% !TEX program = pdflatex
%=====================================================================
% X-F.tex
%
% Standalone source for pdfLaTeX.
%
% Build:
%   pdflatex -interaction=nonstopmode -halt-on-error X-F.tex
%   pdflatex -interaction=nonstopmode -halt-on-error X-F.tex
%
% Frozen:
%   Everything of X-E except two items. (a) Gauge: de Donder condition
%   of the classical background metric qbar = eta + kappa h_cl, with
%   Feynman weight, for the quantum graviton only. (b) Second-order
%   quantum metric: the complete first-order retarded Moller response
%   of the gauge-fixed fluctuation, with the background-field
%   prescription (terms of degree <= 1 in the quantum fields dropped)
%   and the gravitational order-kappa densities switched on by the
%   ramp sigma_eps. Fields: free scalar, free flat gauge-fixed
%   graviton, classical off-shell backgrounds. No ghosts. hbar^0 only.
%
% Result:
%   F1 exact second-order covariance identity
%        2 G_2(K xi, h) = L_xi G_lin[h] - G_lin[L_xi h].
%   F2 background de Donder gauge; kinetic operator
%        Q[qbar] = P_gr + kappa Q^(1)[h_cl] + O(kappa^2),
%        Q^(1)[K xi] = [L_xi, P_gr];  Delta_R P_gr u = u;
%      hence the retarded response to a pure-gauge background is
%        L_xi u  -  Delta_R L_xi P_gr u   (the jet law is dynamical).
%   F3 three trees at kappa^2 hbar^0: matter (X-E), mixed, self.
%      Kernels, supports, admission; the self tree by a three-slot
%      sum-zero lemma.
%   F4 background identity through kappa^2 modulo the equation ideal,
%      graviton held fixed, for the all-tree observable. X-E's defect
%      is a truncation defect.
%   F5 flat-gauge comparison: the residual is
%        kappa^2 A_1[Delta_R [L_xi, F^(0)] h], F^(0) = -(1/4) P K C;
%      displayed, class not decided; X-E's witness does not decide it.
%
% Corrections relative to X-E:
%   - X-E (4.17) carries kappa^2 twice on the tree (once inside
%     T_tree, once outside the bracket). The intended object is
%     kappa^2 times the Wick-ordered affine insertion. Fixed here by
%     defining the trees without the prefactor.
%   - X-E's remark after Prop. 4.21 states G_lin[L_xi h] = -2 G_2[K xi,h].
%     This holds only on G_lin[h] = 0. The exact identity carries the
%     extra term L_xi G_lin[h] (Lemma 4.2 below). The conclusion of the
%     remark stands; its "cure" (a new field) is not needed at hbar^0.
%   - X-E's named self tree has source G_2[h,h] with h a free solution,
%     which is not past-compact; Delta_R of it is undefined. Here the
%     gravitational order-kappa densities are switched by sigma_eps.
%   - X-E Lemma 4.17(i) treats P_m in the scalar slot only; the
%     graviton slot is needed and supplied (Lemma 7.1).
%   - X-E uses K both for supp f and for K xi = L_xi eta. Here
%     supp f = K_f.
%   - X-E's one-tree observable omits the polarized self-interaction
%     term G_2(h_cl, h), which is the term carrying the identity. "One
%     tree" was not a completeness statement; "three trees" is, at
%     kappa^2 hbar^0, for this field content.
%
% Corrections relative to the planning notes:
%   - The shadow condition and the extended pure-gauge representative
%     are not needed for the main theorem. The pure-gauge and
%     constraint pieces cancel between G_2(K xi, .) and the gauge-
%     fixing correction, because both are commutators with L_xi and
%     (1/2) G_lin + F^(0) = P_gr. The bidifferential K_xi of the notes
%     is the Lie derivative on tensor densities.
%   - Ghosts are not needed at hbar^0; no ghost leg can be external.
%   - The three-slot admission is by the sum-zero argument, not by a
%     chain argument.
%   - No claim of successful compilation or independent verification.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

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

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

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\WF}{WF}
\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{\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{\Th}{\Theta}
\newcommand{\Eul}{\mathsf E}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

% New in X-F.
\newcommand{\Edens}{\mathcal E}
\newcommand{\Fop}{\mathcal F}
\newcommand{\Qop}{\mathcal Q}
\newcommand{\kq}{\widehat k}
\newcommand{\hcl}{h_{\mathrm{cl}}}
\newcommand{\kcl}{k_{\mathrm{cl}}}
\newcommand{\qb}{\bar q}
\newcommand{\Tmat}{\mathsf T_{\mathrm{mat}}}
\newcommand{\Tmix}{\mathsf T_{\mathrm{mix}}}
\newcommand{\Tself}{\mathsf T_{\mathrm{self}}}
\newcommand{\Mker}{\mathcal M}
\newcommand{\Tker}{\mathcal T}
\newcommand{\PhiF}{\widehat\Phi^{\mathrm F}}
\newcommand{\Sq}{S_{\mathrm q}}
\newcommand{\Kf}{K_f}
\newcommand{\Gtwo}{\mathcal G_2}
\newcommand{\Lgf}{\mathfrak L_{\mathrm{gf}}}
\newcommand{\qdens}{\mathfrak q}
\newcommand{\Src}{\mathsf S}

% 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-F}\\[0.4em]
The second-order quantum metric as the retarded M\o ller response,\\
and the closure of the \(\kappa^2\) background identity}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We complete the \(\hbar^0\), \(\kappa^2\) sector of Landmark X-E for
the ramp relational scalar, in signature \((+,-,-,-)\), at fixed
regulator width. Two items of the X-E sheet are replaced. The
gauge fixing of the quantum graviton is the de Donder condition of
the classical background metric \(\qb=\eta+\kappa\hcl\), with Feynman
weight. The second-order quantum metric is defined, not postulated:
it is the complete first-order retarded M\o ller response of the
gauge-fixed fluctuation, so that the single tree of X-E becomes
three, the matter tree of X-E, a mixed tree bilinear in
\((\hcl,\hq)\), and a self-interaction tree quadratic in \(\hq\).
No new field is introduced and no ghost is needed.

The mechanism is a commutator identity. Naturality of the
Einstein--Hilbert density and of the background gauge-fixing density
gives, for the kinetic operator of the fluctuation
\(\Qop[\qb]=\Pgr+\kappa\Qop^{(1)}[\hcl]+O(\kappa^2)\),
\[
\Qop^{(1)}[K\xi]=[\Lie_\xi,\Pgr].
\]
Since \(\DR\Pgr u=u\) for compactly supported \(u\), the retarded
response of the fluctuation to a pure-gauge background is
\(\Lie_\xi\hq\) modulo the equation ideal: the transformation law of
the field X-E named and did not construct is a theorem about the
M\o ller map. The mixed tree therefore varies by
\(+\kappa^2\Ahat1[\Lie_\xi\hq]\) and cancels X-E's defect exactly;
the matter and self trees are inert. The all-tree observable is
invariant under the frozen algebra, acting on the classical
backgrounds with the quantum fields held fixed, through \(\kappa^2\)
modulo the equation ideal.

Along the way: the exact second-order covariance identity
\(2\Gtwo(K\xi,h)=\Lie_\xi\Glin[h]-\Glin[\Lie_\xi h]\), which corrects
an on-shell shortcut in X-E; the switching of the gravitational
self-interaction by the ramp, without which the self tree is
undefined; a three-slot admission lemma by a sum-zero argument; and
the flat-gauge residual \(\kappa^2\Ahat1[\DR[\Lie_\xi,\Fop^{(0)}]\hq]\),
\(\Fop^{(0)}=-\tfrac14\Ptr KC\), displayed and not decided. The
one-loop terms, ghosts, the master equation with switched matter,
positivity, and the regulator limit are named and not constructed.
\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) {Ramp sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {This calculation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Off-sheet identification};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The result does not leave the frozen ramp sheet.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The next ending stays on the ramp sheet. 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-F}
\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) {Ramp sheet};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Off-sheet identification};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The result does not leave the frozen ramp sheet.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The next ending stays on the ramp sheet. 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-F}
\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) {Ramp sheet};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {This calculation};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Off-sheet identification};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The result does not leave the frozen ramp sheet.};
\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-F}
\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) {Ramp sheet};
\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) {Off-sheet identification};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {This calculation. The result does not leave the frozen ramp sheet.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Ramp sheet is not Off-sheet identification. 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-F}
\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: Ramp sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: This calculation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Off-sheet identification};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The result does not leave the frozen ramp sheet.};
\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-F}
\end{figure}

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


Figure~\ref{fig:scope-X-F} 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,
through \(\kappa^2\hbar^0\)}{%
Frozen: X-E's sheet with the background de Donder gauge
\(\tfrac14\int\sqrt{-\qb}\,\qb_{\mu\nu}C^\mu_{\qb}C^\nu_{\qb}\),
the background-field prescription, the ramp switch on the
gravitational order-\(\kappa\) densities, and the M\o ller-defined
response \(\kq=\DR\Src\).

\smallskip
Proved.
\textbf{F1}: the exact second-order covariance identity
\eqref{eq:cov2}.
\textbf{F2}: \(\Fop^{(0)}=-\tfrac14\Ptr KC\), \(\Pgr=\tfrac12\Glin+\Fop^{(0)}\),
the commutator identity \eqref{eq:key}, and the dynamical jet law
\eqref{eq:jetlaw}.
\textbf{F3}: three trees, Definition~\ref{def:trees}; kernels,
supports, admission (Lemmas~\ref{lem:kernels}, \ref{lem:sumzero},
\ref{lem:admission}), classical values.
\textbf{F4}: Theorem~\ref{thm:main}, the background identity through
\(\kappa^2\) modulo \(\Ideal\), quantum fields fixed.
\textbf{F5}: the flat-gauge residual, Proposition~\ref{prop:flat};
its class is not decided.
\textbf{F6}: VII-C inapplicable, not contradicted.

\smallskip
Not opened: \(\hbar^1\), ghosts, BRST/BV with switched matter,
positivity, the classical tadpole, gauge-parameter dependence,
\(\eps\to0\), X-Enter, all orders in \(\kappa\).}

Three endings are available:
\begin{enumerate}[label=\textbf{\arabic*.}]
\item \textbf{Constructed.}
The gauge, the response, and all \(\kappa^2\hbar^0\) trees are named,
admitted, and the all-tree observable transforms correctly on the
frozen algebra through \(\kappa^2\).
\item \textbf{Proved obstructed.}
The all-tree identity acquires a nonzero defect at \(\kappa^2\), or a
tree kernel leaves the declared algebra.
\item \textbf{Unresolved.}
A tree is unnamed or its admission is not proved.
\end{enumerate}
The ending reached is (1). The defect of X-E is shown to be a
truncation defect of the one-tree observable in the flat gauge, not
a field-content obstruction.

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}
%=====================================================================

\subsection{Geometry, supports, regulator}

As in X-E \cite{XE}:
\[
M=\R^4,\qquad \eta=\diag(1,-1,-1,-1),\qquad
\Box=\partial_t^2-\Delta,\qquad \Pm=\Box+m^2,\quad m\ge0 .
\]
Indices are moved with \(\eta\) unless \(q\) or \(\qb\) is written.
\(\ER,\EA\) are the Klein--Gordon Green operators, \(\ERzero,\EAzero\)
the massless ones. 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}
\Kf\cup\supp g\cup\overline U\subset\{2\delta<t<T\},\qquad \Kf:=\supp f .
\label{eq:separation}
\end{equation}
Fix \(0<\eps<\delta\). The ramp \(\sig(t)=\chi(t/\eps)\) is X-D's:
smooth, \(\sig=0\) on \(t\le0\), \(\sig=1\) on \(t\ge\eps\). In
particular \(\sig=1\) on a neighbourhood of
\(\{t\ge2\delta\}\supset\overline U\cup\supp g\cup\Kf\).
\(\Gp\) is the group generated formally by \(\kappa\xi+\kappa^2\zeta\),
\(\xi,\zeta\in\ga\). For a vector field \(\xi\), \(F_s\) denotes its
flow and \(K\xi:=\Lie_\xi\eta\), \((K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu\).

\subsection{Inherited statements}
\label{sec:inherited}

The following are copied from X-A to X-E and not re-proved. Equation
numbers in brackets refer to X-E.
\begin{enumerate}[label=\textbf{(XE\arabic*)}]
\item\label{xe:tests}
Density coefficients \(a,b,A,B\) [(2.2)], ramp coordinates
\(\X1,\X2\) [(2.3)], Euler coefficients \(\Lone,\Ltwo\), completed
tests \(d_1,d_2\) [(2.5)], \(\Ahat1[h]=\field(d_1[h])\),
\(\Ahat2=\field(d_2)\), \(\Qhat[h,h]=\,:\!\field(Q[h,h])\!:\), and the
affine-slot lemma \(d_2[h,k]=Q[h,h]+d_1[k]\), \(d_1\) linear
[Lemma 2.1]. The kernel \(D_1\) of \(d_1\) [(4.8)] is a compactly
supported distribution on \(M\times M\) with
\(\supp D_1\subset\Kf\times(J^-(\Kf)\cap\{t\ge0\})\).
\item\label{xe:einstein}
With \(\Edens[q]:=\sqrt{-q}\,G^{\mu\nu}[q]\),
\[
\Edens[\eta+\kappa h+\kappa^2k]=\kappa\Glin[h]
+\kappa^2\bigl(\Glin[k]+\Gtwo[h,h]\bigr)+O(\kappa^3)
\]
[(4.6)], \(\Gtwo(\cdot,\cdot)\) its symmetric bilinear polarization;
\(\Sgr=-(2\kappa^2)^{-1}\int\sqrt{-q}R\), \(\delta\Sgr/\delta h=\tfrac1{2\kappa}\Edens\)
[Lemma 4.1]; \(\Sgr=S_{\mathrm{FP}}[h]+\kappa(2S_{\mathrm{FP}}(h,k)+S_3[h])+O(\kappa^2)\),
\(\delta S_{\mathrm{FP}}/\delta h=\tfrac12\Glin[h]\),
\(\delta S_3/\delta h=\tfrac12\Gtwo[h,h]\) [Lemma 6.1, Sec.~4.6];
\(\Glin[K\xi]=0\); the vertex identity
\(V_{1,g}[h,\phi]=-\tfrac12\int gh_{\mu\nu}T^{\mu\nu}[\phi]\),
\(\delta V_{1,g}/\delta h=-\tfrac12gT[\phi]\),
\(\delta V_{1,g}/\delta\phi=-\Lone[h]\phi\) [Lemma 4.2].
\item\label{xe:gf}
\(\Ptr u=u-\tfrac12\eta\tr_\eta u\), \(\Ptr^2=1\);
\(C^\mu[u]=\partial_\nu(\Ptr u)^{\nu\mu}\);
\(S^{(0)}_{\mathrm{gf}}[u]=\tfrac14\int C_\mu C^\mu\);
\begin{equation}
\Glin[u]=\Ptr\bigl(-\tfrac12\Box u+\tfrac12KC[u]\bigr),\qquad
\frac{\delta(S_{\mathrm{FP}}+S^{(0)}_{\mathrm{gf}})}{\delta u}
=-\tfrac14\Ptr\Box u=:\Pgr u,\qquad
\DR=-4\Ptr\otimes\ERzero
\label{eq:xe-gf}
\end{equation}
[Lemma 6.3]. Here \((KC)_{\mu\nu}=\partial_\mu C_\nu+\partial_\nu C_\mu\).
\item\label{xe:algebra}
The multiplet Wick algebra \(\Wick\) of \((\field,\hq)\) with Hadamard
matrix of zero cross blocks, \(\mathsf H_{\mathrm{gr}}=-4\Ptr\otimes\mathsf H_0\),
and the closed two-sided equation ideal \(\Ideal\) generated by
\(\field(\Pm v)\), \(\hq(\Pgr w)\) [Sec.~6.4]. The admission
criterion of X-B \cite{XB,BF}: a Wick monomial of degree \(n\) in the
multiplet belongs to \(\Wick\) iff its kernel \(u\in\Dspace'(M^n)\)
is compactly supported and
\begin{equation}
\WF(u)\cap\bigl(\overline V_+^{\,n}\cup\overline V_-^{\,n}\bigr)=\emptyset,
\label{eq:microcausal}
\end{equation}
\(\overline V_\pm\) the closed future/past covector cones including
zero. The wavefront bound
\(\WF(\ERzero)\subset\{(x,\xi;y,-\xi):\xi\neq0\}\) [Lemma 4.13,
\cite{DH}], and the facts that elements of \(\Ideal\) vanish on all
pairs of smooth free solutions [Lemma 4.17(ii)].
\item\label{xe:ward}
Theorem 4.18 of X-E: with \(h=\hcl+\hq\), under
\(\hcl'=\hcl+K\xi\), \(\kcl'=\kcl+\Lie_\xi\hcl+K\zeta+\tfrac12\Lie_\xi K\xi\),
\(\field,\hq\) fixed,
\begin{equation}
\Bigl[\field(f)+\kappa\Ahat1[h]+\kappa^2\bigl(\Qhat[h,h]+\Ahat1[\kcl]\bigr)\Bigr]'
-\Bigl[\,\cdots\Bigr]\equiv-\kappa^2\Ahat1[\Lie_\xi\hq]
\pmod{\Ideal,\ \kappa^3}.
\label{eq:xe-defect}
\end{equation}
Proposition 4.21 of X-E: for \(m=0\), \(\xi=\chi\partial_1\),
\(h=x^1\delta^2\otimes\delta^2\), \(\phi_0=tx^1\),
\(\pair{d_1[\Lie_\xi h]}{\phi_0}=2\int f\,\partial_1\partial_t(\ERzero)^2\chi\neq0\)
for suitable \(f,g,\delta\).
\item\label{xe:natural}
Lemma 5.1 and Proposition 5.2 of X-E: for a diffeomorphism \(F\)
equal to the identity outside a compact set, a natural Lagrangian
density integrates to the same value before and after pull-back, and
its Euler--Lagrange tensor density is natural,
\(E[F^*\text{fields}]=F^*E[\text{fields}]\).
\end{enumerate}

\subsection{Lie derivatives and types}
\label{sec:types}

\(\Lie_\xi\) acts on covariant symmetric two-tensors by
\((\Lie_\xi u)_{\mu\nu}=\xi^\lambda\partial_\lambda u_{\mu\nu}
+u_{\lambda\nu}\partial_\mu\xi^\lambda+u_{\mu\lambda}\partial_\nu\xi^\lambda\)
and on contravariant symmetric tensor densities of weight one by
\(\Lie_\xi T^{\mu\nu}=\partial_\lambda(\xi^\lambda T^{\mu\nu})
-T^{\lambda\nu}\partial_\lambda\xi^\mu-T^{\mu\lambda}\partial_\lambda\xi^\nu\).
For the flow \(F_s\) of \(\xi\), \(\frac{d}{ds}F_s^*T=F_s^*\Lie_\xi T\)
for every tensor density \(T\). Metrics and fluctuations are
covariant tensors; Euler derivatives with respect to them, in
particular \(\Edens\), \(\Glin\), \(\Gtwo\), \(\Pgr u\), and the
gauge-fixing derivative \(\Fop\) below, are contravariant tensor
densities. When \(\Lie_\xi\) is written on either side of such an
operator, it acts in the type of its argument. The two-jet law of
\ref{xe:ward} is the expansion
\(F^*(\eta+\kappa\hcl+\kappa^2\kcl)=\eta+\kappa\hcl'+\kappa^2\kcl'+O(\kappa^3)\)
for \(F=\exp(\kappa\xi+\kappa^2\zeta)\).

\subsection{Background de Donder gauge}

\begin{definition}[Background de Donder gauge]
\label{def:bgf}
For a Lorentzian metric \(\qb\) with Levi-Civita connection
\(\bar\nabla\) and a smooth symmetric \(u\),
\[
\bar u^{\nu\mu}_{\qb}:=\qb^{\nu\alpha}\qb^{\mu\beta}u_{\alpha\beta}
-\tfrac12\qb^{\nu\mu}\qb^{\alpha\beta}u_{\alpha\beta},\qquad
C^\mu_{\qb}[u]:=\bar\nabla_\nu\bar u^{\nu\mu}_{\qb},
\]
\[
S_{\mathrm{gf}}[u;\qb]:=\tfrac14\int_M\sqrt{-\qb}\,\qb_{\mu\nu}
C^\mu_{\qb}[u]\,C^\nu_{\qb}[u]\dd^4x ,
\]
and \(\Fop[u;\qb]:=\delta S_{\mathrm{gf}}[u;\qb]/\delta u_{\mu\nu}\) its Euler
derivative with respect to the fluctuation, a contravariant symmetric
tensor density, defined through compactly supported variations of
\(u\). At \(\qb=\eta\), \(C^\mu_\eta=C^\mu\) and \(S_{\mathrm{gf}}[u;\eta]=S^{(0)}_{\mathrm{gf}}[u]\)
of \ref{xe:gf}. For \(\qb=\eta+\kappa\hcl\) write
\begin{equation}
\Fop[u;\eta+\kappa\hcl]=\Fop^{(0)}u+\kappa\,\Fop^{(1)}[\hcl]u+O(\kappa^2),
\label{eq:F-expansion}
\end{equation}
with \(\Fop^{(1)}[w]u\) linear in \(w\) and in \(u\).
\end{definition}

The gauge condition is imposed on the quantum fluctuation only; the
classical backgrounds remain unconstrained. The dependence of the
gauge fixing on \(\kcl\) begins at order \(\kappa^2\) in
\(\Fop\) and enters the observable below at order \(\kappa^3\); it is
dropped.

\subsection{Field content, kinetic operator, switched source, response}

\begin{definition}[Field content and the second-order quantum metric]
\label{def:content}
\begin{equation}
q=\eta+\kappa\bigl(\hcl+\hq\bigr)+\kappa^2\bigl(\kcl+\kq\bigr),\qquad
\qb=\eta+\kappa\hcl,\qquad u:=\hq+\kappa\kq ,
\label{eq:metric}
\end{equation}
with \(\field\) the free Klein--Gordon field, \(\hq\) the free flat
gauge-fixed graviton of \ref{xe:algebra}, \(\hcl,\kcl\) smooth
classical off-shell backgrounds, and \(\kq\) defined as follows.

The \emph{kinetic operator} of the fluctuation at background \(\qb\) is
the Euler derivative of the part of
\(\Sgr[\qb+\kappa u]+S_{\mathrm{gf}}[u;\qb]\) quadratic in \(u\),
\begin{equation}
\Qop[\qb]u:=\tfrac12\,\frac{d}{ds}\Edens[\qb+su]\Big|_{s=0}+\Fop[u;\qb],
\qquad
\Qop[\eta+\kappa\hcl]=\Pgr+\kappa\,\Qop^{(1)}[\hcl]+O(\kappa^2),
\label{eq:Q-def}
\end{equation}
where the expansion and the identification
\(\Qop^{(1)}[\hcl]u=\Gtwo(\hcl,u)+\Fop^{(1)}[\hcl]u\) are
Lemma~\ref{lem:fluct}.

The \emph{switched source} is
\begin{equation}
\Src:=\tfrac12\,g\,\Th-\sig\,\Qop^{(1)}[\hcl]\hq
-\tfrac12\,\sig\,:\!\Gtwo[\hq,\hq]\!:\ ,
\qquad \Th^{\mu\nu}=\,:\!T^{\mu\nu}[\field]\!:\ ,
\label{eq:source}
\end{equation}
and the \emph{second-order quantum metric} is its retarded response,
\begin{equation}
\kq:=\DR\Src=-4\,\Ptr\ERzero\Src .
\label{eq:khat}
\end{equation}
\end{definition}

Two prescriptions are part of the definition. First, the
\emph{background-field prescription}: the terms of the fluctuation
equation that do not depend on \(u\), the Einstein tensor density of
the off-shell background, are dropped (Lemma~\ref{lem:fluct},
Remark~\ref{rem:tadpole}). Second, the \emph{switch}: the order-\(\kappa\)
gravitational and gauge-fixing sources are multiplied by X-D's ramp
\(\sig\), so that \(\Src\) is past-compact and \(\DR\Src\) is defined
by the retarded propagator alone, in the same place where X-A's
coordinate equation \(\Box\X1^\rho=-\partial_\mu(\sig a^{\mu\rho})\)
multiplies its source. The switch acts on the source, not on a
density; the two differ by terms supported in the layer
\(0<t<\eps\), which are not used.

\subsection{The sheet}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.17\linewidth}X@{}}
\toprule
Item & Declaration\\
\midrule
Signature, scalar, ramp, \(\eps\), \(g\), \(U\), \(\ga\) &
As X-E. Separation \eqref{eq:separation}.\\
Gravity & Einstein--Hilbert, X-E's sign. The cubic vertex \(S_3\) is
used through its polarized Euler derivative \(\Gtwo(\hcl,\cdot)\)
and its quadratic part \(\tfrac12\Gtwo[\hq,\hq]\).\\
Gauge & Background de Donder, Definition~\ref{def:bgf}, Feynman
weight, through \(\kappa^1\) in \(\hcl\), quantum graviton only.
Replaces X-E's flat de Donder gauge.\\
Field content & \eqref{eq:metric}. \(\kq\) is derived,
\eqref{eq:khat}; it is not an independent field. No ghosts.\\
Transformation & Complete two-jet law on \((\hcl,\kcl)\);
\(\field,\hq\) held fixed. The same frozen action as X-E's
Theorem~\emph{Background gauge identity}.\\
Order & \(\kappa^2\hbar^0\), all trees. Target: background identity
through \(\kappa^2\) modulo \(\Ideal\).\\
Switch & \(\sig\) on the order-\(\kappa\) gravitational and
gauge-fixing sources; \(g\) on the matter source, as X-E.\\
Not opened & \(\hbar^1\); ghosts; BRST/BV with switched matter;
positivity; the classical self-response; gauge-parameter dependence;
\(\eps\to0\); X-Enter; \(\kappa^3\).\\
\bottomrule
\end{tabularx}
\end{center}

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

No state for the graviton is an input. The graviton Hadamard form
\(\mathsf H_{\mathrm{gr}}\) enters only the definition of Wick
ordering of the graviton-quadratic monomials \(\Qhat[\hq,\hq]\) and
\(\Tself\); no computed quantity, no classical value, and no step of
Theorem~\ref{thm:main} depends on it.

The switch on the gravitational source is a frozen input. It is not
derived from a switched action, and no claim is made about the layer
terms by which a switched density would differ from it.

VII-C \cite{VIIC} is audited in Section~\ref{sec:F6}. Its
assumptions are copied, not inferred. X-D's convergence theorem is
not used. The regulator is not removed.

%=====================================================================
\section{F1: second-order covariance}
\label{sec:F1}
%=====================================================================

\begin{lemma}[Naturality]
\label{lem:natural}
Let \(F\) be a diffeomorphism of \(M\) equal to the identity outside
a compact set. For all smooth Lorentzian \(q,\qb\) and smooth
symmetric \(u\),
\[
\Edens[F^*q]=F^*\Edens[q],\qquad
\Fop[F^*u;F^*\qb]=F^*\Fop[u;\qb],\qquad
\Qop[F^*\qb](F^*u)=F^*\bigl(\Qop[\qb]u\bigr),
\]
as contravariant symmetric tensor densities.
\end{lemma}

\begin{proof}
\(\sqrt{-q}R[q]\) and \(\Lgf[u;\qb]=\tfrac14\sqrt{-\qb}\,\qb_{\mu\nu}C^\mu_{\qb}[u]C^\nu_{\qb}[u]\)
are natural scalar densities: they are built from their arguments by
tensor operations and the Levi-Civita connection of \(\qb\), so
\(\omega[F^*\text{fields}]=F^*\omega[\text{fields}]\). The argument
of X-E's Proposition~\emph{Covariance of the Euler operators}
applies verbatim: differentiate
\(\int\omega[F^*(\text{fields}+s\,\delta)]=\int\omega[\text{fields}+s\,\delta]\)
at \(s=0\) for compactly supported variations \(\delta\) of \(q\),
respectively of \(u\), change variables by \(F\) on the left, and
use that the pairing of a tensor density with a tensor is invariant.
The third identity follows from the first two by differentiating the
first along \(q=\qb+su\) at \(s=0\), since \(F^*\) is linear.
\end{proof}

\begin{lemma}[Exact second-order covariance identity]
\label{lem:cov2}
For every smooth vector field \(\xi\) and smooth symmetric \(u\),
\begin{equation}
2\,\Gtwo(K\xi,u)=\Lie_\xi\Glin[u]-\Glin[\Lie_\xi u],
\qquad\text{and}\qquad \Glin[K\xi]=0,
\label{eq:cov2}
\end{equation}
where \(\Lie_\xi\) acts on \(\Glin[u]\) as on a contravariant tensor
density and on \(u\) as on a covariant tensor.
\end{lemma}

\begin{proof}
Both sides of \eqref{eq:cov2} are bidifferential expressions in
\((\xi,u)\) with constant coefficients, so the identity at a point
\(x\) involves only the germs of \(\xi\) and \(u\) at \(x\); it
suffices to prove it for compactly supported \(\xi\). Let \(F_s\) be
the flow of such a \(\xi\); \(F_s\) is the identity outside a compact
set, \(F_s^*\eta=\eta+sK\xi+O(s^2)\), and
\(\frac{d}{ds}F_s^*T|_{s=0}=\Lie_\xi T\) for every tensor density.
By Lemma~\ref{lem:natural} with \(q=\eta+\lambda u\),
\[
\Edens[\eta+sK\xi+O(s^2)+\lambda\,(u+s\Lie_\xi u+O(s^2))]
=F_s^*\Edens[\eta+\lambda u].
\]
Expand both sides by \ref{xe:einstein} with
\(\kappa h=sK\xi+\lambda u+O(s^2)\), and \(\kappa^2k\) collecting the
\(O(s^2)\) remainder of \(F_s^*\eta\) and the \(s\lambda\) term:
\[
\Glin[sK\xi+\lambda u+s\lambda\Lie_\xi u]
+\Gtwo[sK\xi+\lambda u,sK\xi+\lambda u]+O(s^2)+O(\lambda^3)
=\Glin[\lambda u]+s\Lie_\xi\bigl(\Glin[\lambda u]+\Gtwo[\lambda u,\lambda u]\bigr)+O(s^2).
\]
The \(s\lambda^0\) coefficient gives \(\Glin[K\xi]=0\). The
\(s\lambda\) coefficient gives
\(\Glin[\Lie_\xi u]+2\Gtwo(K\xi,u)=\Lie_\xi\Glin[u]\).
\end{proof}

\begin{remark}
\label{rem:xe-shortcut}
On \(\Glin[u]=0\), \eqref{eq:cov2} is the formula
\(\Glin[\Lie_\xi u]=-2\Gtwo(K\xi,u)\) quoted in X-E's closing
remark of E4. Off it the term \(\Lie_\xi\Glin[u]\) is present. For
the free gauge-fixed graviton, \(\Pgr\hq=0\) gives
\(\Glin[\hq]=\tfrac12\Ptr KC[\hq]\) by \eqref{eq:xe-gf}, which does
not vanish unless \(C[\hq]=0\); the shortcut therefore does not apply
to \(\hq\), and the exact form is used below.
\end{remark}

%=====================================================================
\section{F2: the gauge-fixed kinetic operator and the dynamical jet
law}
\label{sec:F2}
%=====================================================================

\subsection{The flat gauge-fixing derivative}

\begin{lemma}[Flat gauge-fixing derivative]
\label{lem:F0}
\begin{equation}
\Fop^{(0)}u=-\tfrac14\,\Ptr KC[u],\qquad
\tfrac12\Glin[u]+\Fop^{(0)}u=\Pgr u=-\tfrac14\Ptr\Box u,\qquad
C^\mu[K\xi]=\Box\xi^\mu .
\label{eq:F0}
\end{equation}
\end{lemma}

\begin{proof}
\(\delta S^{(0)}_{\mathrm{gf}}=\tfrac12\int C_\mu\partial_\nu(\Ptr\delta u)^{\nu\mu}
=-\tfrac12\int\partial_{(\nu}C_{\mu)}(\Ptr\delta u)^{\nu\mu}
=-\tfrac14\int(\Ptr KC)^{\nu\mu}\delta u_{\nu\mu}\),
using the symmetry of \(\Ptr\). With \eqref{eq:xe-gf},
\(\tfrac12\Glin[u]=\Ptr(-\tfrac14\Box u+\tfrac14KC[u])\), and the
\(KC\) terms cancel. Finally
\((\Ptr K\xi)^{\nu\mu}=\partial^\nu\xi^\mu+\partial^\mu\xi^\nu-\eta^{\nu\mu}\partial\cdot\xi\),
whose divergence is \(\Box\xi^\mu\).
\end{proof}

In particular \(\Pgr K\xi=\tfrac12\Glin[K\xi]+\Fop^{(0)}K\xi=-\tfrac14\Ptr K\Box\xi\),
which is X-E's Proposition~\emph{No compactly supported residual
gauge}; that proposition stands unchanged on this sheet.

\subsection{The fluctuation equation}

\begin{lemma}[Fluctuation equation and the order-\(\kappa\) kinetic
correction]
\label{lem:fluct}
With \(q=\qb+\kappa u+\kappa^2\kcl\), \(\qb=\eta+\kappa\hcl\), and
\(\Sq:=\Sgr[q]+S_{\mathrm{gf}}[u;\qb]+\Ssc[q,\phi]\),
\begin{equation}
\frac{\delta\Sq}{\delta u}
=\tfrac12\Glin[\hcl]+\Pgr u
+\kappa\Bigl(\tfrac12\Glin[\kcl]+\tfrac12\Gtwo[\hcl,\hcl]
+\Qop^{(1)}[\hcl]u+\tfrac12\Gtwo[u,u]-\tfrac12\,g\,T[\phi]\Bigr)
+O(\kappa^2),
\label{eq:fluct}
\end{equation}
\begin{equation}
\Qop^{(1)}[\hcl]u=\Gtwo(\hcl,u)+\Fop^{(1)}[\hcl]u ,
\label{eq:Q1}
\end{equation}
and \eqref{eq:Q-def} holds. For each smooth \(w\), \(\Qop^{(1)}[w]\)
is a linear differential operator of order \(\le2\) in \(u\), with
coefficients linear in \((w,\partial w,\partial^2w)\); in particular
\(\supp\bigl(\Qop^{(1)}[w]u\bigr)\subset\supp w\). Explicitly,
\(\Fop^{(1)}[\hcl]u=\delta\!\int\!\Lgf^{(1)}/\delta u\) with
\begin{equation}
\begin{aligned}
\Lgf^{(1)}[\hcl;u]&=\tfrac14\,b_1[\hcl]\,C_\mu C^\mu
+\tfrac14\,\hcl_{\mu\nu}C^\mu C^\nu
+\tfrac12\,\eta_{\mu\nu}C^\mu C^{(1)\nu}[\hcl;u],\\
C^{(1)\nu}[\hcl;u]&=\partial_\rho\beta^{\rho\nu}
+\tfrac12(\partial_\lambda H_{\mathrm{cl}})\,\bar u^{\lambda\nu}
+\Gamma_1{}^\nu_{\rho\lambda}[\hcl]\,\bar u^{\rho\lambda},\\
\beta^{\rho\nu}&=-\hcl^{\rho\alpha}u_\alpha{}^\nu-\hcl^{\nu\beta}u^\rho{}_\beta
+\tfrac12\hcl^{\rho\nu}\tr_\eta u+\tfrac12\eta^{\rho\nu}\hcl^{\alpha\beta}u_{\alpha\beta},
\end{aligned}
\label{eq:Lgf1}
\end{equation}
\(C^\mu=C^\mu[u]\), \(\bar u=\Ptr u\), \(H_{\mathrm{cl}}=\tr_\eta\hcl\),
\(\Gamma_1\) X-E's linearized Christoffel symbol.
\end{lemma}

\begin{proof}
\(\delta\Sgr/\delta u=\tfrac1{2\kappa}\Edens[q]\) by the chain rule
from \ref{xe:einstein}. With \(\kappa h=\kappa(\hcl+u)\),
\(\kappa^2k=\kappa^2\kcl\),
\(\Edens=\kappa\Glin[\hcl+u]+\kappa^2(\Glin[\kcl]+\Gtwo[\hcl+u,\hcl+u])+O(\kappa^3)\);
divide by \(2\kappa\) and polarize \(\Gtwo\). The matter term is
X-E's \(\delta\Ssc/\delta q=-\tfrac12g\sqrt{-q}T[q,\phi]\) times
\(\kappa\). The gauge-fixing term is \eqref{eq:F-expansion}. Grouping
by powers of \(\kappa\) and using \eqref{eq:F0} gives \eqref{eq:fluct}
and \eqref{eq:Q1}; the \(u\)-linear part of the right side of
\eqref{eq:fluct} is by construction \(\Qop[\qb]u\), which proves
\eqref{eq:Q-def}. For \eqref{eq:Lgf1}:
\(\sqrt{-\qb}=1+\kappa b_1[\hcl]+O(\kappa^2)\),
\(\qb^{\mu\nu}=\eta^{\mu\nu}-\kappa\hcl^{\mu\nu}+O(\kappa^2)\), so
\(\bar u^{\rho\nu}_{\qb}=\bar u^{\rho\nu}+\kappa\beta^{\rho\nu}+O(\kappa^2)\);
\(\bar\nabla_\rho\bar u^{\rho\nu}=\partial_\rho\bar u^{\rho\nu}
+\bar\Gamma^\rho_{\rho\lambda}\bar u^{\lambda\nu}+\bar\Gamma^\nu_{\rho\lambda}\bar u^{\rho\lambda}\)
with \(\bar\Gamma=\kappa\Gamma_1[\hcl]+O(\kappa^2)\) and
\(\Gamma_1{}^\rho_{\rho\lambda}=\tfrac12\partial_\lambda H_{\mathrm{cl}}\).
Insert into \(\tfrac14\sqrt{-\qb}\,\qb_{\mu\nu}C^\mu_{\qb}C^\nu_{\qb}\).
\(\Gtwo(\hcl,u)\) is the polarization of a bidifferential expression
of total order two with constant coefficients; \(\Lgf^{(1)}\) is
quadratic in \((u,\partial u)\) with coefficients linear in
\((\hcl,\partial\hcl)\), so its Euler derivative is of order \(\le2\)
in \(u\) with coefficients linear in \((\hcl,\partial\hcl,\partial^2\hcl)\).
\end{proof}

The terms \(\tfrac12\Glin[\hcl]+\tfrac\kappa2(\Glin[\kcl]+\Gtwo[\hcl,\hcl])\)
of \eqref{eq:fluct} are the Einstein tensor density of the off-shell
background. The background-field prescription of
Definition~\ref{def:content} drops them. The remaining equation,
with \(u=\hq+\kappa\kq\) and \(\Pgr\hq=0\), reads at order \(\kappa\)
\[
\Pgr\kq=-\Qop^{(1)}[\hcl]\hq-\tfrac12\Gtwo[\hq,\hq]+\tfrac12gT[\phi],
\]
whose right side, with \(\phi\to\field\), Wick ordered, and switched
by \(\sig\) on the gravitational terms, is \(\Src\) of
\eqref{eq:source}. The definition \(\kq=\DR\Src\) is therefore the
first-order retarded M\o ller correction of the metric slot, as X-E's
\(k_1\) was its matter part; \(k_1=\DR(\tfrac12g\Th)\) is the first
term of \(\kq\).

\subsection{The commutator identity}

\begin{proposition}[Key identity]
\label{prop:key}
For every smooth vector field \(\xi\) and smooth symmetric \(u\),
\begin{equation}
\Fop^{(1)}[K\xi]\,u=[\Lie_\xi,\Fop^{(0)}]\,u,\qquad
\Qop^{(1)}[K\xi]\,u=[\Lie_\xi,\Pgr]\,u:=\Lie_\xi(\Pgr u)-\Pgr(\Lie_\xi u).
\label{eq:key}
\end{equation}
In the flat gauge of X-E, \(\Fop^{(1)}\equiv0\) and
\(\Qop^{(1)}_{\mathrm{flat}}[K\xi]=\Gtwo(K\xi,\cdot)=\tfrac12[\Lie_\xi,\Glin]
=[\Lie_\xi,\Pgr]-[\Lie_\xi,\Fop^{(0)}]\).
\end{proposition}

\begin{proof}
As in Lemma~\ref{lem:cov2}, reduce to compactly supported \(\xi\)
and let \(F_s\) be its flow. Lemma~\ref{lem:natural} gives
\(\Fop[F_s^*u;F_s^*\eta]=F_s^*\Fop[u;\eta]\). Differentiate at
\(s=0\) using \eqref{eq:F-expansion} with \(\kappa\hcl\to sK\xi\):
\(\Fop^{(1)}[K\xi]u+\Fop^{(0)}\Lie_\xi u=\Lie_\xi\Fop^{(0)}u\). The
second identity follows from the first, \eqref{eq:cov2}, \eqref{eq:Q1},
and \eqref{eq:F0}; or directly from the third identity of
Lemma~\ref{lem:natural}. The flat-gauge statement is \eqref{eq:cov2}
with \eqref{eq:F0}.
\end{proof}

\begin{remark}
\label{rem:killing}
If \(\xi\) is a Killing field of \(\eta\), both sides of
\eqref{eq:key} vanish: \(K\xi=0\), and \(\Pgr\) has constant
coefficients and is Poincar\'e covariant. For \(\xi\in\ga\) neither
side vanishes in general; this is the statement, already in X-E, that
\(\hq\mapsto\hq+\kappa\Lie_\xi\hq\) is not an automorphism of the free
gauge-fixed algebra. The present sheet does not transform \(\hq\).
\end{remark}

\subsection{The dynamical jet law}

\begin{lemma}
\label{lem:DRP}
For past-compact smooth \(u\), \(\DR\Pgr u=u\). For compactly
supported \(w\) and past-compact \(S\),
\(\int w\cdot\DR S=\int\DA w\cdot S\). For a covariant \(v\) and a
density \(T\), one of them compactly supported,
\(\int v\cdot\Lie_\xi T=-\int\Lie_\xi v\cdot T\).
\end{lemma}

\begin{proof}
\(\DR\Pgr u=\Ptr\ERzero\Box\Ptr u=\Ptr\Ptr u=u\), since \(\Ptr\) is
constant and \(\ERzero\Box v=v\) for past-compact \(v\). The second
statement is the transposition of \(\ERzero\), both integrals being
over the compact set \(\supp\DA w\cap\supp S\). The third:
\(v\cdot T\) is a scalar density, \(\Lie_\xi(v\cdot T)=\partial_\lambda(\xi^\lambda v\cdot T)\)
integrates to zero, and \(\Lie_\xi\) is a derivation.
\end{proof}

\begin{proposition}[The jet law is dynamical]
\label{prop:jetlaw}
Let \(\xi\in\ga\). With \(\field\) and \(\hq\) held fixed,
\begin{equation}
\kq[\hcl+K\xi]-\kq[\hcl]=\Lie_\xi\hq-\DR\,\Lie_\xi\Pgr\hq ,
\label{eq:jetlaw}
\end{equation}
and for every compactly supported smooth symmetric \(w\),
\begin{equation}
\int w\cdot\DR\Lie_\xi\Pgr\hq=-\hq\bigl(\Pgr\Lie_\xi\DA w\bigr)\in\Ideal .
\label{eq:jetlaw-ideal}
\end{equation}
Consequently the total second-order metric \(k=\kcl+\kq\) transforms
under the frozen action of \(\Gp\) by the complete two-jet law of
X-A,
\[
k\;\longmapsto\;k+\Lie_\xi(\hcl+\hq)+K\zeta+\tfrac12\Lie_\xi K\xi ,
\]
modulo \(\Ideal\) after smearing, although \(\hq\) itself is not
transformed.
\end{proposition}

\begin{proof}
\(\Src\) depends on \(\hcl\) only through the linear term
\(-\sig\Qop^{(1)}[\hcl]\hq\), so
\(\Src[\hcl+K\xi]-\Src[\hcl]=-\sig\Qop^{(1)}[K\xi]\hq=-\Qop^{(1)}[K\xi]\hq\),
because \(\supp\Qop^{(1)}[K\xi]\hq\subset\supp\xi\subset U\subset\{\sig=1\}\)
by Lemma~\ref{lem:fluct} and \eqref{eq:separation}. By
\eqref{eq:key},
\(-\Qop^{(1)}[K\xi]\hq=\Pgr(\Lie_\xi\hq)-\Lie_\xi(\Pgr\hq)\). Both
terms are compactly supported in \(\supp\xi\). Apply \(\DR\) and
Lemma~\ref{lem:DRP}: \(\DR\Pgr\Lie_\xi\hq=\Lie_\xi\hq\). This is
\eqref{eq:jetlaw}. For \eqref{eq:jetlaw-ideal},
\(\int w\cdot\DR\Lie_\xi\Pgr\hq=\int\DA w\cdot\Lie_\xi\Pgr\hq
=-\int\Lie_\xi\DA w\cdot\Pgr\hq=-\int\Pgr(\Lie_\xi\DA w)\cdot\hq\),
by the last two statements of Lemma~\ref{lem:DRP} and the formal
self-adjointness of \(\Pgr\); \(\Lie_\xi\DA w\) is smooth and
supported in \(\supp\xi\), so \(\hq(\Pgr\Lie_\xi\DA w)\) is a
generator of \(\Ideal\). The last statement combines
\eqref{eq:jetlaw} with the frozen law
\(\kcl\mapsto\kcl+\Lie_\xi\hcl+K\zeta+\tfrac12\Lie_\xi K\xi\).
\end{proof}

The tensor transformation law of the quantum fluctuation under
background diffeomorphisms, which X-E's closing remark of E4 assigned
to a second-order quantum metric ``transforming by \(\Lie_\xi\hq\)'',
is a consequence of \(\Qop^{(1)}[K\xi]=[\Lie_\xi,\Pgr]\) and
\(\DR\Pgr=1\) on compact supports. No field is postulated.

%=====================================================================
\section{F3: three trees}
\label{sec:F3}
%=====================================================================

\subsection{Definition}

\begin{definition}[The three trees and the all-tree observable]
\label{def:trees}
\begin{equation}
\begin{aligned}
\Tmat&:=\,:\!\Ahat1\bigl[\DR(\tfrac12g\,\Th)\bigr]\!:\ ,\\
\Tmix&:=\,:\!\Ahat1\bigl[\DR(-\sig\,\Qop^{(1)}[\hcl]\hq)\bigr]\!:\ ,\\
\Tself&:=\,:\!\Ahat1\bigl[\DR(-\tfrac12\sig\,\Gtwo[\hq,\hq])\bigr]\!:\ ,
\end{aligned}
\label{eq:trees}
\end{equation}
so that \(:\!\Ahat1[\kq]\!:=\Tmat+\Tmix+\Tself\). The all-tree
observable is
\begin{equation}
\PhiF(f)=\field(f)+\kappa\Ahat1[h]
+\kappa^2\Bigl(\Qhat[h,h]+\Ahat1[\kcl]+\Tmat+\Tmix+\Tself\Bigr),
\qquad h=\hcl+\hq .
\label{eq:all-tree}
\end{equation}
\end{definition}

\(\Tmat\) is X-E's tree with the prefactor \(\kappa^2\) removed; X-E's
display of the one-tree observable carried \(\kappa^2\) both inside
the tree and outside the bracket, and \eqref{eq:all-tree} is the
intended object. \(\Tmix\) is of degree \((1,1)\) in \((\field,\hq)\);
with zero cross blocks its Wick ordering is the plain product.
\(\Tself\) is of degree \((1,2)\). Lemma~\ref{lem:affine} of X-E
(the affine slot) is what makes \eqref{eq:all-tree} the completed
observable of X-C at the metric \eqref{eq:metric} through
\(\kappa^2\): \(d_2[h,\kcl+\kq]=Q[h,h]+d_1[\kcl]+d_1[\kq]\).

\subsection{Kernels and supports}

Write \(D_1\) for the kernel of \(d_1\) \ref{xe:tests}, with
\(\supp D_1\subset\Kf\times(J^-(\Kf)\cap\{t\ge0\})\), and set
\begin{equation}
N_0(x,y):=\int D_1(x,z)\,\DR(z,y)\dd z,\qquad
\DR(z,y)=-4\,\Ptr\,\ERzero(z,y),
\label{eq:N0}
\end{equation}
indices suppressed; the \(y\)-slot of \(N_0\) is covariant, paired
with sources. Let \(\mathfrak B(\partial_y,\partial_{y'})\) be the
constant-coefficient bidifferential operator with
\(\Gtwo[u,u](y)=\mathfrak B(\partial_y,\partial_{y'})[u(y)u(y')]|_{y'=y}\),
and \(\mathfrak T\) the analogous operator with
\(T[\phi]=\mathfrak T[\phi(y)\phi(y')]|_{y'=y}\). For a differential
operator \(D\) in \(y\), \(D^\dagger\) is its formal transpose.

\begin{lemma}[Kernels and supports]
\label{lem:kernels}
\(N_0\) is a well-defined distribution on \(M\times M\), and
\begin{align}
\Tmat&=\iiint\Tker_{\mathrm{mat}}(x;y,y')\,:\!\field(x)\field(y)\field(y')\!:\ ,
&\Tker_{\mathrm{mat}}&=\tfrac12\,\mathfrak T(-\partial_y,-\partial_{y'})
\bigl[g(y)N_0(x,y)\,\delta(y-y')\bigr],
\nonumber\\
\Tmix&=\iint\Mker(x,y)\,\field(x)\hq(y),
&\Mker(x,y)&=-\Bigl(\Qop^{(1)}[\hcl]^{\dagger}_{y}\bigl[\sig N_0(x,\cdot)\bigr]\Bigr)(y),
\label{eq:kernels}\\
\Tself&=\iiint\Tker_{\mathrm{self}}(x;y,y')\,:\!\field(x)\hq(y)\hq(y')\!:\ ,
&\Tker_{\mathrm{self}}&=-\tfrac12\,\mathfrak B(-\partial_y,-\partial_{y'})
\bigl[\sig(y)N_0(x,y)\,\delta(y-y')\bigr].
\nonumber
\end{align}
All three kernels are compactly supported, with
\[
\supp\subset\Kf\times\bigl(J^-(\Kf)\cap\{t\ge0\}\bigr)^{\,n-1},
\qquad n=3,2,3,
\]
and \(y=y'\) on the supports of \(\Tker_{\mathrm{mat}}\),
\(\Tker_{\mathrm{self}}\). The coefficients of \(\Mker\) depend on
\(\hcl\) through \((\hcl,\partial\hcl,\partial^2\hcl)\) on
\(J^-(\Kf)\cap\{t\ge0\}\) only, linearly.
\end{lemma}

\begin{proof}
\(D_1\) has compact support and \(z\) ranges over the compact set
\(J^-(\Kf)\cap\{t\ge0\}\); the composition \eqref{eq:N0} exists by
Lemma~\ref{lem:sumzero}(d) below. For each tree,
\(\Ahat1[\DR S]=\int\field(x)\int D_1(x,z)(\DR S)(z)\dd z\dd x
=\iint\field(x)N_0(x,y)S(y)\dd y\dd x\), and \(S\) is read off from
\eqref{eq:trees}: for \(\Tmat\), \(S=\tfrac12g\Th\) and
\(\int\psi\,\Th=\iint\mathfrak T(-\partial_y,-\partial_{y'})[\psi(y)\delta(y-y')]\,:\!\field(y)\field(y')\!:\);
for \(\Tself\) likewise with \(\mathfrak B\); for \(\Tmix\),
\(\int\psi\,\sig\Qop^{(1)}\hq=\int(\Qop^{(1)\dagger}[\sig\psi])\hq\).
Supports: \(x\in\Kf\) from \(D_1\); \(y\in J^-(z)\) from \(\DR\),
and \(g(y)\neq0\) or \(\sig(y)\neq0\) forces \(t_y\ge0\), so
\(y\in J^-(\Kf)\cap\{t\ge0\}\), compact in Minkowski space;
\(\delta(y-y')\) forces \(y'=y\). Differential operators do not
enlarge supports.
\end{proof}

\subsection{Sum-zero calculus}

For \(n\ge2\) let
\[
\Sigma_n:=\Bigl\{(x_1,\xi_1;\dots;x_n,\xi_n)\in T^*M^n\setminus0:\ \sum_{i=1}^n\xi_i=0\Bigr\}.
\]

\begin{lemma}[Sum-zero calculus]
\label{lem:sumzero}
\begin{enumerate}[label=\textup{(\alph*)}]
\item \(\WF(\ERzero)\subset\Sigma_2\); \(\WF(\DR)\subset\Sigma_2\);
\(\WF(\delta(y-y'))\subset\Sigma_2\).
\item If \(\WF(u)\subset\Sigma_n\), then
\(\WF(Du)\subset\Sigma_n\) for every differential operator \(D\) with
smooth coefficients on \(M^n\), and \(\WF(\psi u)\subset\Sigma_n\) for
smooth \(\psi\).
\item If \(u\in\Dspace'(M^2)\) with \(\WF(u)\subset\Sigma_2\), then the
product \(v(x,y,y'):=u(x,y)\,\delta(y-y')\) exists and
\(\WF(v)\subset\Sigma_3\).
\item If \(A\in\Dspace'(X\times Z)\) is compactly supported with
\(\WF(A)\subset\Sigma_2\), and \(B\in\Dspace'(Z\times Y)\) with
\(\WF(B)\subset\Sigma_2\), then the composition
\(A\circ B(x,y)=\int A(x,z)B(z,y)\dd z\) exists and
\(\WF(A\circ B)\subset\Sigma_2\).
\item \(\Sigma_n\cap\bigl(\overline V_+^{\,n}\cup\overline V_-^{\,n}\bigr)=\emptyset\).
\end{enumerate}
\end{lemma}

\begin{proof}
(a) By \ref{xe:algebra}, every element of \(\WF(\ERzero)\) has the
form \((x,\xi;y,-\xi)\), \(\xi\neq0\); \(\DR\) is a constant matrix
times \(\ERzero\). \(\delta(y-y')=(2\pi)^{-4}\int e^{i(y-y')\cdot\eta}\dd\eta\)
has \(\WF=\{(y,\eta;y,-\eta):\eta\neq0\}\).
(b) \(\WF(Du)\subset\WF(u)\) and \(\WF(\psi u)\subset\WF(u)\).
(c) \(\WF(u\otimes1_{y'})\subset\{(x,\xi;y,\eta;y',0):(x,\xi;y,\eta)\in\WF(u)\}\)
and \(\WF(1_x\otimes\delta)\subset\{(x,0;y,\eta;y',-\eta):\eta\neq0\}\)
\cite[Thm.~8.2.9]{Hor}. A sum of an element of the first set and an
element of the second is \((\xi,\eta_1+\eta_2,-\eta_2)\) with
\(\xi+\eta_1=0\); it vanishes only if \(\xi=\eta_1=\eta_2=0\), which
is excluded. So the product exists \cite[Thm.~8.2.10]{Hor}, and its
wavefront set is contained in the set of such sums with one summand
possibly zero; every such sum has total covector
\(\xi+\eta_1+\eta_2-\eta_2=0\).
(d) With \(\WF'(A)=\{(x,\xi;z,-\zeta):(x,\xi;z,\zeta)\in\WF(A)\}=\{(x,\xi;z,\xi)\}\)
and likewise for \(B\), the exceptional sets
\(\WF_X(A)=\{(x,\xi):(x,\xi;z,0)\in\WF(A)\}\) and
\(\WF_Z(A),\WF_Z(B),\WF_Y(B)\) are empty, because a zero second
covector forces a zero first covector. Since \(A\) has compact
support, \cite[Thm.~8.2.14]{Hor} applies and gives
\(\WF'(A\circ B)\subset\WF'(A)\circ\WF'(B)=\{(x,\xi;y,\xi)\}\),
i.e.\ \(\WF(A\circ B)\subset\Sigma_2\).
(e) \(\overline V_+\) is a closed convex cone with
\(\overline V_+\cap(-\overline V_+)=\{0\}\). If \(\xi_i\in\overline V_+\)
and \(\sum\xi_i=0\), then \(\xi_1=-\sum_{i\ge2}\xi_i\in\overline V_+\cap(-\overline V_+)\),
so \(\xi_1=0\); inductively all \(\xi_i=0\), contradicting
\((\xi_i)\neq0\). Same for \(\overline V_-\).
\end{proof}

\subsection{Admission}

\begin{lemma}[Admission of the three trees]
\label{lem:admission}
\(\WF(D_1)\subset\Sigma_2\), \(\WF(N_0)\subset\Sigma_2\),
\(\WF(\Mker)\subset\Sigma_2\), and
\(\WF(\Tker_{\mathrm{mat}}),\WF(\Tker_{\mathrm{self}})\subset\Sigma_3\).
Hence each of the three kernels of Lemma~\ref{lem:kernels} is compactly
supported and satisfies \eqref{eq:microcausal}, and
\(\Tmat,\Tmix,\Tself\in\Wick\) at the fixed \(\eps\).
\end{lemma}

\begin{proof}
The local terms of \(D_1\) are differential operators with smooth
compactly supported coefficients (\(f\), \(g\), \(\EA f\)) applied to
\(\delta(x-z)\); the nonlocal term \(\Lambda\) is a derivative of
\(\ERzero(x,z)\) times the smooth function \(\sig(z)\). Both lie in
\(\Sigma_2\) by Lemma~\ref{lem:sumzero}(a),(b). \(N_0=D_1\circ\DR\)
lies in \(\Sigma_2\) by (d). \(\Mker\) is a differential operator
with smooth coefficients applied to \(\sig N_0\), so (b). The two
cubic kernels are differential operators with constant coefficients
applied to \((g N_0)\,\delta(y-y')\), respectively
\((\sig N_0)\,\delta(y-y')\), so (b) and (c). Compact supports are
Lemma~\ref{lem:kernels}; \eqref{eq:microcausal} is (e). The
admission criterion is \ref{xe:algebra}.
\end{proof}

X-E's admission lemma for its single tree is the \(n=2\) case of the
same argument, with \(\Th\) treated as a composite field; the
three-slot case needed for \(\Tself\) is what X-E named and did not
construct. The sum-zero form is sharper than X-E's chain argument and
needs no case distinction between diagonal and off-diagonal parts of
\(\WF(\ERzero)\).

\subsection{Classical values}

\begin{proposition}[Classical values]
\label{prop:classical}
Let \(\phi_0\), \(h_0\) be smooth solutions of \(\Pm\phi_0=0\),
\(\Pgr h_0=0\), and evaluate the underlying functionals at
\(\field\to\phi_0\), \(\hq\to h_0\). Then \(\PhiF(f)\) evaluates to
X-C's completed classical observable at the metric
\[
\eta+\kappa(\hcl+h_0)+\kappa^2\bigl(\kcl+k^{\mathrm{cl}}_1\bigr),\qquad
k^{\mathrm{cl}}_1=\DR\Bigl(\tfrac12gT[\phi_0]-\sig\Qop^{(1)}[\hcl]h_0
-\tfrac12\sig\Gtwo[h_0,h_0]\Bigr),
\]
through \(\kappa^2\); \(k^{\mathrm{cl}}_1\) is the classical first-order
retarded response of the gauge-fixed fluctuation to the switched
sources. In particular \(\Tmat\) evaluates to X-E's classical tree,
\(\Tmix\) to \(A_1[\DR(-\sig\Qop^{(1)}[\hcl]h_0)]\), and \(\Tself\) to
\(A_1[\DR(-\tfrac12\sig\Gtwo[h_0,h_0])]\), with \(A_1[k]=\pair{d_1[k]}{\phi_0}\)
X-A's first-order relational scalar.
\end{proposition}

\begin{proof}
Evaluation of a Wick-ordered element at a smooth configuration is
evaluation of the underlying polynomial functional. \(d_1\) is linear,
so \(\Ahat1[\kcl]+\Tmat+\Tmix+\Tself\) evaluates to
\(A_1[\kcl+k^{\mathrm{cl}}_1]\), and \(\Qhat[h,h]\) to
\(\pair{Q[\hcl+h_0,\hcl+h_0]}{\phi_0}\). By the affine-slot lemma this
is the \(\kappa^2\) coefficient of X-C's classical observable at the
stated metric.
\end{proof}

\begin{remark}[The classical self-response]
\label{rem:tadpole}
The background-field prescription drops from \eqref{eq:fluct} the
\(u\)-independent terms, whose retarded response would be
\(\DR(-\tfrac12\sig\Glin[\hcl])\) at order \(\kappa^0\) and
\(\DR(-\tfrac12\sig(\Glin[\kcl]+\Gtwo[\hcl,\hcl]))\) at order
\(\kappa\). Including them would make the quantum fluctuation carry a
classical, field-independent shift, and under
\(\hcl\mapsto\hcl+K\xi\) the second would change by
\(\DR(-\Qop^{(1)}[K\xi]\hcl)=\Lie_\xi\hcl-\DR\Lie_\xi\Pgr\hcl\) up to
the gauge-fixing part, where \(\Pgr\hcl\neq0\) off shell is a
c-number: the frozen two-jet law for \(\kcl\) would acquire a second
copy of \(\Lie_\xi\hcl\). The prescription keeps \(\kcl\) off shell
and arbitrary, as in X-A. The classical self-response is named and
not opened.
\end{remark}

%=====================================================================
\section{F4: the background identity through \(\kappa^2\)}
\label{sec:F4}
%=====================================================================

\begin{lemma}[Graviton slot of the equation ideal]
\label{lem:gslot}
Let \(v\in\Dspace'(M\times M)\) be compactly supported with
\(\WF(v)\subset\Sigma_2\), covariant in the second slot. Then the
degree-\((1,1)\) monomial with kernel \((1\otimes\Pgr)v\),
\[
\iint\bigl((1\otimes\Pgr_y)v\bigr)(x,y)\,\field(x)\hq(y)\dd x\dd y ,
\]
is an element of \(\Wick\) and lies in \(\Ideal\). Every element of
\(\Ideal\) vanishes as a functional on all pairs \((\phi_0,h_0)\) of
smooth free solutions.
\end{lemma}

\begin{proof}
\((1\otimes\Pgr)v\) is compactly supported with wavefront set in
\(\Sigma_2\) by Lemma~\ref{lem:sumzero}(b), hence admitted by
\eqref{eq:microcausal} and Lemma~\ref{lem:sumzero}(e). For a finite
sum \(v=\sum_i\psi_i\otimes w_i\) of smooth compactly supported
tensors, the monomial is \(\sum_i\field(\psi_i)\star\hq(\Pgr w_i)\),
because the cross blocks of the Hadamard matrix vanish and no
contraction joins the two factors; each \(\hq(\Pgr w_i)\) is a
generator of the two-sided ideal \(\Ideal\). A general \(v\) is a
limit of such finite sums in H\"ormander's topology on distributions
with wavefront set in a fixed closed cone avoiding
\(\overline V_\pm^{\,2}\) \cite[Thm.~8.2.3]{Hor}; \(1\otimes\Pgr\) is
continuous in that topology, the map from admitted kernels to Wick
monomials is continuous by X-B, and \(\Ideal\) is closed. The last
statement is X-E's Lemma~\emph{Two facts about the equation ideal},
part (ii), which is proved there for both generators.
\end{proof}

\begin{theorem}[Background identity of the all-tree observable]
\label{thm:main}
Let \(\xi,\zeta\in\ga\), and let the classical backgrounds transform
by the complete two-jet law
\[
\hcl'=\hcl+K\xi,\qquad
\kcl'=\kcl+\Lie_\xi\hcl+K\zeta+\tfrac12\Lie_\xi K\xi ,
\]
with \(\field\) and \(\hq\) held fixed. Then
\begin{equation}
\PhiF(f)[\hcl',\kcl']-\PhiF(f)[\hcl,\kcl]\equiv0
\pmod{\Ideal,\ \kappa^3}.
\label{eq:main}
\end{equation}
Term by term: the two-jet part of \eqref{eq:all-tree} varies by
\(-\kappa^2\Ahat1[\Lie_\xi\hq]\); \(\Tmat\) and \(\Tself\) are
invariant; \(\Tmix\) varies by
\(+\kappa^2\Ahat1[\Lie_\xi\hq]-\kappa^2\Ahat1[\DR\Lie_\xi\Pgr\hq]\),
and the last term lies in \(\Ideal\).
\end{theorem}

\begin{proof}
The variation of
\(\field(f)+\kappa\Ahat1[h]+\kappa^2(\Qhat[h,h]+\Ahat1[\kcl])\),
\(h=\hcl+\hq\), is \eqref{eq:xe-defect}: \(-\kappa^2\Ahat1[\Lie_\xi\hq]\)
modulo \(\Ideal\) and \(\kappa^3\). \(\Tmat\) depends on
\((f,g,\eps,\field)\) only and \(\Tself\) on \((f,\eps,\field,\hq)\)
only; neither depends on \((\hcl,\kcl)\). \(\Tmix\) depends on
\(\hcl\) linearly through \(\Qop^{(1)}[\hcl]\) and not on \(\kcl\);
since \(\Ahat1\) and \(\DR\) are linear,
\[
\Tmix[\hcl']-\Tmix[\hcl]
=\Ahat1\bigl[\DR(-\sig\Qop^{(1)}[K\xi]\hq)\bigr]
=\Ahat1\bigl[\kq[\hcl+K\xi]-\kq[\hcl]\bigr]
=\Ahat1[\Lie_\xi\hq]-\Ahat1[\DR\Lie_\xi\Pgr\hq]
\]
by \eqref{eq:jetlaw}. For the last term,
\(d_1[\DR\Lie_\xi\Pgr\hq](x)=\int N_0(x,y)(\Lie_\xi\Pgr\hq)(y)\dd y
=-\int\Pgr_y\bigl[\Lie_\xi N_0(x,\cdot)\bigr](y)\,\hq(y)\dd y\)
by Lemma~\ref{lem:DRP}, so \(\Ahat1[\DR\Lie_\xi\Pgr\hq]\) is the
degree-\((1,1)\) monomial with kernel \((1\otimes\Pgr)v\),
\(v:=-(1\otimes\Lie_\xi)N_0\). The coefficients of \(\Lie_\xi\) are
supported in \(\supp\xi\subset U\), so \(v\) is compactly supported
(in \(\Kf\times\supp\xi\)), and \(\WF(v)\subset\Sigma_2\) by
Lemma~\ref{lem:admission} and Lemma~\ref{lem:sumzero}(b).
Lemma~\ref{lem:gslot} gives \(\Ahat1[\DR\Lie_\xi\Pgr\hq]\in\Ideal\).
Summing, the \(\kappa^2\) variation of \(\PhiF(f)\) is
\(-\Ahat1[\Lie_\xi\hq]+\Ahat1[\Lie_\xi\hq]\equiv0\).
\end{proof}

\begin{corollary}[X-E's defect is a truncation defect]
\label{cor:truncation}
X-E's one-tree observable in the flat gauge is \eqref{eq:all-tree}
without \(\Tmix\) and \(\Tself\). Its \(\kappa^2\) defect
\(-\kappa^2\Ahat1[\Lie_\xi\hq]\) is the negative of the variation of
the omitted mixed tree, modulo \(\Ideal\). The defect is independent
of \((\hcl,\kcl)\) because \(\Tmix\) vanishes identically at
\(\hcl=0\) while its first variation does not.
\end{corollary}

\begin{remark}[Structural route, named]
\label{rem:structural}
A second proof of Theorem~\ref{thm:main} would run: the switched
gauge-fixed system is exactly \(\Gp\)-covariant when
\((\hcl,\kcl,u,\phi)\) all pull back (Lemma~\ref{lem:natural} and
X-E's Lemma~\emph{Exact plateau invariance}, with
\(\sig\circ F^{-1}=\sig\) since \(\sig=1\) on \(U\)); retarded
uniqueness identifies the M\o ller solution at the transformed
background with the pull-back of the M\o ller solution, both having
the same free data on \(t<2\delta\); X-C's off-shell invariance of
the classical completed observable then gives invariance of the
classical value on every solution pair; and the \(\hbar^0\) part of
the retarded products is the classical M\o ller composition
\cite{DF}. This would need the converse of Lemma~\ref{lem:gslot}'s
last statement (vanishing on all solution pairs implies membership in
\(\Ideal\)) and would hold to all orders in \(\kappa\) at
\(\hbar^0\). Neither ingredient is proved here; the direct proof is
the one released.
\end{remark}

%=====================================================================
\section{F5: the flat-gauge residual}
\label{sec:F5}
%=====================================================================

\begin{proposition}[Flat-gauge residual]
\label{prop:flat}
Replace Definition~\ref{def:bgf} by X-E's flat gauge
\(S^{(0)}_{\mathrm{gf}}[u]\) at every background, so that
\(\Fop^{(1)}\equiv0\), \(\Qop^{(1)}_{\mathrm{flat}}[\hcl]=\Gtwo(\hcl,\cdot)\),
and define \(\Tmix^{\mathrm{flat}}\), \(\PhiF_{\mathrm{flat}}\) by
\eqref{eq:trees}, \eqref{eq:all-tree} with this \(\Qop^{(1)}\). Under
the frozen transformation of Theorem~\ref{thm:main},
\begin{equation}
\PhiF_{\mathrm{flat}}(f)[\hcl',\kcl']-\PhiF_{\mathrm{flat}}(f)[\hcl,\kcl]
\equiv\kappa^2\,\Ahat1\bigl[\DR\,[\Lie_\xi,\Fop^{(0)}]\,\hq\bigr]
\pmod{\Ideal,\ \kappa^3},\qquad \Fop^{(0)}=-\tfrac14\Ptr KC .
\label{eq:flat-residual}
\end{equation}
\end{proposition}

\begin{proof}
As in Theorem~\ref{thm:main}, with Proposition~\ref{prop:key}'s
flat-gauge statement:
\(-\Qop^{(1)}_{\mathrm{flat}}[K\xi]\hq=-[\Lie_\xi,\Pgr]\hq+[\Lie_\xi,\Fop^{(0)}]\hq\).
The first term gives \(\Lie_\xi\hq\) modulo \(\Ideal\) after \(\DR\)
and \(\Ahat1\), cancelling \eqref{eq:xe-defect}; the second is the
residual. It is admitted: \([\Lie_\xi,\Fop^{(0)}]\hq\) is compactly
supported in \(\supp\xi\), and the kernel is
\(-(1\otimes[\Lie_\xi,\Fop^{(0)}]^\dagger)N_0\), in \(\Sigma_2\).
\end{proof}

The class of \eqref{eq:flat-residual} modulo \(\Ideal\) is not
decided on this sheet. What is known:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \([\Lie_\xi,\Fop^{(0)}]\hq=-\tfrac14\Lie_\xi\bigl(\Ptr KC[\hq]\bigr)+\tfrac14\Ptr KC[\Lie_\xi\hq]\).
On a configuration with \(C[h_0]=0\) the first term vanishes and
\(\DR\) of the second is \(-K\ERzero C[\Lie_\xi h_0]\), by
\(\DR\Ptr Kv=-4\ERzero Kv=-4K\ERzero v\): a pure-gauge configuration
supported in \(J^+(\supp\xi)\), not compactly supported. X-D's
representative \(d_1[K\xi]=\Pm(\xi\cdot\partial\EA f)\) is proved for
\(\xi\in\ga\) and does not apply to it without an extension off the
plateau, which is not frozen.
\item On a configuration with \(C[h_0]\neq0\) the first term
contributes a constraint term. X-E's witness pair
\(h=x^1\delta^2\otimes\delta^2\) has \(C^\mu[h]=-\tfrac12\delta^\mu_1\)
and evaluates \(\Ahat1[\Lie_\xi\hq]\), a different functional; it
does not decide \eqref{eq:flat-residual}.
\item If the class is nonzero, the \(\kappa^2\) background identity
is gauge-fixing dependent at this order in a non-covariant gauge,
which is the expected behaviour of a background-field identity and
the reason the background gauge is frozen. If it is zero, the flat
gauge would also reach ending (1). Either way
Theorem~\ref{thm:main} is unaffected.
\end{enumerate}

%=====================================================================
\section{F6: VII-C audit with three trees}
\label{sec:F6}
%=====================================================================

X-E's audit covered one tree. The present sheet has three, one of
them graviton-quadratic. The assumption of VII-C is copied as in X-E
and not repeated; the match table is redone.

\begin{center}
\small
\renewcommand{\arraystretch}{1.14}
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.38\linewidth}X@{}}
\toprule
Item & VII-C requires & X-F constructs\\
\midrule
(E1) &
The de Donder multiplet with \(H_{hh}=P\Delta_0^+\) as the fixed
Wick matrix &
The flat gauge-fixed graviton with \(\mathsf H_{\mathrm{gr}}=-4\Ptr\otimes\mathsf H_0\),
opposite signature, used to define Wick ordering of \(\Qhat[\hq,\hq]\)
and \(\Tself\). The background gauge enters through the interaction
\(\Qop^{(1)}[\hcl]\), not through the free algebra. No computed
quantity uses \(\mathsf H_{\mathrm{gr}}\).\\
(E2) &
Admission of the sharp zero-Cauchy insertion &
Ramp insertions at fixed \(\eps\): \(D_1\), \(\Mker\),
\(\Tker_{\mathrm{mat}}\), \(\Tker_{\mathrm{self}}\), all admitted by
the sum-zero lemma. No sharp limit.\\
(E3) &
Normal-topology convergence of mollified kernels &
No approximating family; the regulator is fixed. The only limit used
is the density of finite tensor sums in H\"ormander's topology,
Lemma~\ref{lem:gslot}, at fixed \(\eps\).\\
(E4) &
Hypocontinuous product; extraction of the \(\hbar^3\) coefficient of
\(B_f\star W_g\star B_{f'}\) &
Three \(\hbar^0\) trees. Each is one \(\star\)-contraction along one
graviton line between \(\Ahat1[\hq]\) and an order-\(\kappa\) vertex
linear in that \(\hq\): the matter vertex \(V_{1,g}\), the
background vertex \(\int\sig\,\hq\,\Qop^{(1)}[\hcl]\hq\) (linear in
each of its two \(\hq\)), and the cubic vertex. No \(W_g\), no
\(\hbar^3\) coefficient, no loop.\\
\bottomrule
\end{tabularx}
\end{center}

VII-C does not apply and is not contradicted. The first object on
which it could bear is the mixed loop of X-E's named list, which is
not constructed here.

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

\begin{diagnostic}[Dilation check of the key identity]
\label{diag:dilation}
Let \(\xi^\mu=x^\mu\), so \(K\xi=2\eta\), and \(\Lie_\xi u=(x\cdot\partial+2)u\)
on covariant two-tensors, \(\Lie_\xi T=(x\cdot\partial+2)T\) on
contravariant densities of weight one. Then
\([\Lie_\xi,\Pgr]u=[x\cdot\partial,\Pgr]u=-2\Pgr u\), since \(\Box\)
is homogeneous of degree \(-2\) and \(\Ptr\) is constant; so
\eqref{eq:key} predicts \(\Qop^{(1)}[\eta]=-\Pgr\). Directly:
\(\qb=(1+\kappa)\eta\), and for a constant \(c>0\),
\(\sqrt{-cq'}R[cq']=c\sqrt{-q'}R[q']\), so the part of
\(\Sgr[c\eta+\kappa u]\) quadratic in \(u\) is \(c^{-1}\) times the
flat one; \(C_{c\eta}[u]=c^{-2}C[u]\) and
\(\sqrt{-c\eta}\,c\eta_{\mu\nu}=c^3\eta_{\mu\nu}\), so
\(S_{\mathrm{gf}}[u;c\eta]=c^{-1}S^{(0)}_{\mathrm{gf}}[u]\). Hence
\(\Qop[c\eta]=c^{-1}\Pgr\) and the \(\kappa\)-coefficient at
\(c=1+\kappa\) is \(-\Pgr\). Likewise \eqref{eq:cov2} predicts
\(\Gtwo(\eta,u)=-\tfrac12\Glin[u]\), and
\(\Edens[c\,q']=\Edens[q']\) gives the same. Both agree.
\end{diagnostic}

\begin{diagnostic}[Pure gauge with the graviton held fixed, rerun]
\label{diag:puregauge}
Let \(\xi\in\ga\), \(\zeta=0\), \(\hcl=K\xi\), \(\kcl=\tfrac12\Lie_\xi K\xi\).
By \eqref{eq:jetlaw} at \(\hcl=0\),
\(\Tmix=\Ahat1[\Lie_\xi\hq]-\Ahat1[\DR\Lie_\xi\Pgr\hq]\equiv\Ahat1[\Lie_\xi\hq]\)
modulo \(\Ideal\), and on X-E's witness pair \(h_0=x^1\delta^2\otimes\delta^2\),
\(\phi_0=tx^1\), \(\xi=\chi\partial_1\), its classical value is
\(2\int f\,\partial_1\partial_t(\ERzero)^2\chi\neq0\) for suitable
\(f,g,\delta\) \ref{xe:ward}. The mixed tree is nonzero modulo
\(\Ideal\) on a pure-gauge background, and it is exactly the term
whose absence X-E's Diagnostic~\emph{Pure gauge with the graviton
held fixed} recorded as the order boundary. With it,
\(\PhiF(f)[K\xi,\tfrac12\Lie_\xi K\xi]\equiv\PhiF(f)[0,0]\) modulo
\(\Ideal\) and \(\kappa^3\), by Theorem~\ref{thm:main}.
\end{diagnostic}

\begin{diagnostic}[Sourced tree number, unchanged]
\label{diag:sourced}
At \(\hcl=0\), \(\Tmix=0\) identically. On \(\field\to t\), \(\hq\to0\),
\(m=0\), with \(g=g_0(t)\) near the dependence region of \(\Kf\),
\(\Tself\) and \(\Tmix\) vanish and \(\Tmat\) gives X-E's number
\(\int f\int_0^t(1-g_0(s))G_2(s)\dd s\dd^4x\). X-E's Diagnostic~\emph{Sourced
tree number} is therefore the complete classical \(\kappa^2\hbar^0\)
tree value at trivial background and vanishing graviton.
\end{diagnostic}

\begin{diagnostic}[Killing fields]
\label{diag:killing}
For a Poincar\'e generator \(\xi\) (not in \(\ga\)),
\(\Qop^{(1)}[K\xi]=0\) and \([\Lie_\xi,\Pgr]=0\); \eqref{eq:key} is
\(0=0\). For \(\xi\in\ga\), \(K\xi\neq0\) unless \(\xi=0\), and
\(\Qop^{(1)}[K\xi]\hq\neq0\) in general; Diagnostic~\ref{diag:puregauge}
exhibits a case where its retarded response is not in \(\Ideal\).
\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
F1 covariance &
\(2\Gtwo(K\xi,u)=\Lie_\xi\Glin[u]-\Glin[\Lie_\xi u]\), from
naturality; corrects X-E's on-shell shortcut &
Proved\\
F2 gauge &
Background de Donder; \(\Fop^{(0)}=-\tfrac14\Ptr KC\);
\(\Pgr=\tfrac12\Glin+\Fop^{(0)}\); \(\Qop^{(1)}[K\xi]=[\Lie_\xi,\Pgr]\) &
Frozen; proved\\
F2 jet law &
\(\kq[\hcl+K\xi]-\kq[\hcl]=\Lie_\xi\hq-\DR\Lie_\xi\Pgr\hq\), second
term in \(\Ideal\) &
Proved\\
F3 trees &
\(\Tmat,\Tmix,\Tself\); kernels, supports in
\(\Kf\times(J^-(\Kf)\cap\{t\ge0\})^{n-1}\); sum-zero admission;
classical values &
Constructed at \(\hbar^0\)\\
F4 identity &
All-tree observable invariant through \(\kappa^2\) modulo \(\Ideal\),
\(\field,\hq\) fixed &
Proved\\
F5 flat gauge &
Residual \(\kappa^2\Ahat1[\DR[\Lie_\xi,\Fop^{(0)}]\hq]\) displayed &
Class not decided\\
F6 &
Three lines, \(\hbar^0\), no \(W_g\), no \(\hbar^3\) &
VII-C inapplicable, not contradicted\\
Named &
\(\hbar^1\) terms; ghosts; BV with switched matter; positivity;
classical self-response; \(\kappa^3\); \(\eps\to0\) &
Not constructed\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Fixed-regulator X-F release]
\label{thm:release}
For the frozen sheet of Section~\ref{sec:frozen}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The Einstein tensor density satisfies the exact second-order
covariance identity \eqref{eq:cov2}. The background de Donder
gauge-fixing derivative satisfies \(\Fop^{(1)}[K\xi]=[\Lie_\xi,\Fop^{(0)}]\),
and the kinetic operator of the fluctuation satisfies
\(\Qop[\eta+\kappa\hcl]=\Pgr+\kappa\Qop^{(1)}[\hcl]+O(\kappa^2)\) with
\(\Qop^{(1)}[K\xi]=[\Lie_\xi,\Pgr]\).
\item The second-order quantum metric \(\kq=\DR\Src\) of
Definition~\ref{def:content} transforms under the frozen action of
\(\ga\) by \eqref{eq:jetlaw}; the total second-order metric
\(\kcl+\kq\) obeys X-A's complete two-jet law modulo the equation
ideal, with the graviton held fixed.
\item The three trees of Definition~\ref{def:trees} are elements of
X-B's fixed-regulator multiplet algebra, with compactly supported
kernels of wavefront set in \(\Sigma_n\), and their classical values
are X-A's first-order relational scalar at the classical first-order
retarded response of the gauge-fixed fluctuation.
\item The all-tree observable \eqref{eq:all-tree} is invariant under
the frozen algebra, acting on the classical backgrounds by the
complete two-jet law with \(\field,\hq\) fixed, through \(\kappa^2\)
modulo \(\Ideal\). The \(\kappa^2\) defect of X-E's one-tree
observable is the negative of the variation of the omitted mixed
tree.
\item In X-E's flat gauge the corresponding variation is
\eqref{eq:flat-residual}; its class is not decided.
\end{enumerate}
VII-C does not apply and is not contradicted. The \(\hbar^1\) terms,
ghosts, the BV master equation with switched matter, positivity, the
classical self-response, the order \(\kappa^3\), and the sharp limit
are named and not constructed.
\end{theorem}

\begin{proof}
(i) is Lemmas~\ref{lem:natural}, \ref{lem:cov2}, \ref{lem:F0},
\ref{lem:fluct} and Proposition~\ref{prop:key}.
(ii) is Proposition~\ref{prop:jetlaw}.
(iii) is Lemmas~\ref{lem:kernels}, \ref{lem:sumzero},
\ref{lem:admission} and Proposition~\ref{prop:classical}.
(iv) is Theorem~\ref{thm:main} and Corollary~\ref{cor:truncation}.
(v) is Proposition~\ref{prop:flat}. Section~\ref{sec:F6} audits
VII-C.
\end{proof}

The theorem concerns a fixed regulator, fixed supports separated
from \(\Sigma=\{t=0\}\), fixed classical backgrounds, and the order
\(\kappa^2\hbar^0\). It supplies no statement about \(\hbar^1\), where
the mixed scalar--graviton bubble, the graviton and ghost one-loop
one-point functions in the background \(\hcl\), the switched scalar
loop, and the \(\hbar\)-corrections to the Wick ordering of
\(\Qhat[\hq,\hq]\) and \(\Tself\) all appear at \(\kappa^2\), and
where the gauge-fixed action with a dynamical ghost is not invariant
under the switched matter coupling: \(sS\) acquires terms supported
in \(\supp\dd g\), which meets \(J^-(\Kf)\). That is the boundary of
the next sheet, recorded here and not opened.

%=====================================================================
\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, affine slot, and fixed-regulator identity are from
\texttt{X-C-v3.tex} and \texttt{X-E.tex}. The explicit equation-ideal
representatives are from \texttt{X-D-v3.tex}. The Einstein--Hilbert
sign, the Palatini variation, the flat gauge-fixed operator, the
graviton algebra, the one-tree kernel, the background identity with
its defect, and the witness are from \texttt{X-E.tex}.
\texttt{VII-C.tex} is used only for the copied assumption.

\paragraph{New proof.}
The exact second-order covariance identity; the background de Donder
gauge and its order-\(\kappa\) density; the decomposition
\(\Pgr=\tfrac12\Glin+\Fop^{(0)}\); the commutator identity
\(\Qop^{(1)}[K\xi]=[\Lie_\xi,\Pgr]\); the dynamical jet law; the
switched source and the M\o ller-defined second-order quantum metric;
the three tree kernels and their supports; the sum-zero calculus and
three-slot admission; the graviton-slot ideal lemma; the all-tree
background identity through \(\kappa^2\); the flat-gauge residual.

\paragraph{Corrections to X-E.}
Listed in the header. Two change the sheet: the flat de Donder gauge
is replaced by the background de Donder gauge, and ``one tree'' is
replaced by three. One changes a sentence of X-E's Section~1: a sheet
that freezes the \(\kappa^2\) quantum-metric identity on X-E's field
content reaches ending (1), not (2), once all \(\kappa^2\hbar^0\)
trees are included in the background gauge; X-E's witness is correct
for the one-tree observable and is a truncation witness. X-E's
Proposition~\emph{No compactly supported residual gauge} and its
closing remark that \(\hq\mapsto\hq+\kappa\Lie_\xi\hq\) is not an
automorphism stand; the tensor law is carried by \(\kq\), not by an
automorphism of the free algebra.

\paragraph{Cross-reference corrigendum.}
The X-E item numbers quoted in the header and in
Section~\ref{sec:inherited} were taken from a working copy of X-E in
which E4 was numbered as Section~4. In the compiled \texttt{X-E.tex}
the items are: completed tests (2.4), not (2.5); the
\(\Edens\)-expansion (4.3), not (4.6); Palatini Lemma 4.2, not 4.1;
vertex identity Lemma 4.3, not 4.2; \(\delta S_3/\delta h\) in
Section~7.5, not 4.6; gauge-fixed operator Lemma 6.4, not 6.3;
one-tree observable (7.3), not (4.17); kernel \(D_1\) (7.4), not
(4.8); admission Lemma 7.3, not 4.13; two facts about the ideal
Lemma 7.6, not 4.17; background gauge identity Theorem 7.7, not 4.18;
witness Proposition 7.9, not 4.21. All other quoted numbers agree. In
the body of this manuscript X-E items are cited by title.

\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{XE}
N.~Cheung.
\newblock \emph{Landmark X-E: A specified gravitational action for
the ramp relational scalar, and the first graviton tree}.
\newblock Supplied manuscript \texttt{X-E.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.

\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{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner.
\newblock Quantum gravity from the point of view of locally covariant
quantum field theory.
\newblock \emph{Comm.\ Math.\ Phys.} 345 (2016), 741--779.
\newblock Named for the whole-Lagrangian switch; not used.

\bibitem{DF}
M.~D\"utsch and K.~Fredenhagen.
\newblock The master Ward identity and generalized Schwinger--Dyson
equation in classical field theory.
\newblock \emph{Comm.\ Math.\ Phys.} 243 (2003), 275--314.
\newblock Cited in Remark~\ref{rem:structural} only.

\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. Theorems~8.2.3, 8.2.9, 8.2.10, 8.2.14.

\bibitem{LL}
L.~D.~Landau and E.~M.~Lifshitz.
\newblock \emph{The Classical Theory of Fields}, 4th ed.
\newblock Pergamon, 1975.

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

\end{thebibliography}

\end{document}