% !TEX program = pdflatex
%=====================================================================
% X-E.tex
%
% Standalone source for pdfLaTeX.
%
% Build:
%   pdflatex -interaction=nonstopmode -halt-on-error X-E.tex
%   pdflatex -interaction=nonstopmode -halt-on-error X-E.tex
%
% Frozen:
%   Signature (+,-,-,-). Fixed regulator width epsilon. Fixed f, g,
%   plateau algebra, classical backgrounds h_cl, k_cl (off shell).
%   Gravitational action: Einstein-Hilbert, S_gr = -(2 kappa^2)^{-1}
%   int sqrt(-q) R, sign fixed by positivity of the TT kinetic term.
%   Gauge: de Donder with Feynman weight, quantum graviton only.
%   Quantum fields: free scalar, free gauge-fixed graviton. No second-
%   order quantum metric, no ghosts, no state-dependent statement.
%
% Result:
%   E1 action, metric Euler equation, Noether identity with switch
%   defect. E2 exact plateau invariance of the total Lagrangian,
%   covariance of the metric Euler operator, classical identity
%   through kappa^2, Bianchi obstruction to the coupled jet.
%   E3 P_gr = -(1/4) P Box, Delta_R = -4 P E_0^R, residual algebra
%   meets the plateau algebra in zero. E4 one tree, admitted,
%   supported in J^-(supp f), normalized by the classical retarded
%   response; background identity through kappa^1; kappa^2 defect
%   displayed and proved nonzero. E5 the VII-C word is not formed.
%
% Corrections relative to the planning notes:
%   - A quadratic (Fierz-Pauli) action alone is not invariant under
%     the plateau group at O(kappa); E2 would fail. Einstein-Hilbert
%     is frozen; its quadratic part is derived from the Palatini
%     first variation, not copied.
%   - The gravitational self-interaction tree X_1[h]-S_3 exists at
%     kappa^2 as a single-vertex tree. The leg-count exclusion in the
%     notes was wrong. It is named, not constructed.
%   - The scalar backreaction sits in the kappa^2 (k) slot of the
%     metric; the tree is the affine insertion k -> k + k_1[phi].
%   - The normalization constant of the dressed response is -2:
%     X_1[k_1] = -2 E_0^R E_0^R((d g) T[phi_0]).
%   - The kappa^2 quantum-metric background identity has the defect
%     -kappa^2 A_1[L_xi h]. It is displayed and proved nonzero. The
%     sheet freezes that identity through kappa^1 only.
%   - The Bianchi obstruction is proved from the exact Noether
%     identity, not from an expansion of the second-order Einstein
%     tensor.
%   - The defect witness vanishes when supp f lies in the
%     chronological future of the plateau (Huygens); the witness
%     needs supp f to meet the boundary of J^+(supp chi).
%   - No claim of successful compilation or independent verification.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
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\usepackage{lmodern}
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\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

\allowdisplaybreaks
\setlength{\emergencystretch}{3em}
\setlength{\parskip}{2pt}
\setlist{itemsep=3pt,topsep=5pt,leftmargin=*}
\numberwithin{equation}{section}
\setcounter{tocdepth}{1}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{diagnostic}[theorem]{Diagnostic}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\Lie}{\mathcal L}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\Pm}{P_m}
\newcommand{\MT}{M_T}
\newcommand{\Dspace}{\mathcal D}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\Gp}{\mathcal G_U}
\newcommand{\ER}{\mathsf E_m^{\mathrm R}}
\newcommand{\EA}{\mathsf E_m^{\mathrm A}}
\newcommand{\ERzero}{\mathsf E_0^{\mathrm R}}
\newcommand{\EAzero}{\mathsf E_0^{\mathrm A}}
\newcommand{\Lone}{\mathsf L_{1,g}}
\newcommand{\Ltwo}{\mathsf L_{2,g}}
\newcommand{\field}{\widehat\phi}
\newcommand{\hq}{\widehat h}
\newcommand{\X}[1]{\mathsf X_{#1}}
\newcommand{\Ldens}{\mathfrak L}
\newcommand{\Sgr}{S_{\mathrm{gr}}}
\newcommand{\Ssc}{S^{\mathrm{sc}}_{g}}
\newcommand{\Glin}{G_{\mathrm{lin}}}
\newcommand{\Pgr}{P_{\mathrm{gr}}}
\newcommand{\DR}{\Delta_{\mathrm R}^{\mathrm{gr}}}
\newcommand{\DA}{\Delta_{\mathrm A}^{\mathrm{gr}}}
\newcommand{\Ptr}{\mathcal P}
\newcommand{\Ahat}[1]{\widehat A_{#1}}
\newcommand{\Qhat}{\widehat Q}
\newcommand{\Wick}{\mathfrak W}
\newcommand{\Ideal}{\mathfrak J}
\newcommand{\Ttree}{\mathsf T_{\mathrm{tree}}}
\newcommand{\Th}{\Theta}
\newcommand{\Eul}{\mathsf E}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\formal}{\pmod{\kappa^3}}

% Symbols used only when copying the supplied VII-C assumption.
\newcommand{\Pol}{\mathscr P_{\mathrm{mc}}}
\newcommand{\cA}{\mathcal A}
\newcommand{\ev}{\operatorname{ev}_0}

\newcommand{\scopebox}[2]{%
  \par\medskip\noindent
  \fbox{\begin{minipage}{%
    \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
    \textbf{#1}\par\smallskip #2
  \end{minipage}}
  \par\medskip
}

\title{\textbf{Landmark X-E}\\[0.4em]
A specified gravitational action for the ramp relational scalar,\\
and the first graviton tree}
\author{Ngo Cheung}
\date{}

\usepackage{tikz}
\begin{document}
\maketitle

\begin{abstract}
We freeze one gravitational action, one gauge condition, and one
quadratic graviton operator for the ramp relational scalar of
Landmarks X-A to X-D, in signature \((+,-,-,-)\), at fixed
regulator width.

The action is the Einstein--Hilbert density,
\(\Sgr=-(2\kappa^2)^{-1}\int\sqrt{-q}\,R\), with the sign fixed by
positivity of the transverse-traceless kinetic term and confirmed by
Newtonian attraction.
Its first variation is derived from the Palatini identity.
The metric Euler equation is displayed with every switch factor, and
the Noether identity of the total action is proved with its explicit
switch defect.
On the plateau algebra the total Lagrangian is exactly invariant, so
the metric Euler operator is covariant and the classical identity of
X-A to X-C holds through \(\kappa^2\) on the coupled constraint
surface. That surface is generically empty: the Bianchi identity
forces \((\partial_\mu g)T^{\mu}{}_{\rho}[\phi_0]=0\).

The gauge-fixed operator is \(\Pgr=-\tfrac14\Ptr\Box\) with retarded
inverse \(-4\Ptr\ERzero\), \(\Ptr\) the trace reversal.
Its residual gauge algebra meets the compactly supported plateau
algebra only in zero, so the frozen algebra acts on classical
backgrounds alone.

One tree is constructed: the metric variation of the switched scalar
density, propagated by \(\DR\) and inserted into the affine slot of
the completed observable.
It is a totally Wick-ordered cubic in the scalar, supported in
\(J^-(\supp f)\), microcausally admitted, and normalized by the
classical retarded response, whose relational dressing is
\(-2\ERzero\ERzero\bigl((\partial_\nu g)T^{\nu\mu}[\phi_0]\bigr)\).
The one-tree observable is invariant under the frozen algebra
through \(\kappa^1\). At \(\kappa^2\), the background variation with
the graviton held fixed equals \(-\kappa^2\Ahat1[\Lie_\xi\hq]\)
modulo the equation ideals, and this class is nonzero.

The gravitational self-interaction tree, the mixed one-loop term,
the second-order quantum metric, and VII-C's word
\(B_f\star W_g\star B_{f'}\) are named and not constructed.
The regulator is not removed.
\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) {Inherited ramp data};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {This ending};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A renamed earlier theorem};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The ending is not a relabeling of X-A through X-D.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A further ending is not a relabeling. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-X-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) {Inherited ramp data};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A renamed earlier theorem};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The ending is not a relabeling of X-A through X-D.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A further ending is not a relabeling. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-X-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) {Inherited ramp data};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {This ending};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A renamed earlier theorem};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The ending is not a relabeling of X-A through X-D.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-X-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) {Inherited ramp data};
\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 renamed earlier theorem};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {This ending. The ending is not a relabeling of X-A through X-D.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Inherited ramp data is not A renamed earlier theorem. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-X-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: Inherited ramp data};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: This ending};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A renamed earlier theorem};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The ending is not a relabeling of X-A through X-D.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-X-E}
\end{figure}

Figure~\ref{fig:split-X-E} records one view of the same separation. Figure~\ref{fig:chain-X-E} records one view of the same separation. Figure~\ref{fig:pair-X-E} records one view of the same separation. Figure~\ref{fig:gate-X-E} records one view of the same separation. The drawings add no hypothesis.


Figure~\ref{fig:scope-X-E} records that separation.
The drawing adds no hypothesis.



%=====================================================================
\section{Release and the three endings}
\label{sec:release}
%=====================================================================

\scopebox{Release: Constructed, at fixed regulator and fixed input,
with a stated order boundary}{%
Frozen: \(\Sgr=-(2\kappa^2)^{-1}\int\sqrt{-q}R\);
de Donder gauge fixing \(\tfrac14\int C_\mu C^\mu\) for the quantum
graviton; \(\Pgr=-\tfrac14\Ptr\Box\); \(\DR=-4\Ptr\ERzero\).

\smallskip
Proved.
\textbf{E1}: the metric Euler equation
\eqref{eq:metric-euler}--\eqref{eq:metric-euler-expanded}
and the Noether identity \eqref{eq:noether} with its switch defect.
\textbf{E2}: exact \(\Gp\)-invariance of the total Lagrangian,
covariance of the metric Euler operator, the classical identity
through \(\kappa^2\) on the coupled constraint surface, and the
Bianchi obstruction \eqref{eq:bianchi-constraint}.
\textbf{E3}: the operator, its retarded inverse, the projector,
and the residual algebra.
\textbf{E4}: one tree \(\Ttree\), Definition~\ref{def:tree};
kernel, support, admission, normalization.
The one-tree observable is invariant on \(\ga\) through
\(\kappa^1\). Its \(\kappa^2\) background variation with \(\hq\)
fixed is \(-\kappa^2\Ahat1[\Lie_\xi\hq]\) modulo the ideals,
Theorem~\ref{thm:ward}; this class is nonzero,
Proposition~\ref{prop:witness}.
\textbf{E5}: no loop, no \(W_g\), no subtraction.

\smallskip
Not opened: the self-interaction tree, the mixed loop, a
second-order quantum metric, ghosts, gauge-parameter dependence,
\(\eps\to0\), X-Enter.}

Three endings are available:
\begin{enumerate}[label=\textbf{\arabic*.}]
\item \textbf{Constructed.}
The action, gauge fixing, and one tree are named, and the completed
observable transforms correctly on the frozen algebra through the
frozen order.
\item \textbf{Proved obstructed.}
The gauge identity acquires a nonzero defect on that algebra at the
frozen order, or the tree kernel leaves the declared domain.
\item \textbf{Unresolved.}
The action, the gauge condition, or the propagator is unnamed.
\end{enumerate}
The ending reached is (1) for the gates as frozen in
Section~\ref{sec:frozen}: classical identity through \(\kappa^2\),
quantum-metric background identity through \(\kappa^1\), one tree at
\(\kappa^2\).
The \(\kappa^2\) quantum-metric identity is not frozen; its defect
is displayed. A sheet that froze that identity on this field content
would reach ending (2) with Proposition~\ref{prop:witness} as the
witness. The defect is not removed by shrinking the algebra.

A missing object is recorded as missing, not as a no-go.
Nothing in this manuscript depends on a state for the graviton.

%=====================================================================
\section{Frozen sheet}
\label{sec:frozen}
%=====================================================================

The coordinate prescription, density coefficients, and complete
two-jet gauge law are copied from \texttt{X-A.tex} \cite{XA};
the bare tests, multiplet Wick algebra, and fixed-regulator
admission from \texttt{X-B-v3.tex} \cite{XB}; the completed tests,
recursion, and fixed-regulator identity from \texttt{X-C-v3.tex}
\cite{XC}; the separation sheet and explicit equation-ideal
representatives from \texttt{X-D-v3.tex} \cite{XD}.
Their proofs are not repeated.

\subsection{Geometry, supports, regulator}

\[
M=\R^4,\qquad \eta=\diag(1,-1,-1,-1),\qquad
\Box=\partial_t^2-\Delta,\qquad \Pm=\Box+m^2,\quad m\ge0 .
\]
Indices are moved with \(\eta\) unless \(q\) is written.
\(\ER,\EA\) are the Klein--Gordon Green operators, \(\ERzero,\EAzero\)
the massless ones; \(\pair{v}{\EA f}=\pair{\ER v}{f}\).
Write \(n_\mu=\partial_\mu t\), \(n^\mu=\delta^\mu_0\).

Fix \(f\in C_c^\infty((0,T)\times\R^3;\R)\),
\(g\in C_c^\infty((0,T)\times\R^3;[0,1])\),
a relatively compact plateau \(U\) with
\(\overline U\subset P_g=\Int\{g=1\}\), \(\ga=C_c^\infty(U;TM)\),
and \(\delta>0\) with
\begin{equation}
K\cup\supp g\cup\overline U\subset\{2\delta<t<T\},\qquad K=\supp f .
\label{eq:separation}
\end{equation}
Fix \(0<\eps<\delta\) once and for all.
The ramp \(\sig(t)=\chi(t/\eps)\) is X-D's, flat at both ends.
\(\Gp\) is the group generated formally by \(\kappa\xi+\kappa^2\zeta\),
\(\xi,\zeta\in\ga\); every \(F\in\Gp\) equals the identity outside a
compact subset of \(U\).

\subsection{Metric, density coefficients, coordinates, tests}

\[
q=\eta+\kappa h+\kappa^2k .
\]
With \(H=\tr_\eta h\),
\begin{equation}
\begin{aligned}
a^{\mu\nu}&=a_1^{\mu\nu}[h]=\tfrac12H\eta^{\mu\nu}-h^{\mu\nu},
&\quad b&=b_1[h]=\tfrac12H,\\
b_2[h]&=\tfrac18H^2-\tfrac14h_{\alpha\beta}h^{\alpha\beta},
&\quad a_2^{\mu\nu}[h]&=h^\mu{}_\alpha h^{\alpha\nu}
-\tfrac12Hh^{\mu\nu}+b_2[h]\eta^{\mu\nu},\\
A^{\mu\nu}&=a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h],
&\quad B&=b_1[k]+b_2[h],
\end{aligned}
\label{eq:density}
\end{equation}
so that \(\sqrt{-q}\,q^{\mu\nu}=\eta^{\mu\nu}+\kappa a^{\mu\nu}
+\kappa^2A^{\mu\nu}+O(\kappa^3)\) and
\(\sqrt{-q}=1+\kappa b+\kappa^2B+O(\kappa^3)\).

The ramp coordinates \(\X1,\X2\) have zero Cauchy data at
\(\Sigma=\{t=0\}\) and solve
\begin{equation}
\Box\X1^{\rho}=-\partial_\mu(\sig a^{\mu\rho}),\qquad
\Box\X2^{\rho}=-\partial_\mu
\bigl[\sig\bigl(A^{\mu\rho}+a^{\mu\nu}\partial_\nu\X1^{\rho}\bigr)\bigr].
\label{eq:coordinate-equations}
\end{equation}
For sources vanishing on \(t\le0\) this agrees with
retarded propagation, \(\X1=-\partial_\mu\ERzero(\sig a^{\mu\rho})\).
The bare tests \(c_1=\partial_\mu(f\X1^\mu)\), \(c_2\), the Euler
coefficients
\[
\Lone u=\partial_\mu(ga^{\mu\nu}\partial_\nu u)+gm^2bu,\qquad
\Ltwo u=\partial_\mu(gA^{\mu\nu}\partial_\nu u)+gm^2Bu,
\]
and the completed tests
\begin{equation}
d_1=c_1-\Lone\EA f,\qquad
d_2=c_2-\Lone\EA c_1-\Ltwo\EA f+\Lone\EA\Lone\EA f
\label{eq:completed}
\end{equation}
are X-D's, at the fixed \(\eps\).
The completed scalar-only observable is
\(\widehat\Phi(f)=\field(f)+\kappa\field(d_1)+\kappa^2\field(d_2)\).
We write
\[
\Ahat1[h]:=\field(d_1[h]),\qquad
\Ahat2[h,k]:=\field(d_2[h,k]).
\]

\begin{lemma}[Affine slot]
\label{lem:affine}
\(d_1[h]\) is linear in \(h\). There is a symmetric bilinear
\(Q[h,h']\) with
\[
d_2[h,k]=Q[h,h]+d_1[k].
\]
\end{lemma}

\begin{proof}
\(A\) and \(B\) are linear in \(k\) with coefficients \(a_1[k]\),
\(b_1[k]\); so \(\Ltwo[h,k]=\Ltwo[h,0]+\Lone[k]\).
By \eqref{eq:coordinate-equations} and the zero data,
\(\X2[h,k]=\X2[h,0]+\X1[k]\).
In \(c_2=-\partial_\mu(fB^\mu)+\tfrac12\partial_\mu\partial_\nu
(f\X1^\mu\X1^\nu)\), \(B^\mu=\X1^\nu\partial_\nu\X1^\mu-\X2^\mu\),
the only \(k\)-dependence is \(+\partial_\mu(f\X1^\mu[k])=c_1[k]\).
Collecting the \(k\)-linear terms of \eqref{eq:completed} gives
\(c_1[k]-\Lone[k]\EA f=d_1[k]\). The remainder is quadratic in
\(h\); polarize.
\end{proof}

\subsection{The sheet}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.17\linewidth}X@{}}
\toprule
Item & Declaration\\
\midrule
Signature & \((+,-,-,-)\). VII-C's opposite signature is comparison
only.\\
Scalar & The same switched density
\(\Ldens=(1-g)\Ldens_\eta[\phi]+g\Ldens_q[q,\phi]\), same \(g\),
same ramp, fixed \(\eps\).\\
Gravity & Einstein--Hilbert, \(\Sgr=-(2\kappa^2)^{-1}\int\sqrt{-q}R\),
conventions of Section~\ref{sec:E1}. Sign fixed by
Diagnostic~\ref{diag:sign}. Quadratic part derived in
Lemma~\ref{lem:quadratic}.\\
Gauge & De Donder, \(C^\mu[h]=\partial_\nu\bar h^{\nu\mu}\),
\(\bar h=h-\tfrac12\eta\tr_\eta h\), Feynman weight
\(\tfrac14\int C_\mu C^\mu\), for the quantum graviton only.
Classical backgrounds remain unconstrained.\\
Algebra & The same \(\ga\). It acts on the classical backgrounds
by the complete two-jet law; Proposition~\ref{prop:residual}
shows it cannot act on the gauge-fixed graviton.\\
Field content & \(\field\) free Klein--Gordon; \(\hq\) free
gauge-fixed graviton; \(h_{\mathrm{cl}},k_{\mathrm{cl}}\) classical,
off shell. No second-order quantum metric, no ghosts.\\
Order & Classical identity through \(\kappa^2\). One tree at
\(\kappa^2\), the first order at which \(\hq\) propagates.
Quantum-metric background identity through \(\kappa^1\); the
\(\kappa^2\) defect is displayed.\\
Coordinates & X-A's ramp operator \eqref{eq:coordinate-equations}.
Not replaced by the metric equation.\\
Not opened & Ghost loops; the word \(B_f\star W_g\star B_{f'}\);
a triangle subtraction; X-Enter; \(\eps\to0\) for any kernel;
the self-interaction tree; the mixed loop; gauge-parameter
dependence.\\
\bottomrule
\end{tabularx}
\end{center}

%=====================================================================
\section{Non-inputs}
\label{sec:noninputs}
%=====================================================================

The ramp kernel of X-A is not VII-C's \(B_f\); no identification is
made. No cone, token, or Enter record is an output.
The new propagator \(\DR\) is used at fixed \(\eps\) only; its sharp
limit is not taken. X-D's convergence theorem concerns smooth scalar
tests at classical backgrounds; it is not admission of a mixed
graviton--scalar kernel, and is not used as such.

VII-C \cite{VIIC} is audited in Section~\ref{sec:E5}. Its
assumptions are copied, not inferred.

A gravitational state is not an input. Every statement of this
paper at order \(\hbar^0\) uses only the retarded inverse \(\DR\);
the Hadamard bidistribution named in Section~\ref{sec:algebra}
enters only the definition of Wick ordering of graviton-quadratic
bare terms and no computed quantity.

%=====================================================================
\section{E1: the gravitational action and the metric Euler equation}
\label{sec:E1}
%=====================================================================

\subsection{Conventions}

For a Lorentzian metric \(q_{\mu\nu}\) of signature \((+,-,-,-)\),
\begin{align*}
\Gamma^\rho_{\mu\nu}&=\tfrac12q^{\rho\sigma}
(\partial_\mu q_{\sigma\nu}+\partial_\nu q_{\sigma\mu}
-\partial_\sigma q_{\mu\nu}),\\
R^\rho{}_{\sigma\mu\nu}&=\partial_\mu\Gamma^\rho_{\nu\sigma}
-\partial_\nu\Gamma^\rho_{\mu\sigma}
+\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma},\\
R_{\sigma\nu}&=R^\rho{}_{\sigma\rho\nu}
=\partial_\rho\Gamma^\rho_{\sigma\nu}
-\partial_\nu\Gamma^\rho_{\rho\sigma}
+\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\sigma\nu}
-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\rho\sigma},\\
R&=q^{\sigma\nu}R_{\sigma\nu},\qquad
G_{\mu\nu}=R_{\mu\nu}-\tfrac12q_{\mu\nu}R,\qquad
\sqrt{-q}=\sqrt{-\det q}.
\end{align*}
Under \(q\mapsto-q\) the Christoffel symbols, \(R^\rho{}_{\sigma\mu\nu}\),
\(R_{\mu\nu}\), and \(G_{\mu\nu}\) are unchanged; \(R\) and
\(\sqrt{-q}\,R\) change sign. Signs therefore cannot be copied across
signatures and are fixed below by a positivity check.

\begin{definition}[Gravitational action]
\label{def:action}
\[
\Sgr[q]=-\frac1{2\kappa^2}\int_M\sqrt{-q}\,R[q]\dd^4x ,
\qquad
S=\Ssc[q,\phi]+\Sgr[q],
\]
\[
\Ssc[q,\phi]=\int_M\bigl[(1-g)\Ldens_\eta[\phi]+g\Ldens_q[q,\phi]\bigr]
\dd^4x,\qquad
\Ldens_q=\tfrac12\sqrt{-q}\bigl(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
-m^2\phi^2\bigr),
\]
with \(\Ldens_\eta=\Ldens_q|_{q=\eta}\).
Both actions are Lagrangian densities; their Euler--Lagrange
derivatives are defined through compactly supported variations, and
no decay of \(h,k\) is assumed. \(\kappa\) is formal. The dictionary
\(\kappa^2=8\pi G_{\mathrm N}\) is used in Diagnostic~\ref{diag:sign}
only.
\end{definition}

\subsection{First variation, derived}

\begin{lemma}[Palatini form of the first variation]
\label{lem:palatini}
For a smooth compactly supported variation \(\delta q_{\mu\nu}\),
\begin{equation}
\delta\bigl(\sqrt{-q}R\bigr)
=-\sqrt{-q}\,G^{\mu\nu}\delta q_{\mu\nu}
+\partial_\mu\bigl(\sqrt{-q}\,W^\mu\bigr),\qquad
W^\mu=q^{\alpha\beta}\delta\Gamma^\mu_{\alpha\beta}
-q^{\mu\alpha}\delta\Gamma^\beta_{\beta\alpha}.
\label{eq:palatini}
\end{equation}
Consequently
\[
\frac{\delta\Sgr}{\delta q_{\mu\nu}}=\frac1{2\kappa^2}\sqrt{-q}\,G^{\mu\nu},
\qquad
\frac{\delta\Sgr}{\delta h_{\mu\nu}}=\frac1{2\kappa}\sqrt{-q}\,G^{\mu\nu},
\qquad
\frac{\delta\Sgr}{\delta k_{\mu\nu}}=\frac12\sqrt{-q}\,G^{\mu\nu}.
\]
\end{lemma}

\begin{proof}
\(\delta\Gamma^\rho_{\mu\nu}\) is a tensor. Both sides of
\(\delta R^\rho{}_{\sigma\mu\nu}=\nabla_\mu\delta\Gamma^\rho_{\nu\sigma}
-\nabla_\nu\delta\Gamma^\rho_{\mu\sigma}\) are tensors, and the
identity holds in normal coordinates at any point, where \(\Gamma=0\)
and \(\nabla=\partial\). Contracting,
\(\delta R_{\sigma\nu}=\nabla_\rho\delta\Gamma^\rho_{\nu\sigma}
-\nabla_\nu\delta\Gamma^\rho_{\rho\sigma}\), so
\(q^{\sigma\nu}\delta R_{\sigma\nu}=\nabla_\rho W^\rho\) and
\(\sqrt{-q}\nabla_\rho W^\rho=\partial_\rho(\sqrt{-q}W^\rho)\).
With \(\delta\sqrt{-q}=\tfrac12\sqrt{-q}\,q^{\mu\nu}\delta q_{\mu\nu}\)
and \(\delta q^{\mu\nu}=-q^{\mu\alpha}q^{\nu\beta}\delta q_{\alpha\beta}\),
\[
\delta(\sqrt{-q}R)
=\sqrt{-q}\bigl(\tfrac12q^{\mu\nu}R-R^{\mu\nu}\bigr)\delta q_{\mu\nu}
+\partial_\rho(\sqrt{-q}W^\rho).
\]
The chain rule \(\delta q=\kappa\,\delta h=\kappa^2\delta k\) gives
the remaining formulas.
\end{proof}

\subsection{Expansion through second order in \(q-\eta\)}

Set \(\Gamma=\kappa\Gamma_1[h]+\kappa^2(\Gamma_1[k]+\Gamma_2[h,h])
+O(\kappa^3)\) with
\[
\Gamma_1{}^\rho_{\mu\nu}[h]
=\tfrac12\eta^{\rho\sigma}(\partial_\mu h_{\sigma\nu}
+\partial_\nu h_{\sigma\mu}-\partial_\sigma h_{\mu\nu}),\qquad
\Gamma_2{}^\rho_{\mu\nu}[h,h]=-h^\rho{}_\sigma\,\Gamma_1{}^\sigma_{\mu\nu}[h].
\]
Then
\begin{align}
R^{(1)}_{\mu\nu}[h]&=\partial_\rho\Gamma_1{}^\rho_{\mu\nu}
-\partial_\nu\Gamma_1{}^\rho_{\rho\mu}
=\tfrac12\bigl(\partial_\rho\partial_\mu h^\rho{}_\nu
+\partial_\rho\partial_\nu h^\rho{}_\mu
-\Box h_{\mu\nu}-\partial_\mu\partial_\nu H\bigr),
\nonumber\\
\Glin{}_{\mu\nu}[h]&=R^{(1)}_{\mu\nu}
-\tfrac12\eta_{\mu\nu}\eta^{\alpha\beta}R^{(1)}_{\alpha\beta},
\nonumber\\
R^{(2)}_{\mu\nu}[h,h]&=\partial_\rho\Gamma_2{}^\rho_{\mu\nu}
-\partial_\nu\Gamma_2{}^\rho_{\rho\mu}
+\Gamma_1{}^\rho_{\rho\lambda}\Gamma_1{}^\lambda_{\mu\nu}
-\Gamma_1{}^\rho_{\nu\lambda}\Gamma_1{}^\lambda_{\rho\mu},
\nonumber\\
G_{2,\mu\nu}[h,h]&=R^{(2)}_{\mu\nu}
-\tfrac12\eta_{\mu\nu}\eta^{\alpha\beta}R^{(2)}_{\alpha\beta}
-\tfrac12\bigl(h_{\mu\nu}\eta^{\alpha\beta}-\eta_{\mu\nu}h^{\alpha\beta}\bigr)
R^{(1)}_{\alpha\beta},
\label{eq:G2}
\end{align}
and, raising with \(\eta\),
\begin{equation}
\sqrt{-q}\,G^{\mu\nu}[q]
=\kappa\,\Glin^{\mu\nu}[h]
+\kappa^2\bigl(\Glin^{\mu\nu}[k]+\mathcal G_2^{\mu\nu}[h,h]\bigr)
+O(\kappa^3),
\label{eq:G-expansion}
\end{equation}
\[
\mathcal G_2^{\mu\nu}[h,h]
=\eta^{\mu\alpha}\eta^{\nu\beta}G_{2,\alpha\beta}[h,h]
+b_1[h]\Glin^{\mu\nu}[h]
-h^\mu{}_\alpha\Glin^{\alpha\nu}[h]
-h^\nu{}_\alpha\Glin^{\mu\alpha}[h].
\]
The linearized Bianchi identity \(\partial_\mu\Glin^{\mu\nu}[h]\equiv0\)
holds for every smooth \(h\), by direct computation from
\eqref{eq:G2}. No later proof uses the explicit form of
\(G_2\) or of any higher coefficient; they are displayed so that
\eqref{eq:metric-euler-expanded} is a definite equation.

\subsection{Matter variation and the vertex identity}

With
\begin{equation}
T^{\mu\nu}[q,\phi]
=q^{\mu\alpha}q^{\nu\beta}\partial_\alpha\phi\partial_\beta\phi
-q^{\mu\nu}\bigl(\tfrac12q^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi
-\tfrac12m^2\phi^2\bigr),\qquad
T^{\mu\nu}[\phi]:=T^{\mu\nu}[\eta,\phi],
\label{eq:stress}
\end{equation}
one has \(\partial\Ldens_q/\partial q_{\mu\nu}
=-\tfrac12\sqrt{-q}\,T^{\mu\nu}[q,\phi]\) (no \(\partial q\) enters
\(\Ldens_q\)), hence
\begin{equation}
\frac{\delta\Ssc}{\delta q_{\mu\nu}}
=-\tfrac12 g\sqrt{-q}\,T^{\mu\nu}[q,\phi],\qquad
\sqrt{-q}\,T^{\mu\nu}[q,\phi]
=T^{\mu\nu}[\phi]+\kappa T^{\mu\nu}_{(1)}[h,\phi]+O(\kappa^2),
\label{eq:matter-variation}
\end{equation}
\begin{align}
T^{\mu\nu}_{(1)}[h,\phi]
={}&a^{\mu\alpha}\partial_\alpha\phi\,\partial^\nu\phi
+a^{\nu\beta}\partial^\mu\phi\,\partial_\beta\phi
-b\,\partial^\mu\phi\partial^\nu\phi
-\tfrac12a^{\mu\nu}\partial_\alpha\phi\partial^\alpha\phi
\nonumber\\
&-\tfrac12\eta^{\mu\nu}a^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi
+\tfrac12 b\,\eta^{\mu\nu}\partial_\alpha\phi\partial^\alpha\phi
+\tfrac12m^2a^{\mu\nu}\phi^2 .
\label{eq:T1}
\end{align}
The flat stress tensor obeys
\begin{equation}
\partial_\mu T^{\mu\nu}[\phi]=(\Pm\phi)\,\partial^\nu\phi .
\label{eq:T-divergence}
\end{equation}

\begin{lemma}[Vertex identity]
\label{lem:vertex}
X-A's first-order switched vertex is the metric variation of the
switched density:
\[
V_{1,g}[h,\phi]:=\int g\bigl(\tfrac12a^{\mu\nu}[h]\partial_\mu\phi\partial_\nu\phi
-\tfrac12m^2b_1[h]\phi^2\bigr)
=-\tfrac12\int g\,h_{\mu\nu}T^{\mu\nu}[\phi],
\]
so \(\delta V_{1,g}/\delta h_{\mu\nu}=-\tfrac12gT^{\mu\nu}[\phi]\)
and \(\delta V_{1,g}/\delta\phi=-\Lone[h]\phi\).
\end{lemma}

\begin{proof}
\(-\tfrac12h_{\mu\nu}T^{\mu\nu}=-\tfrac12h^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
+\tfrac14H(\partial\phi\cdot\partial\phi-m^2\phi^2)
=\tfrac12a^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-\tfrac12m^2b\phi^2\)
by \eqref{eq:density}. The \(\phi\)-derivative is X-C's Hessian
ledger \(V''=-\mathsf L\), extended here to the mixed block.
\end{proof}

\subsection{The metric Euler equation}

\begin{definition}[Metric Euler operator]
\label{def:metric-euler}
\begin{equation}
\Eul^{\mu\nu}[q,\phi]:=\frac{\delta S}{\delta h_{\mu\nu}}
=\frac1{2\kappa}\sqrt{-q}\,G^{\mu\nu}[q]
-\frac{\kappa}{2}\,g\sqrt{-q}\,T^{\mu\nu}[q,\phi],
\qquad
\frac{\delta S}{\delta k_{\mu\nu}}=\kappa\,\Eul^{\mu\nu}.
\label{eq:metric-euler}
\end{equation}
The metric equation is \(\Eul^{\mu\nu}=0\); the \(k\)-equation is the
same equation. Expanded,
\begin{align}
\Eul^{\mu\nu}
={}&\tfrac12\Glin^{\mu\nu}[h]
+\kappa\Bigl(\tfrac12\Glin^{\mu\nu}[k]+\tfrac12\mathcal G_2^{\mu\nu}[h,h]
-\tfrac12 g\,T^{\mu\nu}[\phi]\Bigr)
\nonumber\\
&+\kappa^2\Bigl(\tfrac12\bigl[\sqrt{-q}G^{\mu\nu}\bigr]_{(3)}[h,k]
-\tfrac12 g\,T^{\mu\nu}_{(1)}[h,\phi]\Bigr)+O(\kappa^3),
\label{eq:metric-euler-expanded}
\end{align}
where \([\sqrt{-q}G^{\mu\nu}]_{(3)}\) is the cubic coefficient of
the closed form, defined by Lemma~\ref{lem:palatini} and the
conventions above.
\end{definition}

The switch \(g\) multiplies the matter source and nothing else;
\(\partial g\) does not appear in \(\Eul\). It appears in the
divergence of \(\Eul\), which is the content of E1's identity.
The scalar Euler operator \(\Eul_\phi:=\delta S/\delta\phi
=\delta\Ssc/\delta\phi\) is X-A's switched operator, with
\(\Eul_\phi=-\Pm\phi+O(\kappa)\).

\begin{proposition}[Noether identity with switch defect]
\label{prop:noether}
Let \(E^{\mu\nu}=\delta S/\delta q_{\mu\nu}=\kappa^{-1}\Eul^{\mu\nu}\),
a symmetric tensor density, and \(\nabla\) the Levi-Civita
connection of \(q\). For all smooth \((q,\phi)\), pointwise on \(M\),
\begin{equation}
2q_{\rho\nu}\nabla_\mu E^{\mu\nu}
=\Eul_\phi\,\partial_\rho\phi
-(\partial_\rho g)\bigl(\Ldens_\eta-\Ldens_q\bigr)
-(\partial_\mu g)\,\eta_{\rho\nu}T^{\mu\nu}[\phi]
+(1-g)(\Pm\phi)\partial_\rho\phi .
\label{eq:noether}
\end{equation}
On the plateau \(P_g\) the right side is \(\Eul_\phi\partial_\rho\phi\).
The gravitational action contributes nothing to the right side.
\end{proposition}

\begin{proof}
Let \(\xi\in C_c^\infty(M;TM)\) and vary
\(\delta_\xi q=\Lie_\xi q\), \(\delta_\xi\phi=\xi^\mu\partial_\mu\phi\),
with \(g\) held fixed.

\emph{Direct computation.}
\(\sqrt{-q}R\) is a natural density, so
\(\delta_\xi(\sqrt{-q}R)=\partial_\mu(\xi^\mu\sqrt{-q}R)\) and
\(\delta_\xi\Sgr=0\).
Likewise \(\delta_\xi\Ldens_q=\partial_\mu(\xi^\mu\Ldens_q)\), so
\(\delta_\xi\int g\Ldens_q=-\int(\partial_\mu g)\xi^\mu\Ldens_q\).
For the flat term, write naturality of \(\Ldens_q\) at \(q=\eta\):
\[
\frac{\partial\Ldens_q}{\partial q_{\mu\nu}}\Big|_{\eta}(\Lie_\xi\eta)_{\mu\nu}
+\Bigl[\frac{\partial\Ldens_\eta}{\partial\phi}\xi^\mu\partial_\mu\phi
+\frac{\partial\Ldens_\eta}{\partial(\partial_\mu\phi)}
\partial_\mu(\xi^\nu\partial_\nu\phi)\Bigr]
=\partial_\mu(\xi^\mu\Ldens_\eta).
\]
The bracket is \(\delta_\xi\Ldens_\eta[\phi]\), and
\(\partial\Ldens_q/\partial q_{\mu\nu}|_\eta=-\tfrac12T^{\mu\nu}[\phi]\),
\((\Lie_\xi\eta)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu\).
Hence
\(\delta_\xi\Ldens_\eta=\partial_\mu(\xi^\mu\Ldens_\eta)
+T^{\mu\nu}[\phi]\partial_\mu\xi_\nu\), and
\[
\delta_\xi S
=\int(\partial_\mu g)\xi^\mu(\Ldens_\eta-\Ldens_q)
+\int(1-g)T^{\mu\nu}[\phi]\partial_\mu\xi_\nu .
\]

\emph{Euler--Lagrange form.}
\(\delta_\xi S=\int E^{\mu\nu}\Lie_\xi q_{\mu\nu}+\int\Eul_\phi\xi^\rho\partial_\rho\phi\),
and for a symmetric tensor density
\(\int E^{\mu\nu}\Lie_\xi q_{\mu\nu}=-2\int\xi^\rho q_{\rho\nu}\nabla_\mu E^{\mu\nu}\),
by \(\Lie_\xi q_{\mu\nu}=\xi^\lambda\partial_\lambda q_{\mu\nu}
+2q_{\lambda(\nu}\partial_{\mu)}\xi^\lambda\), one integration by
parts, and \(2q_{\rho\nu}\Gamma^\nu_{\mu\sigma}E^{\mu\sigma}
=E^{\mu\sigma}(2\partial_\mu q_{\rho\sigma}-\partial_\rho q_{\mu\sigma})\).

\emph{Comparison.}
Integrate the last term of the direct computation by parts,
use \eqref{eq:T-divergence}, and equate coefficients of the
arbitrary \(\xi^\rho\).
\end{proof}

\begin{remark}
The tensor \((\partial_\mu g)T^{\mu\nu}[\phi]\) is the same tensor
that controls the varied-action defect of X-A's Route II off the
plateau. Proposition~\ref{prop:noether} is its Bianchi form: it
measures the failure of the switched matter source to be conserved,
and it is independent of any gauge parameter and of \(\eps\).
\end{remark}

%=====================================================================
\section{E2: classical identity}
\label{sec:E2}
%=====================================================================

\subsection{Route II rerun: the varied total action}

\begin{lemma}[Exact plateau invariance of the total Lagrangian]
\label{lem:invariance}
Let \(F\in\Gp\), and let \(\mathscr L[q,\phi]
=(1-g)\Ldens_\eta[\phi]+g\Ldens_q[q,\phi]-(2\kappa^2)^{-1}\sqrt{-q}R[q]\).
For all smooth \((q,\phi)\), the density
\(\mathscr L[F^*q,F^*\phi]-\mathscr L[q,\phi]\) is supported in a
compact subset of \(U\) and integrates to zero.
\end{lemma}

\begin{proof}
\(F\) is the identity outside a compact \(L\subset U\), and
\(F(L)=L\). Outside \(L\) all pulled-back fields and their
derivatives coincide with the originals, which gives the support
statement.

\(\sqrt{-q}R\) and \(\Ldens_q\) are natural:
\(\omega[F^*q,F^*\phi]=F^*(\omega[q,\phi])\) as densities.
Hence \(\int(F^*\omega-\omega)=0\) for \(\omega=\sqrt{-q}R\),
since \(F\) is a diffeomorphism of \(M\) onto itself, and
\(\int gF^*\omega=\int(g\circ F^{-1})\,\omega=\int g\omega\)
for \(\omega=\Ldens_q\), since \(g\circ F^{-1}=g\):
on \(L\) both sides equal \(1\), and off \(L\) \(F^{-1}=\mathrm{id}\).
Finally \(\supp(1-g)\subset M\setminus P_g\subset M\setminus L\),
where \(F^*\phi=\phi\) with all derivatives; so the
\((1-g)\Ldens_\eta\) term is unchanged pointwise.
\end{proof}

\begin{proposition}[Covariance of the Euler operators on the plateau]
\label{prop:covariance}
For \(F\in\Gp\) and all smooth \((q,\phi)\),
\[
E^{\mu\nu}[F^*q,F^*\phi]=F^*\bigl(E^{\mu\nu}[q,\phi]\bigr),\qquad
\Eul_\phi[F^*q,F^*\phi]=F^*\bigl(\Eul_\phi[q,\phi]\bigr),
\]
as a contravariant symmetric tensor density and a scalar density.
\end{proposition}

\begin{proof}
Apply Lemma~\ref{lem:invariance} to \((q+s\,\delta q,\phi+s\,\delta\phi)\)
with compactly supported \((\delta q,\delta\phi)\) and differentiate
at \(s=0\):
\[
\int E^{\mu\nu}[F^*q,F^*\phi](F^*\delta q)_{\mu\nu}
+\int\Eul_\phi[F^*q,F^*\phi]F^*\delta\phi
=\int E^{\mu\nu}[q,\phi]\delta q_{\mu\nu}+\int\Eul_\phi[q,\phi]\delta\phi .
\]
Change variables by \(F\) on the left; the pairing of a tensor
density with a tensor is invariant. Since \(\delta q,\delta\phi\)
are arbitrary, the densities agree.
\end{proof}

This is X-A's plateau covariance lemma for the scalar Euler
operator, rerun with the gravitational term and extended to the
metric Euler operator. The gravitational term contributes no new
defect: Lemma~\ref{lem:invariance} is exact, not an inheritance.

\subsection{The classical identity on the coupled constraint surface}

\begin{definition}[Coupled constraint surface]
\(\mathcal C\) is the set of formal triples \((h,k,\phi)\),
\(\phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2+\dots\), with
\(\Eul^{\mu\nu}[q,\phi]=0\) and \(\Eul_\phi[q,\phi]=0\) through
\(\kappa^2\) in \eqref{eq:metric-euler-expanded}, with the ramp
coordinates \eqref{eq:coordinate-equations} unchanged.
\end{definition}

\begin{theorem}[Classical identity through \(\kappa^2\)]
\label{thm:classical}
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\mathcal C\) is \(\Gp\)-invariant: if \((h,k,\phi)\in\mathcal C\)
and \((h',k')\) is the complete two-jet transform
\(h'=h+K\xi\), \(k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi\),
\(K\xi=\Lie_\xi\eta\), with \(\phi'=F^*\phi\), then
\((h',k',\phi')\in\mathcal C\).
\item X-A's relational scalar and X-C's completed classical observable
are \(\Gp\)-invariant through \(\kappa^2\) for all smooth \((h,k)\),
in particular on \(\mathcal C\). The gravitational action enters
neither object.
\item The varied-action identity of X-A's Route II holds for the
total action with zero defect on the plateau. Off the plateau the
defect is the right side of \eqref{eq:noether}, recorded for all
\((h,k)\), on or off \(\mathcal C\).
\end{enumerate}
\end{theorem}

\begin{proof}
(i) is Proposition~\ref{prop:covariance}: \(F^*\) is invertible
order by order and maps zero to zero. (ii) is X-A (iv) and X-C's
fixed-regulator identity, both proved off shell in the metric; X-D's
Corollary on the fixed algebra gives the explicit equation-ideal
representatives. (iii) is Lemma~\ref{lem:invariance} together with
Proposition~\ref{prop:noether}.
\end{proof}

The relational scalar has no on-shell/off-shell distinction on
\(\ga\). The new content of E2 is that the constraint surface itself
is invariant, and the following fact about it.

\begin{proposition}[Bianchi obstruction to the coupled jet]
\label{prop:bianchi}
If \((h,k,\phi)\in\mathcal C\), then
\begin{equation}
(\partial_\mu g)\,T^{\mu}{}_{\rho}[\phi_0]=0\quad\text{on }M,
\qquad T^\mu{}_\rho=\eta_{\rho\nu}T^{\mu\nu},
\label{eq:bianchi-constraint}
\end{equation}
where \(\phi_0\) is the free solution \(\Pm\phi_0=0\) at order
\(\kappa^0\). For \(m=0\), \(\phi_0=t\), and any nonconstant \(g\),
\((\partial_\mu g)T^\mu{}_\rho[t]=(\partial_tg)n_\rho-\tfrac12\partial_\rho g\)
does not vanish on \(\supp\dd g\); hence \(\mathcal C=\emptyset\)
for such data.
\end{proposition}

\begin{proof}
Insert \(E^{\mu\nu}=0\), \(\Eul_\phi=0\) into \eqref{eq:noether}:
the right side vanishes identically in \(\kappa\). At order
\(\kappa^0\), \(\Ldens_\eta-\Ldens_q=O(\kappa)\), \(\phi=\phi_0\),
and \(\Eul_\phi=0\) at \(\kappa^0\) is \(\Pm\phi_0=0\), which kills
the last term. What remains is \eqref{eq:bianchi-constraint}.
For \(\phi_0=t\), \(T^{\mu\nu}[t]=n^\mu n^\nu-\tfrac12\eta^{\mu\nu}\).
\end{proof}

\begin{remark}
Equivalently, at the first order of
\eqref{eq:metric-euler-expanded} at which the source appears,
\(\tfrac12\Glin[k]+\tfrac12\mathcal G_2[h,h]=\tfrac12gT[\phi_0]\) on
\(\Glin[h]=0\); the divergence of the left side vanishes by the
Bianchi identity, that of the right side is
\((\partial_\mu g)T^{\mu\nu}[\phi_0]\).
The obstruction lives in \(\supp\dd g\), outside \(U\); it is not
removed by any choice of \(\ga\). The known resolution switches the
entire Lagrangian and recovers gauge invariance only in the
algebraic adiabatic limit; it is named and not opened here.
The relational observable is untouched: it was never defined by
solving the metric equation.
\end{remark}

%=====================================================================
\section{E3: the quadratic operator, its inverse, and the residual
algebra}
\label{sec:E3}
%=====================================================================

\subsection{The quadratic part of the Einstein--Hilbert density}

\begin{lemma}[Quadratic part, derived]
\label{lem:quadratic}
Up to a total derivative,
\[
\int\sqrt{-q}R[\eta+\kappa h]
=-\tfrac12\kappa^2\int h_{\mu\nu}\Glin^{\mu\nu}[h]+O(\kappa^3),
\qquad
\Sgr=S_{\mathrm{FP}}[h]+\kappa\bigl(2S_{\mathrm{FP}}(h,k)+S_3[h]\bigr)+O(\kappa^2),
\]
\[
S_{\mathrm{FP}}[h]=\tfrac14\int h_{\mu\nu}\Glin^{\mu\nu}[h],\qquad
\frac{\delta S_{\mathrm{FP}}}{\delta h_{\mu\nu}}=\tfrac12\Glin^{\mu\nu}[h],
\]
where \(S_{\mathrm{FP}}(h,k)\) is the polarization and
\(S_3[h]=-\tfrac12\int[\sqrt{-q}R]_{(3)}[h]\) the cubic vertex.
\end{lemma}

\begin{proof}
Let \(\varphi(s)=\int\sqrt{-q}R[\eta+sh]\) for compactly supported
\(h\) (the density statement follows). By \eqref{eq:palatini},
\(\varphi'(s)=-\int\sqrt{-q}G^{\mu\nu}[\eta+sh]h_{\mu\nu}\),
so \(\varphi'(0)=0\) and
\(\varphi''(0)=-\int\Glin^{\mu\nu}[h]h_{\mu\nu}\).
The Euler derivative follows from the symmetry of \(\Glin\), which is
the symmetry of the second variation.
\end{proof}

\begin{diagnostic}[Sign: transverse-traceless positivity]
\label{diag:sign}
For \(\partial_\mu h^{\mu\nu}=0\), \(H=0\), \eqref{eq:G2} gives
\(\Glin{}_{\mu\nu}[h]=-\tfrac12\Box h_{\mu\nu}\), so
\(S_{\mathrm{FP}}=-\tfrac18\int h_{\mu\nu}\Box h^{\mu\nu}
=\tfrac18\int\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu}\),
whose spatial components read
\(\tfrac18\int(\partial_th_{ij}\partial_th_{ij}-\partial_kh_{ij}\partial_kh_{ij})\),
the same sign as \(\tfrac12\int\dot\phi^2\).
Independently, with the dictionary \(\kappa^2=8\pi G_{\mathrm N}\),
Lemma~\ref{lem:gf} below and a static source \(T^{00}=\varrho>0\)
give \(\Delta(\kappa\bar h_{00})=2\kappa^2\varrho=16\pi G_{\mathrm N}\varrho\),
i.e.\ \(\bar h_{00}=4\Phi_{\mathrm N}\) with
\(\Delta\Phi_{\mathrm N}=4\pi G_{\mathrm N}\varrho\), \(\Phi_{\mathrm N}<0\):
attraction. Both checks fix the sign of Definition~\ref{def:action};
it agrees with the Landau--Lifshitz convention \cite{LL} and is not
copied from it.
\end{diagnostic}

\subsection{De Donder gauge fixing and the operator}

\begin{definition}
\label{def:gf}
\(\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}H\),
\(\Ptr h:=\bar h\),
\(C^\mu[h]=\partial_\nu\bar h^{\nu\mu}=\partial_\nu h^{\nu\mu}-\tfrac12\partial^\mu H\),
\(S_{\mathrm{gf}}[h]=\tfrac14\int C_\mu C^\mu\).
Note \(\bar h^{\mu\nu}=-a_1^{\mu\nu}[h]\) and
\(C^\mu[h]=-\partial_\nu a^{\nu\mu}[h]\): the de Donder vector is
X-A's coordinate source without the ramp.
\end{definition}

\begin{lemma}[Gauge-fixed operator]
\label{lem:gf}
\(\Ptr^2=1\), and for every smooth symmetric \(h\),
\begin{equation}
\Glin{}_{\mu\nu}[h]=-\tfrac12\Box\bar h_{\mu\nu}
+\partial_{(\mu}C_{\nu)}[h]-\tfrac12\eta_{\mu\nu}\partial_\rho C^\rho[h],
\qquad
\frac{\delta(S_{\mathrm{FP}}+S_{\mathrm{gf}})}{\delta h_{\mu\nu}}
=-\tfrac14\Box\bar h^{\mu\nu}=:(\Pgr h)^{\mu\nu}.
\label{eq:Pgr}
\end{equation}
Thus \(\Pgr=-\tfrac14\Ptr\Box=-\tfrac14\Box\Ptr\), a diagonal
normally hyperbolic operator on symmetric tensors. Its retarded and
advanced inverses are
\[
\DR=-4\,\Ptr\otimes\ERzero,\qquad \DA=-4\,\Ptr\otimes\EAzero,
\]
and \(\supp\DR S\subset J^+(\supp S)\) for past-compact \(S\).
\end{lemma}

\begin{proof}
\(\tr_\eta\bar h=-H\), so \(\Ptr\bar h=\bar h+\tfrac12\eta H=h\).
Substitute \(h=\bar h-\tfrac12\eta\tr\bar h\) into
\eqref{eq:G2}: the terms
\(\partial_\mu\partial^\rho h_{\rho\nu}+\partial_\nu\partial^\rho h_{\rho\mu}
-\partial_\mu\partial_\nu H\) become \(\partial_\mu C_\nu+\partial_\nu C_\mu\),
and \(-\eta_{\mu\nu}\partial^\rho\partial^\sigma h_{\rho\sigma}+\eta_{\mu\nu}\Box H
-\Box h_{\mu\nu}\) becomes
\(-\Box\bar h_{\mu\nu}-\eta_{\mu\nu}\partial\cdot C\); halve.
For the gauge term, \(\delta C^\rho=\partial_\sigma\delta h^{\sigma\rho}
-\tfrac12\partial^\rho\eta_{\alpha\beta}\delta h^{\alpha\beta}\) gives
\(\delta S_{\mathrm{gf}}=\int\delta h^{\mu\nu}
(-\tfrac12\partial_{(\mu}C_{\nu)}+\tfrac14\eta_{\mu\nu}\partial\cdot C)\),
which cancels the two \(C\)-terms of \(\tfrac12\Glin\).
Since \(\Ptr\) is constant and invertible, inverses of \(\Pgr\) are
\(-4\Ptr\) times those of \(\Box\).
\end{proof}

\begin{lemma}[De Donder violation of a retarded response]
\label{lem:violation}
For a smooth symmetric source \(S^{\mu\nu}\) vanishing on \(t\le0\),
\begin{equation}
C^\mu[\DR S]=-4\,\ERzero\bigl(\partial_\nu S^{\nu\mu}\bigr).
\label{eq:violation}
\end{equation}
In particular \(\DR S\) is in de Donder gauge iff \(S\) is conserved.
\end{lemma}

\begin{proof}
\(\Ptr\DR S=-4\ERzero S\), so
\(C^\mu=\partial_\nu(\Ptr\DR S)^{\nu\mu}=-4\partial_\nu\ERzero S^{\nu\mu}\),
and \(\ERzero\) commutes with constant-coefficient derivatives.
\end{proof}

\subsection{Residual algebra and the action of \(\ga\)}

\begin{proposition}[No compactly supported residual gauge]
\label{prop:residual}
For a vector field \(\xi\), \(\Pgr(K\xi)=-\tfrac14\Ptr K(\Box\xi)\).
Hence \(K\xi\in\ker\Pgr\) iff \(\Box\xi\) is a Killing field of
\(\eta\). If \(\xi\) is compactly supported, this forces \(\xi=0\).
Consequently
\[
\ga\cap\{\xi:\ K\xi\in\ker\Pgr\}=\{0\},
\]
and the frozen algebra acts on the gauge-fixed free graviton only as
a background transformation of \((h_{\mathrm{cl}},k_{\mathrm{cl}})\),
never as a residual symmetry of \(\hq\).
\end{proposition}

\begin{proof}
\(\Ptr K\xi=K\xi-\eta\,\partial\cdot\xi\) and \(\Box\) commutes with
\(K\), \(\partial\cdot\), and \(\eta\); so
\(\Box\Ptr K\xi=\Ptr K\Box\xi\). \(\Ptr\) is invertible, so the
kernel condition is \(K\Box\xi=0\). A compactly supported Killing
field of Minkowski space vanishes, so \(\Box\xi=0\) with \(\xi\)
compactly supported, and Cauchy uniqueness gives \(\xi=0\).
\end{proof}

\begin{lemma}[De Donder configurations dress only through the layer]
\label{lem:layer}
If \(C^\mu[h]=0\), then
\[
\X1^\mu[h]=-\ERzero\bigl(\sig'\,a^{0\mu}[h]\bigr),
\]
a functional of \(h\) on \(0<t<\eps\) alone.
\end{lemma}

\begin{proof}
\(-\partial_\nu(\sig a^{\nu\mu})=-\sig'a^{0\mu}+\sig C^\mu[h]\).
\end{proof}

X-D proved that such layer functionals converge, as \(\eps\to0\), to
the homogeneous solution with velocity \(-a^{0\mu}[h]|_\Sigma\).
That limit is not taken here. It is recorded because it is the place
where a sharp limit of a quantum-metric dressing will differ in kind
from X-D's smooth scalar tests: it is a trace of \(\hq\) on
\(\Sigma\).

\subsection{The free graviton algebra, Wick ordering, and what is
not an input}
\label{sec:algebra}

The free gauge-fixed graviton algebra is generated by \(\hq(w)\),
\(w\) a compactly supported smooth symmetric tensor density, linear
in \(w\), with \(\hq(\Pgr w)=0\) and commutator \(\mathrm i\hbar\)
times the Peierls bracket of \(\Pgr\), i.e.\ the pairing with
\(\Delta^{\mathrm{gr}}=\DR-\DA\), in the same convention in which X-B's
scalar commutator uses the Peierls bracket of \(-\Pm\).
The multiplet Wick algebra \(\Wick\) of \((\field,\hq)\) and its
microcausal admission are X-B's; the equation ideal \(\Ideal\) is
generated by \(\field(\Pm v)\) and \(\hq(\Pgr w)\).
Wick ordering of graviton-quadratic monomials uses a fixed Hadamard
bidistribution of the form \(\mathsf H_{\mathrm{gr}}=-4\Ptr\otimes\mathsf H_0\),
\(\mathsf H_0\) Hadamard for \(\Box\) in \((+,-,-,-)\).
It is indefinite, as \(\Ptr\) is; no positivity is claimed or used.

VII-C's \(H_{hh}=P\Delta_0^+\) is a bisolution in the opposite
signature, attached to a sharp insertion and a different algebra.
It is not \(\mathsf H_{\mathrm{gr}}\), not \(\DR\), and not an input.
Every quantity computed below is at \(\hbar^0\) and uses \(\DR\)
only.

%=====================================================================
\section{E4: one tree}
\label{sec:E4}
%=====================================================================

\subsection{Field content and the one-tree observable}

The metric is
\[
q=\eta+\kappa\bigl(h_{\mathrm{cl}}+\hq\bigr)+\kappa^2k_{\mathrm{cl}},
\]
with \(\hq\) the free gauge-fixed graviton, \(\field\) the free
scalar, and \(h_{\mathrm{cl}},k_{\mathrm{cl}}\) classical and off shell.
Write \(h=h_{\mathrm{cl}}+\hq\). The bare and completed coefficient
functionals of X-B/X-C are polynomial in the metric perturbation
with distributional coefficients; substituting \(\hq\) and Wick
ordering gives elements of \(\Wick\), admitted at the fixed \(\eps\)
by X-B's joint admission of the multiplet.

Let \(\Th^{\mu\nu}:=\,:\!T^{\mu\nu}[\field]\!:\) be the Wick stress
tensor. By \eqref{eq:T-divergence},
\(\partial_\mu\Th^{\mu\nu}=\,:\!(\Pm\field)\partial^\nu\field\!:\in\Ideal\).

\begin{definition}[Retarded metric response and the tree]
\label{def:tree}
The first-order retarded response of the metric to the switched
scalar source is
\begin{equation}
k_1[\field]:=\DR\Bigl(-\frac{\delta V_{1,g}}{\delta h}[\field]\Bigr)
=\DR\bigl(\tfrac12 g\,\Th\bigr)
=-2\,\Ptr\ERzero\bigl(g\,\Th\bigr),
\label{eq:k1}
\end{equation}
so that \(\Pgr k_1+\delta V_{1,g}/\delta h=0\) with retarded support.
The tree is the insertion of \(k_1\) into the affine slot of
Lemma~\ref{lem:affine}:
\begin{equation}
\Ttree:=\kappa^2\,:\!\Ahat1\bigl[k_1[\field]\bigr]\!:
=\kappa^2\,:\!\field\bigl(d_1[k_1[\field]]\bigr)\!:\ ,
\label{eq:tree}
\end{equation}
totally Wick ordered. The one-tree completed observable is
\begin{equation}
\widehat\Phi^{\mathrm{tree}}(f)
=\field(f)+\kappa\Ahat1[h]
+\kappa^2\Bigl(\Qhat[h,h]+\Ahat1[k_{\mathrm{cl}}]+\Ttree\Bigr),
\qquad \Qhat[h,h]:=\,:\!\field(Q[h,h])\!:\ ,
\label{eq:one-tree}
\end{equation}
with \(h=h_{\mathrm{cl}}+\hq\) in \(\Ahat1\) and \(\Qhat\).
\end{definition}

\(\Ttree\) is the \(\kappa^2\) coefficient of
\(\Ahat1[\hq+\kappa k_1]-\Ahat1[\hq]\): the retarded M\o ller map of
the metric slot at first order, at \(\hbar^0\), in the same sense in
which X-C's \(\Lone\EA f\) is the retarded M\o ller correction of the
scalar slot. The graviton line is the single contraction between
\(\kappa\Ahat1[\hq]\) and \(\kappa V_{1,g}[\hq,\field]\); no
\(\hq\) survives in \(\Ttree\).

\subsection{Kernel and support}

Write \(d_1[k](x)=\int D_1^{\alpha\beta}(x,z)k_{\alpha\beta}(z)\dd z\),
with
\begin{align}
D_1^{\alpha\beta}(x,z)
&=\partial^x_\mu\bigl(f(x)\Lambda^{\mu,\alpha\beta}(x,z)\bigr)
-\partial^x_\mu\bigl(g(x)\,\mathfrak a^{\mu\nu,\alpha\beta}
\partial_\nu(\EA f)(x)\,\delta(x-z)\bigr)
-\tfrac12m^2g(x)(\EA f)(x)\eta^{\alpha\beta}\delta(x-z),
\nonumber\\
\Lambda^{\mu,\alpha\beta}(x,z)
&=-\partial^x_\nu\ERzero(x,z)\,\sig(z)\,\mathfrak a^{\nu\mu,\alpha\beta},
\qquad
\mathfrak a^{\nu\mu,\alpha\beta}
=\tfrac12\eta^{\nu\mu}\eta^{\alpha\beta}
-\tfrac12\bigl(\eta^{\nu\alpha}\eta^{\mu\beta}+\eta^{\nu\beta}\eta^{\mu\alpha}\bigr),
\label{eq:D1}
\end{align}
so that \(a^{\nu\mu}[k]=\mathfrak a^{\nu\mu,\alpha\beta}k_{\alpha\beta}\).

\begin{lemma}[Tree kernel and support]
\label{lem:kernel}
\[
\Ttree=\kappa^2\iint\mathcal T^{\alpha\beta}(x,y)\,
:\!\field(x)\,(\Ptr\Th)_{\alpha\beta}(y)\!:\ \dd x\dd y,
\qquad
\mathcal T^{\alpha\beta}(x,y)
=-2\,g(y)\int D_1^{\alpha\beta}(x,z)\,\ERzero(z,y)\dd z,
\]
where \((\Ptr\Th)_{\alpha\beta}=\Th_{\alpha\beta}-\tfrac12\eta_{\alpha\beta}\Th^\lambda{}_\lambda\).
The \(z\)-integration runs over the compact set
\(\{t\ge0\}\cap J^-(K)\cap J^+(\supp g)\), and
\[
\supp\mathcal T\subset K\times\bigl(\supp g\cap J^-(K)\bigr)
\subset J^-(K)\times J^-(K).
\]
\end{lemma}

\begin{proof}
Insert \eqref{eq:k1} into \eqref{eq:tree}; \(\Ptr\) is symmetric
and constant. The \(x\)-slot carries \(f\), the \(y\)-slot carries
\(g\); \(\ERzero(x,z)\) forces \(z\in J^-(x)\), \(\sig(z)\) forces
\(t_z\ge0\), and \(\ERzero(z,y)\) forces \(y\in J^-(z)\).
\(J^-(K)\cap J^+(\supp g)\) is compact in Minkowski space.
\end{proof}

\subsection{Admission}

\begin{lemma}[Microcausal admission of the tree kernel]
\label{lem:admission}
\(\mathcal T\) is a compactly supported distribution on \(M\times M\)
with
\[
\WF(\mathcal T)\cap\bigl(\overline V_+\times\overline V_+\ \cup\
\overline V_-\times\overline V_-\bigr)=\emptyset ,
\]
where \(\overline V_\pm\) are the closed future/past covector cones
including zero. Hence \(\Ttree\in\Wick\): it is a totally
Wick-ordered cubic in \(\field\) with a microcausal kernel, in the
algebra X-B already used.
\end{lemma}

\begin{proof}
The retarded propagator of a normally hyperbolic operator on
Minkowski space satisfies \cite{DH}
\[
\WF(\ERzero)\subset
\{(x,\xi;y,-\eta):\ (x,\xi)\sim(y,\eta),\ y\in J^-(x)\}
\cup\{(x,\xi;x,-\xi):\ \xi\ne0\},
\]
where \((x,\xi)\sim(y,\eta)\) means that \(\xi=\eta\) is null,
nonzero, and \(x-y\) is parallel to \(\xi^\sharp\).
In every element, both covectors are nonzero and
\((\xi,-\eta)\) has one entry in \(V_+\) and the other in \(V_-\)
whenever either is causal. Derivatives and multiplication by smooth
functions do not enlarge the wavefront set, so the local part of
\(D_1\) composed with \(\ERzero\) and the kernel \(\Lambda\) have the
same property.
For the nonlocal part, \(\mathcal T\) is the composition
\(\Lambda\circ(\ERzero g)\) with compact \(z\)-range by
Lemma~\ref{lem:kernel}. H\"ormander's composition theorem
\cite[Theorem~8.2.14]{Hor} applies because neither factor has a
zero-section component: the exceptional sets
\(\WF_X\times0\) and \(0\times\WF_Y\) are empty, and
\[
\WF(\Lambda\circ\ERzero g)\subset
\{(x,\xi;y,-\eta):\ \exists(z,\zeta)\ (x,\xi;z,-\zeta)\in\WF(\Lambda),\
(z,\zeta;y,-\eta)\in\WF(\ERzero)\}.
\]
Along such a chain \(\xi=\zeta=\eta\) up to the diagonal parts, so
\(\xi\) and \(\eta\) lie in the same half cone when causal, and
\((\xi,-\eta)\) is never in \(\overline V_\pm\times\overline V_\pm\).
Compact support is Lemma~\ref{lem:kernel}. The microcausal condition
on the kernel of a Wick monomial is X-B's admission criterion
\cite{BF}.
\end{proof}

X-D's warning concerned the sharp limit of a mixed kernel.
No limit is taken here; the regulator \(\eps\) is fixed.

\subsection{Normalization: the classical retarded response}

\begin{proposition}[Classical value and the dressing formula]
\label{prop:normalization}
Replace \(\field\) by a smooth free solution \(\phi_0\) and
\(\hq\) by \(0\). Then \(\Ttree\) becomes
\(\kappa^2A_1[k_1[\phi_0]]\), where \(A_1[k]=\pair{d_1[k]}{\phi_0}\)
is X-A's first-order relational scalar at background \(k\), and
\(k_1[\phi_0]=-2\Ptr\ERzero(gT[\phi_0])\) is the classical retarded
response of the metric to the switched stress tensor.
Its coordinate part is
\begin{equation}
\X1^\mu\bigl[k_1[\phi_0]\bigr]
=\ERzero\bigl(C^\mu[k_1]\bigr)
=-2\,\ERzero\ERzero\bigl((\partial_\nu g)\,T^{\nu\mu}[\phi_0]\bigr),
\label{eq:dressing}
\end{equation}
and its scalar part is \(\rho_1=-\ER\Lone[k_1]\phi_0\). Hence
\begin{equation}
\Ttree\big|_{\field\to\phi_0,\ \hq\to0}
=\kappa^2\int f\Bigl(2\,\partial_\mu\phi_0\,
\ERzero\ERzero\bigl((\partial_\nu g)T^{\nu\mu}[\phi_0]\bigr)
-\ER\Lone[k_1]\phi_0\Bigr).
\label{eq:classical-tree}
\end{equation}
\end{proposition}

\begin{proof}
\(\supp k_1\subset J^+(\supp g)\subset\{t>2\delta\}\), so
\(\sig a[k_1]=a[k_1]\) and
\(\X1[k_1]=\ERzero(-\partial_\nu a^{\nu\mu}[k_1])=\ERzero(C^\mu[k_1])\).
Lemma~\ref{lem:violation} with \(S=\tfrac12gT[\phi_0]\) and
\eqref{eq:T-divergence} with \(\Pm\phi_0=0\) give
\(C^\mu[k_1]=-2\ERzero((\partial_\nu g)T^{\nu\mu})\).
Then \(\pair{d_1[k]}{\phi_0}=-\int f\X1^\mu[k]\partial_\mu\phi_0
-\pair{\EA f}{\Lone[k]\phi_0}\) and
\(\pair{\EA f}{\Lone\phi_0}=\pair{f}{\ER\Lone\phi_0}\).
\end{proof}

The relational dressing of the backreaction is the double retarded
integral of the de Donder violation sourced by \(\partial g\).
It vanishes if and only if the switched source is conserved, that is,
if and only if \((\partial_\nu g)T^{\nu\mu}[\phi_0]=0\): the same
tensor as in Proposition~\ref{prop:bianchi}. The tree therefore
measures, at the level of the observable, exactly the quantity whose
vanishing the coupled jet would require. The \(\rho_1\) term of
\eqref{eq:classical-tree} is X-A's scalar correction at the
background \(k_1\); it does not vanish for conserved sources and is
the part of the tree that survives when the dressing does not.

\begin{diagnostic}[Two normalization checks]
\label{diag:norm}
(i) If \(\partial_\nu S^{\nu\mu}=0\) for a smooth source supported in
\(t>2\delta\), then \(\X1[\DR S]=\ERzero(C[\DR S])=0\) by
Lemma~\ref{lem:violation}: a conserved source is not dressed.
(ii) If \(h_{\mathrm{cl}}=K\xi\), \(k_{\mathrm{cl}}=\tfrac12\Lie_\xi K\xi\),
\(\xi\in\ga\), then \(\Glin[K\xi]=0\), so the pure-gauge background
is not a source of the metric equation at first order, and the
classical completed observable is unchanged by X-D's Corollary;
\(\Ttree\) does not depend on \((h_{\mathrm{cl}},k_{\mathrm{cl}})\)
at all. Both checks are consistent with
Proposition~\ref{prop:normalization}; neither replaces it.
\end{diagnostic}

\subsection{Named and not constructed}
\label{sec:named}

Two further terms live at \(\kappa^2\) and are not constructed.

\emph{The gravitational self-interaction tree.}
By Lemma~\ref{lem:quadratic}, \(\delta S_3/\delta h_{\mu\nu}
=\tfrac12\mathcal G_2^{\mu\nu}[h,h]\). The complete first-order
retarded response of the metric is
\(\DR(-\delta(V_{1,g}+S_3)/\delta h)\), and the second summand
inserted in the affine slot gives
\[
\mathsf T_{\mathrm{self}}
=\kappa^2\,:\!\Ahat1\bigl[\DR(-\tfrac12\mathcal G_2[\hq,\hq])\bigr]\!:\ ,
\]
a Wick monomial of degree \((1,2)\) in \((\field,\hq)\). It exists
at \(\kappa^2\hbar^0\); the leg count in earlier planning that
excluded it was wrong. It is not constructed because its kernel is
a retarded propagator composed with a bilinear differential operator
at coincident points, Wick ordered with \(\mathsf H_{\mathrm{gr}}\),
and X-B's admission lemmas are two-slot statements. The one-tree
observable \eqref{eq:one-tree} omits it by declaration.

\emph{The mixed loop.}
Since \(\Ahat1[\hq]\) is bilinear in \((\hq,\field)\) and
\(V_{1,g}[\hq,\field]\) is trilinear, a second cross-contraction
exists along the scalar line. It is linear in \(\field\), of order
\(\hbar\kappa^2\), with kernel
\(f(x)g(y)\int\Lambda(x,z)\DR(z,y)\dd z\;\partial\partial\Delta_m(x,y)\),
whose extension to \(y=x\) is a genuine renormalization problem.
In X-B and X-C tree-exactness was a theorem because the insertion was
linear in the quantum field. Here it is not. ``Tree'' in this
manuscript means the \(\hbar^0\) coefficient by definition.

\subsection{The background gauge identity at fixed graviton}

\begin{lemma}[Two facts about the equation ideal]
\label{lem:ideal}
Let \(\Ideal\subset\Wick\) be the closed two-sided ideal generated by
\(\field(\Pm v)\) and \(\hq(\Pgr w)\).
\begin{enumerate}[label=\textup{(\roman*)}]
\item If \(W[\hq]\) is a Wick polynomial in the graviton alone and
\(u=\Pm u'\), then \(:\!\field(u)W[\hq]\!:=\field(u)\star:\!W[\hq]\!:\in\Ideal\).
Consequently a Wick monomial whose kernel lies in
\((\Pm\otimes1\otimes\cdots\otimes1)\Dspace'(M^{n})\), with the scalar
in the first slot, belongs to \(\Ideal\).
\item Every element of \(\Ideal\) vanishes as a functional on every
pair \((\phi_0,h)\) of smooth solutions \(\Pm\phi_0=0\), \(\Pgr h=0\).
Hence a degree-\((1,1)\) monomial \(:\!\field\,\hq\!:(B)\) with
\(B(\phi_0,h)\neq0\) for one such pair is not in \(\Ideal\).
\end{enumerate}
\end{lemma}

\begin{proof}
(i) The cross blocks of the Hadamard matrix vanish, so no contraction
joins \(\field(u)\) to \(W\), and the \(\star\)-product equals the
Wick-ordered product. \(\Ideal\) is a two-sided ideal and closed.
(ii) For any \(F\in\Wick\), the \(\star_{\mathsf H}\)-product
\(F\star\field(\Pm v)\) equals the pointwise product \(F\cdot\field(\Pm v)\),
because every term of positive order in \(\hbar\) pairs a derivative
of \(F\) with \(\mathsf H(\cdot,\Pm v)=0\); \(\mathsf H\) is a
bisolution. Likewise \(\Pgr\mathsf H_{\mathrm{gr}}=0\) since
\(\Ptr^2=1\) and \(\Box\mathsf H_0=0\). So \(\Ideal\) is the closure
of the pointwise ideal generated by functionals vanishing on
solutions, and evaluation at a fixed smooth configuration is
continuous \cite{FR,Rej}.
\end{proof}

\begin{theorem}[Background gauge identity of the one-tree observable]
\label{thm:ward}
Let \(\xi,\zeta\in\ga\), and let the classical backgrounds transform
by the complete two-jet law
\[
h_{\mathrm{cl}}'=h_{\mathrm{cl}}+K\xi,\qquad
k_{\mathrm{cl}}'=k_{\mathrm{cl}}+\Lie_\xi h_{\mathrm{cl}}+K\zeta
+\tfrac12\Lie_\xi K\xi ,
\]
with \(\field\) and \(\hq\) held fixed. Then, modulo \(\Ideal\) and
modulo \(\kappa^3\),
\begin{equation}
\widehat\Phi^{\mathrm{tree}}(f)[h_{\mathrm{cl}}',k_{\mathrm{cl}}']
-\widehat\Phi^{\mathrm{tree}}(f)[h_{\mathrm{cl}},k_{\mathrm{cl}}]
\equiv-\kappa^2\,\Ahat1\bigl[\Lie_\xi\hq\bigr].
\label{eq:defect}
\end{equation}
In particular the one-tree observable is invariant through
\(\kappa^1\); the tree \(\Ttree\) contributes nothing to the
variation at any order; and the \(\kappa^2\) defect is bilinear in
\((\hq,\field)\), independent of \(h_{\mathrm{cl}},k_{\mathrm{cl}}\)
and of \(\zeta\).
\end{theorem}

\begin{proof}
\(\Ttree\) depends on \(f,g,\eps,\field\) only, so it is invariant.
Write \(h=h_{\mathrm{cl}}+\hq\) and
\(k'_{\mathrm{full}}=k_{\mathrm{cl}}+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi\).
X-C's fixed-regulator identity, in X-D's explicit form, is a
polynomial identity in the smooth tensor variables \((h,k)\):
\[
d_1[h+K\xi]-d_1[h]=\Pm(X_\xi\EA f),\qquad
d_2[h+K\xi,k'_{\mathrm{full}}]-d_2[h,k_{\mathrm{cl}}]
=\Pm\bigl(X_\xi\EA d_1[h]+T_2\EA f\bigr),
\]
\(X_\xi=\xi^\mu\partial_\mu\), \(T_2=X_\zeta+\tfrac12X_\xi^2\),
with all functions inside \(\Pm\) compactly supported. Substituting
\(\hq\) for the quantum part of \(h\) and Wick ordering preserves the
identity of kernels, which are polynomial in \(h\) with
distributional coefficients. By Lemma~\ref{lem:ideal}(i) the right
sides are in \(\Ideal\). This gives invariance at \(\kappa^1\), since
\(\Ahat1[h']-\Ahat1[h]=\field(d_1[K\xi])\), and at \(\kappa^2\) for
the \emph{full} transform \((h',k'_{\mathrm{full}})\).

The frozen transform has \(k'_{\mathrm{cl}}=k'_{\mathrm{full}}-\Lie_\xi\hq\).
By Lemma~\ref{lem:affine}, \(d_2\) is affine in \(k\) with slope
\(d_1\), so
\(d_2[h',k'_{\mathrm{cl}}]=d_2[h',k'_{\mathrm{full}}]-d_1[\Lie_\xi\hq]\).
Wick ordering and \eqref{eq:one-tree} give \eqref{eq:defect}.
\end{proof}

\begin{lemma}[The defect is a tensor in first derivatives of the graviton]
\label{lem:gamma}
With \(\Gamma^{(1)}_{\lambda\mu\nu}[h]
=\tfrac12(\partial_\mu h_{\lambda\nu}+\partial_\nu h_{\lambda\mu}
-\partial_\lambda h_{\mu\nu})\),
\[
-\Ahat1[\Lie_\xi\hq]\equiv
2\,\Ahat1\bigl[\xi^\lambda\Gamma^{(1)}_{\lambda\mu\nu}[\hq]\bigr]
\pmod{\Ideal}.
\]
\end{lemma}

\begin{proof}
For any smooth symmetric \(h\) and vector field \(\xi\),
\[
(\Lie_\xi h)_{\mu\nu}
=\bigl(K(h\cdot\xi)\bigr)_{\mu\nu}
-2\xi^\lambda\Gamma^{(1)}_{\lambda\mu\nu}[h],\qquad
(h\cdot\xi)_\nu=h_{\nu\lambda}\xi^\lambda ,
\]
by expanding \(K(h\cdot\xi)=\partial_\mu(h_{\nu\lambda}\xi^\lambda)
+\partial_\nu(h_{\mu\lambda}\xi^\lambda)\) and comparing with
\(\xi^\lambda\partial_\lambda h_{\mu\nu}+h_{\lambda\nu}\partial_\mu\xi^\lambda
+h_{\mu\lambda}\partial_\nu\xi^\lambda\).
Since \(\supp(h\cdot\xi)\subset\supp\xi\subset U\), X-D's
representative gives \(d_1[K(h\cdot\xi)]=\Pm((h\cdot\xi)^\mu\partial_\mu\EA f)\),
linearly in \(h\). Substituting \(\hq\), the kernel of
\(\Ahat1[K(\hq\cdot\xi)]\) lies in \((\Pm\otimes1)\Dspace'_c\), and
Lemma~\ref{lem:ideal}(i) applies.
\end{proof}

The pure-gauge part of \(\Lie_\xi\hq\) is absorbed by the plateau
identity; what remains is the linearized Christoffel symbol of
\(\hq\) contracted with \(\xi\). This is not a gauge artefact of the
de Donder choice: \(\Gamma^{(1)}[\hq]\) is defined for every
configuration, and the statement is modulo the full ideal.

\begin{proposition}[The defect class is nonzero]
\label{prop:witness}
Let \(m=0\). For every nonzero \(\chi\in C_c^\infty(U)\) there exist
\(f,g,\delta\) satisfying \eqref{eq:separation} with the given \(U\)
such that, for \(\xi=\chi\,\partial_1\in\ga\),
\[
\Ahat1[\Lie_\xi\hq]\notin\Ideal .
\]
Explicitly, the smooth solutions
\[
h_{\mu\nu}=x^1\,\delta^2_\mu\delta^2_\nu,\qquad \Pgr h=0,
\qquad\qquad
\phi_0=t\,x^1,\qquad \Box\phi_0=0,
\]
give \(\Lie_\xi h=\chi\,\delta^2_\mu\delta^2_\nu\) and
\begin{equation}
\pair{d_1[\Lie_\xi h]}{\phi_0}
=2\int f\;\partial_1\partial_t\,(\ERzero)^2\chi\dd^4x\neq0 .
\label{eq:witness}
\end{equation}
Consequently the \(\kappa^2\) identity of Theorem~\ref{thm:ward} fails
on the frozen field content for every nonempty plateau: shrinking
\(\ga\) does not remove the defect.
\end{proposition}

\begin{proof}
\(\Box h=0\), so \(\Pgr h=-\tfrac14\Ptr\Box h=0\); \(\Box(tx^1)=0\).
Directly, \((\Lie_\xi h)_{\mu\nu}=\xi^1\partial_1h_{\mu\nu}
+h_{1\nu}\partial_\mu\chi+h_{\mu1}\partial_\nu\chi
=\chi\delta^2_\mu\delta^2_\nu\), since \(h_{1\cdot}=0\).

Put \(k=\chi\,\delta^2\otimes\delta^2\). On \(\supp k\subset U\) both
\(g\) and \(\sig\) equal \(1\). Then \(\tr_\eta k=-\chi\),
\(k^{22}=\chi\), and \eqref{eq:density} gives the diagonal
\(a^{00}=-\tfrac12\chi\), \(a^{11}=\tfrac12\chi\), \(a^{22}=-\tfrac12\chi\),
\(a^{33}=\tfrac12\chi\). Let \(w^\mu=\partial_\nu a^{\nu\mu}\), so
\(w^0=-\tfrac12\partial_t\chi\), \(w^1=\tfrac12\partial_1\chi\), and
\(\X1^\mu[k]=-\ERzero w^\mu\).
X-A's first-order relational scalar is
\[
\pair{d_1[k]}{\phi_0}=\int f\bigl(\rho_1-\X1^\mu\partial_\mu\phi_0\bigr),
\qquad \rho_1=-\ERzero\bigl(\partial_\mu(a^{\mu\nu}\partial_\nu\phi_0)\bigr),
\]
by the symmetry of \(\Lone\) and \(\pair{\EA f}{v}=\pair{f}{\ERzero v}\).
With \(\partial_t\phi_0=x^1\), \(\partial_1\phi_0=t\):
\(\partial_\mu(a^{\mu\nu}\partial_\nu\phi_0)=x^1w^0+tw^1\) and
\(\X1^\mu\partial_\mu\phi_0=-x^1\ERzero w^0-t\ERzero w^1\).
The commutator identities
\(\ERzero(x^1v)=x^1\ERzero v+2\ERzero\partial_1\ERzero v\) and
\(\ERzero(tv)=t\ERzero v-2\ERzero\partial_t\ERzero v\), valid for
\(v\) with past-compact support because
\(\Box(x^1u)=x^1\Box u-2\partial_1u\) and
\(\Box(tu)=t\Box u+2\partial_tu\), give
\[
\rho_1-\X1^\mu\partial_\mu\phi_0
=2(\ERzero)^2\bigl(\partial_tw^1-\partial_1w^0\bigr)
=2(\ERzero)^2\partial_1\partial_t\chi ,
\]
which is \eqref{eq:witness}.

Let \(\Phi:=\partial_1\partial_t(\ERzero)^2\chi\), smooth, supported in
\(J^+(\supp\chi)\subset\{t>2\delta\}\). If \(\Phi\) vanished on
\(\{t<T\}\), then \(\Box^2\Phi=\partial_1\partial_t\chi\) would vanish
there, hence on \(\supp\chi\), and a compactly supported \(\chi\) with
\(\partial_1\partial_t\chi=0\) is zero. So \(\Phi\neq0\) on an open
set \(O\subset\{2\delta<t<T\}\). Choose \(f\geq0\) supported in \(O\)
with \(\int f\Phi\neq0\), then \(g\) equal to \(1\) on a neighbourhood
of \(\overline U\), supported in \(\{2\delta<t<T\}\), and shrink
\(\delta\) if needed. All of \eqref{eq:separation} holds.
By Lemma~\ref{lem:ideal}(ii), \(\Ahat1[\Lie_\xi\hq]\notin\Ideal\).
\end{proof}

\begin{remark}
The defect cannot be cancelled by letting \(\hq\) co-transform:
\(\hq\mapsto\hq+\kappa\Lie_\xi\hq\) is not an automorphism of the free
gauge-fixed algebra, since \([\Pgr,\Lie_\xi]\neq0\) for non-Killing
\(\xi\), and \(\Lie_\xi\hq\) is not a solution even when \(\hq\) is:
\(\Glin[\Lie_\xi h]=-2G_2[K\xi,h]\) by covariance of the Einstein
tensor at second order. The cure is a second-order quantum metric
\(\hat k\) transforming by \(\Lie_\xi\hq\), i.e.\ the BRST jet.
That is Branch~B and is not opened. The mechanism is the one X-B
recorded for the scalar pullback, now in the metric slot.
\end{remark}

%=====================================================================
\section{E5: VII-C audit with a quantum graviton present}
\label{sec:E5}
%=====================================================================

X-D copied VII-C's Assumption (E1)--(E4) and found no graviton block
on that sheet. This sheet has one. The audit must therefore be redone
rather than inherited.

\subsection{VII-C's hypotheses, copied}

In VII-C's signature \(\eta_{\mathrm{VII}}=-\eta\),
\(\square_{\mathrm{VII}}=-\Box\), \(\mathsf E_{0,\mathrm{VII}}=-\ERzero\).
Its insertion is \(B_f[h,\phi]=-\int fw^\mu[h]\partial_\mu\phi\),
\(w=\mathsf E_{0,\mathrm{VII}}C_{\mathrm{VII}}h\), sharp and
zero-Cauchy. Its Wick matrix has
\(H_{hh}=P\Delta_0^+\), \(P_{\alpha\beta\rho\sigma}
=\eta_{\mathrm{VII}\,\alpha(\rho}\eta_{\mathrm{VII}\,\sigma)\beta}
-\tfrac12\eta_{\mathrm{VII}\,\alpha\beta}\eta_{\mathrm{VII}\,\rho\sigma}\),
\(H_{\phi\phi}=\Delta_m^+\), zero cross blocks.

\begin{assumption}[VII-C's Normal-topology Wick extension, copied]
\label{ass:VII}
A proposed extension is a Hausdorff locally convex
\(\C[[\hbar]]\)-algebra \(\cA\) such that:
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item \(\Pol[[\hbar]]\) embeds into \(\cA\), with the fixed Wick
product.
\item \(\cA\) admits the sharp insertions \(Q_\Sigma(B_f)\) with
\([Q_\Sigma(B_f)]_{\hbar^0}=B_f\) on all smooth configurations.
\item On the degree-\((1,1)\) kernel stages, admission is
sequentially continuous for the normal topology and agrees with the
old inclusion on regular kernels; in particular
\(B_{f,\eps}\to Q_\Sigma(B_f)\) in \(\cA\).
\item Multiplication is hypocontinuous for bounded sets, and the
field-independent \(\hbar^3\) coefficient map
\(a\mapsto\ev[\hbar^3]a\) is continuous and agrees with the usual
extraction on the old domain.
\end{enumerate}
\end{assumption}

\subsection{Match table}

\begin{center}
\small
\renewcommand{\arraystretch}{1.14}
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.40\linewidth}X@{}}
\toprule
Item & VII-C requires & X-E constructs\\
\midrule
(E1) &
The de Donder multiplet with \(H_{hh}=P\Delta_0^+\) as the fixed
Wick matrix &
A free gauge-fixed graviton with Hadamard form
\(\mathsf H_{\mathrm{gr}}=-4\Ptr\otimes\mathsf H_0\) in the opposite
signature, used only to define Wick ordering of
graviton-quadratic bare terms. \(\Ptr\) and \(P\) agree under the
dictionary; \(\mathsf H_0\) and \(\Delta_0^+\) are not identified,
and no computed quantity uses either.\\
(E2) &
Admission of the sharp zero-Cauchy insertion \(Q_\Sigma(B_f)\) &
The ramp insertion at fixed \(\eps\). Its kernel is
\(\Lambda\) of \eqref{eq:D1}, smooth-coefficient and admitted by
X-B. The sharp limit is not taken; X-D's limit concerns classical
backgrounds only.\\
(E3) &
Normal-topology convergence of mollified sharp kernels &
No approximating family. The regulator is fixed.\\
(E4) &
Hypocontinuous product and extraction of the fully contracted
\(\hbar^3\) coefficient of \(B_f\star W_g\star B_{f'}\) &
One \(\star\)-contraction along one graviton line, at \(\hbar^0\),
between \(\Ahat1[\hq]\) and \(V_{1,g}[\hq,\field]\). The vertex
\(V_{1,g}=-\tfrac12\int gh_{\mu\nu}T^{\mu\nu}\) is linear in the
graviton; it is not \(W_g=\int gh_{00}^2\). The graviton-quadratic
terms \(\Qhat[\hq,\hq]\) are Wick ordered, not \(\star\)-multiplied,
and no \(\hbar^3\) coefficient is computed.\\
\bottomrule
\end{tabularx}
\end{center}

VII-C's theorem excludes the named extension with the stated
admission and product requirements. The one-tree algebra of this
manuscript is a smaller object: fixed regulator, \(\hbar^0\), one
line, no \(W_g\). VII-C therefore does not apply and is not
contradicted. Its reflected light-ray residue is not subtracted
because no loop coefficient is formed; the mixed loop of
Section~\ref{sec:named} is the first place such a coefficient would
appear, and it is not constructed.

%=====================================================================
\section{Diagnostics}
\label{sec:diagnostics}
%=====================================================================

\begin{diagnostic}[Sourced tree number]
\label{diag:sourced}
Let \(m=0\), \(\phi_0=t\), and near a causal cylinder containing the
dependence region of \(K\) let \(g=g_0(t)\), with the usual spatial
cutoffs outside it. Then \(T^{\mu\nu}[t]=n^\mu n^\nu-\tfrac12\eta^{\mu\nu}\),
\(\Ptr T=n\otimes n\), and \eqref{eq:k1} gives
\[
k_1=-2\,G_2(t)\,n\otimes n,\qquad
G_1(t)=\int_0^tg_0,\quad G_2(t)=\int_0^tG_1,\quad G_3(t)=\int_0^tG_2 .
\]
Hence \(a^{00}[k_1]=G_2\), \((\partial_\nu g)T^{\nu\mu}=\tfrac12g_0'n^\mu\),
\(C^\mu[k_1]=-G_1n^\mu\), and \(\X1^0[k_1]=-G_3\), in agreement with
\eqref{eq:dressing}. The scalar part is
\(\rho_1=-\ERzero\partial_t(g_0G_2)=-\int_0^tg_0G_2\), so
\[
\Ttree\big|_{\field\to t,\ \hq\to0}
=\kappa^2\int f\Bigl(G_3-\int_0^tg_0G_2\Bigr)\dd^4x
=\kappa^2\int f\int_0^t\bigl(1-g_0(s)\bigr)G_2(s)\dd s\dd^4x .
\]
This agrees with \eqref{eq:classical-tree} and is nonnegative for
\(f\ge0\). It vanishes if \(g_0\equiv1\) on \([0,t]\), consistent
with the fact that for \(\phi_0=t\) the first-order relational
scalar is identically zero on plateau-supported backgrounds:
\(t\) is itself a harmonic coordinate. The witness of
Proposition~\ref{prop:witness} therefore uses \(\phi_0=tx^1\), not
\(\phi_0=t\).
\end{diagnostic}

\begin{diagnostic}[Pure gauge with the graviton held fixed]
\label{diag:puregauge}
Let \(\xi\in\ga\), \(\zeta=0\), \(h_{\mathrm{cl}}=K\xi\),
\(k_{\mathrm{cl}}=\tfrac12\Lie_\xi K\xi\). The classical completed
observable equals its value at \((0,0)\) modulo \(\Ideal\) by X-D's
Corollary. With \(\hq\) present and fixed, Theorem~\ref{thm:ward}
gives the additional term \(-\kappa^2\Ahat1[\Lie_\xi\hq]\), which by
Lemma~\ref{lem:gamma} is \(2\kappa^2\Ahat1[\xi^\lambda\Gamma^{(1)}_{\lambda\cdot\cdot}[\hq]]\)
modulo \(\Ideal\), and by Proposition~\ref{prop:witness} is nonzero
for suitable inputs. The two-jet check of X-D passes; the
quantum-metric check at \(\kappa^2\) does not. That is the order
boundary.
\end{diagnostic}

%=====================================================================
\section{Decision and final release}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.14\linewidth}p{.49\linewidth}X@{}}
\toprule
Gate & Evidence & Status\\
\midrule
E1 action &
Einstein--Hilbert, sign by TT positivity and Newtonian attraction;
Palatini first variation derived &
Frozen\\
E1 equation &
\eqref{eq:metric-euler-expanded}; Noether identity
\eqref{eq:noether} with switch defect \((\partial g)T\) &
Proved\\
E2 invariance &
Total Lagrangian exactly \(\Gp\)-invariant; Euler operators
covariant; classical identity through \(\kappa^2\) &
Proved\\
E2 jet &
Bianchi identity forces \((\partial_\mu g)T^\mu{}_\rho[\phi_0]=0\) &
Coupled jet generically empty; observable unaffected\\
E3 operator &
\(\Pgr=-\tfrac14\Ptr\Box\), \(\DR=-4\Ptr\ERzero\), \(\Ptr^2=1\) &
Constructed\\
E3 residual &
Compactly supported de Donder residuals vanish &
\(\ga\) acts on backgrounds only\\
E4 tree &
Kernel, support in \(J^-(K)\), microcausal admission, classical
normalization &
Constructed at \(\hbar^0\)\\
E4 identity &
Invariant through \(\kappa^1\); \(\kappa^2\) defect
\(-\kappa^2\Ahat1[\Lie_\xi\hq]\) displayed and nonzero &
Order boundary at \(\kappa^1\)\\
E5 &
(E1)--(E4) copied; one line, no \(W_g\), no \(\hbar^3\) &
Inapplicable, not contradicted\\
Named &
Self-interaction tree; mixed loop; second-order quantum metric;
ghosts; \(\eps\to0\) &
Not constructed\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Fixed-regulator X-E release]
\label{thm:release}
For the frozen sheet of Section~\ref{sec:frozen}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The gravitational action \(\Sgr=-(2\kappa^2)^{-1}\int\sqrt{-q}R\)
has metric Euler operator \eqref{eq:metric-euler}, expanded in
\eqref{eq:metric-euler-expanded}, and the total action satisfies the
Noether identity \eqref{eq:noether}, whose only off-plateau terms
are the switch defects. On the plateau algebra the total Lagrangian
is exactly invariant, the Euler operators are covariant, and the
classical completed observable is invariant through \(\kappa^2\).
The coupled constraint surface is \(\Gp\)-invariant and generically
empty, by \eqref{eq:bianchi-constraint}.
\item The de Donder gauge-fixed quadratic operator is
\(\Pgr=-\tfrac14\Ptr\Box\) with retarded inverse \(-4\Ptr\ERzero\).
Its residual algebra meets \(\ga\) only in zero. A de Donder
configuration dresses the relational scalar only through the ramp
layer.
\item The tree \(\Ttree\) of Definition~\ref{def:tree} is a totally
Wick-ordered cubic in the scalar with microcausal kernel supported in
\(J^-(K)\times J^-(K)\), an element of the fixed-regulator multiplet
algebra of X-B. Its classical value is X-A's first-order relational
scalar at the retarded backreaction \(k_1=-2\Ptr\ERzero(gT[\phi_0])\),
whose coordinate dressing is
\(-2\ERzero\ERzero((\partial_\nu g)T^{\nu\mu}[\phi_0])\).
\item The one-tree observable is invariant under the frozen algebra,
acting on classical backgrounds with the graviton fixed, through
\(\kappa^1\). At \(\kappa^2\) its variation is
\(-\kappa^2\Ahat1[\Lie_\xi\hq]\equiv
2\kappa^2\Ahat1[\xi^\lambda\Gamma^{(1)}_{\lambda\cdot\cdot}[\hq]]\)
modulo the equation ideal, and this class is nonzero for suitable
fixed inputs on every nonempty plateau.
\end{enumerate}
VII-C does not apply and is not contradicted.
The self-interaction tree, the mixed loop, a second-order quantum
metric, ghosts, gauge-parameter dependence, and the sharp limit are
named and not constructed.
\end{theorem}

\begin{proof}
(i) is Lemma~\ref{lem:palatini}, Definition~\ref{def:metric-euler},
Propositions~\ref{prop:noether}, \ref{prop:covariance},
\ref{prop:bianchi}, and Theorem~\ref{thm:classical}.
(ii) is Lemmas~\ref{lem:quadratic}, \ref{lem:gf}, \ref{lem:layer}
and Proposition~\ref{prop:residual}.
(iii) is Lemmas~\ref{lem:kernel}, \ref{lem:admission} and
Proposition~\ref{prop:normalization}.
(iv) is Theorem~\ref{thm:ward}, Lemma~\ref{lem:gamma}, and
Proposition~\ref{prop:witness}.
Section~\ref{sec:E5} audits VII-C; Section~\ref{sec:named} names the
omitted terms.
\end{proof}

The theorem concerns a fixed regulator, fixed supports separated
from \(\Sigma\), and fixed classical backgrounds. It supplies no
statement about \(\eps\to0\) for any kernel containing \(\hq\), no
statement about a state, and no statement about the complete
\(\kappa^2\hbar^0\) observable, which also contains the
self-interaction tree.

%=====================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}

\paragraph{Dependencies.}
Coordinates, density coefficients, the two-jet gauge law, and the
first-order relational scalar are from \texttt{X-A.tex}.
The multiplet Wick algebra, joint admission, and the pullback
non-automorphism are from \texttt{X-B-v3.tex}.
The completed tests, recursion, and fixed-regulator identity are
from \texttt{X-C-v3.tex}.
The separation sheet, the layer lemma, and the explicit
equation-ideal representatives are from \texttt{X-D-v3.tex}.
\texttt{VII-C.tex} is used only for the copied assumption.

\paragraph{New proof.}
The Palatini variation and its expansion; the Noether identity with
switch defect; exact plateau invariance of the total Lagrangian and
covariance of the metric Euler operator; the Bianchi obstruction;
the gauge-fixed operator, its inverse, and the residual algebra; the
affine-slot lemma; the tree kernel, its support, admission, and
classical normalization; the background identity through
\(\kappa^1\); the explicit \(\kappa^2\) defect, its reduction to the
linearized Christoffel symbol, and the nonvanishing witness.

\paragraph{Corrections.}
Listed in the header. The two that change the sheet: the quadratic
action alone cannot carry E2, so Einstein--Hilbert is frozen; and
the self-interaction tree exists at \(\kappa^2\), so ``one tree'' is
a declaration, not a completeness statement.

\paragraph{Verification status.}
No successful \texttt{pdflatex} execution, proof-assistant check,
numerical verification, or independent referee review is claimed.
The source requires no external figures, bibliography database, or
executable code.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}

\bibitem{XA}
N.~Cheung.
\newblock \emph{Landmark X-A: Classical covariance of one
fixed-regulator switched relational scalar through order
\(\kappa^2\)}.
\newblock Supplied manuscript \texttt{X-A.tex}.

\bibitem{XB}
N.~Cheung.
\newblock \emph{Landmark X-B: Fixed-regulator admission and one
retarded correction for the ramp relational scalar, with the metric
kept classical}.
\newblock Supplied manuscript \texttt{X-B-v3.tex}.

\bibitem{XC}
N.~Cheung.
\newblock \emph{Landmark X-C: The missing second-order retarded maps
for the ramp relational scalar, with the metric kept classical}.
\newblock Supplied manuscript \texttt{X-C-v3.tex}.

\bibitem{XD}
N.~Cheung.
\newblock \emph{Landmark X-D: Convergence of the completed ramp
relational scalar as the regulator width tends to zero}.
\newblock Supplied manuscript \texttt{X-D-v3.tex}.

\bibitem{VIIC}
\emph{Landmark VII-C: A mixed-Wick obstruction to a normal-topology
boundary extension for sharp Cauchy harmonic insertions}.
\newblock Supplied manuscript \texttt{VII-C.tex}.
\newblock Used for the copied assumption only.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle.
\newblock \emph{Wave Equations on Lorentzian Manifolds and
Quantization}.
\newblock ESI Lectures in Mathematics and Physics. European
Mathematical Society, 2007. arXiv:0806.1036.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen.
\newblock Microlocal analysis and interacting quantum field theories:
renormalization on physical backgrounds.
\newblock \emph{Comm.\ Math.\ Phys.} 208 (2000), 623--661.

\bibitem{DH}
J.~J.~Duistermaat and L.~H\"ormander.
\newblock Fourier integral operators II.
\newblock \emph{Acta Math.} 128 (1972), 183--269.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner.
\newblock Batalin--Vilkovisky formalism in the functional approach to
classical field theory.
\newblock \emph{Comm.\ Math.\ Phys.} 314 (2012), 93--127.

\bibitem{Hor}
L.~H\"ormander.
\newblock \emph{The Analysis of Linear Partial Differential
Operators I}, 2nd ed.
\newblock Springer, 1990. Theorem~8.2.14 (composition of
distribution kernels).

\bibitem{LL}
L.~D.~Landau and E.~M.~Lifshitz.
\newblock \emph{The Classical Theory of Fields}, 4th ed.
\newblock Pergamon, 1975. Signature \((+,-,-,-)\),
\(S_g=-(16\pi G)^{-1}\int R\sqrt{-g}\); cited for agreement, not
as the source of the sign.

\bibitem{Rej}
K.~Rejzner.
\newblock \emph{Perturbative Algebraic Quantum Field Theory}.
\newblock Mathematical Physics Studies. Springer, 2016.

\end{thebibliography}

\end{document}