% !TeX program = pdflatex %===================================================================== % XI-E.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XI-E.tex % pdflatex -interaction=nonstopmode -halt-on-error XI-E.tex % % Scope: % Three separately gated results at fixed regulator width: % 1. Polynomial Hadamard admission on the free shell. % 2. A localized first-order coefficient identity. % 3. Cancellation of the held-fixed split defect, followed by % an explicit classical variation of the omitted cubic tree. % % Corrections to the proposed memo: % 1. Leaving the finite-generator algebra does not prove that an % element cannot belong to an algebraic bimodule extension. % 2. Keeping the causal commutation relations is different from % extending the raw causal-Weyl contraction prescription. % 3. A Hadamard kernel used as product data is not a selected state. % 4. The simple mixed auxiliary blocks require translation invariance. % For general product data they must be defined by derivatives in % their actual slots. % 5. Exact inclusion of old algebra elements is distinguished from % unchanged Weyl-symbol formulas. Normal ordering changes the % latter by lower-degree quantum terms. % 6. The switch-localized support claim requires an observation % plateau. Otherwise supp((1-g)f) is an additional support term. % 7. Compact plateau ghosts are off-shell test parameters. On the % free ghost shell their class contains only zero. % 8. The single-field finite transformation includes the second % flow parameter and the quadratic flow term. % 9. The cubic Einstein response has an explicitly computed, % generally nonzero free variation. The split cancellation is % not a complete second-order interacting Ward identity. % % No expectation value, mixed-loop value, interacting BRST class, % quantum master equation, sharp-width limit, or Einstein-jet repair. % % No successful compilation, mechanized proof 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} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\MT}{M_T} \newcommand{\MTo}{M_T^\circ} \newcommand{\sig}{\sigma_\eps} \newcommand{\PP}{\mathcal P} \newcommand{\RR}{\mathcal R} \newcommand{\Lie}{\mathcal L} \newcommand{\ga}{\mathfrak g_U} \newcommand{\Afree}{\mathscr A_0} \newcommand{\Pfin}{\mathscr P_{\mathrm{fin}}} \newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\Csol}{\mathscr C_{\mathrm{sol}}} \newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}} \newcommand{\ER}{\mathsf E_0^{\mathrm R}} \newcommand{\EA}{\mathsf E_0^{\mathrm A}} \newcommand{\ERm}{\mathsf E_m^{\mathrm R}} \newcommand{\EAm}{\mathsf E_m^{\mathrm A}} \newcommand{\Ggr}{G_{\mathrm{gr}}^{\mathrm R}} \newcommand{\Pfree}{P_m} \newcommand{\sfree}{s_0} \newcommand{\Xone}{\mathsf X_1} \newcommand{\Xtwo}{\mathsf X_2} \newcommand{\Bcal}{\mathcal B} \newcommand{\Tcal}{\mathcal T} \newcommand{\Aone}{\mathcal A_1} \newcommand{\Atwo}{\mathcal A_2} \newcommand{\Obs}{\mathcal O} \newcommand{\Wcal}{\mathcal W} \newcommand{\Jcal}{\mathcal J} \newcommand{\Lag}{\mathfrak L} \newcommand{\Vone}{\mathsf V_{1,g}} \newcommand{\Vtwo}{\mathsf V_{2,g}} \newcommand{\Hcal}{\mathcal H} \newcommand{\Cthree}{\mathcal C_3} \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{\gh}{gh} \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 XI-E}\\[0.4em] On-shell admission of the fixed-width tree,\\ a localized first-order identity,\\ and the held-fixed defect} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We decide three calculations separately, retaining the graviton inverse $G_{\mathrm{gr}}^{\mathrm R}=-\mathcal P\mathsf E_0^{\mathrm R}$. First, the fixed-width insertion and classical graviton tree are admitted to a polynomial on-shell Hadamard algebra extending the finite-generator causal algebra of XI-B. The full auxiliary and ghost blocks are derived, their graded antisymmetric parts are checked, and smooth changes of product data give algebra isomorphisms. No state or expectation value is selected. Second, the complete coordinate word has a first-order coefficient identity for compact ghosts above the ramp. Its ghost-slot defect is localized in the ramp. The scalar-response frame adds switching terms; their advertised support bound requires an observation plateau. The affine ghost witness leaves precisely $\varepsilon/2$ after the complete-word cancellation, whereas the bare insertion retains its larger response. Third, the held-fixed second-order defect cancels against the order-$\kappa$ co-transformation of the metric in the first coefficient. The positive witness pairing verifies the sign. The omitted cubic Einstein response is then named and its classical variation is computed, including a nonzero evaluation. These are admission and coefficient theorems, not one interacting BRST-closure theorem. Raw causal-Weyl multiplication has a separate obstruction; the Hadamard extension preserves the causal commutation relations, not that undefined contraction prescription. The mixed loop, quantum master equation, sharp-width limit, and empty Einstein jet remain outside the release. \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) {On-shell Hadamard route}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {A second route}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A third route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The three calculations are separate. They are not one quantum theory.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Three calculations are not one theory. 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-XI-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) {On-shell Hadamard route}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A third route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The three calculations are separate. They are not one quantum theory.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Three calculations are not one theory. 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-XI-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) {On-shell Hadamard route}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {A second route}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A third route}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The three calculations are separate. They are not one quantum theory.}; \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-XI-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) {On-shell Hadamard route}; \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) {A third route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {A second route. The three calculations are separate. They are not one quantum theory.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{On-shell Hadamard route is not A third route. 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-XI-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: On-shell Hadamard route}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: A second route}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A third route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The three calculations are separate. They are not one quantum theory.}; \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-XI-E} \end{figure} Figure~\ref{fig:split-XI-E} records one view of the same separation. Figure~\ref{fig:chain-XI-E} records one view of the same separation. Figure~\ref{fig:pair-XI-E} records one view of the same separation. Figure~\ref{fig:gate-XI-E} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XI-E} records that separation. The drawing adds no hypothesis. \scopebox{Release: three separately constructed calculations}{% \begin{enumerate}[label=\textup{(\arabic*)}] \item \textbf{Route 1: Constructed.} A polynomial on-shell Hadamard extension admits $\Bcal_f$ and $\Tcal$ and embeds XI-B's algebra. The smaller raw causal-Weyl branch is obstructed under its specified continuity interface. \item \textbf{Route 2: Constructed.} The plateau coefficient identity, affine recomputation, and corrected localized support statement are proved. This is not an interacting differential. \item \textbf{Route 3: Constructed at the classical coefficient level.} The split defect cancels. The cubic Einstein tree has an explicit variation, not an asserted vanishing variation. \end{enumerate} The combined release is constructed for these named calculations. A complete interacting quantum Ward identity is not constructed. No loop value, state, sharp limit, or Einstein-jet repair is claimed.} \tableofcontents %===================================================================== \section{Scope, interfaces, and essential corrections} \label{sec:scope} %===================================================================== XI-D stopped before quantum composite admission and distinguished bare closure from closure of a complete observable \cite{XID}. This paper takes the admission step by specifying Hadamard product data. It does not turn the three routes into a single BRST theorem. Three distinctions prevent incorrect conclusions. \paragraph{Causal relations versus causal multiplication.} An extension may retain every causal commutation relation of XI-B while using a different symbol product. The obstruction below is to extending the \emph{raw causal-Weyl contraction prescription} to singular polynomial kernels with the stated continuity. It is not an obstruction to every algebra having those commutators. \paragraph{Old elements versus old symbol formulas.} An exact algebra embedding preserves old elements and products. It need not preserve their Weyl-symbol formulas after a change to normal-ordering coordinates. Agreement on smooth finite-rank symbols is stated with that ordering convention made explicit. \paragraph{Coefficient identities versus differentials.} The expression $\sfree\Bcal_f+\frac12\int f\,c\cdot\partial\phi$ is a first-order coefficient calculation. The second term is the scalar co-transformation in that calculation. Naming it does not construct a differential on the on-shell algebra. The extension is restricted to polynomials with fixed compact distributional kernels. We do not assert that a pointwise wavefront condition alone gives a closed algebra of every nonpolynomial smooth functional. This restriction also avoids the broader functional-analytic issue described in \cite{HRV}. The three endings are applied to each named route: \emph{Constructed}, \emph{Proved obstructed}, or \emph{Unresolved}. Excluded calculations are not silently incorporated into a constructed route. %===================================================================== \section{Copied sheet and frozen conventions} \label{sec:sheet} %===================================================================== Use \[ \MT=[0,T]\times\R^3,\qquad \MTo=(0,T)\times\R^3,\qquad \Sigma=\{t=0\}, \] \[ \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta,\qquad \Pfree=\Box+m^2,\qquad m\geq0. \] Parentheses include the factor $1/2$. All classical fields are smooth on a neighbourhood of the closed slab. Fix $0<\eps0$. The second contractions therefore have no distributional limit. In wavefront language, both signs of the conormal occur and the diagonal pullback of the tensor square is not defined by the multiplication criterion. A prescription requiring this limit cannot assign the proposed product. Renormalized or reordered products are different prescriptions and are not excluded by the argument. \end{proof} The quantifier includes all local quadratic symbols, as does the proposed microcausal extension. It does not exclude an abstract extension preserving the linear causal commutators. The Hadamard construction below is precisely such an extension. The example also shows why disjoint endpoint supports are not a universal multiplication theorem: distinct null-related points already exhibit the problem. %===================================================================== \section{Route 1: a named polynomial Hadamard extension} \label{sec:hadamard} %===================================================================== \subsection{Ambient realization and scalar product data} We use the surrounding Minkowski space $M=\R^4$ to define distributional kernels. The observation and switch remain on the slab. The ramp is smooth on $M$ and zero for $t\leq0$. All insertion kernels below are compact on $M^n$. Smooth free slab solutions extend to global free solutions by their Cauchy data. Thus restricting these compact kernel functionals to global solutions gives the same interpretation on $\Csol$ as the original slab prescription. No boundary test is silently replaced by an interior test. For $\mu\geq0$ set \[ \omega_\mu(\mathbf p)=\sqrt{|\mathbf p|^2+\mu^2}, \] and define the scalar bidistribution \begin{equation} W_\mu(x,y) =\int_{\R^3} \frac{e^{-i\omega_\mu(\mathbf p)(x^0-y^0) +i\mathbf p\cdot(\mathbf x-\mathbf y)}} {(2\pi)^3\,2\omega_\mu(\mathbf p)} \dd^3\mathbf p. \label{eq:W} \end{equation} It is interpreted distributionally, including the massless case. The spatial Fourier formula for $\delta_\mu$ gives \[ W_\mu-W_\mu^{\mathsf T}=-i\delta_\mu. \] To match the proposed block notation put \begin{equation} h_\mu=-W_\mu,\qquad h_\mu-h_\mu^{\mathsf T}=i\delta_\mu. \label{eq:h} \end{equation} These are bisolutions with one common oriented Hadamard wavefront relation. Their normalization follows directly from \eqref{eq:W}; it is not copied from another metric action. We use these distributions as product data. No functional $\omega$ on the resulting algebra is selected, and no expectation value is computed. \subsection{Full auxiliary and ghost blocks} Let $A=2G_{\mathrm{lin}}$. The metric--auxiliary Hessian is \[ \begin{pmatrix}A&2D^*\\2D&-2\id\end{pmatrix}, \qquad A+2D^*D=-\Box\PP, \] with \[ (D^*v)_{\mu\nu} =-\partial_{(\mu}v_{\nu)} +\tfrac12\eta_{\mu\nu}\partial_\rho v^\rho. \] Flat tensor and vector pairings are understood throughout. Define the nonzero blocks of a bidistribution matrix $H$ by \begin{align} H_{\gamma\gamma}&=-\PP h_0, & H_{\phi\phi}&=-h_m, \label{eq:Heven}\\ H_{\gamma b}&=D_yH_{\gamma\gamma}, & H_{b\gamma}&=D_xH_{\gamma\gamma}, & H_{bb}&=D_xD_yH_{\gamma\gamma}=0, \label{eq:Haux}\\ H_{c\bar c}&=-ih_0, & H_{\bar c c}&=ih_0. \label{eq:Hghost} \end{align} Vector identities accompany the ghost blocks. All other cross-blocks vanish. \begin{lemma}[Auxiliary transposition and normalization] \label{lem:blocks} For the translation-invariant data \eqref{eq:h}, \[ H_{\gamma b}=\RR_xh_0,\qquad H_{b\gamma}=D_x(-\PP h_0). \] The matrix $H$ solves the full free equations in both slots. Its graded antisymmetric part is \begin{equation} H_{AB}(x,y)-(-1)^{|A||B|}H_{BA}(y,x) =i\Delta_{AB}(x,y), \label{eq:antisym} \end{equation} where $\Delta$ is exactly XI-B's causal matrix. \end{lemma} \begin{proof} Differentiating $-\PP h_0$ in its second tensor slot and using $\partial_yh_0=-\partial_xh_0$ gives $\RR_xh_0$. In contrast, differentiation in the first slot gives the displayed $D_x$ formula. Applying both divergences produces a constant multiple of $\Box_xh_0$, hence zero. The auxiliary equations hold because $b=D\gamma$ in each slot and $-\Box\PP H_{\gamma\gamma}=0$. The scalar and ghost equations follow from the corresponding bisolution equations. Equation \eqref{eq:h} gives \[ H_{\gamma\gamma}-H_{\gamma\gamma}^{\mathsf T} =-i\PP\delta_0,\qquad H_{\phi\phi}-H_{\phi\phi}^{\mathsf T} =-i\delta_m. \] Differentiation gives the mixed auxiliary blocks $i\RR\delta_0$ and $iD(-\PP\delta_0)$. For the odd pair, \[ H_{c\bar c}+H_{\bar c c}^{\mathsf T} =-i(h_0-h_0^{\mathsf T})=\delta_0, \] which is $i\Delta_{c\bar c}$ since $\Delta_{c\bar c}=-i\delta_0$. The reversed pair similarly gives $-\delta_0$. \end{proof} In particular, \[ [\phi(x),\phi(y)]=-i\hbar\delta_m(x,y) \] has the supplied sign. The indefinite tensor factor causes no problem for this algebraic construction; positivity is not a hypothesis. For non-translation-invariant product data, the safe definitions are the actual slot derivatives in \eqref{eq:Haux}. Replacing $D_yH_{\gamma\gamma}$ by $\RR_xh_0$ without translation invariance is not valid. \subsection{The polynomial domain and product} \begin{definition}[Polynomial microcausal domain] \label{def:domain} Let $\Fpol$ consist of finite graded polynomials \[ F(\Phi)=\sum_{n=0}^N\pair{u_n}{\Phi^{\otimes n}}, \] where the kernels are compact, graded symmetric distributions on $M^n$, with finite field indices, and satisfy \begin{equation} \WF(u_n)\cap \bigl(\widetilde V_+^{\,n}\cup \widetilde V_-^{\,n}\bigr)=\varnothing. \label{eq:mc} \end{equation} The cones include zero covectors; a total zero covector is never a wavefront point. Define $\Fpol^{\mathrm{sol}}$ by restriction to the linear solution space \eqref{eq:shell}. \end{definition} This fixed-kernel polynomial class is sufficient here. Wick polynomials also belong to the more restrictive equicausal class discussed in \cite[Theorem~5.9]{HRV}. Let $\Gamma_H$ be the graded contraction operator between two functional factors, normalized so that \[ \Phi_A(u)\star_H\Phi_B(v) =\Phi_A(u)\Phi_B(v)+\hbar\pair{u}{H_{AB}v}. \] Define \begin{equation} F\star_HG =m\exp(\hbar\Gamma_H)(F\otimes G),\qquad \Ealg=\bigl(\Fpol^{\mathrm{sol}}[[\hbar,\kappa]],\star_H\bigr). \label{eq:star} \end{equation} Each contraction series terminates for fixed polynomials. The graded derivative convention includes the Koszul signs for odd slots. \begin{theorem}[Existence and on-shell descent] \label{thm:star} Equation \eqref{eq:star} defines an associative unital graded algebra. All its polynomial contractions exist. The product descends to the free shell and has the causal relations \eqref{eq:antisym}. \end{theorem} \begin{proof} Every nonzero wavefront pair of $H$ has the same orientation: one endpoint is in one causal cone and the other in the opposite cone. Derivatives and constant field matrices do not alter this fact. In a contraction, cancellation of the propagator covectors would require the contracted slots of the first kernel all to point in one cone, and those of the second all to point in the opposite cone. If every slot is contracted, this is excluded by \eqref{eq:mc}. With uncontracted slots the same argument excludes the dangerous zero covector in a pushforward. It also excludes an output wavefront tuple wholly in either causal cone. Compact kernel supports make the integrations proper. Thus contraction gives another compact kernel satisfying \eqref{eq:mc}. Associativity follows from the exponential formula: on three factors the contraction operators $\Gamma_{12},\Gamma_{13},\Gamma_{23}$ have constant coefficients and commute with the graded signs already included. Both bracketings give their common exponential followed by multiplication. For descent, the ranges of $H$ satisfy the free equations in both slots. On the reduced variables $(\gamma,\phi,c,\bar c)$, with $b=D\gamma$, all contraction directions are therefore tangent to the linear solution space. A polynomial vanishing on that space has all tangential derivatives zero there. Its contractions consequently vanish there as well. The kernel version follows by the same distributional contractions just established. Hence the vanishing ideal is a two-sided star ideal. Finally, the graded commutator of linear generators is $\hbar$ times \eqref{eq:antisym}. \end{proof} The scalar version of this construction and its normal-ordering interpretation are described in \cite[Section~5]{FR}; the block normalization and graded auxiliary calculation above are specific to this sheet. \subsection{Exact embedding and the ordering interface} Map each XI-B linear generator to its linear functional in $\Ealg$. The equations and commutators agree, so the map extends to a homomorphism \begin{equation} j_H:\Afree[[\kappa]]\longrightarrow\Ealg. \label{eq:j} \end{equation} It is injective. Indeed, PBW gives the polynomial presentation of the old central graded relations. Independent old test classes are separated by smooth solutions of the reduced wave equations. On any finite-dimensional span this gives the ordinary symmetric/exterior polynomial algebra as the classical limit. No nonzero old polynomial can acquire a zero leading symbol; the same argument, coefficientwise in $\hbar$, proves injectivity. Here is the precise finite-rank interface. Put \[ C=\frac{i}{2}\Delta,\qquad S=H-C. \] On regular finite-rank kernels, $S$ is graded symmetric and \[ \alpha_S=\exp\!\left(\frac{\hbar}{2} \pair{S}{\frac{\delta^2}{\delta\Phi^2}}\right) \] intertwines the causal-Weyl and $H$ products. Thus an old Weyl-symbol polynomial $F$ has image $\alpha_SF$. Equivalently, the old $H$-normally ordered polynomial $\alpha_S^{-1}F$ has image $F$. Admission below uses these $H$-normal symbol coordinates. It agrees exactly with old inclusion in that presentation and has the required unchanged classical symbol. We do not assert the stronger, generally divergent claim that $\alpha_S$ extends unchanged to every singular kernel. That stronger statement would confuse normal ordering with an extension of raw causal multiplication. %===================================================================== \section{Route 1: membership, change of product data, and the loop stop} \label{sec:membership} %===================================================================== \subsection{The actual fixed-width kernels} Let $G_0^{\mathrm R}(x-z)$ be the full retarded wave kernel and put \[ \mathfrak a^{\alpha\mu,ab} =\tfrac12\eta^{\alpha\mu}\eta^{ab} -\eta^{\alpha(a}\eta^{b)\mu}. \] Then \[ \Lambda_\eps^{\mu,ab}(x,z) =\sig(z)\mathfrak a^{\alpha\mu,ab} \partial_\alpha^zG_0^{\mathrm R}(x-z), \] and the mixed insertion kernel is \begin{equation} b_f^{ab}(x,z) =\partial_\mu^x\bigl(f(x)\Lambda_\eps^{\mu,ab}(x,z)\bigr). \label{eq:bf} \end{equation} Its support is compact: $x\in\supp f$ and $z\in J^-(x)\cap\{t\geq0\}$. The ramp is flat at $t=0$, so extension to the ambient space adds no boundary conormal. \begin{proposition}[Admission of the insertion and tree] \label{prop:membership} The kernels of $\Bcal_f$ and the polarized cubic $\Tcal$ belong to the polynomial domain of Definition~\ref{def:domain}. Their free-shell classes therefore define elements \[ Q_H(\Bcal_f)=[\Bcal_f],\qquad Q_H(\Tcal)=[\Tcal]\quad\text{in }\Ealg. \] Their leading symbols are the supplied classical functionals restricted to $\Csol$. \end{proposition} \begin{proof} The wave kernel has opposite endpoint covectors, including its diagonal conormals. Smooth multiplication and differentiation give \[ \WF(b_f)\subset \{(x,p;z,-p):(x,p;z,-p)\in\WF(G_0^{\mathrm R})\}. \] No nonzero opposite pair lies entirely in either causal cone. For the tree, compose \begin{equation} Q_\eps^\mu{}_{ab}(x,y) =-\int\Lambda_\eps^{\mu,cd}(x,z) \PP_{cd,ab}G_0^{\mathrm R}(z-y)\dd^4z. \label{eq:Q} \end{equation} Before integrating, the two kernels share only $z$. A violation of the multiplication criterion would force the unshared $x$ and $y$ covectors to vanish, and then force the shared covector to vanish. The product is defined. Its pushforward is proper in the compact causal region selected by $f$, $g$, and the finite slab. Attach the stress kernel at $y$ with its two scalar slots, and the final scalar slot at $x$. At each attachment the local diagonal conormal has total covector sum zero. The same one-shared-slot argument applies. Every resulting external triple satisfies \[ p_1+p_2+p_3=0 \] in the fixed Cartesian trivialization. A nonzero triple entirely in either closed causal cone cannot satisfy that identity. Polarization and slot symmetrization preserve the bound. Lower derivatives are obtained by smearing slots with smooth configurations and obey the corresponding exclusion. Ghost slots introduced by the linear free rule replace a metric slot by a differentiated ghost slot; they do not enlarge the wavefront set. Thus the graded polynomial domain also covers the free variation of $\Bcal_f$. \end{proof} For compact kernel stages with a fixed allowed wavefront cone, we use the normal distribution topology. The operations just used satisfy the applicable continuity or hypocontinuity statements of \cite{BDH}. In $H$-normal coordinates admission is the kernel inclusion followed by the on-shell quotient. This supplies fixed-width kernel-limit continuity on those stages. It is not VII-C's sharp-admission package. \subsection{Independence up to isomorphism} Let $H'$ be another full bisolution matrix with the same graded antisymmetric part and the same Hadamard singularity class. Its auxiliary blocks obey the actual slot equations. Then $d=H'-H$ is smooth and graded symmetric. \begin{proposition}[Change of product data] \label{prop:alpha} The map \[ \alpha_d =\exp\!\left(\frac{\hbar}{2} \pair{d}{\frac{\delta^2}{\delta\Phi^2}}\right) \] is an algebra isomorphism $\Ealg\to\mathscr E_{H'}$, fixing the linear generators and having inverse $\alpha_{-d}$. \end{proposition} \begin{proof} Smooth self-contractions against compact kernels exist and preserve the polynomial domain. The second-order operator has the graded product rule whose cross term is twice $\Gamma_d$. Exponentiation gives \[ \alpha_d(F\star_HG) =\alpha_dF\star_{H'}\alpha_dG. \] The bisolution property makes the map descend to the free shell. The two exponentials with opposite $d$ compose to the identity. \end{proof} This proves independence of the algebra up to isomorphism, not equality of all normal-ordering formulas. In particular, a cubic normally ordered symbol can acquire an order-$\hbar$ linear term under $\alpha_d$. A transported admission is $\alpha_dQ_H(F)$; the separately written $Q_{H'}(F)$ need not be that same abstract element without a normalization convention. At $\hbar^0$ their symbols agree. The scalar extended-Wick comparison is also expressed in \cite[Lemma~2.1]{HW}. No state was needed for these statements. Conversely, a loop coefficient can depend on the smooth product data even when no expectation value is taken. \subsection{The loop is not evaluated} The schematic mixed retarded loop contains a scalar Hadamard line together with retarded metric and coordinate lines. If $\supp f\cap\supp g=\varnothing$, the endpoint coincidence $x=y$ is absent. This removes that particular diagonal, not every singular configuration. At distinct null-related endpoints, a retarded kernel has both conormal orientations while a Hadamard kernel has one. Their naive product can therefore contain opposite covectors whose sum is zero. Failing this criterion is not by itself a universal nonexistence theorem, but it invalidates the claimed automatic multiplication. A properly defined retarded quantum product would need its ordering combinations and extension data specified. The existence of $\star_H$ does not supply those data for this nonlocal retarded loop. Its value remains unresolved and is not used in admission, nor made finite merely by disjoint supports. %===================================================================== \section{Route 2: the complete word and the affine recomputation} \label{sec:localized} %===================================================================== \subsection{The word before the solution map} For a $\kappa$-independent scalar define \[ F_f=\int f\phi\dd^4x, \] \begin{align} \Obs_f &=F_f+\kappa\Bcal_f+\kappa^2\Obs_{2,f}, \label{eq:word}\\ \Obs_{2,f} &=\int f\Bigl[ (\Xone[\gamma]\cdot\partial\Xone[\gamma] -\Xtwo[\gamma,0])\cdot\partial\phi +\tfrac12\Xone^\mu[\gamma]\Xone^\nu[\gamma] \partial_\mu\partial_\nu\phi\Bigr]\dd^4x. \label{eq:O2} \end{align} This is the coordinate expansion of $\phi\circ\mathsf X^{-1}$. The scalar responses $\rho_1,\rho_2$ and the graviton response $-\Tcal$ are solution-map outputs, not terms added to \eqref{eq:word}. No admission of \eqref{eq:O2} is needed for the first-order identity below. For this coefficient calculation let \[ d_1F_f=\frac12\int f\,c\cdot\partial\phi\dd^4x. \] This is notation for the proposed first scalar co-transformation. We do not thereby define $d_1$ on every generator or construct $\widehat s$. \begin{theorem}[Above-ramp coefficient identity] \label{thm:plateau} For every odd $c\in C_c^\infty(\{t>\eps\};TM)$, \begin{equation} L_\eps\RR c=\tfrac12c,\qquad \sfree\Bcal_f+d_1F_f=0 \label{eq:plateau} \end{equation} off shell in $\gamma$ and $\phi$. \end{theorem} \begin{proof} The tensor $a_1[\RR c]$ is supported where $\sig=1$. Consequently \[ \Box L_\eps\RR c =-\partial_\alpha a_1^{\alpha\cdot}[\RR c] =D\RR c=\tfrac12\Box c. \] Both fields have zero initial data because $c$ vanishes near $\Sigma$. Cauchy uniqueness gives the first identity. The second follows from $\sfree\Bcal_f=-\int f(L_\eps\RR c)\cdot\partial\phi$. \end{proof} No free ghost equation is used. The plateau condition involving $g$ is unnecessary for this coordinate identity. \subsection{The affine witness retains a layer term} Let $\vartheta$ be an odd Grassmann generator and take \[ c=2\vartheta t\,\partial_t,\qquad \phi=\phi_m(t)= \begin{cases}\sin(mt)/m,&m>0,\\t,&m=0.\end{cases} \] Then \[ (\RR c)_{00}=2\vartheta,\qquad a_1^{00}[\RR c]=-\vartheta, \] and \[ (L_\eps\RR c)^0 =\vartheta S_\eps(t),\qquad S_\eps(t)=\int_0^t\sig(s)\dd s. \] Ramp symmetry $\chi(1-s)=1-\chi(s)$ gives \begin{equation} S_\eps(t)=t-\eps/2,\qquad t\geq\eps. \label{eq:Seps} \end{equation} Therefore \begin{align} \sfree\Bcal_f &=-\vartheta\int fS_\eps(t)\cos(mt)\dd^4x, \label{eq:bareaffine}\\ \sfree\Bcal_f+d_1F_f &=\vartheta\frac{\eps}{2}\int f\cos(mt)\dd^4x, \qquad \supp f\subset\{t\geq\eps\}. \label{eq:completeaffine} \end{align} A nonnegative test supported where $\cos(mt)$ has a fixed nonzero sign detects both formulas. The complete-word coefficient does not vanish on this witness. It has exactly the stated width-dependent factor, while the bare insertion retains $S_\eps(t)$. There is no limit statement. The witness is not in the compact above-ramp ghost class of Theorem~\ref{thm:plateau}. %===================================================================== \section{Route 2: localized support and the scalar-response frame} \label{sec:frames} %===================================================================== \subsection{The coordinate ghost-slot kernel} Put $a_\mu=f\partial_\mu\phi$ and let $L_\eps^*$ be the formal transpose into the metric slot. The complete first coefficient is \begin{equation} \Wcal_f(c;\phi) =\int f\bigl(c/2-L_\eps\RR c\bigr)\cdot\partial\phi\dd^4x =\pair{a/2-\RR^*L_\eps^*a}{c}. \label{eq:Wlocal} \end{equation} \begin{proposition}[Ramp localization] \label{prop:ramp} At fixed smooth $\phi$, the ghost-slot coefficient in \eqref{eq:Wlocal} is a smooth compact kernel on the ambient space, with support contained in \[ \{0\leq t\leq\eps\}\cap J^-(\supp f). \] \end{proposition} \begin{proof} Transposition of the divergence-form coordinate prescription gives \[ L_\eps^*a=\sig C^*(\EA a), \] where $C r=Dr$. The advanced response is smooth and has compact support in the relevant closed-slab causal past. Multiplication by the ramp makes the kernel flat at $t=0$. Applying the differential operator $\RR^*$ gives another smooth compact kernel. Theorem~\ref{thm:plateau} says that its pairing with every compact ghost supported above $\eps$ is zero. Below $t=0$ both terms in \eqref{eq:Wlocal} vanish. The asserted support follows. \end{proof} A decomposition into homogeneous initial-data terms and layer sources is possible, but it must not be mistaken for an additional delta term in this final kernel. For the action-level ramp, the transpose above is smooth; artificial initial-surface terms in an integrated decomposition cancel appropriately. \subsection{Scalar completion and its equation term} Define \[ b_1[r]=\tfrac12\tr_\eta r,\qquad \Vone[r]u =\partial_\mu(ga_1^{\mu\nu}[r]\partial_\nu u) +gm^2b_1[r]u, \] \[ \Aone[r;\phi] =\int f\bigl(-\ERm\Vone[r]\phi -\Xone[r]\cdot\partial\phi\bigr)\dd^4x. \] For $\xi=c/2$ supported in $U$, \[ \Vone[K\xi]\phi =D_\xi\Pfree\phi-\Pfree X_\xi\phi, \] where $D_\xi u=\partial_\mu(\xi^\mu u)$ and $X_\xi=\xi^\mu\partial_\mu$. Zero data for $X_\xi\phi$ and the preflight identity give \begin{equation} \Aone[K\xi;\phi] =-\pair{\EAm f}{D_\xi\Pfree\phi} =\pair{X_\xi\EAm f}{\Pfree\phi}. \label{eq:KT} \end{equation} Thus it vanishes for a free scalar. If one names an external scalar antifield slot and the single Koszul--Tate rule $\delta_{\mathrm{KT}}\phi^\ddagger=-\Pfree\phi$, the last expression is \[ \delta_{\mathrm{KT}} \left(-\int(X_\xi\EAm f)\phi^\ddagger\dd^4x\right). \] This identifies the equation term without constructing a full antibracket or an interacting differential. The relation between \eqref{eq:plateau} and \eqref{eq:KT} is explicit: the variation of the scalar response supplies $X_\xi\phi$ and an equation term, whereas the coordinate word displays its scalar co-transformation separately. No full quantum solution-map intertwiner is inferred from this coefficient comparison. \subsection{The correct switching support} For a compact interior $\xi$, the free scalar identity gives \begin{align} \Aone[K\xi;\phi] ={}&\Wcal_f(2\xi;\phi) +\int f(g-1)X_\xi\phi\dd^4x \nonumber\\ &-\int f\,\ERm\left( [\Pfree,g]X_\xi\phi +(\partial_\mu g)a_1^{\mu\nu}[K\xi] \partial_\nu\phi\right)\dd^4x, \qquad \Pfree\phi=0, \label{eq:generaldefect} \end{align} where \[ [\Pfree,g]u=(\Box g)u+2(\partial^\mu g)\partial_\mu u. \] \begin{proof} Use $\Vone[K\xi]\phi=-g\Pfree X_\xi\phi+ (\partial g)a_1[K\xi]\partial\phi$ on the free scalar shell. Since $gX_\xi\phi$ has zero initial data, \[ \ERm(g\Pfree X_\xi\phi) =gX_\xi\phi-\ERm[\Pfree,g]X_\xi\phi. \] Subtract the coordinate term and group $\xi-L_\eps K\xi$. \end{proof} After transposition and integration by parts in the ghost, \eqref{eq:generaldefect} has ghost-slot support contained in \begin{equation} \{0\leq t\leq\eps\} \ \cup\ \supp\dd g \ \cup\ \supp((1-g)f), \label{eq:supportgeneral} \end{equation} intersected with its compact causal dependence region. The support of derivatives of $g$ is contained in the closed support of $\dd g$. \begin{corollary}[Observation-plateau localization] If $\supp f\subset P_g$, the last term in \eqref{eq:supportgeneral} is absent. The scalar-response first-order defect is localized in \[ \{0\leq t\leq\eps\}\cup\supp\dd g. \] \end{corollary} Without an observation plateau the shorter statement is false. For example, with $m=0$, $\phi=t$, and an above-ramp observation outside $\supp g$, a compact ghost supported there gives the nonzero local dressing variation, even though it misses both the ramp and $\supp\dd g$. In a spatially constant massless diagnostic, $c=2\vartheta t\partial_t$ gives \[ \rho_1[K(c/2);t]=\vartheta\int_0^t g_0(s)\dd s. \] Thus the scalar-response coefficient, after the ramp, is \[ \vartheta\int f\left( \int_0^t g_0(s)\dd s-t+\eps/2\right)\dd^4x. \] A sufficiently large spatial cutoff realizes this formula in an observation dependence cylinder. It is a diagnostic of the different frame, not \eqref{eq:completeaffine}. Finally, \[ \Box c=0,\qquad \supp c\Subset U \quad\Longrightarrow\quad c=0. \] This is XI-B's Cauchy-uniqueness lemma \cite{XIB}. The plateau identity is nontrivial for off-shell ghost test parameters; its restriction to compact plateau free ghosts has only the zero parameter. Unrestricted free ghosts, including the affine witness, are not thereby made closed. %===================================================================== \section{Route 3: the held-fixed sign cancels} \label{sec:split} %===================================================================== Let $\Atwo^{\mathrm{sc}}[H,k;\phi]$ be the complete classical scalar-only second coefficient from X-A: it includes the second scalar response, mixed response--coordinate term, inverse-coordinate term, and quadratic Taylor term \cite{XA}. Its dependence on the second metric coefficient is \begin{equation} \Atwo^{\mathrm{sc}}[H,k+r;\phi] -\Atwo^{\mathrm{sc}}[H,k;\phi] =\Aone[r;\phi]. \label{eq:affine} \end{equation} Indeed, the $k$ change in $\Xtwo$ is $\Xone[r]$, and the $k$ change in the second scalar response is $-\ERm\Vone[r]\phi$. Set $H=h+\gamma$ and \[ \Atwo^{\mathrm{cand}}[h,k,\gamma;\phi] =\Atwo^{\mathrm{sc}}[h+\gamma,k;\phi]-\Tcal[\phi]. \] For a free scalar, XI-C's forward held-fixed difference is \begin{equation} \Atwo^{\mathrm{cand}}[h',k',\gamma;\phi] -\Atwo^{\mathrm{cand}}[h,k,\gamma;\phi] =-\Aone[\Lie_\xi\gamma;\phi], \label{eq:held} \end{equation} where \[ h'=h+K\xi,\qquad k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi, \qquad \xi,\zeta\in\ga. \] \begin{theorem}[Cancellation in the complete coefficient expansion] \label{thm:cancel} In \[ F_f+\kappa\Aone[h+\gamma;\phi] +\kappa^2\Atwo^{\mathrm{cand}}[h,k,\gamma;\phi], \] restore the order-$\kappa$ shift $\gamma\mapsto\gamma+\kappa\Lie_\xi\gamma$ while applying the stated background transformation. The order-$\kappa^2$ held-fixed defect cancels exactly. \end{theorem} \begin{proof} Linearity gives \[ \kappa\Aone[h'+\gamma+\kappa\Lie_\xi\gamma;\phi] =\kappa\Aone[h'+\gamma;\phi] +\kappa^2\Aone[\Lie_\xi\gamma;\phi]. \] The first coefficient's $K\xi$ difference vanishes for a free scalar by \eqref{eq:KT}. Changes of the second coefficient caused by this additional order-$\kappa$ shift start at $\kappa^3$. Its remaining difference is \eqref{eq:held}, which has exactly the opposite sign. The tree has no external metric argument to vary here. \end{proof} For a single metric field $q=\eta+\kappa\gamma$, $k=0$ describes the original parametrization, not a prohibition on $\kappa$ dependence after transformation. The genuine flow of $\kappa\xi+\kappa^2\zeta$ gives \[ \gamma' =\gamma+K\xi +\kappa\left(\Lie_\xi\gamma+K\zeta +\tfrac12\Lie_\xi K\xi\right) +O(\kappa^2). \] Its infinitesimal law is $\delta\gamma=K\xi+\kappa\Lie_\xi\gamma$. No independent second metric field is needed for this bookkeeping identity. \subsection{The evaluated sign check} Take \[ m=0,\qquad \gamma=t\eta,\qquad \phi=t^2+\tfrac13|\mathbf x|^2,\qquad \xi=a(t)\partial_t \] in a sufficiently large causal cylinder, with $a\geq0$, $a\not\equiv0$, compactly supported above the ramp. A compact spatial cutoff and plateau make $\xi$ admissible without changing the calculation. For $r=\Lie_\xi\gamma$, \[ r_{00}=a+2ta',\qquad r_{ij}=-a\delta_{ij}, \] \[ a_1^{00}[r]=a-ta',\qquad a_1^{ij}[r]=-(a+ta')\delta^{ij},\qquad D^0r=ta''. \] Hence \[ \Xone^0[r]=ta-2\int_0^t a(s)\dd s,\qquad \Vone[r]\phi=-4ta'-2t^2a'', \] and \[ -\ER\Vone[r]\phi =2t^2a-4\int_0^t s\,a(s)\dd s. \] For a nonnegative observation after the support of $a$, \begin{equation} I:=\Aone[r;\phi] =4\int f(t,\mathbf x) \int_0^t(t-s)a(s)\dd s\,\dd^4x>0. \label{eq:I} \end{equation} The held-fixed change contributes $-I$; the restored first coefficient contributes $+I$. Their sum is zero. The same tensor $r$ is not generally free. In the cylinder its spatial wave operator contains $-a''\delta_{ij}$, which is not identically zero for a nonzero compact bump. Thus the co-transformation is a legitimate off-shell coefficient operation, not an automorphism of XI-B's free solution algebra. %===================================================================== \section{Route 3: the cubic Einstein tree and its computed variation} \label{sec:cubic} %===================================================================== The preceding cancellation is not the cubic Einstein calculation. We now name that missing contribution and compute its classical variation. \subsection{The local source is explicit} Retain XI-A's curvature convention and \[ S_{\mathrm{EH}}=-\frac{2}{\kappa^2} \int\sqrt{-q}\,R[q]\dd^4x, \qquad q=\eta+\kappa\gamma. \] Write \[ \sqrt{-q}\,G^{\mu\nu}(q) =\kappa\Hcal_1^{\mu\nu}[\gamma] +\kappa^2\Hcal_2^{\mu\nu}[\gamma]+O(\kappa^3), \qquad A\gamma=2\Hcal_1[\gamma]. \] For completeness, a finite prescription for $\Hcal_2$ is as follows: \[ \Gamma_1^\rho{}_{\mu\nu} =\tfrac12\eta^{\rho\sigma} (\partial_\mu\gamma_{\nu\sigma} +\partial_\nu\gamma_{\mu\sigma} -\partial_\sigma\gamma_{\mu\nu}), \] \[ \Gamma_2^\rho{}_{\mu\nu} =-\tfrac12\gamma^{\rho\sigma} (\partial_\mu\gamma_{\nu\sigma} +\partial_\nu\gamma_{\mu\sigma} -\partial_\sigma\gamma_{\mu\nu}), \] \[ R_{1,\mu\nu} =\partial_\rho\Gamma_1^\rho{}_{\mu\nu} -\partial_\nu\Gamma_1^\rho{}_{\mu\rho}, \] \[ R_{2,\mu\nu} =\partial_\rho\Gamma_2^\rho{}_{\mu\nu} -\partial_\nu\Gamma_2^\rho{}_{\mu\rho} +\Gamma_1^\rho{}_{\rho\lambda}\Gamma_1^\lambda{}_{\mu\nu} -\Gamma_1^\rho{}_{\nu\lambda}\Gamma_1^\lambda{}_{\mu\rho}. \] Put \[ r_1=\eta^{\mu\nu}R_{1,\mu\nu},\qquad r_2=\eta^{\mu\nu}R_{2,\mu\nu} -\gamma^{\mu\nu}R_{1,\mu\nu}, \] \[ C_{1,\mu\nu}=R_{1,\mu\nu}-\tfrac12\eta_{\mu\nu}r_1, \qquad C_{2,\mu\nu}=R_{2,\mu\nu} -\tfrac12\eta_{\mu\nu}r_2-\tfrac12\gamma_{\mu\nu}r_1. \] Then, with $\tau=\tr_\eta\gamma$, \begin{equation} \Hcal_2^{\mu\nu} =C_2^{\mu\nu}+\tfrac12\tau C_1^{\mu\nu} -\gamma^{\mu\alpha}C_{1,\alpha}{}^\nu -\gamma^{\nu\alpha}C_{1,\alpha}{}^\mu. \label{eq:H2} \end{equation} The cubic Einstein vertex has metric Euler derivative \[ J_3[\gamma]=2\Hcal_2[\gamma]. \] Its actual classical retarded metric response is \begin{equation} u_3[\gamma]=-\Ggr J_3[\gamma] =\PP\ER(2\Hcal_2[\gamma]). \label{eq:u3} \end{equation} The sign follows from the response rule and the frozen inverse. Define its affine scalar-completed insertion by \begin{equation} \Cthree(f,g)[\gamma,\phi] =\Aone[u_3[\gamma];\phi]. \label{eq:C3} \end{equation} This has field degree $(2,1)$, unlike the scalar-cubic tree $\Tcal$. It is a named classical sourced evaluation, not a quantum retarded contraction constructed by this paper. \subsection{Variation from tensor-density covariance} For a contravariant rank-two density define \[ (\Lie_\xi^{\mathrm{dens}}v)^{\mu\nu} =\xi^\rho\partial_\rho v^{\mu\nu} -v^{\rho\nu}\partial_\rho\xi^\mu -v^{\mu\rho}\partial_\rho\xi^\nu +(\partial_\rho\xi^\rho)v^{\mu\nu}. \] \begin{theorem}[Cubic source and tree variation] \label{thm:cubicvar} Let $\delta_0\gamma=K\xi$, with $\xi\in\ga$, and hold the free scalar fixed. Then \begin{align} \delta_0J_3[\gamma] &=\Lie_\xi^{\mathrm{dens}}(A\gamma) -A(\Lie_\xi\gamma), \label{eq:dJ}\\ \delta_0u_3 &=-\Ggr\left( \Lie_\xi^{\mathrm{dens}}(A\gamma) -A(\Lie_\xi\gamma)\right), \label{eq:du}\\ \delta_0\Cthree &=\Aone[\delta_0u_3;\phi]. \label{eq:dC} \end{align} If $A\gamma=0$ and $r=\Lie_\xi\gamma$, the response simplifies to \begin{equation} \delta_0u_3=\Ggr Ar =r-2\RR\ER Dr. \label{eq:duvac} \end{equation} \end{theorem} \begin{proof} Naturality of the invariant Einstein Euler tensor density under the infinitesimal flow of $\kappa\xi$ is exact. Its expansion is $A\gamma+\kappa J_3[\gamma]+\cdots$. The metric variation is $K\xi+\kappa\Lie_\xi\gamma$. At order $\kappa$, covariance gives \[ A(\Lie_\xi\gamma)+\delta_0J_3[\gamma] =\Lie_\xi^{\mathrm{dens}}(A\gamma), \] proving \eqref{eq:dJ}. Applying the fixed linear response operator proves \eqref{eq:du}. Linearity of $\Aone$ proves \eqref{eq:dC}. If $A\gamma=0$, only $-Ar$ remains in \eqref{eq:dJ}. Since $r$ is compact inside $U$, its data vanish initially. Use \[ A+2D^*D=-\Box\PP,\qquad \Ggr D^*=\RR\ER \] and the left inverse identity on compact tests to obtain \[ \Ggr Ar=r-2\Ggr D^*Dr=r-2\RR\ER Dr. \] \end{proof} The source identity is compatible with XI-A's nonlinear Noether identity. If \[ Q_\nu(v) =2\partial_\mu(v^{\mu\rho}\gamma_{\rho\nu}) -v^{\mu\rho}\partial_\nu\gamma_{\mu\rho}, \] its gravitational order-one coefficient is \[ 2\partial_\mu(J_3^{\mu\rho}\eta_{\rho\nu}) +Q_\nu(A\gamma)=0. \] For the invariant coupled action, the switching breaking vanishes on $U$. Gauge fixing nevertheless has its separate contribution, whose leading divergence is $-\Box D^\nu$. Neither fact permits replacing the full variation by zero. \subsection{The cubic tree is not individually closed} The witness of Section~\ref{sec:split} also evaluates \eqref{eq:dC}. For $\gamma=t\eta$, $A\gamma=0$. In the causal cylinder, \[ Dr=(ta'',0,0,0),\qquad (\ER Dr)^0=ta-2\int_0^t a(s)\dd s. \] Consequently \eqref{eq:duvac} gives \[ (\delta_0u_3)_{00}=3a,\qquad (\delta_0u_3)_{ij}=-a\delta_{ij},\qquad (\delta_0u_3)_{0i}=0. \] For this tensor, \[ a_1^{00}[\delta_0u_3]=0,\qquad a_1^{ij}[\delta_0u_3]=-2a\delta^{ij}, \qquad \Xone[\delta_0u_3]=0 \] in the cylinder. With the massless scalar used there, \[ \Vone[\delta_0u_3]\phi=-4a. \] It follows that \begin{equation} \delta_0\Cthree =4\int f(t,\mathbf x) \int_0^t(t-s)a(s)\dd s\,\dd^4x =I>0. \label{eq:cubicwitness} \end{equation} Thus $\sfree\Tcal=0$ does not imply closure of this cubic Einstein response. Equation \eqref{eq:cubicwitness} is a computed variation, not a failed split cancellation: the latter already canceled $-I$ against $+I$ in Theorem~\ref{thm:cancel}. A complete interacting second-order identity would have to include the other gravitational, scalar, ghost, gauge-fixing, and antifield terms with their proper response maps. We do not claim that identity from \eqref{eq:dJ}. In particular, membership of $\Bcal_f$ and $\Tcal$ in $\Ealg$ is not a quantum retarded-product theorem for the unswitched cubic Einstein source. %===================================================================== \section{Decision table, exclusions, and handoff} \label{sec:decision} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}X@{}} \toprule Route & Decision and exact scope\\ \midrule 1, raw branch & \textbf{Proved obstructed} for continuous extension of unmodified causal-Weyl contractions to the declared local quadratic class. No universal bimodule or CCR-extension nonexistence theorem.\\[0.4em] 1, Hadamard branch & \textbf{Constructed}. Full normalized product data, polynomial star algebra, on-shell descent, old algebra embedding, admission, and smooth-change isomorphisms.\\[0.4em] 2 & \textbf{Constructed}. Above-ramp coefficient identity, affine $\eps/2$ remainder, and localized support. The switching support statement has the required observation-plateau hypothesis.\\[0.4em] 3 & \textbf{Constructed as classical calculations}. Held-fixed pairing cancels. The omitted cubic response and its generally nonzero variation are explicit.\\ \bottomrule \end{tabularx} \end{center} The worst ending among the three named affirmative routes is therefore \textbf{Constructed}. This does not reclassify unconstructed interacting quantum closure as constructed. Its antibracket, master identity, response products, and renormalization conditions remain separate gates. The following exclusions persist. \begin{itemize} \item No mixed-loop value is computed. Disjoint supports remove one coincidence locus but are not a loop theorem. \item No state, representation, positivity requirement for the full multiplet, or expectation value is selected. \item No sharp-regulator limit or identification with a sharp Cauchy insertion is made. \item The empty fixed-input Einstein jet $\Jcal_g(\phi_m)=\varnothing$ remains XI-A's result. The graviton tree propagates the switched source's de Donder violation; it does not repair conservation. \item The word $B_f\star W_g\star B_{f'}$, reflected residue, and VII-C's four topological interfaces are not constructed. \item The order-$\kappa$ metric co-transformation is not declared an automorphism of the free-shell algebra. \end{itemize} A successor receives a genuine admitted polynomial algebra, the two distinct affine numbers \eqref{eq:bareaffine} and \eqref{eq:completeaffine}, the corrected support formula \eqref{eq:supportgeneral}, the exact split cancellation, and the cubic variation \eqref{eq:dC}. It does not receive an interacting BRST class merely because these calculations now have names. \subsection*{Conclusion} At fixed width, the three proved answers are: \[ \boxed{ \begin{gathered} \Bcal_f,\Tcal\text{ are admitted in a named on-shell polynomial Hadamard extension};\\ L_\eps\RR c=c/2\text{ for compact above-ramp ghosts}, \quad \text{the affine complete-word remainder is }\eps/2;\\ \text{the held-fixed split defect cancels}, \quad \text{the cubic Einstein variation is explicit and need not vanish}. \end{gathered}} \] These statements preserve XI-B's coefficient $-1$ and separate admission, localization, and nonlinear response. They are not one closure theorem. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Construction level.} The quantum construction is a formal polynomial Hadamard algebra with compact distributional kernels and an on-shell quotient. It is not a construction of every smooth microcausal functional or an interacting quantum theory. \paragraph{Ordering convention.} The old causal algebra is embedded exactly on its generators and products. Its Weyl symbols are related to the chosen normal symbols by the explicitly stated ordering map. No singular extension of that regular ordering map is assumed. \paragraph{Classical identity level.} The localized ghost calculation and metric co-transformation are coefficient identities on their stated off-shell parameter classes. The cubic source variation is a classical sourced response calculation. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant verification, executed symbolic computation, independent referee review, or journal acceptance is asserted. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \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}. Used for the coordinate prescription, scalar completion, Cauchy facts, and plateau intertwiners. \bibitem{XIA} N.~Cheung, \emph{Landmark XI-A: A switched Einstein--scalar action for the ramp relational scalar: sign conventions, Noether identities, and an empty fixed-input Einstein jet}. Supplied manuscript \texttt{XI-A.tex}. Used for the signed Einstein action, tensor-density identities, and finite curvature coefficients. \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}. Used for the causal matrix, auxiliary multiplet, normalization, and restricted free algebra. \bibitem{XIC} N.~Cheung, \emph{Landmark XI-C: One fixed-regulator graviton tree for the ramp relational scalar: the retarded symbol, its normalization, and the boundary of the free-algebra handoff}. Supplied manuscript \texttt{XI-C.tex}. Used for the tree sign and held-fixed defect. \bibitem{XID} N.~Cheung, \emph{Landmark XI-D: Admission of the fixed-width graviton tree and one ghost-number-zero identity: the admission interface and a free-order closure obstruction}. Supplied manuscript \texttt{XI-D.tex}. Used for the admission stop and the distinction between bare and complete-word closure. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}. arXiv:1208.1428v2, 2013. Section~5 supplies background for regular ordering equivalences and Hadamard polynomial products. \bibitem{HW} S.~Hollands and R.~M.~Wald, ``Local Wick polynomials and time ordered products of quantum fields in curved spacetime,'' \emph{Communications in Mathematical Physics} \textbf{223} (2001), 289--326. arXiv:gr-qc/0103074. Used for the extended-Wick comparison and smooth-change algebra isomorphism. \bibitem{BDH} C.~Brouder, N.~V.~Dang, and F.~H\'elein, ``Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker),'' \emph{Studia Mathematica} \textbf{232} (2016), 201--226. arXiv:1409.7662. Used for normal-topology operations on specified wavefront kernel stages. \bibitem{HRV} E.~Hawkins, K.~Rejzner, and B.~Visser, \emph{A novel class of functionals for perturbative algebraic quantum field theory}. arXiv:2312.15203. Used to distinguish fixed-kernel Wick polynomials from unrestricted pointwise-microcausal smooth functional classes; in particular, Theorem~5.9 identifies Wick polynomials as equicausal. \end{thebibliography} \end{document}