(a,-V_i,0)$,\quad $\dot{\bar j}=(-\partial_i V_i,0,0)$}; \draw[arr] (Q.east) -- (C.west); \draw[arr] (Q.east) -- (N.west); \node[gbox, right=4mm of C, text width=4.4cm, font=\scriptsize] (eq) {$\Box\bar j=-2Q$,\ $D(j)=0$,\ $G^{(1)}(j)=Q$}; \node[cbox, below=8mm of eq, text width=5.2cm, font=\scriptsize] (diff) {$\cR^{\NW}Q-\cR^{\mathrm C}Q=K\xi$\\ $\xi_0=0$,\quad $\xi_i=-\dfrac{\sin((t-t_0)D_0)}{D_0}V_i$}; \node[pbox, below=2mm of diff, font=\scriptsize] (b1) {same linearized curvature}; \node[pbox, below=2mm of b1, font=\scriptsize] (b2) {same compact conserved-current pairings}; \node[gbox, below=2mm of b2, text width=5.4cm, font=\scriptsize] {not identical metric maps; $\xi$ generally noncompact}; \end{tikzpicture} \caption{Executed comparison of the classical response maps. The two constraint-compatible initial-data prescriptions produce responses differing by the displayed homogeneous pure gauge. Their curvatures and compact conserved-current pairings agree, but arbitrary metric smearings need not. The comparison gauge is not generally compactly supported.} \label{fig:response-comparison} \end{figure} \subsection{Coulomb obstruction and scalar compatibility} For a classical free scalar, \[ T_{\mu\nu}(\phi) =\partial_\mu\phi\,\partial_\nu\phi -\frac12\eta_{\mu\nu} (\partial^\alpha\phi\,\partial_\alpha\phi+m^2\phi^2), \] and direct differentiation gives \(\partial^\mu T_{\mu\nu}=(P\phi)\partial_\nu\phi\). Compact smooth scalar data therefore give \(T(\phi)\in\QQ\). At the initial slice, integration of \[ G^{(1)}_{00}(j) =\frac12(\partial_i\partial_jj_{ij}-\Delta j_{ii}) \] shows \[ \int Q_{00}\dd x =\frac12\lim_{r\to\infty}\int_{|x|=r} (\partial_jj_{ij}-\partial_ij_{jj})n_i\dd S. \] For \(Q=T(\phi)\) the left side is \[ E_\phi=\frac12\int (|\dot\phi|^2+|\nabla\phi|^2+m^2\phi^2)(t_0,x)\dd x>0 \] unless the data vanish. Hence compactly supported or Schwartz spatial metric data cannot solve the sourced constraint. For the displayed prescriptions, \[ j_{ij}(t_0,x)=\frac{E_\phi}{4\pi|x|}\delta_{ij} +O(|x|^{-2}),\qquad j_{00}(t_0,x)=\frac{E_\phi}{4\pi|x|}+O(|x|^{-2}). \] The derivative energy is finite but these components are not generally in \(L^2\). Nevertheless \(S_{t_0}(j)\phi\) exists and has finite scalar energy on every finite slab. Indeed, \(L_j\phi\) is supported where \(\phi\) is supported; all needed derivatives of the smooth \(j\) are bounded on that compact region. Lemma~\ref{lem:wave} applies at every Sobolev order. The map \(\phi\mapsto T(\phi)\), and hence \(\phi\mapsto\cR T(\phi)\), is quadratic. The scalar correction \(\phi\mapsto S_{t_0}(\cR T(\phi))\phi\) is cubic. Neither this cubic nor a second-order complete scalar observable is quantized in the following section. The classical coefficient construction itself makes no quantum assertion. Figure~\ref{fig:Coulomb-and-degree} separates the sourced metric tail from the distinct quadratic and cubic constructions. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries] at (2.3,3.4) {(a) sourced constraint and tail}; \node[sbox, text width=4.8cm, font=\scriptsize] at (2.3,2.5) {compact scalar data, $E_\phi>0$}; \node[mbox, text width=4.8cm, font=\scriptsize] at (2.3,1.55) {boundary flux of $G^{(1)}_{00}(j)$}; \draw[->, sgray] (0.4,0.2) -- (3.6,0.2) node[right, font=\scriptsize] {$|x|$}; \draw[mblue, thick] (0.7,1.05) .. controls (1.4,0.85) and (2.0,0.45) .. (3.3,0.32); \node[gbox, font=\scriptsize] at (2.2,0.7) {schematic $1/|x|$ tail}; \node[mbox, text width=5.2cm, font=\scriptsize] at (2.3,-0.85) {$j_{ij}(t_0,x)=\dfrac{E_\phi}{4\pi|x|}\delta_{ij}+O(|x|^{-2})$\\ $j_{00}(t_0,x)=\dfrac{E_\phi}{4\pi|x|}+O(|x|^{-2})$}; \node[gbox, text width=5.2cm, font=\scriptsize] at (2.3,-2.05) {finite derivative energy; metric need not lie in $L^2$}; \node[font=\small\bfseries] at (9.0,3.4) {(b) polynomial degree}; \node[sbox, font=\scriptsize] (phi) at (7.6,2.45) {$\phi$}; \node[sbox, font=\scriptsize] (T) at (9.2,2.45) {$T(\phi)$}; \node[mbox, font=\scriptsize] (j) at (10.8,2.45) {$j=\cR T(\phi)$}; \draw[arr] (phi)--(T); \draw[arr] (T)--(j); \node[gbox, font=\scriptsize] at (9.2,1.75) {quadratic}; \node[pbox, text width=5.4cm, font=\scriptsize] at (9.2,0.75) {$\int Jj$: quadratic response pairing, Model 3}; \node[cbox, text width=5.4cm, font=\scriptsize] at (9.2,-0.35) {$S_{t_0}(j)\phi$: cubic scalar correction, not represented}; \node[gbox, text width=5.4cm, font=\scriptsize] at (9.2,-1.55) {$L_j\phi$ is supported where $\phi$ is supported, so the classical scalar response still exists on finite slabs.}; \end{tikzpicture} \caption{Sourced tails and polynomial degree. Nonzero scalar energy requires a noncompact constraint-compatible metric response, although its derivatives have finite energy. The reconstructed metric and its curvature pairing are quadratic in the scalar preparation. Inserting that metric into a scalar correction produces a different cubic functional, not quantized in Model 3.} \label{fig:Coulomb-and-degree} \end{figure} %===================================================================== \section{Model 3: scalar stress form and exterior record} \label{sec:model3} %===================================================================== By translating time, take \(t_0=0\) in this section. All formulas can be translated back. \subsection{Normal-ordered source and the reconstructed form} There is only scalar Fock space: \[ \HH_3=\Gamma_s(\hh_\phi),\qquad \hh_\phi=L^2(\R^3,\dd p). \] Put \[ \omega_p=\sqrt{|p|^2+m^2},\qquad n_p=(2\pi)^{-3/2}(2\omega_p)^{-1/2},\qquad u_p(t,x)=n_pe^{-i\omega_pt+ip\cdot x}. \] Let \(\DD_3=\Gamma_s^{\mathrm{alg}}(\Sch(\R^3))\), and use the minimal vacuum-normal-ordered stress \[ :T_{\mu\nu}: =:\partial_\mu\widehat\phi\,\partial_\nu\widehat\phi: -\frac12\eta_{\mu\nu}: (\partial^\alpha\widehat\phi\,\partial_\alpha\widehat\phi +m^2\widehat\phi^2):. \label{eq:Wick3} \] This is a fixed prescription. No improvement or constant counterterm is left unspecified. \begin{lemma}[Source matrix elements] \label{lem:stress3} For \(\Psi,\Chi\in\DD_3\), \[ t_{\mu\nu}^{\Psi,\Chi}(x) =\ip{\Psi}{:T_{\mu\nu}(x):\Chi} \] belongs to the complexified \(\QQ\), and \(\overline{t^{\Psi,\Chi}}=t^{\Chi,\Psi}\). \end{lemma} \begin{proof} Expand the finite creation-operator products defining the two vectors. Commuting the annihilators leaves finitely many products of scalar wave packets and their derivatives, multiplied by one-particle inner products. For a Schwartz label the wave packet is Schwartz in space, uniformly with all derivatives on finite time intervals: spatial weights are obtained by momentum integrations by parts, and \(m>0\) makes all derivatives of \(\omega_p\) smooth and polynomially bounded. For polarized solutions define \[ \cT_{\mu\nu}(v,w) =\partial_{(\mu}v\,\partial_{\nu)}w -\frac12\eta_{\mu\nu} (\partial^\alpha v\,\partial_\alpha w+m^2vw). \] Its divergence is \[ \partial^\mu\cT_{\mu\nu}(v,w) =\frac12\left((Pv)\partial_\nu w+(Pw)\partial_\nu v\right)=0. \] This proves conservation term by term. Hermitian symmetry follows from the normal-ordered expression. \end{proof} For compact smooth symmetric contravariant \(J\), define \begin{equation} j_{\NW}(J)[\Psi,\Chi] =\int J^{\mu\nu} \bigl(\cR^{\NW}t^{\Psi,\Chi}\bigr)_{\mu\nu}\dd^4x. \label{eq:jform} \end{equation} \begin{proposition}[Model 3 metric-valued form] \label{prop:form3} Equation \eqref{eq:jform} is a well-defined sesquilinear form on \(\DD_3\), symmetric for real \(J\). Its unsmeared smooth matrix elements satisfy \[ G^{(1)}(j^{\Psi,\Chi}_{\NW}) =t^{\Psi,\Chi},\qquad D(j^{\Psi,\Chi}_{\NW})=0 \] with the explicitly prescribed Newton--wave initial data. The form is continuous for an algebraic Schwartz Fock topology. \end{proposition} \begin{proof} The first assertions follow from Lemma~\ref{lem:stress3} and Proposition~\ref{prop:response3}. Here is an explicit meaning of the final assertion. For an integer \(L\), let \[ s_L(u)=\norm u_2+ \sum_{|\alpha|\le L}\norm{\br p^L\partial_p^\alpha u}_2. \] Write \(e_n(u_1,\ldots,u_n)= a^\dagger(u_1)\cdots a^\dagger(u_n)\Omega/\sqrt{n!}\), and define \[ p_L(\Psi)=\inf_{\Psi=\sum_\ell c_\ell e_{n_\ell} (u_{\ell1},\ldots,u_{\ell n_\ell})} \sum_\ell|c_\ell|(n_\ell+2)^2 \prod_i s_L(u_{\ell i}). \] These are finite seminorms dominating the Hilbert norm. The annihilation formula \[ a(v)e_n(u_1,\ldots,u_n) =\frac1{\sqrt n}\sum_i\ip v{u_i}\, e_{n-1}(u_1,\ldots,\widehat u_i,\ldots,u_n) \] shows that any fixed source seminorm is bounded by \(Cp_L(\Psi)p_L(\Chi)\), with sufficiently large \(L\). There are at most two field factors, and the factors \((n+2)^2\) absorb the finite annihilation choices. Momentum integrations by parts control the uncontracted wave packets. Continuity of the classical response on a compact set containing \(\supp J\) therefore gives \[ |j_{\NW}(J)[\Psi,\Chi]| \le C_Jp_L(\Psi)p_L(\Chi). \] \end{proof} This is already a form-valued quantization of one specified reconstructed response. It is not an independent quantum metric field, a self-adjoint unregulated metric smearing, or a quantum Einstein--scalar constraint algebra. The distinction is substantive, not terminological. \subsection{Explicit scalar kernels} For \(\sigma,\tau\in\{+1,-1\}\), let \(u_p^+=u_p,\ u_p^-=\bar u_p\), and define \[ \ell_\mu(p)=(-\omega_p,p),\qquad k=\sigma p+\tau q,\quad r=|k|,\quad \Omega_{\sigma\tau}=\sigma\omega_p+\tau\omega_q. \] The polarized stress amplitude is \begin{equation} C_{\mu\nu}^{\sigma\tau}(p,q) =n_pn_q\left[ -\sigma\tau\ell_{(\mu}(p)\ell_{\nu)}(q) +\frac12\eta_{\mu\nu} \left(\sigma\tau\eta^{\alpha\beta} \ell_\alpha(p)\ell_\beta(q)-m^2\right) \right]. \label{eq:Cstress} \end{equation} It multiplies \(e^{-i\Omega_{\sigma\tau}t+ik\cdot x}\). For \(r>0\), the Newton--wave amplitude has initial tensors \[ A_{00}=\frac{2C_{00}}{r^2},\qquad A_{0i}=\frac{2C_{0i}}{r^2},\qquad A_{ij}=0, \] \[ B_{00}=ik_iA_{i0},\qquad B_{0i}=B_{ij}=0, \] and evolution \begin{equation} F_{\mu\nu}^{\sigma\tau}(t;p,q) =\cos(rt)A_{\mu\nu} +\frac{\sin(rt)}rB_{\mu\nu} +2C_{\mu\nu}^{\sigma\tau} \int_0^t\frac{\sin(r(t-s))}{r} e^{-i\Omega_{\sigma\tau}s}\dd s. \label{eq:transfer3} \end{equation} Define \[ \mathcal K_J^{\sigma\tau}(p,q) =\int J^{\mu\nu}(t,x)e^{ik\cdot x} (\mathsf rF^{\sigma\tau})_{\mu\nu}(t;p,q) \dd t\dd x. \] Plane waves are used here to compute a kernel, not as members of the original Schwartz source class. Local integrability is proved below. Choose a real even smooth compact mask \(\chi_\Lambda(p)= \chi(p/\Lambda)\), equal to one on \(|p|\le\Lambda\) and zero on \(|p|\ge2\Lambda\). Apply it to each scalar field leg. The cutoff field still solves \(P\widehat\phi_\Lambda=0\), so its normal-ordered stress is conserved. Set \[ h_\Lambda(p,q) =2\chi_\Lambda(p)\chi_\Lambda(q)\mathcal K_J^{-+}(p,q), \] \[ g_\Lambda(p,q) =2\chi_\Lambda(p)\chi_\Lambda(q)\mathcal K_J^{--}(p,q). \] \begin{lemma}[Transfer-momentum cancellation] \label{lem:kernels3} At fixed finite \(\Lambda\), \(g_\Lambda\) is a symmetric Hilbert--Schmidt pair tensor, and \(h_\Lambda\) is a bounded self-adjoint one-particle operator. On the cutoff support, \[ |\mathcal K_J^{++}|+|\mathcal K_J^{--}| \le C_{J,\Lambda}(1+|p+q|^{-1}), \] \[ |\mathcal K_J^{-+}|+|\mathcal K_J^{+-}| \le C_{J,\Lambda}(1+|p-q|^{-2}). \] No separate Newton infrared cutoff is needed. \end{lemma} \begin{proof} For the pair sector, \[ C_{00}^{++} =\frac{n_pn_q}{2}(-\omega_p\omega_q-p\cdot q+m^2), \] \[ C_{0i}^{++} =\frac{n_pn_q}{2}(\omega_pq_i+\omega_qp_i). \] The first numerator and its first \(q\)-derivatives vanish at \(q=-p\); the second numerator vanishes there. Taylor's theorem on the bounded cutoff support gives \[ C_{00}^{++}=O(|p+q|^2),\qquad C_{0i}^{++}=O(|p+q|). \] The same holds for \(--\). Thus \(A_{00}=O(1)\), \(A_{0i}=O(r^{-1})\), and \(B_{00}=O(1)\). Use \[ |\sin(rt)/r|\le |t|,\qquad \left|\int_0^t\frac{\sin(r(t-s))}{r}e^{-i\Omega s}\dd s\right| \le t^2/2 \] on the compact time support of \(J\). This proves the pair bound. Its square is locally integrable in the three-dimensional variable \(p+q\), since \(\int_{|k|<1}|k|^{-2}\dd k<\infty\). For mixed signs use \(A=O(r^{-2})\), \(B=O(r^{-1})\), without assuming a pair cancellation. The same estimates prove the mixed bound. Its Schur integrals are finite uniformly on the bounded support, because \(|p-q|^{-2}\) is locally integrable in three dimensions. Therefore it defines a bounded operator. Reality of \(J\) and the response gives \(h_\Lambda(p,q)=\overline{h_\Lambda(q,p)}\); polarized symmetry gives symmetry of \(g_\Lambda\). \end{proof} The distinction here is not the same as the large-momentum distinction of Model 2. These are scalar stress transfer kernels with \(\Lambda\) fixed. Figure~\ref{fig:source-transfer-cancellation} summarizes the pair cancellation and mixed-sector Schur bounds. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries] at (2.6,3.6) {Pair sector, $k=p+q$}; \node[cbox, font=\scriptsize] at (2.6,2.85) {singular locus $q=-p$}; \node[sbox, text width=5.2cm, font=\scriptsize] at (2.6,1.9) {$C_{00}^{++}=O(|k|^2)$,\quad $C_{0i}^{++}=O(|k|)$\\ same orders for $--$}; \node[mbox, text width=5.2cm, font=\scriptsize] at (2.6,0.85) {Newton factors $1/|k|^2$:\\ $A_{00}=O(1)$,\ $A_{0i}=O(|k|^{-1})$,\ $B_{00}=O(1)$}; \node[pbox, text width=5.2cm, font=\scriptsize] at (2.6,-0.3) {$|\mathcal K_J^{\mathrm{pair}}|\le C_{J,\Lambda}(1+|p+q|^{-1})$\\ square integrable in three-dimensional transfer momentum}; \node[font=\small\bfseries] at (9.2,3.6) {Mixed sector, $k=q-p$}; \node[pbox, text width=5.4cm, font=\scriptsize] at (9.2,2.4) {$|\mathcal K_J^{\mathrm{mixed}}|\le C_{J,\Lambda}(1+|p-q|^{-2})$\\ finite Schur integrals on bounded cutoff support}; \node[pbox, text width=5.4cm, font=\scriptsize] at (9.2,1.15) {$g_\Lambda$ Hilbert--Schmidt\\ $h_\Lambda$ bounded self-adjoint}; \node[gbox, text width=5.4cm, font=\scriptsize] at (9.2,0.15) {fixed finite $\Lambda$; no separate Newton IR mask}; \node[cbox, text width=5.4cm, font=\scriptsize] at (9.2,-0.9) {Not an identification with the Model 2 ultraviolet estimates.}; \end{tikzpicture} \caption{Sourced transfer-kernel estimates at fixed \(\Lambda\). Pair-sector stress numerators cancel the strongest Newton singularities, leaving a square-integrable transfer bound. The mixed-sector singularity instead has finite Schur integrals. These estimates justify the pair and number coefficients without an additional Newton infrared cutoff.} \label{fig:source-transfer-cancellation} \end{figure} \subsection{Regulated generators and coherent correspondence} Define on \(\Dom\NN\subset\HH_3\) \begin{equation} B_\Lambda(J) =\dG(h_\Lambda) +\frac12\bigl(a^\dagger(g_\Lambda)+a(g_\Lambda)\bigr). \label{eq:B3} \end{equation} The sector proof of Lemma~\ref{lem:quadratic2} gives \begin{equation} \norm{B_\Lambda(J)\Psi} \le C_\Lambda\norm{(\NN+2)\Psi},\qquad C_\Lambda=\norm{h_\Lambda}+\norm{g_\Lambda}_{\HS}. \label{eq:bound3} \end{equation} It is symmetric. Expanding the normally ordered stress and the response formula shows on \(\DD_3\) that its matrix elements equal \eqref{eq:jform} with the cutoff scalar field. The factors two above count the two mixed polarized terms. Fubini is justified by the locally integrable transfer bounds. Thus \[ B_{\Lambda,M}(J)=\Pi_MB_\Lambda(J)\Pi_M,\qquad \Pi_M=I_{\{\NN\le M\}}, \] is bounded self-adjoint, even though \(\Pi_M\) need not have finite rank. The momentum-kernel estimate, not number compression alone, supplies boundedness. Figure~\ref{fig:model3-form-hierarchy} distinguishes the unregulated response form from the regulated measured operator. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries, align=center] at (3.4,0.4) {unregulated metric-valued sesquilinear form}; \node[sbox, text width=7.2cm] (s1) at (3.4,-0.6) {1. Algebraic Schwartz Fock vectors $\Psi,\Chi$}; \node[sbox, text width=7.2cm, font=\scriptsize] (s2) at (3.4,-1.7) {2. $t^{\Psi,\Chi}_{\mu\nu}=\ip{\Psi}{:T_{\mu\nu}:\Chi}$}; \node[mbox, text width=7.2cm, font=\scriptsize] (s3) at (3.4,-2.8) {3. $j^{\Psi,\Chi}_{\NW}=\cR^{\NW}t^{\Psi,\Chi}$}; \node[pbox, text width=7.2cm, font=\scriptsize] (s4) at (3.4,-3.9) {4. $j_{\NW}(J)[\Psi,\Chi]=\int J j^{\Psi,\Chi}_{\NW}$}; \draw[arr] (s1)--(s2); \draw[arr] (s2)--(s3); \draw[arr] (s3)--(s4); \node[cbox, anchor=west, text width=5.6cm, font=\scriptsize] (cut) at (8.2,-0.6) {$\chi_\Lambda$ on each scalar leg}; \node[pbox, anchor=west, text width=5.6cm, font=\scriptsize] (B) at (8.2,-1.7) {$B_\Lambda(J)$}; \node[pbox, anchor=west, text width=5.6cm, font=\scriptsize] (BM) at (8.2,-2.8) {$\Pi_MB_\Lambda(J)\Pi_M$}; \node[pbox, anchor=west, text width=5.6cm, font=\scriptsize] (U) at (8.2,-3.9) {$e^{i\tau\lambda B_{\Lambda,M}}$\\ bounded self-adjoint}; \draw[arr] (cut)--(B); \draw[arr] (B)--(BM); \draw[arr] (BM)--(U); \node[cbox, text width=5.6cm, font=\scriptsize] at (11.0,-5.1) {Boundedness needs the kernel estimates, not number compression alone.}; \node[gbox, font=\scriptsize] at (2.2,-5.2) {no graviton factor}; \node[gbox, font=\scriptsize] at (5.6,-5.2) {no independent quantum metric field}; \node[gbox, font=\scriptsize, text width=4.2cm] at (9.6,-6.15) {no unregulated metric instrument}; \end{tikzpicture} \caption{Model 3 representation hierarchy. Conserved normal-ordered stress matrix elements feed a specified classical response and define a metric-valued form. Momentum-kernel estimates produce a regulated quadratic operator, whose number compression is the bounded measured generator. No independent gravitational quantum degree of freedom is introduced.} \label{fig:model3-form-hierarchy} \end{figure} For a Schwartz coherent label \(z\), let \[ \phi_z(t,x)=\int(u_pz(p)+\bar u_p\bar z(p))\dd p, \qquad \phi_{\Lambda,z}=\phi_{\chi_\Lambda z}. \] Normal ordering gives the exact identity \begin{equation} \ip{\Coh(z)}{B_\Lambda(J)\Coh(z)} =\int J^{\mu\nu} \bigl(\cR^{\NW}T(\phi_{\Lambda,z})\bigr)_{\mu\nu} \dd^4x. \label{eq:coherent3} \end{equation} This follows either by coherent contraction of \eqref{eq:B3} or by substituting the classical one-point function in the normal-ordered stress. The number estimate justifies the coherent expectations. As \(\Lambda\to\infty\), \(\chi_\Lambda z\to z\) in Schwartz topology. Wave-packet estimates and classical response continuity therefore give convergence of the right side to the uncut classical pairing. This is an expectation limit, not an operator or instrument limit. For \(\Psi=\Coh(z)\), \(\nu=\norm z^2\), define \[ t_M(\nu)^2 =e^{-\nu}\sum_{n>M}(n+2)^2\frac{\nu^n}{n!}. \] Then \[ \norm{(\NN+2)\Psi}^2=\nu^2+5\nu+4, \] and \begin{equation} \left| \ip\Psi{B_{\Lambda,M}\Psi} -\ip\Psi{B_\Lambda\Psi}\right| \le E_M^{(3)} :=2C_\Lambda\sqrt{\nu^2+5\nu+4}\,t_M(\nu)\longrightarrow0. \label{eq:tail3} \end{equation} To verify it, write \(P=\Pi_M\), \(r=(I-P)\Psi\), and expand the difference as \(\ip r{B_\Lambda\Psi}+\ip{P\Psi}{B_\Lambda r}\). Apply \eqref{eq:bound3}. The vacuum expectation of every compression is exactly zero. \subsection{A compact exterior curvature current} Use the convention \[ R^{(1)}_{1212}(h) =\partial_1\partial_2h_{12} -\frac12\partial_1^2h_{22} -\frac12\partial_2^2h_{11}. \] For real compact \(\psi\), define the only nonzero entries of \(J_\psi\) by \begin{equation} J_\psi^{11}=-\tfrac12\partial_2^2\psi,\qquad J_\psi^{22}=-\tfrac12\partial_1^2\psi,\qquad J_\psi^{12}=J_\psi^{21}=\tfrac12\partial_1\partial_2\psi. \label{eq:Jpsi} \end{equation} Direct differentiation gives \(\partial_\mu J_\psi^{\mu\nu}=0\), and integration by parts gives \[ \int J_\psi^{\mu\nu}h_{\mu\nu}\dd^4x =\int\psi R^{(1)}_{1212}(h)\dd^4x. \label{eq:curvpair} \] Thus this current annihilates all smooth pure gauges. Choose nonzero real radial \(F\in C_c^\infty(\R^3)\), supported in \(|x|\le r_0\), and take \[ \phi_*(0)=F,\qquad \dot\phi_*(0)=0. \] Then \[ \rho=\frac12(|\nabla F|^2+m^2F^2),\qquad J_i=T_{0i}(0)=0,\qquad E=\int\rho>0. \] Both response prescriptions consequently have identical initial data for this particular source. Newton's radial formula gives, for \(|x|>r_0\), \[ j_{ij}(0,x)=\frac{E}{4\pi|x|}\delta_{ij}. \] At \(x_*=(0,0,r_*)\), \(r_*>r_0\), \[ R^{(1)}_{1212}(j)(0,x_*) =\frac{E}{4\pi r_*^3}>0. \] The scalar support at time \(t\) is contained in \(|x|\le r_0+|t|\). Smoothness therefore permits a small future neighborhood of \((0,x_*)\) on which the curvature remains positive and the scalar vanishes. Choose a nonzero nonnegative \(\psi\) supported there. Then \begin{equation} d=\int\psi R^{(1)}_{1212} (\cR^{\NW}T(\phi_*))\dd^4x>0,\qquad \supp\psi\cap\supp\phi_*=\varnothing. \label{eq:d3} \end{equation} The coherent label is \(z_*(p)=\sqrt{\omega_p/2}\widehat F(p)\), which is Schwartz and produces the stated data. By \eqref{eq:coherent3}, one may fix a sufficiently large finite \(\Lambda\) such that \begin{equation} d_\Lambda =\ip{\Coh(z_*)}{B_\Lambda(J_\psi)\Coh(z_*)}\ge d/2>0. \label{eq:dLambda} \end{equation} A momentum cutoff destroys exact compact spatial support. Thus support separation in \eqref{eq:d3} is a classical uncut statement; \eqref{eq:dLambda} transfers its compact pairing, not its pointwise support property. Figure~\ref{fig:exterior-curvature-witness} shows the uncut exterior certificate and the more limited transfer to a finite momentum cutoff. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[font=\small\bfseries] at (1.8,3.7) {(a) uncut support}; \draw[->, sgray] (0.2,0.3) -- (0.2,3.2) node[above, font=\scriptsize] {$t$}; \draw[->, sgray] (0.2,1.3) -- (3.5,1.3) node[right, font=\scriptsize] {$|x|$}; \fill[sgreen!18] (0.2,1.3) -- (1.05,1.3) -- (2.15,3.0) -- (0.2,3.0) -- cycle; \fill[sgreen!18] (0.2,1.3) -- (1.05,1.3) -- (2.15,-0.2) -- (0.2,-0.2) -- cycle; \draw[sgreen] (1.05,1.3) -- (2.15,3.0); \draw[sgreen] (1.05,1.3) -- (2.15,-0.2); \node[font=\scriptsize, sgreen, align=center] at (0.7,2.45) {$|x|\le r_0+|t|$}; \fill[mblue] (1.85,1.3) circle (1.5pt); \node[font=\scriptsize, mblue, below] at (1.85,1.25) {$x_*=(0,0,r_*)$}; \node[sbox, font=\scriptsize] at (2.7,2.15) {$\psi$}; \node[gbox, font=\scriptsize, text width=3.4cm] at (1.8,0.15) {schematic; $\psi$ outside the scalar cone}; \node[font=\small\bfseries] at (6.6,3.7) {(b) curvature certificate}; \node[mbox, text width=4.8cm, font=\scriptsize] at (6.6,2.7) {$j_{ij}(0,x)=\dfrac{E}{4\pi|x|}\delta_{ij}$}; \node[mbox, text width=4.8cm, font=\scriptsize] at (6.6,1.75) {$R^{(1)}_{1212}(j)(0,x_*)=\dfrac{E}{4\pi r_*^3}>0$}; \node[pbox, text width=4.8cm, font=\scriptsize] at (6.6,0.8) {$d=\int\psi R^{(1)}_{1212}(j)>0$}; \node[font=\small\bfseries] at (6.6,-0.15) {(c) conserved current}; \node[pbox, text width=4.8cm, font=\scriptsize] at (6.6,-1.0) {nonzero entries of $J_\psi$ only\\ $\partial_\mu J_\psi^{\mu\nu}=0$\\ $\int J_\psi h=\int\psi R^{(1)}_{1212}(h)$}; \node[cbox, text width=9.2cm, font=\scriptsize] at (4.6,-2.35) {Momentum cutoff destroys exact compact spatial support. Only the pairing transfers: $d_\Lambda\ge d/2$ for a selected finite $\Lambda$.}; \end{tikzpicture} \caption{Exterior curvature witness for Model 3. A radial compact scalar preparation produces positive initial exterior curvature, giving a positive compact conserved-current pairing on a short future slab. Coherent expectation convergence transfers this pairing to a sufficiently large finite momentum cutoff, but not the uncut pointwise support separation.} \label{fig:exterior-curvature-witness} \end{figure} \subsection{The source-sensitive record} \begin{theorem}[Model 3 record: fixed scalar cutoff and finite \(M\)] \label{thm:model3} Fix \(J_\psi,z_*,\Lambda\) as above, and take \[ \Psi_0=\Omega,\qquad \Psi_1=\Coh(z_*),\qquad A^{(3)}_{\Lambda,M}(\lambda) =\lambda B_{\Lambda,M}(J_\psi). \] Choose \(\lambda>0\), let \(\nu=\norm{z_*}^2\), and put \[ S_3=\lambda^2C_\Lambda^2(\nu^2+5\nu+8)>0. \] For the common instrument \eqref{eq:Kraus}, choose \[ \theta=\pi/2,\qquad 0<\tau\le\lambda d_\Lambda/S_3. \] Then for every finite \(M\), \[ p^{(3)}_{+,M}(\Psi_1)-p^{(3)}_{+,M}(\Psi_0) \ge c_3-e_M^{(3)}, \] where \[ c_3=\frac{\tau\lambda d_\Lambda}{4}>0,\qquad e_M^{(3)}=\frac{\tau\lambda}{2}E_M^{(3)}\longrightarrow0. \] The detector, states, \(\Lambda,\lambda,\tau,\theta\) are independent of \(M\). In particular the contrast is positive for every sufficiently large finite \(M\). At \(\lambda=0\), both probabilities are exactly \(1/2\). Replacing the stress source by zero also makes both probabilities exactly \(1/2\). No graviton factor is present, and neither \(\Lambda\) nor the measured number cutoff is removed in this theorem. \end{theorem} \begin{proof} The vacuum mean vanishes. Equations \eqref{eq:dLambda} and \eqref{eq:tail3} give a mean difference at least \(\lambda(d_\Lambda-E_M^{(3)})\). The number bound gives \[ \sum_{s=0,1}\norm{A^{(3)}_{\Lambda,M}(\lambda)\Psi_s}^2 \le S_3, \] since the vacuum value of \(\norm{(\NN+2)\Omega}^2\) is four. At \(\theta=\pi/2\), Lemma~\ref{lem:Ramsey} gives \[ \Delta p\ge \frac{\tau\lambda}{2}(d_\Lambda-E_M^{(3)}) -\frac{\tau^2S_3}{4} \ge c_3-e_M^{(3)}. \] The tail tends to zero. At \(\lambda=0\), the generator is zero and \(U=I\); the chosen phase gives \(p_+=1/2\) in every state. If \(T\) is replaced by zero, linearity of the response makes every kernel and generator zero, proving the same control. Setting the scalar mass to zero is not this source-off operation. \end{proof} Figure~\ref{fig:model3-record-controls} displays the source-sensitive record and its two exact controls. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[sbox] (vac) {$\Psi_0=\Omega$}; \node[sbox, right=8mm of vac] (coh) {$\Psi_1=\Coh(z_*)$}; \node[pbox, below=4mm of vac, text width=8.4cm] (A) {$A^{(3)}_{\Lambda,M}=\lambda\Pi_MB_\Lambda(J_\psi)\Pi_M$}; \draw[arr] (vac.south) -- (A.north west); \draw[arr] (coh.south) -- (A.north east); \node[gbox, text width=8.4cm, font=\scriptsize, below=2mm of A] {$J_\psi$, $z_*$, $\Lambda$, $\lambda$, $\tau$, $\theta=\pi/2$\\ independent of $M$}; \node[pbox, text width=8.4cm, font=\scriptsize, below=2mm of A, yshift=-12mm] {$p^{(3)}_{+,M}(\Psi_1)-p^{(3)}_{+,M}(\Psi_0)\ge c_3-e_M^{(3)}$,\quad $e_M^{(3)}\to 0$}; \node[gbox, font=\scriptsize, below=14mm of A, yshift=-8mm] {positive for sufficiently large finite $M$}; \node[sbox, text width=4.2cm, font=\scriptsize, below left=22mm and -4mm of A] (offl) {interpolation off: $\lambda=0$}; \node[sbox, text width=4.6cm, font=\scriptsize, below right=22mm and -4mm of A] (offs) {source off: replace $T$ by zero}; \node[pbox, text width=9.2cm, font=\scriptsize, below=32mm of A] {both give $A=0$, $U=I$, and $p_+(\Psi_0)=p_+(\Psi_1)=1/2$}; \node[cbox, text width=9.2cm, font=\scriptsize, below=2mm of offl, yshift=-16mm] {The massless scalar is not the source-off operation.}; \node[cbox, font=\scriptsize, below=46mm of A] {$\Lambda$ and measured $M$ stay retained.}; \end{tikzpicture} \caption{Model 3 record and controls. At a selected finite momentum cutoff, the coherent preparation has a positive curvature-pairing mean relative to the vacuum. Number-tail estimates give positive contrast for sufficiently large finite compressions. Interpolation-off and explicit source-off operations both give the exact probability \(1/2\) at the chosen Ramsey phase.} \label{fig:model3-record-controls} \end{figure} %===================================================================== \section{Comparison ledger and non-identifications} \label{sec:comparison} %===================================================================== The common apparatus pattern does not identify its generators. \begin{center} \small \renewcommand{\arraystretch}{1.15} \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}XXX@{}} \toprule & Model 1 & Model 2 & Model 3\\ \midrule Independent gravity & One canonical oscillator & Vacuum TT Fock space & None\\ Classical quantity & \(\ell_0+q\ell_e+\pi\ell_r\) with formal coefficient bookkeeping & Bilinear \(\cB_f(k,\phi)\) & Quadratic \(\int J\,\cR^{\NW}T(\phi)\)\\ State comparison & Gravitational displacement; scalar fixed & Gravitational displacement; scalar fixed & Scalar vacuum versus scalar coherent\\ Generator & ESA polynomial & Bounded number compression & Bounded number compression\\ Spectral removal & Initial-time edge removed & \(R\to\infty\) at fixed \(M\) & No measured \(\Lambda\)-removal\\ Off control & Scalar-only evolution & Scalar-only evolution & Zero generator\\ Positive record & \(c_1-e_\varepsilon\) & \(c_2-e_M-e_R(M)\) & \(c_3-e_M^{(3)}\)\\ \bottomrule \end{tabularx} \end{center} \subsection{Classical response freedom} For any two harmonic response maps solving \(G^{(1)}(\cR_iQ)=Q\), their difference is a homogeneous harmonic linearized vacuum solution. Equality of trace-reversed Cauchy data implies equality of the maps by wave uniqueness. Without equality of data, no equality follows from the sourced equation alone. For the two actual prescriptions in Section~\ref{sec:classical3}, the comparison was executed: their difference is the explicit \(K\xi\) in Proposition~\ref{prop:mapcomparison}. They therefore agree on compact conserved currents, including \(J_\psi\), but not on arbitrary metric test tensors. This is the precise transfer available here. \subsection{Formal coefficients versus numerical measurements} Equation \eqref{eq:formalCCR} is a first-order coefficient identity for a declared initial-data bracket. The spectral calculus in Theorem~\ref{thm:model1} belongs to a numerical polynomial model. No estimate compares it with an exact nonlinear gravitational observable. The dense-TT kernel estimate concerns a vacuum bilinear. The stress-kernel estimate concerns a scalar quadratic composite. Their transfer variables, ultraviolet estimates, and representations differ. One cannot feed the stress source into the dense-TT Fock factor without defining a new model and proving its constraints and domains. \subsection{A metric form is not a cubic scalar observable} Model 3 represents matrix elements of a prescribed sourced response and certain regulated curvature smearings. It therefore establishes a restricted, form-level quantization of that response. It does not establish an unregulated self-adjoint metric field, a canonical metric algebra, or a quantization of \[ S_{t_0}(\cR^{\NW}T(\phi))\phi. \] The latter is cubic even at the classical level. Its ordering, distributional kernels, and domain estimates are separate tasks. \subsection{Global apparatus is the surviving implementation boundary} The normal instruments are explicitly defined on \(\mathcal B(\HH_i)\) or its trace-class predual, and admit a qubit controlled-unitary dilation. Those facts are proved. Compact support of \(f\) or \(J_\psi\) identifies a smearing in a classical or form-valued functional. It does not show that the resulting controlled unitary is generated by a compact relativistic system--probe interaction. Likewise, finite active-mode support does not mean spacetime-local apparatus access. No external background-anomaly calibration or two-probe mixed-mode disturbance is needed for any of the three proofs. Those experiments concern different measured quantities. %===================================================================== \section{Successor tasks and conclusion} \label{sec:successors} %===================================================================== The following four arrows are outside the measurement theorems. They are stated as extensions of scope, not as claims that every abstract operator-existence subquestion is mathematically undecided. \scopebox{Four successor tasks}{% \textbf{1. Uncompressed dense-TT dynamics.} Select and analyze the self-adjoint realization, invariant domains, and physical interpretation needed for a specified uncompressed measurement problem. The symmetric operator constructed here is already closable; asking whether it is closable would not be a remaining problem. \medskip \textbf{2. Number-cutoff instrument removal.} Prove a specified \(M\to\infty\) instrument limit, with its topology, input class, references, and resource restrictions. Agreement of finite-particle form matrix elements is not that theorem. \medskip \textbf{3. Beyond the sourced quadratic form.} Construct an unregulated operator-valued sourced metric, or the cubic scalar correction \(S_{t_0}(j)\phi\), with the required ordering and domain theory. Model 3 supplies a particular quadratic response form and regulated curvature measurements, not these stronger objects. \medskip \textbf{4. Compact probe coupling.} Replace the admitted global Ramsey control by a specified compact system--probe field interaction and prove equality or a controlled approximation of its selective instrument. Compact smearing alone does not provide such a coupling.} Figure~\ref{fig:endpoints-successors} separates the completed measurement endpoints from the four stated extensions of scope. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[mbox, text width=4.2cm, font=\scriptsize] (m1) {Model 1\\ fixed plane; essentially self-adjoint numerical generator; edge removed}; \node[mbox, right=3mm of m1, text width=4.3cm, font=\scriptsize] (m2) {Model 2\\ evaluated TT kernels; operator-norm spectral removal at fixed finite $M$}; \node[sbox, right=3mm of m2, text width=4.3cm, font=\scriptsize] (m3) {Model 3\\ prescribed stress-response form; regulated exterior curvature record on scalar Fock space}; \node[pbox, below=5mm of m2, text width=8.6cm] (common) {global normal binary instrument with exact off-control}; \draw[arr] (m1.south) -- (common.north west); \draw[arr] (m2.south) -- (common.north); \draw[arr] (m3.south) -- (common.north east); \node[gbox, font=\scriptsize, below=2mm of common] {The three generators and Hilbert spaces are not merged.}; \draw[sgray, dashed] (-6.6,-3.15) -- (6.8,-3.15); \node[font=\scriptsize, sgray] at (0,-3.4) {scope boundary}; \node[gbox, text width=3.3cm, font=\scriptsize] (s1) at (-5.0,-4.5) {1. specified uncompressed dense-TT dynamics}; \node[gbox, text width=3.3cm, font=\scriptsize] (s2) at (-1.5,-4.5) {2. number-cutoff instrument removal}; \node[gbox, text width=3.5cm, font=\scriptsize] (s3) at (2.1,-4.5) {3. unregulated sourced metric or cubic scalar correction}; \node[gbox, text width=3.3cm, font=\scriptsize] (s4) at (5.7,-4.5) {4. compact system--probe coupling}; \draw[arrg] (common.south) -- (s1.north); \draw[arrg] (common.south) -- (s2.north); \draw[arrg] (common.south) -- (s3.north); \draw[arrg] (common.south) -- (s4.north); \node[gbox, font=\scriptsize, text width=12.4cm] at (0,-5.8) {additional definitions and proofs required\\ dense-TT symmetric operator already closable\\ compact smearing does not imply compact apparatus}; \end{tikzpicture} \caption{Completed endpoints and successor tasks. The three models establish different represented records using a common global apparatus pattern. Uncompressed dynamics, measured number-cutoff removal, stronger sourced observables, and compact probe implementation require separate constructions. The diagram does not identify the models or recast established closability as an open question.} \label{fig:endpoints-successors} \end{figure} The three completed endpoints are different. Model 1 has a fixed plane and an essentially self-adjoint numerical polynomial, with an edge regulator removed. Model 2 has evaluated vacuum TT kernels and a spectral limit in bounded generators at each fixed number cutoff. Model 3 has a scalar stress response form, an exterior classical curvature certificate, and a regulated scalar-Fock instrument. Every contrast includes its exact off-switch. Every convergence statement specifies the regulator that remains. None of these results turns the three models into a single sourced quantum measurement theory. %===================================================================== \appendix \clearpage \section{Cutoff ledger: Model 1} \label{app:one} %===================================================================== \subsection*{Objects and domains} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}} \toprule Object & Definition and status\\ \midrule Gravitational sector & The fixed real plane \(\Span\{e,r\}\), with \(\sigma_{\mathrm{plane}}(e,r)=1\). It is the model, not a regulator.\\ Quantum space & \(L^2(\R)\otimes\Gamma_s(L^2(\R^3))\).\\ Scalar coefficients & \(u_{0,f},u_{e,f},u_{r,f}\in\Sch(\R^3)\), obtained from the zero-data map and the signed effective smearing \eqref{eq:gplane}.\\ Initial operator domain & All finite total-particle vectors, with arbitrary one-particle labels.\\ Closure & Essentially self-adjoint; \(\Dom\NN\) is contained in its domain.\\ Active scalar space & At most three detector-dependent one-particle directions.\\ Formal parameter & \(\kappa^2=0\), used only for coefficient correspondence.\\ Numerical interpolation & \(\lambda\in\R\), defining the separately specified operator.\\ \bottomrule \end{tabularx} \end{center} \subsection*{Edge error and measured limit} The only removed regulator is \(\varepsilon\): \[ \norm{u_{a,f,\varepsilon}-u_{a,f}} \le B_{a,f}\varepsilon,\qquad a=e,r. \] For \(2\overline C|\tau|<1\), \[ \norm{(U_\varepsilon-U)\Psi_{\le n}} \le \frac{2D_\varepsilon|\tau|(n+1)} {(1-2\overline C|\tau|)^{n+2}}\norm{\Psi_{\le n}}. \] The coherent-state remainder is \[ \mathcal E_\varepsilon(\mu;\tau) = \inf_n\left[ 2\sqrt{\Pr_{\mathrm{Pois}(\mu)}(N>n)} +\frac{2D_\varepsilon|\tau|(n+1)} {(1-2\overline C|\tau|)^{n+2}}\right]. \] It tends to zero by choosing \(n\) first and then \(\varepsilon\). Strong convergence for arbitrary fixed time follows by subdivision, not by extrapolating the small-time bound. Figures~\ref{fig:edge-removal} and \ref{fig:model1-record} summarize the removed regulator and the associated record control. \subsection*{Record and control} The two states have the same scalar coherent factor. Their oscillator covariances agree and their means differ by \((\Delta Q,\Delta\Pi_0)=(1,0)\). The fixed detector satisfies \(d_*\ge d_0>0\). For the fixed calibration of Theorem~\ref{thm:model1}, \[ |\Delta p_\varepsilon|\ge c_1-e_\varepsilon,\qquad e_\varepsilon\to0. \] At \(\lambda=0\), the generator acts on the scalar factor alone, so equality of the probabilities is exact. No graviton-sector removal, sourced Einstein constraint, or coupled reduced symplectic-form identification appears in this theorem. \clearpage \section{Cutoff ledger: Model 2} \label{app:two} %===================================================================== \subsection*{The spectral family} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.28\linewidth}X@{}} \toprule Quantity & Meaning\\ \midrule One-particle measure & \(\dd p\) on TT tensor functions and \(\dd q\) on scalar functions. The mode functions contain the frequency denominators.\\ \(\alpha_R\) & Gravitational IR and UV mask: one on \(R^{-1}\le|p|\le R\), supported in \((2R)^{-1}<|p|<2R\).\\ \(\beta_R\) & Scalar UV mask: one on \(|q|\le R\), supported in \(|q|<2R\).\\ \(M\) & Total particle-number cutoff on the mixed quadratic.\\ Free-term cutoff & Scalar number cutoff only; it must not depend on gravitational particle number.\\ \(N\) & Finite-rank projection of gravitational kernel legs.\\ \bottomrule \end{tabularx} \end{center} \subsection*{Kernel topology} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.31\linewidth}X@{}} \toprule Coefficient & Proven result\\ \midrule Pair \(P_R\) & Hilbert--Schmidt convergence: \(\norm{P_R-P}_{\HS}\le C_PR^{-1/2}\).\\ Number leg \(L_R\) & Operator-norm convergence: \(\norm{L_R-L}\le C_LR^{-1}\); \(L\) is compact.\\ Number Hilbert--Schmidt norm & For nonzero \(f\), \(\norm{L_R}_{\HS}^2\) is comparable to \(R\). The limit is not Hilbert--Schmidt.\\ Consolidated pair & \(\norm{K_R-K}=\sqrt2\norm{P_R-P}_{\HS}\).\\ Constant coefficient & Zero; cross-species reordering produces no constant.\\ Uncut pairing & Agrees with the symmetric \(\Dom\NN\) operator form on the original dense test domain.\\ \bottomrule \end{tabularx} \end{center} \subsection*{Dynamical limit and record} For every fixed finite \(M\ge2\), \[ \norm{B_{R,M}-B_{\infty,M}} \le MC_LR^{-1} +\sqrt{2M(M-1)}C_PR^{-1/2}. \] Thus \[ \norm{U_{R,M}-U_{\infty,M}} \le|\tau\lambda|\overline\delta_M(R). \] The probability cost is half this norm per state, and the two-state contrast cost is at most the full norm. The witness has compact gravitational momentum support but only Schwartz scalar momentum decay. Accordingly \(d_R\to d_*\) superpolynomially; it is not generally eventually constant. Uniform number estimates justify a single time calibration for all spectral cutoffs. At \(\lambda=0\), the identical scalar factors give exact equality. The \(R,N\) limits may be combined at fixed \(M\) using the explicit compact-operator and Hilbert-tensor remainder. No measured \(M\to\infty\) limit is included. Figures~\ref{fig:model2-kernel-ideals} and \ref{fig:fixed-M-convergence} distinguish the coefficient topologies and the limits available at fixed number cutoff. \clearpage \section{Cutoff ledger: Model 3} \label{app:three} %===================================================================== \subsection*{Classical map and quantum object} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}} \toprule Object & Definition and status\\ \midrule First vacuum coefficient & \(k=0\).\\ Classical map used & The specified \(\cR^{\NW}_{t_0}\), with nonzero constraint-compatible trace-reversed initial data.\\ Comparison map & \(\cR^{\mathrm C}_{t_0}\); the actual data comparison gives \(\cR^{\NW}Q-\cR^{\mathrm C}Q=K\xi\), not equality.\\ Classical agreement & Linearized curvatures and compact conserved-current pairings agree for the two maps.\\ Hilbert space & Scalar Fock space only.\\ Unregulated object & The metric-valued sesquilinear form on algebraic Schwartz Fock vectors.\\ Normal ordering & Minimal vacuum-normal-ordered massive scalar stress, with no unspecified improvement term.\\ \bottomrule \end{tabularx} \end{center} \subsection*{Cutoff kernel estimates} At fixed scalar momentum cutoff \(\Lambda\), \[ |\mathcal K_J^{\mathrm{pair}}(p,q)| \le C_{J,\Lambda}(1+|p+q|^{-1}), \] \[ |\mathcal K_J^{\mathrm{mixed}}(p,q)| \le C_{J,\Lambda}(1+|p-q|^{-2}). \] The first is square integrable on the cutoff support. The second has finite Schur integrals there. The cancellation \[ C_{00}^{++}=O(|p+q|^2),\qquad C_{0i}^{++}=O(|p+q|) \] is essential for the pair estimate. No Newton infrared mask is used. The resulting symmetric operator obeys \[ \norm{B_\Lambda(J)\Psi} \le C_\Lambda\norm{(\NN+2)\Psi}, \] and \(B_{\Lambda,M}\) is bounded self-adjoint for each finite \(M\). This is not a consequence of number compression alone. \subsection*{Witness, limit boundary, and controls} The compact curvature current \(J_\psi\) is conserved. Its classical support is exterior to the uncut compact scalar preparation on a short slab. The scalar momentum cutoff does not preserve that exact support statement. Only the compact curvature pairing is transferred by coherent expectation convergence, allowing one finite \(\Lambda\) to be fixed. With \(\nu=\norm{z_*}^2\), \[ E_M^{(3)} =2C_\Lambda\sqrt{\nu^2+5\nu+4} \left[e^{-\nu}\sum_{n>M}(n+2)^2\frac{\nu^n}{n!}\right]^{1/2}. \] Theorem~\ref{thm:model3} gives \[ \Delta p_M\ge c_3-\frac{\tau\lambda}{2}E_M^{(3)}. \] The scalar vacuum is compared with a scalar coherent state. At \(\lambda=0\), or when \(T\) is explicitly replaced by zero, both probabilities are \(1/2\). Neither the \(\Lambda\)-expectation argument nor the \(M\)-tail bound supplies an unregulated metric instrument. The cubic \(S_{t_0}(\cR T(\phi))\phi\) is not represented by these kernels. Figures~\ref{fig:exterior-curvature-witness} and \ref{fig:model3-record-controls} separate the classical support certificate from the regulated record and its controls. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Mathematical scope.} All three represented models, their preparations, kernels, operators, and global instruments are defined in this manuscript. The proofs do not require companion manuscript citations. The single cited article supplies contextual terminology concerning measurement semantics and represented implementations; it is not used to establish any of the analytic estimates. \paragraph{Data and materials.} This is a theoretical study. No empirical data are used. The manuscript source is self-contained apart from standard \LaTeX{} packages. \paragraph{Verification status.} No proof-assistant verification, executed compiler check, or independent review is claimed. \begin{thebibliography}{1} \addcontentsline{toc}{section}{References} \bibitem{EnterCalculus} 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:10.3389/fphy.2026.1954943. \newblock \url{https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2026.1954943/}. \end{thebibliography} \end{document}