% !TEX program = pdflatex %===================================================================== % X-D.tex % % Standalone source for pdfLaTeX. % % Build: % pdflatex -interaction=nonstopmode -halt-on-error X-D.tex % pdflatex -interaction=nonstopmode -halt-on-error X-D.tex % % Combined and corrected manuscript: % X-D-v2 supplies the release, reduction, and identification. % X-D-v1 supplies the normal-flux estimate and explicit limit % representatives for the equation ideal. % % Frozen: % Fixed f, g, h, k; fixed plateau gauge algebra. % Scalar quantum; metric coefficients classical and off shell. % Completed X-C-v3 observable; no new retarded map. % % Result: % Completed tests converge in one fixed D_C(M), at rate O(epsilon) % in every smooth seminorm, hence in the quotient test topology. % The limit is the sharp action-level transmission prescription. % % Corrections incorporated: % - The switched Euler coefficients do not depend on epsilon. % - The rate proof integrates the normal flux before estimating; % it does not differentiate the ramp to estimate the solution. % - Green-operator continuity is used with a finite, unspecified % seminorm loss, rather than an unproved fixed derivative count. % - Limiting initial positions vanish; limiting initial velocities % are generally nonzero. % - The action-level scalar-background kernel has a possible jump % and no hypersurface delta. This is not joint sharp admission. % - VII-C is supplied and its actual (E1)--(E4) are audited. % Its multiplet, insertion, approximants, topology, and mandatory % Wick word are not constructed in X-D. % - No claim of successful compilation or independent verification. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlength{\parskip}{2pt} \setlist{itemsep=3pt,topsep=5pt,leftmargin=*} \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} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{diagnostic}[theorem]{Diagnostic} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Int}{int} \DeclareMathOperator{\WF}{WF} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\Lie}{\mathcal L} \newcommand{\sig}{\sigma_\eps} \newcommand{\Pm}{P_m} \newcommand{\MT}{M_T} \newcommand{\Dspace}{\mathcal D} \newcommand{\Afree}{\mathfrak A_0} \newcommand{\Jfree}{\mathfrak J_0} \newcommand{\ga}{\mathfrak g_U} \newcommand{\Gp}{\mathcal G_U} \newcommand{\ER}{\mathsf E_m^{\mathrm R}} \newcommand{\EA}{\mathsf E_m^{\mathrm A}} \newcommand{\ERzero}{\mathsf E_0^{\mathrm R}} \newcommand{\EAzero}{\mathsf E_0^{\mathrm A}} \newcommand{\Lone}{\mathsf L_{1,g}} \newcommand{\Ltwo}{\mathsf L_{2,g}} \newcommand{\Rq}{\mathscr R_q} \newcommand{\field}{\widehat\phi} \newcommand{\Xe}[1]{\mathsf X_{#1,\eps}} \newcommand{\Xz}[1]{\mathsf X_{#1,0}} \newcommand{\ce}[1]{c_{#1,\eps}} \newcommand{\cz}[1]{c_{#1,0}} \newcommand{\de}[1]{d_{#1,\eps}} \newcommand{\dz}[1]{d_{#1,0}} \newcommand{\WC}{\mathsf C} \newcommand{\WS}{\mathsf S} \newcommand{\formal}{\pmod{\kappa^3}} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} % Symbols used only when copying the supplied VII-C assumption. \newcommand{\Pol}{\mathscr P_{\mathrm{mc}}} \newcommand{\cA}{\mathcal A} \newcommand{\ev}{\operatorname{ev}_0} \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{Landmark X-D}\\[0.4em] Convergence of the completed ramp relational scalar\\ as the regulator width tends to zero} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We decide the sharp limit of the completed scalar-only observable of Landmark X-C-v3 for fixed observation test, interaction switch, classical metric coefficients, and plateau gauge algebra. The observation and interaction supports remain separated from the initial surface. Both completed coefficient tests converge in the smooth topology of one regulator-independent compact test space, with rate \(O(\eps)\) in every seminorm, and hence in the quotient test topology of the free scalar algebra. The proof integrates the coordinate normal flux before estimating the boundary layer. The limiting positive-order initial positions vanish. The limiting velocities are \(-a_1^{0\rho}[h]|_\Sigma\) and \((a_1^{00}[h]a_1^{0\rho}[h]-A_2^{0\rho}[h,k])|_\Sigma\). The quadratic contribution is retained; profile-dependent position moments vanish at order \(\eps\). These data identify the sharp action-level transmission problem: continuous coordinates and continuous normal flux. Its smeared linear background kernel is \(\theta C^*(\mathsf E_0^{\mathrm A}a)\), with a possible jump but no hypersurface delta. It differs from the multiplied-equation prescription with zero positive-order Cauchy velocities. The fixed-algebra Ward identity passes to the limit. The supplied VII-C hypotheses are copied and audited: its quantum metric block, mixed insertion, normal-topology approximants, and mandatory triple product are absent here. VII-C therefore does not apply and is not contradicted. No new product, quantum metric, or subtraction is constructed. Neither the moving edge witness nor a crude inverse-regulator upper bound is used as the decision. \end{abstract} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {X-C output}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {This ramp calculation}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {IX sharp product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The ramp is not a repair of IX and not a repair of X-A-conv.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The ramp step is not the sharp product. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:scope-X-D} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {X-C output}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {IX sharp product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The ramp is not a repair of IX and not a repair of X-A-conv.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The ramp step is not the sharp product. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:split-X-D} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {X-C output}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {This ramp calculation}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {IX sharp product}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The ramp is not a repair of IX and not a repair of X-A-conv.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (c) -- (s); \end{tikzpicture} \caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:chain-X-D} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {X-C output}; \node[align=center,font=\small] (m) at (5.2,0) {not the same object}; \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {IX sharp product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {This ramp calculation. The ramp is not a repair of IX and not a repair of X-A-conv.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{X-C output is not IX sharp product. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:pair-X-D} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: X-C output}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: This ramp calculation}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: IX sharp product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The ramp is not a repair of IX and not a repair of X-A-conv.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:gate-X-D} \end{figure} Figure~\ref{fig:split-X-D} records one view of the same separation. Figure~\ref{fig:chain-X-D} records one view of the same separation. Figure~\ref{fig:pair-X-D} records one view of the same separation. Figure~\ref{fig:gate-X-D} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-X-D} records that separation. The drawing adds no hypothesis. %===================================================================== \section{Release and the three endings} \label{sec:release} %===================================================================== \scopebox{Release: Constructed, at fixed input through \(\kappa^2\)}{% Fix \(f,g,h,k\) and the plateau algebra as in Section~\ref{sec:frozen}. For one compact set \[ C=K\cup\bigl(\supp g\cap J^-(K)\bigr), \qquad K=\supp f, \] the completed X-C-v3 tests satisfy \[ \de{j}\longrightarrow\dz{j} \quad\text{in }\Dspace_C(M),\qquad p_N(\de{j}-\dz{j})\leq A_N\eps,\quad j=1,2. \] Consequently, \[ \widehat\Phi_\eps(f) =\field(f)+\kappa\field(\de{1}) +\kappa^2\field(\de{2}) \] converges coefficientwise in the original quotient test topology. The explicit limits are built from the sharp action-level transmission coordinates, not the multiplied-equation variant. The fixed-algebra identity passes to the limit. VII-C's named normal-topology extension is not constructed or contradicted. \smallskip Not opened: a new \(R_n\), a quantum metric, a triangle subtraction, a convolution repair, or joint sharp admission.} Three endings are available: \begin{enumerate}[label=\textbf{\arabic*.}] \item \textbf{Constructed.} Explicit limiting tests exist in a named topology and identify the limiting prescription. \item \textbf{Proved obstructed.} For fixed input a named seminorm diverges, or the proposed representative limit leaves the declared test space. \item \textbf{Unresolved.} The topology, boundary-layer estimate, propagation, or identification is unfinished. \end{enumerate} The ending reached is (1). The question is formal in \(\kappa\). It concerns convergence of coefficient tests, not convergence of a numerical perturbation series. A crude \(\eps^{-r}\) upper bound is not a divergence theorem. The moving edge witness is not fixed input. Neither argument is used to decide the limit. %===================================================================== \section{Frozen tests and regulator-independent Euler coefficients} \label{sec:frozen} %===================================================================== The classical coordinate prescription is copied from \texttt{X-A.tex} \cite{XA}. The bare insertion representatives and free-field quotient topology are copied from \texttt{X-B-v3.tex} \cite{XB}. The completed tests, Yang--Feldman recursion, and fixed-regulator Ward identity are copied from \texttt{X-C-v3.tex} \cite{XC}. Their covariance and product proofs are not repeated. \subsection{Fixed geometry, supports, and backgrounds} Let \[ M=\R^4,\qquad \eta=\diag(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta,\qquad \Pm=\Box+m^2,\quad m\geq0. \] The quantum scalar is defined on \(M\). The slab \(\MT=[0,T]\times\R^3\) and \(\Sigma=\{t=0\}\) belong to the coordinate prescription. Write \(\ER,\EA\) for the Klein--Gordon Green operators and \(\ERzero,\EAzero\) for the massless ones. The backgrounds \(h,k\) are fixed smooth real symmetric tensors on \(M\), without field equations or decay assumptions. Only their restrictions to finite causal regions will be used. Set \[ q=\eta+\kappa h+\kappa^2k. \] With \(H=\operatorname{tr}_\eta h\), abbreviate \begin{align} a^{\mu\nu}&=a_1^{\mu\nu}[h] =\tfrac12H\eta^{\mu\nu}-h^{\mu\nu}, & b&=\tfrac12H, \nonumber\\ b_2[h]&=\tfrac18H^2-\tfrac14h_{\alpha\beta}h^{\alpha\beta}, & a_2^{\mu\nu}[h] &=h^\mu{}_\alpha h^{\alpha\nu} -\tfrac12Hh^{\mu\nu}+b_2[h]\eta^{\mu\nu}, \nonumber\\ A^{\mu\nu}&=a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h], & B&=b_1[k]+b_2[h]. \label{eq:density} \end{align} Indices are moved with \(\eta\). Fix \[ f\in C_c^\infty((0,T)\times\R^3;\R),\qquad g\in C_c^\infty((0,T)\times\R^3;[0,1]), \] and the same relatively compact plateau \(U\) as upstream: \[ \overline U\subset P_g,\qquad P_g=\Int\{g=1\},\qquad \ga=C_c^\infty(U;TM). \] Choose \(\delta>0\) such that \begin{equation} K\cup\supp g\cup\overline U \subset\{2\delta3\). \subsection{Integrating before estimating} Let \[ \WC(t)=\cos(t|D|),\qquad \WS(t)=\frac{\sin(t|D|)}{|D|},\qquad |D|=\sqrt{-\Delta}. \] The sine quotient has its continuous value \(t\) at frequency zero. \begin{lemma}[Normal-flux form] \label{lem:flux} Suppose \[ \Box u=-\partial_tF-H,\qquad u(0)=\partial_tu(0)=0,\qquad F(0)=0. \] Then \begin{align} u(t)&=-\int_0^t\WC(t-s)F(s)\,\dd s -\int_0^t\WS(t-s)H(s)\,\dd s, \nonumber\\ \partial_tu(t)+F(t) &=-\int_0^t\WC'(t-s)F(s)\,\dd s -\int_0^t\WC(t-s)H(s)\,\dd s. \label{eq:flux-form} \end{align} If \(F,H\) are uniformly bounded in every spatial smooth seminorm on \(0\leq t\leq\eps\), then \[ u=O(\eps),\qquad \partial_tu+F=O(\eps) \] in every local spatial smooth seminorm, uniformly on that layer. No time derivative of \(F\) is needed for these bounds. \end{lemma} \begin{proof} Integrate the zero-data Duhamel formula once by parts in time. The initial term vanishes because \(F(0)=0\). Fourier multipliers give \[ \|\WC(\tau)b\|_r\leq\|b\|_r,\quad \|\WS(\tau)b\|_r\leq\tau\|b\|_r,\quad \|\WC'(\tau)b\|_r\leq\tau\|b\|_{r+2}. \] Consequently, if \(M_r(F)\) and \(M_r(H)\) denote the corresponding suprema on the layer, \begin{align*} \|u(t)\|_r &\leq tM_r(F)+\tfrac12t^2M_r(H),\\ \|\partial_tu(t)+F(t)\|_r &\leq\tfrac12t^2M_{r+2}(F)+tM_r(H). \end{align*} Spatial Sobolev embedding proves the stated smooth estimates. The bounds integrate the flux rather than differentiate the ramp before estimating. \end{proof} \subsection{Application to the two coordinate coefficients} Define \begin{align} F_{1,\eps}^{\rho}&=\sig a^{0\rho}, & H_{1,\eps}^{\rho}&=\sig\,\partial_i a^{i\rho}, \nonumber\\ F_{2,\eps}^{\rho} &=\sig\bigl(A^{0\rho} +a^{00}\partial_t\Xe{1}^{\rho} +a^{0i}\partial_i\Xe{1}^{\rho}\bigr), \nonumber\\ H_{2,\eps}^{\rho} &=\sig\,\partial_i \bigl(A^{i\rho} +a^{i0}\partial_t\Xe{1}^{\rho} +a^{ij}\partial_j\Xe{1}^{\rho}\bigr). \label{eq:fluxes} \end{align} Spatial indices range from \(1\) to \(3\). Both normal flux terms vanish initially, and \[ \Box\Xe{j}^{\rho} =-\partial_tF_{j,\eps}^{\rho}-H_{j,\eps}^{\rho}. \] The first application of Lemma~\ref{lem:flux} gives \begin{equation} \Xe{1}=O(\eps),\qquad \partial_t\Xe{1}=-\sig a^{0\cdot}+O(\eps). \label{eq:first-layer} \end{equation} Thus \(F_{2,\eps},H_{2,\eps}\) are uniformly bounded in every spatial seminorm, without a bound on \(\partial_t^2\Xe{1}\). The second application gives \begin{equation} \Xe{2}=O(\eps),\qquad \partial_t\Xe{2} =-\sig A^{0\cdot} +\sig^2a^{00}a^{0\cdot}+O(\eps). \label{eq:second-layer} \end{equation} These are also exact normal-current estimates. For \(\mathsf X_\eps=x+\kappa\Xe{1}+\kappa^2\Xe{2}\), the positive-order normal currents are \[ \Pi_{1,\eps}^\rho=\partial_t\Xe{1}^\rho+F_{1,\eps}^\rho, \qquad \Pi_{2,\eps}^\rho=\partial_t\Xe{2}^\rho+F_{2,\eps}^\rho. \] They satisfy \(\partial_t\Pi_{j,\eps}=-\partial_iJ_{j,\eps}^i\) and start at zero. The spatial currents are uniformly bounded, so their layer integrals are \(O(\eps)\). \begin{lemma}[Rescaled layer, exit data, and profile independence] \label{lem:profile} Write \[ a_\Sigma=a(0,\cdot),\qquad A_\Sigma=A(0,\cdot),\qquad I_\ell(s)=\int_0^s\chi(r)^\ell\,\dd r,\quad \ell=1,2. \] Uniformly for \(0\leq s\leq1\), in every local spatial seminorm, \begin{align} \Xe{1}^{\rho}(\eps s) &=-\eps I_1(s)a_\Sigma^{0\rho}+O(\eps^2), \nonumber\\ \partial_t\Xe{1}^{\rho}(\eps s) &=-\chi(s)a_\Sigma^{0\rho}+O(\eps), \nonumber\\ \Xe{2}^{\rho}(\eps s) &=\eps\bigl[-I_1(s)A_\Sigma^{0\rho} +I_2(s)a_\Sigma^{00}a_\Sigma^{0\rho}\bigr]+O(\eps^2), \nonumber\\ \partial_t\Xe{2}^{\rho}(\eps s) &=-\chi(s)A_\Sigma^{0\rho} +\chi(s)^2a_\Sigma^{00}a_\Sigma^{0\rho}+O(\eps). \label{eq:rescaled} \end{align} The exit positions tend to zero, and the exit velocities tend to \begin{equation} v_1^\rho=-a_\Sigma^{0\rho},\qquad v_2^\rho=-A_\Sigma^{0\rho} +a_\Sigma^{00}a_\Sigma^{0\rho}. \label{eq:velocities} \end{equation} The limiting data depend only on the profile endpoints. \end{lemma} \begin{proof} Replace the smooth coefficients at \(\eps s\) by their initial traces in \eqref{eq:first-layer}--\eqref{eq:second-layer}. The error is \(O(\eps)\). Integrating the velocities gives the position formulas. To display every ramp-derivative contribution, the expanded sources are \begin{align} \Box\Xe{1}^{\rho} ={}&-\sig'a^{0\rho} -\sig\,\partial_ta^{0\rho} -\sig\,\partial_i a^{i\rho}, \nonumber\\ \Box\Xe{2}^{\rho} ={}&-\sig'A^{0\rho} -\sig\,\partial_tA^{0\rho} -\sig\,\partial_iA^{i\rho} \nonumber\\ &-\sig'a^{0\nu}\partial_\nu\Xe{1}^{\rho} -\sig\,\partial_t(a^{0\nu}\partial_\nu\Xe{1}^{\rho}) -\sig\,\partial_i(a^{i\nu}\partial_\nu\Xe{1}^{\rho}). \label{eq:ramp-terms} \end{align} The first equation implies \[ \partial_t^2\Xe{1}^{\rho}(\eps s) =-\eps^{-1}\chi'(s)a_\Sigma^{0\rho}+O(1). \] The singular part of the second source is therefore \[ \eps^{-1}\bigl[ -\chi'(s)A_\Sigma^{0\rho} +2\chi(s)\chi'(s)a_\Sigma^{00}a_\Sigma^{0\rho} \bigr]. \] One quadratic contribution comes from \(-\sig'a^{00}\partial_t\Xe{1}\); the other comes from \(-\sig a^{00}\partial_t^2\Xe{1}\). Their integrals each give half of the quadratic velocity: \[ \int_0^1\chi'(s)\,\dd s=1,\qquad \int_0^1\chi(s)\chi'(s)\,\dd s=\tfrac12. \] Their sum agrees with the exact flux calculation. Discarding either term gives an incorrect exit velocity. The moments \(I_1(1),I_2(1)\) multiply \(\eps\) in the exit positions. Other profile-dependent moments similarly occur in remainders, not in \eqref{eq:velocities}. \end{proof} The same limiting data follow for any fixed smooth profile with the same constant extensions and endpoint values. The proof does not assert uniformity for profiles themselves varying with \(\eps\). Retarded propagation has not erased the layer: the finite quadratic contribution in \(v_2\) survives. Profile independence concerns that retained datum, not the absence of all profile-dependent corrections. %===================================================================== \section{Propagation and convergence} \label{sec:propagation} %===================================================================== \subsection{Candidate coordinates determined by the layer} Define the right-hand limiting coordinates by \begin{align} \Box\Xz{1}^{\rho} &=-\partial_\mu a^{\mu\rho}, & \Xz{1}^{\rho}(0)&=0, & \partial_t\Xz{1}^{\rho}(0)&=v_1^\rho, \nonumber\\ \Box\Xz{2}^{\rho} &=-\partial_\mu \bigl(A^{\mu\rho}+a^{\mu\nu}\partial_\nu\Xz{1}^{\rho}\bigr), & \Xz{2}^{\rho}(0)&=0, & \partial_t\Xz{2}^{\rho}(0)&=v_2^\rho. \label{eq:limit-cauchy} \end{align} These are ordinary smooth Cauchy problems on \(t\geq0\), locally by finite propagation. For an explicit statement of the data at \(t=\delta\), let \(s_j^\rho\) be the right-hand sources in \eqref{eq:limit-cauchy}. Then \begin{align} \Xz{j}^{\rho}(\delta) &=\WS(\delta)v_j^\rho +\int_0^\delta\WS(\delta-t)s_j^\rho(t)\,\dd t, \nonumber\\ \partial_t\Xz{j}^{\rho}(\delta) &=\WC(\delta)v_j^\rho +\int_0^\delta\WC(\delta-t)s_j^\rho(t)\,\dd t. \label{eq:delta-data} \end{align} All layer contributions are included through the derived velocities. \begin{proposition}[Propagation of the exit-data limit] \label{prop:coordinate-convergence} For every spatial Sobolev order \(r\), \[ \sup_{\eps\leq t\leq T} \left( \|\Xe{j}(t)-\Xz{j}(t)\|_r+ \|\partial_t\Xe{j}(t)-\partial_t\Xz{j}(t)\|_r \right)=O(\eps),\qquad j=1,2, \] after the fixed spatial localization. All spacetime derivatives converge at rate \(O(\eps)\) on compact subsets of \([\delta,T]\times\R^3\). In particular, the ramp data at \(t=\delta\) converge to \eqref{eq:delta-data}. \end{proposition} \begin{proof} For \(t\geq\eps\), \(\sig=1\) and every positive-order ramp derivative vanishes. Put \(w_j=\Xe{j}-\Xz{j}\). The layer lemma and smoothness of the right-hand solutions give \(O(\eps)\) Cauchy data for both differences at \(t=\eps\) in every spatial Sobolev norm. The first difference satisfies \(\Box w_1=0\). The homogeneous wave formula bounds \(w_1,\partial_tw_1\) by those data uniformly on the fixed interval. For the second difference, \[ \Box w_2^\rho =-\partial_\mu(a^{\mu\nu}\partial_\nu w_1^\rho). \] The source is \(O(\eps)\) in every spatial Sobolev norm: use the first estimate and \(\partial_t^2w_1=\Delta w_1\). The inhomogeneous Duhamel formula gives the second estimate. Higher time derivatives follow inductively from these equations, using higher spatial norms and derivatives of the fixed smooth coefficients. Sobolev embedding gives the asserted smooth convergence. The constants are independent of the initial time \(\eps\in(0,\delta)\). \end{proof} \subsection{Limiting tests and the rate theorem} Define \[ B_0^\mu =\Xz{1}^{\nu}\partial_\nu\Xz{1}^{\mu}-\Xz{2}^{\mu}, \] and \begin{align} \cz{1}&=\partial_\mu(f\Xz{1}^{\mu}), \nonumber\\ \cz{2}&=-\partial_\mu(fB_0^\mu) +\tfrac12\partial_\mu\partial_\nu (f\Xz{1}^{\mu}\Xz{1}^{\nu}). \label{eq:limit-bare} \end{align} They are smooth compact tests on \(M\), because \(f\) vanishes near \(\Sigma\). Define \(\dz{j}\) by \eqref{eq:limiting-completed}. \begin{theorem}[Convergence of the completed tests] \label{thm:convergence} For each fixed input satisfying the sheet, \(\dz{1},\dz{2}\in\Dspace_C(M)\), and for every \(N\), \[ p_N(\ce{j}-\cz{j})+ p_N(\de{j}-\dz{j})\leq A_N\eps,\qquad j=1,2. \] Thus \(\de{j}\to\dz{j}\) in \(\Dspace_C(M)\), and their free-field classes converge in \(Q_m\). \end{theorem} \begin{proof} Proposition~\ref{prop:coordinate-convergence} controls every coordinate derivative on a neighbourhood of \(K\). The formulas \eqref{eq:bare} are finite smooth products and derivatives, multiplied by \(f\); their differences are \(O(\eps)\) in every compact-test seminorm. Lemma~\ref{lem:reduction} gives the same rate for the completed tests. Their supports lie in the fixed compact \(C\), and the quotient map is continuous. \end{proof} The representatives converge before taking equation classes. No quotient cancellation is needed to rescue a divergent representative family. X-B's crude bound differentiated the ramp before integrating the solution and consequently contained powers of \(\eps^{-1}\). Lemma~\ref{lem:flux} explains why that bound does not describe the present fixed-input limit: \(\partial_tu+F\) is estimated without differentiating \(F\). The theorem supplies boundedness and the actual rate on the declared supports. %===================================================================== \section{Identification of the sharp prescription} \label{sec:identification} %===================================================================== \subsection{Transmission equation} Let \(\theta\) be the characteristic function of \(t>0\), and set \[ \mathfrak A_{\mathrm{sh}}^{\mu\nu} =\eta^{\mu\nu} +\theta(t)(\kappa a^{\mu\nu}+\kappa^2A^{\mu\nu}). \] Extend the coordinate map by \(x\) for \(t<0\), and by \(x+\kappa\Xz{1}+\kappa^2\Xz{2}\) for \(t>0\). \begin{theorem}[Sharp action-level identification] \label{thm:identification} The limiting coordinates are the unique formal piecewise smooth continuous coordinates equal to \(x\) for \(t<0\) and satisfying \begin{equation} \partial_\mu \bigl(\mathfrak A_{\mathrm{sh}}^{\mu\nu} \partial_\nu\mathsf X_{\mathrm{sh}}^\rho\bigr)=0 \formal \label{eq:transmission} \end{equation} distributionally. Their normal flux is continuous across \(\Sigma\). \end{theorem} \begin{proof} The flux is a locally integrable piecewise smooth function. Its distributional divergence has a surface delta with coefficient equal to the normal-flux jump. The negative-side flux is \(\delta_0^\rho\). The positive-side initial flux is \[ \delta_0^\rho +\kappa(v_1^\rho+a_\Sigma^{0\rho}) +\kappa^2(v_2^\rho+A_\Sigma^{0\rho} +a_\Sigma^{00}v_1^\rho). \] Tangential positive-order coordinate derivatives vanish initially because the initial positions vanish. The jump is zero exactly for \eqref{eq:velocities}. The positive-time bulk equations are \eqref{eq:limit-cauchy}, so \eqref{eq:transmission} holds. Conversely, continuity and normal-flux matching force the same initial data. Coefficientwise Cauchy uniqueness gives uniqueness. \end{proof} This argument defines the flux first; it does not multiply a delta by an unspecified discontinuous velocity. The limiting positive-order positions vanish, but the velocities generally do not. \subsection{Kernel formula and the multiplied-equation variant} Write \[ C^\rho[h]=-\partial_\mu a^{\mu\rho}[h],\qquad (C^*b)^{\rho\sigma} =-\partial^{(\rho}b^{\sigma)} +\tfrac12\eta^{\rho\sigma}\partial_\mu b^\mu. \] For a compact covector test \(a\) supported in the open slab, set \(b=\EAzero a\). X-A's ramp kernel identity and dominated convergence give \begin{equation} \int_{\MT}a_\rho\Xz{1}^{\rho}\,\dd^4x =\int_{\MT}h_{\rho\sigma}\, \theta(t)(C^*b)^{\rho\sigma}\,\dd^4x. \label{eq:limit-kernel} \end{equation} The background kernel is therefore \[ q_{a,0}=\theta C^*(\EAzero a). \] It is smooth on either side, locally bounded, and compactly supported after restriction to \(t\geq0\). It can have a jump, but contains no hypersurface delta. The multiplied-equation prescription instead solves \[ \Box\widetilde{\mathsf X}_{1,0}=C[h],\qquad \widetilde{\mathsf X}_{1,0}(0) =\partial_t\widetilde{\mathsf X}_{1,0}(0)=0. \] If \(\mathsf H_0[u_0,u_1]\) denotes the homogeneous wave solution, then \begin{equation} \Xz{1}-\widetilde{\mathsf X}_{1,0} =-\mathsf H_0[0,a_\Sigma^{0\cdot}]. \label{eq:homogeneous-difference} \end{equation} Thus the prescriptions differ in general, although they can agree for special backgrounds with vanishing relevant trace. With \(n=\dd t\), define \[ \mathsf B^{\rho\sigma}(b) =-n^{(\rho}b^{\sigma)} +\tfrac12\eta^{\rho\sigma}n_\mu b^\mu. \] The multiplied-equation kernel is \[ C^*(\theta b)=\theta C^*b +\delta(t)\mathsf B(b|_\Sigma). \] This distinction identifies the candidate; it does not prove joint sharp microcausal admission or define a quantum metric product. Smooth coordinate convergence through \(t=0\) generally fails, as anticipated in the topology sheet. The completed scalar tests nevertheless converge because their supports remain separated from that surface. %===================================================================== \section{VII-C checklist: copied hypotheses and failed match} \label{sec:viic} %===================================================================== The supplied \texttt{VII-C.tex} \cite{VIIC} proves a continuity obstruction for a particular boundary extension. We now copy its actual assumptions rather than infer them from their labels. \subsection{VII-C's own signature and fixed model} In this subsection only, VII-C uses \[ \eta_{\mathrm{VII}}=\diag(-1,1,1,1),\qquad \square_{\mathrm{VII}}=-\partial_t^2+\Delta. \] Its insertion is \[ B_f[h,\phi] =-\int f\,w^\mu[h]\partial_\mu\phi\,\dd^4x,\qquad w[h]=\mathsf E_{0,\mathrm{VII}}C_{\mathrm{VII}}h, \] with the sharp zero-Cauchy inverse and unrestricted smooth off-shell configurations. Its Wick matrix includes a quantum graviton block \[ H_{hh\,\alpha\beta\rho\sigma} =P_{\alpha\beta\rho\sigma}\Delta_0^+,\qquad P_{\alpha\beta\rho\sigma} =\eta_{\mathrm{VII}\,\alpha(\rho} \eta_{\mathrm{VII}\,\sigma)\beta} -\tfrac12\eta_{\mathrm{VII}\,\alpha\beta} \eta_{\mathrm{VII}\,\rho\sigma}, \] and a scalar block \(H_{\phi\phi}=\Delta_m^+\), with zero cross blocks. The signature dictionary is \(\eta_{\mathrm{VII}}=-\eta\), \(\square_{\mathrm{VII}}=-\Box\), and \(\mathsf E_{0,\mathrm{VII}}=-\mathsf E_0\) for the same source. Raised-index formulas must be recomputed accordingly. This dictionary does not identify either the coordinate prescription or the quantum field content. \subsection{The four assumptions, copied} The symbols in the next assumption belong to VII-C: \(\Pol\) is its multiplet polynomial domain, \(\cA\) its proposed extension, \(b_f\) its degree-\((1,1)\) sharp kernel, and \(B_{f,\eps}\) its compact-mollified sharp insertion. They are not X-D objects. \begin{assumption}[VII-C's Normal-topology Wick extension, copied for comparison] \label{ass:VII} A proposed extension consists of a Hausdorff locally convex \(\C[[\hbar]]\)-algebra \(\cA\), with the following properties. \begin{enumerate}[label=\textup{(E\arabic*)}] \item \(\Pol[[\hbar]]\) embeds into \(\cA\), and its product is the fixed Wick product. \item The algebra admits the sharp insertions \(Q_\Sigma(B_f)\), with \[ [Q_\Sigma(B_f)]_{\hbar^0}=B_f \] as functionals on all smooth configurations. \item On the degree-\((1,1)\) kernel stages used above, admission is sequentially continuous for the normal topology and agrees with the old inclusion on regular kernels. In particular, \[ B_{f,\eps}\longrightarrow Q_\Sigma(B_f) \quad\hbox{in }\cA. \] \item Multiplication is hypocontinuous with respect to bounded sets. The field-independent \(\hbar^3\) coefficient map \[ a\longmapsto \ev[\hbar^3]a \] is continuous and agrees with the usual coefficient extraction on the old polynomial domain. \end{enumerate} \end{assumption} The cross-references in the original wording refer to VII-C's fixed product and normal-topology stages \(\mathcal E'_{\Gamma_B}(K)\). No such stage is being substituted for \(\Dspace_C(M)\). \begin{center} \small \renewcommand{\arraystretch}{1.14} \begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.40\linewidth}X@{}} \toprule Item & VII-C requires & X-D constructs\\ \midrule (E1) & Its de Donder multiplet polynomial domain, with the fixed Wick matrix & The free scalar algebra only. The backgrounds \(h,k\) remain classical; no \(H_{hh}\) block is present.\\ (E2) & Admission of \(Q_\Sigma(B_f)\), retaining the exact sharp insertion on all smooth configurations & Smooth compact scalar tests \(\dz{j}\) at fixed backgrounds. No VII-C mixed quantum kernel \(b_f\) is admitted.\\ (E3) & Compact-mollified sharp kernels converging in the specified normal topology of \(\mathcal E'_{\Gamma_B}\) & Ramp coefficient tests converging in \(\Dspace_C(M)\). They are neither those kernels nor those approximants.\\ (E4) & Hypocontinuous extension and continuous extraction of the fully contracted triple coefficient & The word involving quantum \(h_{00}\) is not formed. No extension of that coefficient is asserted.\\ \bottomrule \end{tabularx} \end{center} This is failure to match the required construction, not a claim that the scalar Wick product fails continuity. The assumptions are not established as a package on this sheet, and the mandatory objects needed for VII-C's contradiction are absent. \subsection{The logarithm is not a limit obstruction here} VII-C's named word is \[ B_f\star_H W_g\star_H B_{f'},\qquad W_g=\int g\,h_{00}^{\,2}\,\dd^4x. \] For its mollified sharp kernels, its fully contracted \(\hbar^3\) coefficient has a nonzero logarithmic term for suitable tests. That theorem is about a quantum multiplet product, not the seminorms of \(\de{j}\). X-D neither constructs that word nor subtracts its reflected light-ray coefficient. VII-C's bulk representation, after integration by parts, and X-A's multiplied-equation kernel carry a boundary delta; X-D's limit kernel does not. This identifies the candidate and does not repair VII-C. VII-C itself excludes only the named normal-topology extension with the stated admission and product requirements. An algebra omitting \(W_g\), or a different construction not meeting those requirements, is outside that conclusion. The present scalar-only limit is such a smaller object. Therefore VII-C does not apply and is not contradicted. %===================================================================== \section{D6: gauge corollary with explicit representatives} \label{sec:gauge} %===================================================================== For \(F\in\Gp\), generated formally by \(\kappa\xi+\kappa^2\zeta\), \(\xi,\zeta\in\ga\), the complete metric law is \[ h'=h+K\xi,\qquad k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi. \] The transformed backgrounds are fixed smooth inputs, independent of \(\eps\). \begin{corollary}[The fixed-algebra identity passes to the limit] \label{cor:gauge} With the free field held fixed, \[ \field(\dz{j}[h',k'])=\field(\dz{j}[h,k]), \qquad j=1,2. \] Thus the limiting completed observable is background-gauge invariant through \(\kappa^2\) on the unchanged algebra. \end{corollary} \begin{proof} Convergence holds for both background pairs. Use X-C's existing three-term variation on the plateau, where its coordinate and switch defects vanish: \[ e_\eps[q']-e_\eps[q] =\Pm((F^*-1)w_\eps),\qquad w_\eps=\EA e_\eps. \] The coordinate and interaction plateau conditions hold for every \(\eps<\delta\). Green-operator continuity gives smooth convergence of \(w_\eps\) on the compact generator support. Every positive-order coefficient of \((F^*-1)w_\eps\) has that fixed compact support and converges as a compact test. Hence \[ e_0[q']-e_0[q]=\Pm((F^*-1)w_0),\qquad w_0=\EA(f+\kappa\dz{1}+\kappa^2\dz{2}). \] Explicitly, with \(X=X_\xi\) and \(T_2=X_\zeta+\tfrac12X_\xi^2\), \[ \dz{1}'-\dz{1}=\Pm(X\EA f), \] and \[ \dz{2}'-\dz{2} =\Pm\bigl(X\EA\dz{1}+T_2\EA f\bigr). \] The functions inside \(\Pm\) are compactly supported. These are explicit representatives in the unchanged equation ideal, so their free-field classes vanish. \end{proof} Closedness of \(\Pm\Dspace(M)\) gives the same conclusion by taking limits of the differences. The displayed representatives make the passage explicit without redoing X-C's variation proof. This remains background covariance at fixed free field. It is not a scalar pullback automorphism of the free shell, a quantum-metric identity, or an Enter readout. %===================================================================== \section{Diagnostics and the excluded edge argument} \label{sec:diagnostics} %===================================================================== \begin{diagnostic}[Pure gauge above the layer] Let \(\xi\in\ga\), \(\zeta=0\), and \[ h=K\xi,\qquad k=\tfrac12\Lie_\xi K\xi. \] The backgrounds vanish on \(0\leq t\leq\delta\). The rescaled layer therefore has zero source and zero solution, and both limiting data at \(t=\delta\) are zero. The upstream pure-gauge laws give \(\Xe{1}=\xi\) and \(\Xe{2}=\tfrac12\xi\cdot\partial\xi\) for every small \(\eps\). This checks the complete two-jet and the zero-layer regime, not the general layer estimate. \end{diagnostic} \begin{diagnostic}[Constant-near-zero conformal layer] Near a causal cylinder containing the observation dependence region, take \(h=\psi(t)\eta\), \(k=0\), with \(\psi=\psi_*\) near zero. Spatial cutoffs outside the cylinder leave this evaluation unchanged. Then \(a=\psi\eta\), \(A=a_2[\psi\eta]=0\), and \[ \Xe{1}^0(t)=-\int_0^t\sig\psi\,\dd s,\qquad \Xe{2}^0(t)=\int_0^t(\sig\psi)^2\,\dd s. \] For sufficiently small \(\eps\), the exit positions are \[ -\eps\psi_*\int_0^1\chi\,\dd s,\qquad \eps\psi_*^2\int_0^1\chi^2\,\dd s, \] while the exit velocities are \(-\psi_*\) and \(\psi_*^2\). The profile moments vanish with the positions. The quadratic velocity is retained and agrees with \eqref{eq:velocities}. \end{diagnostic} \begin{diagnostic}[Sourced number, including \(\psi(0)\neq0\)] Let \(m=0\), \(h=\psi(t)\eta\), \(k=0\), with smooth temporal data near the relevant causal cylinder and spatial cutoffs outside it. Let \(g=g_0(t)\) there. Define \[ \Psi_\eps=\int_0^t\sig\psi\,\dd s,\quad \Psi_{2,\eps}=\int_0^t(\sig\psi)^2\,\dd s,\quad \Gamma=\int_0^tg_0\psi\,\dd s,\quad \Gamma_2=\int_0^t(g_0\psi)^2\,\dd s. \] Since \(a_2[\psi\eta]=0\), the massless second scalar vertex vanishes. On the smooth free solution \(u=t\), X-C's completed representatives give \[ \pair{\de{1}}{t} =\int f(\Psi_\eps-\Gamma)\,\dd^4x, \] and \[ \pair{\de{2}}{t} =\int f\bigl[ \Gamma_2-\Psi_{2,\eps} +\Psi_\eps\psi(\sig-g_0) \bigr]\,\dd^4x. \] If \(\psi\) is supported in \(t>\delta\), these numbers are already independent of \(\eps<\delta\). For a genuine layer with \(\psi(0)\neq0\), put \[ \Psi_0=\int_0^t\psi\,\dd s,\qquad \Psi_{2,0}=\int_0^t\psi^2\,\dd s. \] Then \[ |\Psi_\eps-\Psi_0|\leq\eps\sup_{[0,\eps]}|\psi|, \qquad |\Psi_{2,\eps}-\Psi_{2,0}| \leq\eps\sup_{[0,\eps]}\psi^2. \] On \(\supp f\), \(\sig=1\), so \[ \pair{\dz{2}}{t} =\int f\bigl[ \Gamma_2-\Psi_{2,0}+\Psi_0\psi(1-g_0) \bigr]\,\dd^4x, \] with \(O(\eps)\) error in the ramp evaluation. The first limiting number is \(\int f(\Psi_0-\Gamma)\,\dd^4x\). The limiting time-coordinate velocities are \(-\psi(0)\) and \(\psi(0)^2\). The multiplied-equation first velocity instead tends to \(-\psi(t)+\psi(0)\), displaying the homogeneous difference. These are classical numbers, not quantum expectation values. They check the rate and signs but do not replace the proof. \end{diagnostic} The excluded edge witness \[ \xi_\eps=\alpha_\eps(t)\beta(\mathbf x)\partial_1, \qquad \supp\alpha_\eps\subset(0,\eps), \] moves with the ramp. Its integral \(-\int_0^\eps\sig'\alpha_\eps\,\dd t\) is neither a seminorm of the completed fixed-input tests nor a direction in the fixed algebra. It is not used as an obstruction. No smooth coordinate limit through the initial surface is claimed. No fixed-input completed-test seminorm diverges: Theorem~\ref{thm:convergence} proves the opposite. %===================================================================== \section{Decision and final release} \label{sec:decision} %===================================================================== \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{.14\linewidth}p{.49\linewidth}X@{}} \toprule Gate & Evidence & Status\\ \midrule D1 & One \(\Dspace_C(M)\), then the original quotient topology & Named before estimates\\ D2, layer & Normal-flux estimate without differentiating \(F\); explicit \(v_1,v_2\) & Constructed\\ Profile & Endpoint fluxes determine the limit; position moments are \(O(\eps)\) & Independent limiting data\\ Propagation & Ordinary wave equations above the layer & \(O(\eps)\) in every seminorm\\ Completed tests & Explicit \(\dz{1},\dz{2}\), fixed support \(C\) & Convergent representatives\\ D3 & Continuous coordinates and normal flux; kernel \(\theta C^*b\), no delta & Action-level transmission\\ D4 & Edge parameter moves with \(\eps\) & Not the decision\\ D5 & VII-C's (E1)--(E4) copied; multiplet, mixed kernel, topology, and word unmatched & Inapplicable, not contradicted\\ D6 & Explicit limiting equation-ideal representatives & Same fixed algebra\\ Crude bound & Differentiates the ramp before propagation & Not a blow-up conclusion\\ Successors & Quantum metric and reflected-residue subtraction & Not opened\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[Fixed-input X-D release] \label{thm:release} For the frozen scalar-only X-C-v3 observable and fixed inputs satisfying \eqref{eq:separation}: \begin{enumerate}[label=\textup{(\roman*)}] \item The completed tests converge to \eqref{eq:limiting-completed} in \(\Dspace_C(M)\), at rate \(O(\eps)\) in every seminorm, and hence in \(Q_m\). \item The limits are built from the sharp action-level transmission coordinates with zero positive-order initial positions and velocities \eqref{eq:velocities}. Their smeared linear background kernel is \(\theta C^*(\EAzero a)\), with a possible jump and no hypersurface delta. \item The limiting data and tests are independent of the choice of fixed smooth ramp profile with the same endpoints. \item The completed limiting observable is background-gauge invariant through \(\kappa^2\) on the unchanged plateau algebra. \end{enumerate} The limit is generally different from the multiplied-equation prescription. VII-C does not apply and is not contradicted. No moving-edge argument, crude inverse-\(\eps\) estimate, or residue subtraction supplies the decision. \end{theorem} \begin{proof} Lemma~\ref{lem:profile} constructs the layer data. Proposition~\ref{prop:coordinate-convergence} propagates them. Theorem~\ref{thm:convergence} proves the compact-test limit. Theorem~\ref{thm:identification} and \eqref{eq:limit-kernel} identify the prescription. Corollary~\ref{cor:gauge} passes the identity after convergence. Section~\ref{sec:viic} audits the separate VII-C construction. \end{proof} The theorem concerns fixed backgrounds and fixed supports. It supplies no uniform theorem for arbitrary \(\eps\)-dependent backgrounds or observation supports approaching \(\Sigma\). Those are different limit questions. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Dependencies.} The ramp coordinate equations, density coefficients, finite-propagation prescription, kernel identity, pure-gauge law, and edge witness are premises from \texttt{X-A.tex}. The bare tests, scalar algebra, quotient topology, and crude upper-bound interpretation are from \texttt{X-B-v3.tex}. The completed tests, recursion, variation, Ward identity, and sourced diagnostic are from \texttt{X-C-v3.tex}. The supplied \texttt{VII-C.tex} is used only for the copied extension assumptions and comparison. \paragraph{New proof.} The rate argument integrates the normal flux before estimating, derives the two exit velocities, propagates their limit, and proves convergence of the completed representatives. The finite second-order layer contribution is retained. \paragraph{Corrections.} The Euler coefficients are regulator independent. Green continuity is used with a justified finite seminorm loss, not an assumed universal count. Initial positions and velocities are distinguished. A possibly jumping background kernel is not a singular scalar output test. VII-C's actual quantum product hypotheses are audited without importing a quantum metric. \paragraph{Verification status.} No successful \texttt{pdflatex} execution, proof-assistant check, numerical verification, or independent referee review is claimed. The source requires no external figures, bibliography database, or executable code. %===================================================================== \begin{thebibliography}{9} \addcontentsline{toc}{section}{References} \bibitem{XA} N.~Cheung. \newblock \emph{Landmark X-A: Classical covariance of one fixed-regulator switched relational scalar through order \(\kappa^2\)}. \newblock Supplied manuscript \texttt{X-A.tex}. \newblock Classical coordinate prescription, density coefficients, kernel identity, covariance, and analytic diagnostics. \bibitem{XB} N.~Cheung. \newblock \emph{Landmark X-B: Fixed-regulator admission and one retarded correction for the ramp relational scalar, with the metric kept classical}. \newblock Supplied manuscript \texttt{X-B-v3.tex}. \newblock Bare tests, scalar algebra and quotient topology, off-shell warning, and regulator-estimate boundary. \bibitem{XC} N.~Cheung. \newblock \emph{Landmark X-C: The missing second-order retarded maps for the ramp relational scalar, with the metric kept classical}. \newblock Supplied manuscript \texttt{X-C-v3.tex}. \newblock Completed tests, Yang--Feldman recursion, variation, fixed-regulator identity, and sourced evaluation. \bibitem{VIIC} \emph{Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions}. \newblock Supplied manuscript \texttt{VII-C.tex}. \newblock The Normal-topology Wick extension assumption \texttt{ass:extension}, hypotheses (E1)--(E4), and the named mixed-triple obstruction. Used for comparison only. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle. \newblock \emph{Wave Equations on Lorentzian Manifolds and Quantization}. \newblock ESI Lectures in Mathematics and Physics. European Mathematical Society, 2007. \newblock arXiv:0806.1036. \newblock Chapter~3 supplies Cauchy and Green-operator facts; Theorem~3.4.7 supplies the exact sequence used for the linear free-field quotient. \end{thebibliography} \end{document}