% !TeX program = pdflatex %===================================================================== % XII-E.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XII-E.tex % pdflatex -interaction=nonstopmode -halt-on-error XII-E.tex % % Assembled object: % Phi^(2)_D(f,g)[gamma,phi] = A_2^sc[f,g](gamma, r_D[gamma]; phi), % r_D = -u_D = P E_0^R J_D (the gauge-fixed second-order metric % coefficient generated by the density-weighted cubic source). % % Debugging recorded explicitly: % 1. SIGN. With XII-D's forcing convention u_D = G^R J_D, the % gauge-fixed second metric coefficient is -u_D. By the affine % slot identity the admitted insertion therefore enters the % relational coefficient as -C_D, not +C_D: % Phi^(2)_D = A_2^sc[gamma,0;phi] - C_D. % The literal sum A_2^sc + C_D carries a doubled two-jet % defect -2 A_1(s_xi) that no plateau hypothesis removes. % XI-E's evaluated witness I>0 detects it. % 2. SLOT. The second metric slot is determined by gamma. Its % two-jet row IS the co-transformation. Listing a separate % co-transformation row and a two-jet row is the double count. % 3. FIELDS. XII-A's carrier has one metric field. XI-E's split % H = h + gamma is bookkeeping for a held-fixed test; no % background h is quantized or separately varied here. % 4. INPUTS. The held-fixed row needs X-A's classical on-shell % covariance of the scalar completion. It is a classical input % on the same sheet and is listed as such; X-A's quantum claims % are not used. % 5. SCOPE. "Complete" means the scalar-linear sector. The % scalar-sourced slot r_g (XI-C/XI-E tree and its scalar leg) % is recorded, has zero metric variation, and is admitted by % the same tree lemma; it is not part of the named target. % % Proved here: % affine slot identity; single-entry decomposition; held-fixed % defect; exact assembled plateau variation; explicit companion % remainder with support; Einstein row as a compact plateau test % of A gamma, equal to a b-test on the gauge-fixed shell; % tree-composition cone lemma; admission of the scalar-only % completion; class, ordering correction, sector stability. % % Not constructed: % a quantum retarded product of J_D, an interacting differential, % a state, a loop, a sharp limit, an Einstein-jet repair. % % No successful compilation, proof-assistant verification, % independent referee review, or journal acceptance is asserted. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \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} \newtheorem{inputstatement}[theorem]{Input statement} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\sig}{\sigma_\eps} \newcommand{\MT}{M_T} \newcommand{\MTo}{M_T^\circ} \newcommand{\PP}{\mathcal P} \newcommand{\RR}{\mathcal R} \newcommand{\Lie}{\mathcal L} \newcommand{\ga}{\mathfrak g_U} \newcommand{\Gp}{\mathcal G_U} \newcommand{\Pmass}{P_m} \newcommand{\ER}{E_0^{\mathrm R}} \newcommand{\EA}{E_0^{\mathrm A}} \newcommand{\ERm}{E_m^{\mathrm R}} \newcommand{\EAm}{E_m^{\mathrm A}} \newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}} \newcommand{\Csol}{\mathscr C_{\mathrm{sol}}} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Fsol}{\mathscr F^{\mathrm{sol}}_{\mathrm{mc,pol}}} \newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\Sect}{\mathscr S_H} \newcommand{\Aone}{\mathcal A_1} \newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}} \newcommand{\CD}{\mathcal C_{3,\mathrm D}} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\DS}{\mathcal D_\sig} \newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}} \newcommand{\PhiP}{\Phi^{(2)}_{+}} \newcommand{\rD}{r_{\mathrm D}} \newcommand{\rg}{r_g} \newcommand{\Tcal}{\mathcal T} \newcommand{\Jcal}{\mathcal J} \newcommand{\Vp}{\widetilde V_+} \newcommand{\Vm}{\widetilde V_-} \newcommand{\Edist}{\mathcal E'} \newcommand{\alphad}{\alpha_d} \newcommand{\balphad}{\bar\alpha_d} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Int}{int} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Sym}{Sym} \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 XII-E}\\[0.4em] Assembly of the second-order relational coefficient\\ after admission of the density-weighted cubic insertion} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} XII-D admitted the completed density-weighted cubic insertion $\CD$ to XII-A's carrier and proved its signed plateau variation. We assemble the second-order relational coefficient that XII-D named and withheld, on the scalar-linear sector: X-A's scalar completion with its second metric slot filled by the gauge-fixed gravitational coefficient generated by the actual source. Two corrections to the production brief are proved rather than assumed. With XII-D's forcing convention $u_{\mathrm D} =G^{\mathrm R}_{\mathrm{gr}}\JD$, the gauge-fixed second-order metric coefficient is $-u_{\mathrm D}$, so the admitted insertion enters the relational coefficient with coefficient $-1$: $\PhiD=\Asc[\gamma,0;\phi]-\CD$. The literal sum with $+\CD$ carries a doubled two-jet defect $-2\Aone(s_\xi;\phi)$ that survives every plateau hypothesis; XI-E's evaluated witness detects it. Moreover the slot is determined by the metric, so its two-jet row is the co-transformation; a separate co-transformation row would be the double count. The exact plateau variation is the sum of a gauge-fixing companion and an Einstein row. The companion is displayed with its factors $1-g$ and $\dd g$, lives above the ramp, and vanishes under the causal plateau condition. The Einstein row is a compact plateau test of $A\gamma$; on the gauge-fixed shell it is a compact test of the auxiliary field, $-2\int(D\Lie_\xi\beta)\cdot b$, and the carrier does not delete it. A tree-composition lemma proves that the scalar-only completion kernel is compact with a covector-sum-zero bound, so $Q_H\PhiD\in\Ealg$ and the enlarged named sector is ordering-stable. No quantum retarded product of $\JD$, interacting differential, state, loop, sharp-width limit, or Einstein-jet repair is constructed. XI-G's exclusion and XII-C's empty jet are retained. \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) {Named plateau coefficient}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Companion remainder}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Unqualified invariance}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The coefficient does not establish unqualified relational invariance.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One coefficient is not relational invariance. 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-XII-E} \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) {Named plateau coefficient}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Unqualified invariance}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The coefficient does not establish unqualified relational invariance.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One coefficient is not relational invariance. 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-XII-E} \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) {Named plateau coefficient}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Companion remainder}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Unqualified invariance}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The coefficient does not establish unqualified relational invariance.}; \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-XII-E} \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) {Named plateau coefficient}; \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) {Unqualified invariance}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Companion remainder. The coefficient does not establish unqualified relational invariance.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Named plateau coefficient is not Unqualified invariance. 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-XII-E} \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: Named plateau coefficient}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Companion remainder}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Unqualified invariance}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The coefficient does not establish unqualified relational invariance.}; \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-XII-E} \end{figure} Figure~\ref{fig:split-XII-E} records one view of the same separation. Figure~\ref{fig:chain-XII-E} records one view of the same separation. Figure~\ref{fig:pair-XII-E} records one view of the same separation. Figure~\ref{fig:gate-XII-E} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XII-E} records that separation. The drawing adds no hypothesis. \scopebox{Release and proof boundary}{% \textbf{Constructed:} the named coefficient $\PhiD$, its single-entry decomposition, the exact assembled plateau variation, the explicit companion remainder and its support, the Einstein row in metric and auxiliary form, admission of the scalar-only completion, and ordering stability of the enlarged sector. \smallskip \textbf{Corrections to the brief:} (1) the admitted insertion enters with coefficient $-1$, forced by the affine slot identity and the gauge-fixed second-order equation; the ``$+$'' sum is detected as defective; (2) the second slot is determined, so no separate co-transformation row is added; (3) one metric field, no quantized background; (4) X-A's classical on-shell covariance is a premise and is listed as such; (5) ``complete'' is the scalar-linear sector; the scalar-sourced slot is recorded and admitted, not part of the target. \smallskip \textbf{Not claimed:} unqualified invariance of $\PhiD$ (the Einstein row is a named remainder), a quantum retarded product of $\JD$, an interacting BRST class, a loop, a state, or an $\eps\to0$ limit. \smallskip The XI-G classes remain excluded. The XII-C fixed-input jet remains empty. The inverse coefficient remains $-1$.} \tableofcontents %===================================================================== \section{Release criteria and copied inputs} \label{sec:inputs} %===================================================================== The endings are \textbf{Constructed}, \textbf{Proved obstructed}, and \textbf{Unresolved}, applied to the object actually named. A missing map is not a no-go. A remainder already displayed in XII-D is not repaired by shrinking $\ga$ after the calculation. \[ \boxed{\text{XII-E assembles the coefficient XII-D withheld. It does not re-apply }-\PP\ER\text{ to }\JD.} \] \subsection{XII-D's handoff} \begin{inputstatement}[XII-D: the six handed objects] \label{input:XIID} On the sheet of Section~\ref{sec:sheet} the following are constructed in \cite{XIID} and used here without re-proof. \begin{enumerate} \item The response \begin{equation} u_{\mathrm D}[\gamma]=-\PP\ER\bigl(\JD[\gamma]\bigr),\qquad \JD=\sig J_3+\DS,\qquad Lu_{\mathrm D}=\JD,\quad L=-\Box\PP . \label{eq:uD} \end{equation} It is smooth, cutoff independent, and supported in $J^+(\supp\JD[\gamma])\cap\MT$. \item The completed two-leg insertion \begin{equation} \CD(f,g)[\gamma,\phi]=\Aone[f,g]\bigl(u_{\mathrm D}[\gamma];\phi\bigr), \label{eq:CD} \end{equation} with both the coordinate and the switched scalar leg (Section~\ref{sec:sheet}). \item The first root $\alpha_{f,g}[\phi]$, its transpose $\beta_{f,g}[\phi]=-\EA(\PP\alpha_{f,g}[\phi])$, and the compact collar root $a_{f,g}[\phi]=\vartheta\beta_{f,g}[\phi]$, with the derived identity $\CD=\int a_{f,g}[\phi]\cdot\JD[\gamma]$. \item The admission theorem: the polarized kernel $t_{\mathrm D}$ of $\CD$ is compact and $\WF(t_{\mathrm D})\subset Z_3$, hence $\CD(f,g)\in\Fpol$. \item The class $Q_H\CD(f,g)\in\Ealg$, the regular ordering correction $\alphad\CD=\CD+\hbar L_d$, and ordering stability of the enlarged named sector. \item The signed ledger. For $\xi\in\ga$ and every smooth $\gamma,\phi$, with $s_\xi[\gamma]=\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi$ and $\rho_\xi[\gamma]=\ER Ds_\xi[\gamma]$, \begin{equation} \begin{aligned} u_{\mathrm D}[\gamma+K\xi]-u_{\mathrm D}[\gamma] &=-s_\xi[\gamma]+K\rho_\xi[\gamma] -\PP\ER\Lie_\xi^{\mathrm{dens}}(A\gamma),\\ \Delta_\xi\CD+\Aone(s_\xi;\phi) &=\Aone(K\rho_\xi;\phi) -\Aone\bigl(\PP\ER\Lie_\xi^{\mathrm{dens}}(A\gamma);\phi\bigr), \end{aligned} \label{eq:ledgerD} \end{equation} together with the causal plateau proposition: if $\Pmass\phi=0$ and $g=1$ on a neighbourhood of $J^-(\supp f)\cap\supp(K\rho_\xi)$, then $\Aone(K\rho_\xi;\phi)=0$. \end{enumerate} XII-D refused: an interacting BRST class, the complete second-order relational construction, a loop value, a quantum retarded product, a state, and a sharp limit. \end{inputstatement} \begin{inputstatement}[XII-C, XII-A, XI-B, XII-B] \label{input:rest} XII-C supplies the cubic-only action, the actual source \eqref{eq:uD}, the expansion $e_{\mathrm D}=A\gamma+\kappa(\JD-\tfrac12gT_0)+O(\kappa^2)$ for coupling-independent fields, the gauge-fixing Euler term $e_{\mathrm{gf}}=2D^*D\gamma$, and the empty fixed-input jet $\Jcal^{\mathrm D}_{g,\sig}(\phi_m)=\varnothing$, used only as a boundary \cite{XIIC}. XII-A supplies $\Ealg=(\Fpol[[\hbar,\kappa]],\star_H)/\Nsol[[\hbar,\kappa]]$, the shell $\Csol=\{\Box\gamma=0,\ \Pmass\phi=0,\ \Box c=\Box\bar c=0,\ b=D\gamma\}$, the ordering maps, and the exclusion of the XI-G classes with covectors $\pm(k_1,k_2,0)$, $k_1+k_2=2\dd t$ \cite{XIIA,XIG}. XI-B supplies \[ \Gret=-\PP\ER,\qquad A+2D^*D=-\Box\PP,\qquad \PP D^*=-\RR,\qquad K=2\RR, \] with inverse coefficient $-1$ \cite{XIB}. From XII-B only the sentence that density weighting is not source weighting is used \cite{XIIB}. \end{inputstatement} \subsection{A classical input not on XII-D's list} The held-fixed row of the ledger needs one theorem that XII-D did not hand over because it is not XII-D's: X-A's classical on-shell covariance of the regulated, switched scalar completion \cite{XA}. It is proved on exactly the present sheet (same ramp, switch, plateau, Cauchy prescription, signature, and inverse normalization). We list it as a premise rather than cite it as comparison. \begin{inputstatement}[X-A: scalar completion and its covariance] \label{input:XA} For the formal metric $q=\eta+\kappa H+\kappa^2k$ with smooth symmetric $H,k$, a free scalar input $\Pmass\phi_0=0$, and the regulated coordinates and switched scalar jet of X-A, the smeared relational scalar has second coefficient $\Asc[f,g](H,k;\phi_0)$, displayed in \eqref{eq:Asc} below. For every $\xi\in\ga$ (X-A's group element $\exp(\kappa\Lie_\xi)$), \begin{equation} \Asc[f,g]\bigl(H+K\xi,\ k+\Lie_\xi H+\tfrac12\Lie_\xi K\xi;\phi_0\bigr) =\Asc[f,g](H,k;\phi_0). \label{eq:XAinv} \end{equation} Also $\Aone[f,g](K\zeta;\phi_0)=0$ for $\zeta\in\ga$, and the pure-gauge two-jet $(K\xi,\tfrac12\Lie_\xi K\xi)$ of the flat configuration has vanishing second coefficient. \end{inputstatement} X-A's quantum, admission, and sharp-limit statements are not used. \subsection{E0: status table} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.36\linewidth}X@{}} \toprule XII-D object & Use in XII-E\\ \midrule 1. $u_{\mathrm D}=-\PP\ER\JD$ & Used as $\rD=-u_{\mathrm D}$, the gauge-fixed second metric coefficient\\ 2. $\CD=\Aone(u_{\mathrm D};\phi)$ & Enters $\PhiD$ once, with coefficient $-1$\\ 3. Roots $\alpha,\beta,a_{f,g}$ & $\beta_{f,g}$ displays the Einstein row\\ 4. Admission $\WF(t_{\mathrm D})\subset Z_3$ & Membership of $\CD$; not cited for any other kernel\\ 5. Class and ordering correction & Extended to $\PhiD$ by the same argument\\ 6. Signed ledger and causal plateau & Copied verbatim; supplies three rows\\ \midrule XII-D refusal & Status here\\ \midrule Interacting BRST class & Not opened\\ Complete second-order relational construction & Assembled on the scalar-linear sector (this paper)\\ Loop value & Not opened\\ Quantum retarded product of $\JD$ & Not opened\\ State & Not opened\\ Sharp limit & Not taken\\ \bottomrule \end{tabularx} \end{center} Beyond this list, Section~\ref{sec:ledger} uses Input statement~\ref{input:XA}, and Section~\ref{sec:carrier} proves a new cone lemma. No other predecessor object is consumed. %===================================================================== \section{E2: the sheet, the two legs, and the scalar completion} \label{sec:sheet} %===================================================================== Use $M=\R^4$, $\eta=\operatorname{diag}(1,-1,-1,-1)$, $\Box=\partial_t^2-\Delta$, $\Pmass=\Box+m^2$, $m\ge0$, $\MT=[0,T]\times\R^3$, $\MTo=(0,T)\times\R^3$, $0<\eps0$. Since $s_{\xi_\lambda}=\lambda\Lie_{a\partial_t}(t\eta)+\tfrac{\lambda^2}2\Lie_{a\partial_t}K(a\partial_t)$, linearity gives $\Aone(s_{\xi_\lambda};\phi)=\lambda I+\lambda^2I'$ with $I'$ independent of $\lambda$, which is nonzero for all sufficiently small $\lambda\ne0$. \end{proof} The structural form of the same diagnostic is XII-D's: at $\gamma=0$, $L\rD[K\xi]=-J_3[K\xi]=As_\xi$, the $L$-image of the X-A slot change $s_\xi-K\rho_\xi$, whereas $Lu_{\mathrm D}[K\xi]=-As_\xi$ is the $L$-image of a slot change no element of $\Gp$ produces. The witness value $I$ is cited from XI-E, not recomputed. %===================================================================== \section{E4: the plateau decision} \label{sec:plateau} %===================================================================== \subsection{The companion row, displayed} \begin{proposition}[Explicit companion remainder] \label{prop:companion} Let $\rho$ be a smooth vector field on the slab, spatially compact on every slice, with $\supp\rho\subset\{t\ge\eps\}$, and let $\Pmass\phi=0$. Then \begin{equation} \begin{aligned} \Aone[f,g](K\rho;\phi) ={}&\int f\,(g-1)\,\rho^\mu\partial_\mu\phi\dd^4x\\ &-\int f\,\ERm\Bigl([\Pmass,g](\rho^\mu\partial_\mu\phi) +(\partial_\mu g)\,a_1^{\mu\nu}[K\rho]\,\partial_\nu\phi\Bigr)\dd^4x, \end{aligned} \label{eq:companion} \end{equation} with $[\Pmass,g]u=(\Box g)u+2(\partial^\mu g)\partial_\mu u$. In particular $\Aone(K\rho;\phi)=0$ whenever \begin{equation} \supp\rho\ \cap\ \Bigl[\bigl(\supp f\setminus P_g\bigr) \cup\bigl(\supp\dd g\cap J^-(\supp f)\bigr)\Bigr]=\varnothing . \label{eq:companionsupport} \end{equation} For $\rho=\rho_\xi[\gamma]$, $\xi\in\ga$, one has $\supp\rho_\xi\subset J^+(\overline U)\cap\MT$, disjoint from the ramp layer $\{0\le t\le\eps\}$; and \eqref{eq:companionsupport} holds under the causal plateau condition \begin{equation} \text{(CP)}\qquad J^-(\supp f)\cap J^+(\overline U)\cap\MT\subset P_g . \label{eq:CP} \end{equation} \end{proposition} \begin{proof} On $\supp\rho$ one has $\sig=1$ and $\sig'=0$, so \eqref{eq:legs} gives $\Box X_1[K\rho]=-\partial_\lambda a_1^{\lambda\mu}[K\rho] =D^\mu K\rho=\Box\rho^\mu$; both sides have zero initial data, hence $X_1[K\rho]=\rho$. Next, X-A's pure-gauge identity for the unswitched first variation, $\mathsf Q_{K\rho}\psi=D_\rho\Pmass\psi-\Pmass(\rho\cdot\partial\psi)$, gives on the free shell $V_{1,g}[K\rho]\phi=-g\Pmass(\rho\cdot\partial\phi)+(\partial_\mu g)a_1^{\mu\nu}[K\rho]\partial_\nu\phi$. Since $g\rho\cdot\partial\phi$ is compactly supported in $\MTo$, $\ERm\Pmass(g\rho\cdot\partial\phi)=g\rho\cdot\partial\phi$, and $g\Pmass u=\Pmass(gu)-[\Pmass,g]u$. Therefore \[ \phi_1[K\rho;\phi] =g\rho\cdot\partial\phi-\ERm[\Pmass,g](\rho\cdot\partial\phi) -\ERm\bigl((\partial g)a_1[K\rho]\partial\phi\bigr). \] Insert into \eqref{eq:Aone} with $X_1[K\rho]\cdot\partial\phi=\rho\cdot\partial\phi$. The first integrand of \eqref{eq:companion} vanishes outside $\supp f\cap\{g\ne1\}\cap\supp\rho$; the second, after transposition onto $\EAm f$, vanishes unless some point of $J^-(\supp f)\cap\supp\dd g$ lies in $\supp\rho$. This proves \eqref{eq:companionsupport}. For $\rho_\xi=\ER Ds_\xi$ the source $Ds_\xi$ is compact in $U$, so finite propagation gives $\supp\rho_\xi\subset J^+(\overline U)\cap\MT$, which lies in $\{t>\eps\}$ because $\overline U\subset\{t>\eps\}$. Under (CP) the compact set $J^-(\supp f)\cap J^+(\overline U)\cap\MT$ lies in the open set $P_g$, where $1-g=0$ and $\dd g=0$; both sets in \eqref{eq:companionsupport} meet $\supp\rho_\xi$ only inside it. \end{proof} Equation \eqref{eq:companion} is XI-E's switching display with the ghost parameter replaced by the propagated companion. Every term carries $1-g$ or $\partial g$; none carries a ramp derivative. The remainder is not a layer term, and it is not deleted by compactness of $\xi$. \subsection{The Einstein row, displayed} \begin{proposition}[The Einstein row is a compact plateau test of $A\gamma$] \label{prop:einstein} For every smooth $\gamma,\phi$ and $\xi\in\ga$, \begin{equation} \Aone[f,g](e_\xi[\gamma];\phi) =\int\bigl(\Lie_\xi\beta_{f,g}[\phi]\bigr)_{\mu\nu}(A\gamma)^{\mu\nu}\dd^4y, \qquad \Lie_\xi\beta_{f,g}[\phi]\in C_c^\infty(U;\Sym^2T^*M). \label{eq:einsteinrow} \end{equation} It vanishes if $A\gamma=0$ on $U$. On the gauge-fixed free shell $\Csol$, \begin{equation} \Aone[f,g](e_\xi[\gamma];\phi)\big|_{\Csol} =-2\int\bigl(\Lie_\xi\beta_{f,g}[\phi]\bigr)\cdot D^*b\dd^4y =-2\int\bigl(D\Lie_\xi\beta_{f,g}[\phi]\bigr)_\nu b^\nu\dd^4y . \label{eq:shellrow} \end{equation} \end{proposition} \begin{proof} By \eqref{eq:exi} and Lemma~\ref{lem:betapair}, $\Aone(e_\xi;\phi)=-\int\beta\cdot\Lie^{\mathrm{dens}}_\xi(A\gamma)$. For a covariant tensor $\beta$ and a contravariant density $v$, $\Lie_\xi(\beta\cdot v)=(\Lie_\xi\beta)\cdot v+\beta\cdot\Lie^{\mathrm{dens}}_\xi v$, and the left side is $\partial_\rho(\xi^\rho\beta\cdot v)$, which integrates to zero for compact $\xi$. This gives \eqref{eq:einsteinrow}; $\beta$ is smooth, so $\Lie_\xi\beta$ is smooth and supported in $\supp\xi\subset U$. On $\Csol$, $\Box\gamma=0$ and $b=D\gamma$, so $A\gamma=-\Box\PP\gamma-2D^*D\gamma=-2D^*b$; the second equality in \eqref{eq:shellrow} is the flat adjoint of $D^*$ on a compact test. \end{proof} The free gauge-fixed shell does not impose $D^*b=0$. The Einstein row therefore stays named on the carrier. Its vanishing on the smaller Einstein subshell is XII-A's undecided question and is not decided here. \subsection{The decision} \begin{theorem}[Plateau decision for the named coefficient] \label{thm:plateau} Let $\xi\in\ga$ and $\Pmass\phi=0$. \begin{enumerate}[label=\textup{(\alph*)}] \item Always: $\Delta_\xi\PhiD=-\Aone(K\rho_\xi[\gamma];\phi) +\int(\Lie_\xi\beta_{f,g}[\phi])\cdot A\gamma$, with the companion given by \eqref{eq:companion}. \item Under (CP): $\Delta_\xi\PhiD=\int(\Lie_\xi\beta_{f,g}[\phi])\cdot A\gamma$. \item Under (CP) and $A\gamma=0$ on $U$: \[ \boxed{\ \Delta_\xi\PhiD=0.\ } \] \item Without (CP) the companion is the explicit remainder \eqref{eq:companion}, supported as in \eqref{eq:companionsupport} inside $J^+(\overline U)$, with no ramp-layer contribution. Without $A\gamma=0$ on $U$ the Einstein row is the explicit remainder \eqref{eq:einsteinrow}. \end{enumerate} \end{theorem} \begin{proof} (a) is Theorem~\ref{thm:assembly} with Propositions~\ref{prop:companion} and \ref{prop:einstein}. (b) and (c) apply the two vanishing statements. (d) restates the displays. \end{proof} \begin{corollary}[On the carrier] \label{cor:carrierplateau} For $\xi\in\ga$, the functional $\Delta_\xi\PhiD$ belongs to $\Fpol$, and in $\Ealg$ \[ Q_H\bigl(\Delta_\xi\PhiD\bigr) =Q_H\bigl(-\Aone(K\rho_\xi;\phi)\bigr) -2\,Q_H\Bigl(\int(D\Lie_\xi\beta_{f,g}[\phi])_\nu b^\nu\Bigr). \] Under (CP) the first class vanishes. The second is the class of a compact microcausal degree-two kernel in $(b,\phi)$; it vanishes on configurations with $D^*b=0$ on $U$ and is not asserted to vanish in $\Ealg$. \end{corollary} \begin{proof} Shifting the metric slots of a compact microcausal kernel by the smooth configuration $K\xi$ smears slots against a smooth function and preserves compactness and the cone bound (XII-A), so $\Delta_\xi\PhiD\in\Fpol$ once $\PhiD\in\Fpol$ (Theorem~\ref{thm:admission}). On $\Csol$, $\Pmass\phi=0$ and \eqref{eq:shellrow} hold, so Theorem~\ref{thm:plateau}(a) is an identity of restrictions to the shell, hence of classes. The kernel of the $b$-row is $\beta$'s kernel $B_{f,g}$ (XII-D, $\WF\subset Z_2$) acted on by differential operators and localized in $U$, hence compact with $\WF\subset Z_2$. Under (CP) the companion vanishes on every configuration with $\Pmass\phi=0$, in particular on $\Csol$. \end{proof} Neither hypothesis is a shrinkage of $\ga$. (CP) is a condition on the observation and the switch; $A\gamma=0$ on $U$ is a condition on the configuration. The compact shift $K\xi$ does not preserve $\Csol$ (XI-B: $\ga\cap\{\Box\xi=0\}=\{0\}$), so no action of $\ga$ by automorphisms of $\Ealg$ is inferred, and no interacting differential or quantum Ward identity is hidden in Theorem~\ref{thm:plateau}. %===================================================================== \section{E5: carrier check of the assembled coefficient} \label{sec:carrier} %===================================================================== $Q_H\CD\in\Ealg$ is XII-D's. The scalar-only completion $\Asc(\gamma,0;\phi)$ is a different kernel, not in XII-A's named sector ($\mathcal B_f$ has degree $(1,1)$, $\Tcal$ has degree $(0,3)$). Its admission is proved here by an explicit calculation; XII-D's bound for $t_{\mathrm D}$ is not cited for it. \subsection{A tree-composition lemma} Covectors are compared in the fixed Minkowski trivialization. For $n\ge2$, $Z_n=\{(x_1,p_1;\ldots;x_n,p_n):\sum p_i=0\}\setminus0$; put $Z_1=\varnothing$. A distribution $k\in\mathcal D'(M\times M)$ with finitely many fiber components is a \emph{balanced line} if \[ \WF(k)\subset\{(x,p;x',-p)\}. \] Examples: $\partial^\alpha E^{\mathrm R/A}_\mu(x-x')$ and $\partial^\alpha\delta(x-x')$, as pullbacks of distributions in $x-x'$ under a submersion \cite[Theorem~8.2.4]{Hormander}, and their products with smooth functions of $x$ and $x'$. \begin{definition}[Tree datum] A tree datum consists of a finite vertex set $V=I\sqcup X$ with $|X|=n\ge1$ external and $I$ internal vertices, a tree $T$ on $V$ (connected, $|E(T)|=|V|-1$) with an orientation of each line, a balanced line $k_\ell$ for each $\ell=(a,b)\in E(T)$, placed on the variables $(x_a,x_b)$, and a smooth weight $w_a$ for each $a\in V$. The support hypothesis is \begin{equation} \text{(S)}\qquad S:=\bigl\{(x_a)_{a\in V}\in M^V:\ x_a\in\supp w_a\ \forall a,\ (x_a,x_b)\in\supp k_\ell\ \forall\ell=(a,b)\bigr\} \ \text{is compact}. \label{eq:S} \end{equation} \end{definition} \begin{lemma}[Tree composition] \label{lem:tree} Under (S), the product $\Pi=\prod_{a\in V}w_a(x_a)\prod_{\ell\in E(T)}k_\ell(x_a,x_b)$ exists in $\mathcal D'(M^V)$ and is compactly supported. Every point of $\WF(\Pi)$ has covectors $\zeta_a=\sum_{\ell\ni a}\epsilon_{\ell,a}p_\ell$ for some $p_\ell\in\R^4$, where $\epsilon_{\ell,a}=\pm1$ according to the orientation of $\ell$ at $a$. The pushforward $t=\pi_{X*}\Pi\in\Edist(M^X)$ satisfies \[ \WF(t)\subset Z_n . \] \end{lemma} \begin{proof} Pull each $k_\ell$ back to $M^V$ along the projection to $(x_a,x_b)$; its wavefront set consists of tuples with $(p,-p)$ at the endpoints of $\ell$ and zero elsewhere \cite[Theorem~8.2.4]{Hormander}. Order the lines $\ell_1,\dots,\ell_{N}$ and let $\Pi_j$ be the product of the weights and the first $j$ lines. Claim: $\WF(\Pi_j)\subset\Sigma_j$, the set of nonzero tuples of the form $\zeta_a=\sum_{i\le j,\ \ell_i\ni a}\epsilon_{\ell_i,a}p_{\ell_i}$. For $j=0$ this is empty. Suppose it holds for $j$. The product $\Pi_j\cdot k_{\ell_{j+1}}$ is defined if there is no $(x,\zeta)\in\WF(\Pi_j)$ with $(x,-\zeta)\in\WF(k_{\ell_{j+1}})$ \cite[Theorem~8.2.10]{Hormander}. Such a pair would give an assignment $(p_{\ell_i})_{i\le j+1}$, with $p_{\ell_{j+1}}=q\neq0$ the line covector, satisfying $\sum_{i\le j+1,\ \ell_i\ni a}\epsilon_{\ell_i,a}p_{\ell_i}=0$ at every vertex $a$. The subgraph formed by $\ell_1,\dots,\ell_{j+1}$ is a forest. In a forest with at least one line there is a vertex of degree one; its unique line has zero covector; remove it and repeat. Hence all $p_{\ell_i}=0$, contradicting $q\neq0$. So the product exists and, by the same theorem, its wavefront set lies in the sums of the two wavefront sets with zero allowed, which is $\Sigma_{j+1}$. Smooth weights do not enlarge wavefront sets. This proves the claim for $\Pi=\Pi_N$ and the covector description. The support of $\Pi$ lies in $S$, compact by (S), so the projection to $M^X$ is proper on it and $\WF(t)\subset\{(x_X,\zeta_X)\ne0:\exists x_I,\ (x_X,\zeta_X;x_I,0)\in\WF(\Pi)\}$ \cite[Theorem~8.2.12]{Hormander}. A point of $\WF(\Pi)$ with zero covectors at all internal vertices is a line assignment balanced at every internal vertex. Then $\sum_{a\in X}\zeta_a=\sum_{a\in V}\zeta_a-\sum_{a\in I}\zeta_a =\sum_\ell(p_\ell-p_\ell)-0=0$. Hence $\WF(t)\subset Z_n$. \end{proof} A nonzero tuple wholly in $\Vp^{\,n}$ or $\Vm^{\,n}$ has nonzero sum, because the closed causal cones are pointed. Hence every kernel bounded by $Z_n$ is microcausal in XII-A's sense. The lemma is special to tree compositions on Minkowski space with translation-invariant lines; it is not a statement about loops. \subsection{Admission of the scalar-only completion} \begin{theorem}[The scalar completion is a microcausal compact polynomial] \label{thm:admission} The functional $(\gamma,\phi)\mapsto\Asc[f,g](\gamma,0;\phi)$ has a compact kernel $t^{\mathrm{sc}}_{f,g}\in\Edist(M^3)$, symmetric in its two metric slots, with $\WF(t^{\mathrm{sc}}_{f,g})\subset Z_3$. Hence $\Asc(\cdot,0;\cdot)\in\Fpol$, and $\PhiD(f,g)=\Asc(\gamma,0;\phi)-\CD(f,g)\in\Fpol$. \end{theorem} \begin{proof} Write $E_0=\ER$, $E_m=\ERm$ for the retarded kernels and $\Lambda^{\mu,ab}(x,z)=\sig(z)\mathfrak a^{\alpha\mu,ab}\partial^z_\alpha E_0(x-z)$, so that $X_1^\mu[\gamma](x)=\int\Lambda^{\mu,ab}(x,z)\gamma_{ab}(z)\dd z$ \cite{XA,XIID}; this is the line $\partial E_0(x-z)$ with weight $\sig$ at $z$ and a constant fiber map. Every local vertex $V_{1,g}[\gamma]\psi$, $V_{2,g}[\gamma,0]\psi$, $\mathcal V[\gamma]\psi$, $\mathcal W[\gamma,0]x^\mu$ is a finite sum of constant tensors times derivatives of $g$ or $\sig$ at the vertex point times products of differentiated deltas joining the vertex point to each of its field slots; each such delta is a balanced line. Thus each of the five terms of \eqref{eq:Asc} at $k=0$ is a finite sum of tree compositions with external vertices $z_1,z_2$ (metric) and $y$ (scalar): \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.30\linewidth}X@{}} \toprule Term & Internal vertices, weights; lines\\ \midrule $\tfrac12\int fX_1^\mu X_1^\nu\partial_\mu\partial_\nu\phi$ & $x$ ($f$); $\partial E_0(x{-}z_1)$ [$\sig$ at $z_1$], $\partial E_0(x{-}z_2)$ [$\sig$ at $z_2$], $\partial^2\delta(x{-}y)$\\[0.3em] $\int f(X_1\!\cdot\!\partial X_1)\!\cdot\!\partial\phi$ & as above with one line $\partial^2E_0(x{-}z_2)$\\[0.3em] $-\int fX_1^\mu\partial_\mu\phi_1[\gamma;\phi]$ & $x$ ($f$), $x'$ ($g,\partial g$); $\partial E_0(x{-}z_1)$ [$\sig$ at $z_1$], $\partial E_m(x{-}x')$, $\partial\delta(x'{-}z_2)$, $\partial\delta(x'{-}y)$\\[0.3em] $-\int fX_2^\mu[\gamma,0]\partial_\mu\phi$, part $\mathcal V X_1$ & $x$ ($f$), $x'$ ($\sig,\sig'$); $E_0(x{-}x')$, $\partial\delta(x'{-}z_1)$, $\partial E_0(x'{-}z_2)$ [$\sig$ at $z_2$], $\partial\delta(x{-}y)$\\[0.3em] same, part $\mathcal W[\gamma,0]x^\mu$ & $x$ ($f$), $x'$ ($\sig,\sig'$); $E_0(x{-}x')$, $\partial\delta(x'{-}z_1)$, $\partial\delta(x'{-}z_2)$, $\partial\delta(x{-}y)$\\[0.3em] $\int f\phi_2[\gamma,0;\phi]$, part $V_{1,g}\phi_1$ & $x$ ($f$), $x'$ ($g,\partial g$), $x''$ ($g,\partial g$); $E_m(x{-}x')$, $\partial\delta(x'{-}z_1)$, $\partial E_m(x'{-}x'')$, $\partial\delta(x''{-}z_2)$, $\partial\delta(x''{-}y)$\\[0.3em] same, part $V_{2,g}[\gamma,0]\phi$ & $x$ ($f$), $x'$ ($g,\partial g$); $E_m(x{-}x')$, $\partial\delta(x'{-}z_1)$, $\partial\delta(x'{-}z_2)$, $\partial\delta(x'{-}y)$\\ \bottomrule \end{tabularx} \end{center} Each graph is a tree. Hypothesis (S) holds in every row: $x\in\supp f$ is compact; an internal vertex weighted by $g$ or $\partial g$ lies in $\supp g$, one weighted by $\sig$ or $\sig'$ lies in $\{t\ge0\}$, and each lies in the causal past of its retarded predecessor by the support of $E^{\mathrm R}_\mu$; an external metric vertex reached by $\partial E_0$ carries the weight $\sig$ and lies in $J^-(\cdot)\cap\{t\ge0\}$; an external vertex reached by a delta coincides with its internal vertex. Finite intersections of causal pasts of compact sets with $\{t\ge0\}$ in the slab are compact. Lemma~\ref{lem:tree} applies to every row; finite sums and the symmetrization over $z_1\leftrightarrow z_2$ preserve compactness and the bound $Z_3$. Membership in $\Fpol$ is XII-A's definition. The last assertion adds XII-D's $\CD\in\Fpol$. \end{proof} The kernel identity between the iterated integrals and the pushforward is Fubini on the compact support for smooth fields; no approximation is used. \begin{corollary}[Class, ordering correction, enlarged sector] \label{cor:class} $Q_H\PhiD(f,g)\in\Ealg$ has even parity and ghost number zero. For an admissible smooth change $d=H'-H$, \[ \alphad\PhiD=\PhiD+\hbar L_d,\qquad L_d(\Phi)=\pair{\ell_d}{\Phi},\qquad \ell_d^C=3\,\pi_{3*}\bigl[d_{AB}(x_1,x_2)T^{ABC}(x_1,x_2,x_3)\bigr], \] with $T$ the species-symmetric kernel of $\PhiD$ and $\ell_d$ compact and smooth. The coefficientwise $(\hbar,\kappa)$-adic completion of the algebra generated by XII-D's enlarged named sector and all $Q_H\PhiD(f,g)$ is mapped by $\balphad$ onto its $\star_{H'}$ counterpart generated by the unchanged formulas. \end{corollary} \begin{proof} XII-A's free-shell descent applies to any element of $\Fpol$. The ordering computation is XII-D's Proposition on the regular ordering correction, whose proof uses only that the cubic kernel is compact with $\WF\subset Z_3$: a nonzero wavefront point of $\ell_d$ would need $(0,0,p)$ in the cubic wavefront set, forcing $p=0$. Sector stability follows as in XII-D: each new generator is transported to itself plus a regular linear field, which already belongs to the old sector; the inverse map gives the reverse inclusion; formal order is preserved. \end{proof} \subsection{The scalar-sourced slot, recorded} \begin{corollary}[Scalar-sourced slot] \label{cor:scalarsourced} The functional $\Aone[f,g](\rg[\phi];\phi)$, with $\rg[\phi]=-\tfrac12\PP\ER(gT_0[\phi])$, is a compact microcausal polynomial of degree $(0,3)$; its coordinate leg is $-\Tcal(f,g)$ of XII-A's named sector and its scalar leg is $-\int f\,\ERm(V_{1,g}[\rg[\phi]]\phi)$. The full gauge-fixed second-order coefficient on the sheet is \[ \PhiD(f,g)[\gamma,\phi]+\Aone[f,g](\rg[\phi];\phi), \] and its variation under $\gamma\mapsto\gamma+K\xi$ at fixed $\phi$ equals $\Delta_\xi\PhiD$: the added term has no metric argument. \end{corollary} \begin{proof} Both legs are tree compositions: the coordinate leg has vertices $x$ ($f$), $x'$ ($\sig$), $x''$ ($g$) with lines $\partial E_0(x-x')$, $E_0(x'-x'')$, two deltas from $x''$ to the stress slots, and $\partial\delta(x-y_3)$; the scalar leg has vertices $x$ ($f$), $x'$ ($g,\partial g$), $x''$ ($g$) with lines $E_m(x-x')$, $E_0(x'-x'')$, two deltas at $x''$, and $\partial\delta(x'-y_3)$. Hypothesis (S) holds as in Theorem~\ref{thm:admission}. The identification of the coordinate leg is XI-E's \cite{XIE}; the decomposition of the second slot is Proposition~\ref{prop:gfcoeff} with Lemma~\ref{lem:affine}. \end{proof} The named target of this paper is $\PhiD$. The scalar-sourced slot is recorded so that the word ``complete'' has a definite meaning: the second slot of Proposition~\ref{prop:gfcoeff} is $\rD+\rg$, both summands are admitted, and only $\rD$ varies under the metric shift. %===================================================================== \section{E6: boundaries, decision, and successor package} \label{sec:decision} %===================================================================== \subsection*{Boundaries recorded, not re-proved} \begin{itemize} \item The XI-G classes $\pm(k_1,k_2,0)$, $k_1+k_2=2\dd t$, remain excluded. $\CD$ and $\PhiD$ are different, model-dependent insertions and supply no representative of the original class. \item $\Jcal^{\mathrm D}_{g,\sig}(\phi_m)=\varnothing$ stands. The slot $\rD+\rg$ solves the gauge-fixed forcing equation with zero data; it is not an invariant coupled Einstein solution, and the Einstein row of Theorem~\ref{thm:plateau} is where that distinction appears. \item No quantum retarded product of $\JD$: $Q_H\CD$ and $Q_H\PhiD$ are classes of classical insertions. \item No loop, state, antibracket, master identity, interacting differential, or $\eps\to0$ limit. \item No identification with an X-track mixed map; the inverse is $-\PP\ER$, not $-4\PP\ER$; no triangle residue is subtracted; the initial surface is not moved; $J_3$ is not ramped a second time. \end{itemize} \subsection*{Decision table} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.08\linewidth} p{.24\linewidth}X@{}} \toprule Gate & Ending & Exact scope\\ \midrule E0 & Recorded & XII-D's six objects and six refusals; X-A's classical covariance flagged as the one additional premise.\\[0.4em] E1 & Constructed, with sign correction & $\PhiD=\Asc(\gamma,\rD;\phi)=\Asc(\gamma,0;\phi)-\CD$; one entry, coefficient $-1$.\\[0.4em] E2 & Constructed & Both legs with $\partial g$; affine slot identity proved.\\[0.4em] E3 & Constructed & Four rows printed before addition; the slot's two-jet row is the co-transformation; the ``$+$'' sum is detected.\\[0.4em] E4 & Constructed as a signed identity & $\Delta_\xi\PhiD=-\Aone(K\rho_\xi)+\int\Lie_\xi\beta\cdot A\gamma$; companion explicit and above the ramp; zero under (CP) and $A\gamma=0$ on $U$.\\[0.4em] E5 & Constructed & Tree lemma; $\Asc(\cdot,0;\cdot)\in\Fpol$; $Q_H\PhiD\in\Ealg$; ordering-stable enlarged sector; scalar-sourced slot admitted.\\[0.4em] E6 & Recorded & No old-class repair, jet repair, quantum product, loop, state, or limit.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[Assembly release] \label{thm:release} For the sheet of Section~\ref{sec:sheet} and the inputs of Section~\ref{sec:inputs}: the coefficient $\PhiD(f,g)$ of Definition~\ref{def:PhiD} exists, contains the admitted insertion exactly once with coefficient $-1$, belongs to $\Fpol$, and has class $Q_H\PhiD(f,g)\in\Ealg$; the sector it generates with XII-D's is ordering-stable. For $\xi\in\ga$ and free $\phi$ its plateau variation is \eqref{eq:assembly}; it vanishes under (CP) together with $A\gamma=0$ on $U$, and otherwise equals the displayed remainders \eqref{eq:companion} and \eqref{eq:einsteinrow}. \end{theorem} \begin{proof} Propositions~\ref{prop:gfcoeff} and \ref{prop:single}; Theorems~\ref{thm:assembly}, \ref{thm:plateau}, and \ref{thm:admission}; Corollaries~\ref{cor:class} and \ref{cor:carrierplateau}. \end{proof} The ending for the named coefficient is \textbf{Constructed}. An alternative requirement of unqualified invariance of $\PhiD$ is not part of this release: the Einstein row is a named remainder on the gauge-fixed shell, and the companion is a named remainder off (CP). The brief's ``$+$'' assembly is \textbf{Proved obstructed} as a candidate, by Proposition~\ref{prop:signdetect}; it is not the object released. \subsection*{Successor package} \begin{enumerate} \item $\PhiD=\Asc(\gamma,0;\phi)-\CD$, with the slot $\rD=-u_{\mathrm D}=\PP\ER\JD$ and the scalar-sourced slot $\rg$; \item the affine slot identity and the held-fixed lemma as the double-count test; \item the exact variation \eqref{eq:assembly} with the explicit companion \eqref{eq:companion}, its support \eqref{eq:companionsupport}, and the Einstein row \eqref{eq:einsteinrow}--\eqref{eq:shellrow}; \item the tree-composition lemma and the admission of $\Asc(\cdot,0;\cdot)$ and of $\Aone(\rg;\phi)$; \item the classes $Q_H\PhiD$, their ordering correction, and the enlarged ordering-stable sector; \item the detected sign: the two-jet law of the relational slot is positive, the forcing response's is negative, and the affine identity converts one into the other. \end{enumerate} It receives no interacting BRST class, quantum retarded product, loop value, state, decision on the Einstein subshell, or sharp limit. \subsection*{Conclusion} \[ \boxed{ \begin{gathered} \PhiD=\Asc(\gamma,\rD[\gamma];\phi)=\Asc(\gamma,0;\phi)-\CD, \qquad \rD=-u_{\mathrm D}=\PP\ER\JD,\\[2pt] \Delta_\xi\PhiD =-\Aone(K\rho_\xi;\phi)+\int(\Lie_\xi\beta_{f,g}[\phi])\cdot A\gamma \ \overset{\Csol}{=}\ -\Aone(K\rho_\xi;\phi)-2\int(D\Lie_\xi\beta_{f,g}[\phi])\cdot b,\\[2pt] \text{(CP)}\ \&\ A\gamma|_U=0\ \Longrightarrow\ \Delta_\xi\PhiD=0; \qquad \Delta_\xi\PhiP=-2\Aone(s_\xi;\phi)+\cdots\ne0;\\[2pt] \WF(t^{\mathrm{sc}}_{f,g})\subset Z_3,\qquad Q_H\PhiD\in\Ealg . \end{gathered}} \] A reader can see one named second-order coefficient, can see that $\CD$ entered it once and with which sign, and can see that the plateau identity holds only under the hypotheses XII-D already stated. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Classical scope.} All variations are classical identities of polynomial lifts on smooth configurations, restricted to the free shell where stated. No quantum Ward identity, quantum retarded product, or interacting differential is constructed or implied. \paragraph{Sign correction.} The coefficient $-1$ on $\CD$ is derived from the gauge-fixed second-order equation and the affine slot identity, and is confirmed by the detection of the opposite sign. It changes no inverse normalization and no object handed over by XII-D. \paragraph{Admission scope.} The tree-composition lemma proves membership of the scalar-only completion and of the scalar-sourced slot in XII-A's domain. It is not an admission of loops or of nonlocal non-tree kernels, and it does not reopen the XI-G exclusion. \paragraph{Dependency scope.} XII-D supplies the response, insertion, roots, admission of $\CD$, class, ordering correction, and ledger. XII-C supplies the gauge-fixed second-order equation. XII-A supplies the carrier. XI-B supplies the inverse normalization. X-A supplies the classical completion formula and its on-shell covariance, used as a premise and listed as such. XI-E supplies the evaluated witness $I>0$ and the slot $\rg$, used as cited values. The affine identity, single entry, held-fixed lemma, assembly, remainders, tree lemma, and admission are proved here. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant check, independently executed symbolic calculation, independent referee review, or journal acceptance is asserted. The nonzero witness is cited from XI-E and not recomputed. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{XIID} N.~Cheung, \emph{Landmark XII-D: Retarded response and carrier admission for the cubic-only density-weighted Einstein source}. Supplied manuscript \texttt{XII-D.tex}. Response, two-leg insertion, roots, admission, class, ordering correction, signed ledger, and causal plateau proposition. \bibitem{XIIC} N.~Cheung, \emph{Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question}. Supplied manuscript \texttt{XII-C.tex}. Action, source expansion, gauge-fixing Euler term, empty jet. \bibitem{XIIB} N.~Cheung, \emph{Landmark XII-B: Admission and one retarded response of the ramped cubic Einstein source, at fixed regulator width}. Supplied manuscript \texttt{XII-B.tex}. Density versus source weighting only. \bibitem{XIIA} N.~Cheung, \emph{Landmark XII-A: One graded polynomial carrier for the finite-ramp multiplet, with ordering-stable composites and a robust observation exclusion}. Supplied manuscript \texttt{XII-A-v2.tex}. Carrier, shell, descent, ordering transport, named sector, exclusion. \bibitem{XIG} N.~Cheung, \emph{Landmark XI-G: The free-shell class of the cubic Einstein kernel: the onset covector is inherited by every representative}. Supplied manuscript \texttt{XI-G.tex}. Retained exclusion; not reopened. \bibitem{XIE} N.~Cheung, \emph{Landmark XI-E: On-shell admission of the fixed-width tree, a localized first-order identity, and the held-fixed defect}. Supplied manuscript \texttt{XI-E.tex}. Scalar-sourced slot $r_g$, identification of the coordinate leg with $-\mathcal T$, switching display, and the evaluated witness $I>0$. \bibitem{XIB} N.~Cheung, \emph{Landmark XI-B: The free de Donder multiplet for the ramp Einstein--scalar sheet: causal propagators, residual gauge, and the layer dressing}. Supplied manuscript \texttt{XI-B.tex}. Inverse normalization, free identities, trivial plateau residual algebra. \bibitem{XA} N.~Cheung, \emph{Landmark X-A: Classical covariance of one fixed-regulator switched relational scalar through order $\kappa^2$}. Supplied manuscript \texttt{X-A.tex}. Scalar completion formula, affine $k$-dependence, on-shell covariance with fixed free input, pure-gauge identity for the scalar operator, transposition lemma. Classical statements only. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, arXiv:0806.1036. Smooth Cauchy theory, causal Green operators, finite propagation. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I}, second edition, Springer, 1990. Theorems~8.2.4, 8.2.10, and 8.2.12: pullback, product, and pushforward wavefront bounds. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}, arXiv:1208.1428. Comparison background for classical response functionals and their kernels; no quantum retarded product is imported. \end{thebibliography} \end{document}