% !TeX program = pdflatex
%=====================================================================
% XI-A.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XI-A.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XI-A.tex
%
% Mathematical corrections to the proposed production sheet:
% 1. A quadratic Fierz--Pauli action is not the nonlinear
%    Einstein--Hilbert action through coupling order kappa^2.
%    The classical cubic and quartic coefficients cannot be deleted
%    by an unspecified graph-counting argument.
% 2. De Donder gauge fixing breaks the diffeomorphism Noether identity.
%    The invariant and gauge-fixed Euler operators are distinguished.
% 3. Minimal scalar coupling produces no derivative of g in the
%    metric Euler equation. Such derivatives occur in the scalar
%    equation and in divergences of the switched stress.
% 4. The nonlinear Noether identity contains the coupling normalization
%    and the metric-dependent adjoint gauge operator.
% 5. Metric Euler derivatives are tensor densities, not collections
%    of scalar densities.
% 6. A fixed-input Einstein jet can be empty. The unrestricted
%    collection of jets is not empty, since it contains the vacuum.
%
% No propagator, free quantum multiplet, ghost, antifield, tree,
% BRST class, quantum product, or sharp-ramp limit is constructed.
%
% No successful compilation, mechanized proof verification, independent
% review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=27mm]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{booktabs,array,tabularx}
\usepackage{enumitem}
\usepackage{microtype}
\usepackage[hidelinks]{hyperref}

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

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

\newcommand{\R}{\mathbb R}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\Lie}{\mathcal L}
\newcommand{\eps}{\varepsilon}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\Gp}{\mathcal G_U}
\newcommand{\Lag}{\mathfrak L}
\newcommand{\Adens}{\mathfrak A}
\newcommand{\EH}{\mathrm{EH}}
\newcommand{\FP}{\mathrm{FP}}
\newcommand{\gf}{\mathrm{gf}}
\newcommand{\inv}{\mathrm{inv}}
\newcommand{\dens}{\mathrm{dens}}
\newcommand{\TT}{\mathrm{TT}}
\newcommand{\Ephi}{E_\phi}
\newcommand{\Egamma}{e}
\newcommand{\Hcal}{\mathcal H}
\newcommand{\Ncal}{\mathcal N}
\newcommand{\Bcal}{\mathcal B}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Obs}{\mathcal O}
\newcommand{\Coord}{\mathsf X}
\newcommand{\Pfree}{P_m}
\newcommand{\Em}{\mathsf E_m}
\newcommand{\formal}{\pmod{\kappa^3}}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\id}{id}

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

\title{\textbf{Landmark XI-A}\\[0.4em]
A switched Einstein--scalar action for the ramp relational scalar:\\
sign conventions, Noether identities,\\
and an empty fixed-input Einstein jet}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We specify a classical Einstein--scalar model compatible with the
signature, scalar switching prescription, and plateau gauge algebra
of Landmark X-A. The Einstein--Hilbert density is not switched.
Its sign is fixed by a positive transverse-traceless kinetic term
and an attractive Newtonian comparison. We display the quadratic
Fierz--Pauli action, one de Donder gauge-fixing term, and the metric
Euler derivative through coupling order $\kappa^2$.

Several distinctions are indispensable. Minimal scalar coupling
introduces the switch itself, but no switch derivatives, into the
metric Euler derivative. Switch derivatives appear in the scalar
equation and in the divergence of the metric source. Nonlinear
Noether identities require the classical higher gravitational
coefficients; the quadratic action alone is insufficient.
Gauge fixing supplies a separate breaking term and is not
diffeomorphism invariant on the plateau.

We prove the exact fixed-background Noether identity and the tensor-density
covariance of the invariant metric Euler operator on the admitted
plateau group. X-A's varied-action formula is used as a premise,
not presented as a new relational covariance theorem. Finally, for
a specified free scalar input, the ungauge-fixed coupled Einstein
jet is empty for every nonzero compact switch of the declared class.
This is not an emptiness theorem for all inputs, nor for the
gauge-fixed wave equations. The manuscript stops at the classical
sheet: no propagator, quantum multiplet, tree, or BRST class is
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) {Classical Einstein-scalar sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Newtonian sign check};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Vacuum-energy computation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The sign check is a linear diagnostic, not a cosmological-constant result.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A sign check is not the vacuum energy. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-XI-A}
\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) {Classical Einstein-scalar sheet};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Vacuum-energy computation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The sign check is a linear diagnostic, not a cosmological-constant result.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A sign check is not the vacuum energy. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-XI-A}
\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) {Classical Einstein-scalar sheet};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Newtonian sign check};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Vacuum-energy computation};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The sign check is a linear diagnostic, not a cosmological-constant result.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-XI-A}
\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) {Classical Einstein-scalar 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) {Vacuum-energy computation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Newtonian sign check. The sign check is a linear diagnostic, not a cosmological-constant result.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Classical Einstein-scalar sheet is not Vacuum-energy computation. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-XI-A}
\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: Classical Einstein-scalar sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Newtonian sign check};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Vacuum-energy computation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The sign check is a linear diagnostic, not a cosmological-constant result.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-XI-A}
\end{figure}

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


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



\scopebox{Release: a corrected classical sheet, with the on-shell
obstruction separated}{%
The signed action, quadratic operator, metric Euler coefficients,
Noether identity with its switching and gauge-fixing terms, and
plateau covariance of the \emph{invariant} metric Euler derivative
are constructed.

For the fixed free input of Theorem~\ref{thm:empty}, the coupled
\emph{Einstein} jet is empty. The unrestricted jet family is not
declared empty.

Neither invariance of the gauge-fixed action nor nonlinear covariance
of a Fierz--Pauli-only truncation is asserted. Those versions of the
proposed sheet are false. The metric remains off shell throughout
the construction of the sheet.}

\tableofcontents

%=====================================================================
\section{Scope and the corrections required for a coherent sheet}
\label{sec:scope}
%=====================================================================

The objective is to give a classical action and its identities before
constructing any quantum metric. The relational prescription of X-A
is retained, but no quantum conclusion is inferred from its classical
covariance.

A production plan cannot require both a background de Donder term
and an unbroken diffeomorphism identity for the resulting action.
It also cannot identify a quadratic gravitational action with the
nonlinear Einstein action at the next two coupling orders.
Accordingly, the model below has one declared gauge-fixed action,
together with its explicitly identified invariant part:
\[
 S_{\mathrm{tot}}=S_{\inv}+S_{\gf},
 \qquad
 S_{\inv}=S_{\EH}+S_g^{\mathrm{sc}}.
\]
The identities for these two functionals are different and are printed
separately.

Five corrections determine the release.

\begin{enumerate}
\item
The Einstein--Hilbert action is specified before expansion. Its
quadratic coefficient is Fierz--Pauli. Its cubic and quartic
coefficients are retained in the classical sheet when required by
coupling order. They are not evaluated as graphs.

\item
The scalar action contains no metric derivatives. Its metric variation
therefore contains $g$ but no $\partial g$, including at second order.
Adding such terms would change the stated minimally coupled model.

\item
The nonlinear Noether identity uses the formal adjoint of the
metric pullback action. A bare partial divergence is only its
leading flat contribution.

\item
Covariance of a metric Euler derivative means covariance as a
rank-two tensor density. Gauge fixing contributes a noncovariant
background-dependent term.

\item
An empty jet must have a specified input. The zero scalar and flat
metric provide an immediate counterexample to universal emptiness.
\end{enumerate}

The broader perturbative treatment of gravity distinguishes classical
gauge symmetry, gauge fixing, and subsequent BV quantization
\cite{BFR}. Here those distinctions are enforced without opening a
BV complex. Effective-field-theory discussions likewise do not justify
deleting the classical Einstein self-interactions
\cite{Donoghue}.

%=====================================================================
\section{X-A premises, conventions, and non-inputs}
\label{sec:premises}
%=====================================================================

\subsection{Frozen geometric data}

Use the X-A conventions:
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad
 \Sigma=\{t=0\},
\]
\[
 \eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1),
 \qquad
 \Box=\partial_t^2-\Delta,
 \qquad
 \Pfree=\Box+m^2,
 \qquad m\geq0.
\]
The mass is fixed for each instance of the model. All coefficient
fields are real and smooth on a neighbourhood of the slab.

Fix $g\in C_c^\infty(\MTo;[0,1])$ with nonempty plateau
$P_g=\Int\{g=1\}$. Fix $0<\eps<T$ and
\[
 \sig(t)=\chi(t/\eps),
 \qquad
 \chi(s)=
 \begin{cases}
 0,&s\leq0,\\[2pt]
 \displaystyle\frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
 1,&s\geq1.
 \end{cases}
\]
Choose an open relatively compact $U$ such that
\[
 \overline U\subset P_g\cap\{\eps<t<T\},
 \qquad
 \ga=C_c^\infty(U;TM).
\]
The group $\Gp$ is generated by the corresponding compactly
supported flows. Every admitted transformation is the identity near
$\Sigma$ and preserves $g$ and $\sig$.

For coefficient calculations, genuine flows of
$\kappa\xi+\kappa^2\zeta$, $\xi,\zeta\in\ga$, provide all Taylor
coefficients needed below. They lift X-A's second-order formal
transformations without enlarging its support class.

\subsection{Metric and inherited density coefficients}

The field variable is one classical symmetric tensor $\gamma$:
\begin{equation}
 q_{\mu\nu}=\eta_{\mu\nu}+\kappa\gamma_{\mu\nu}.
 \label{eq:q}
\end{equation}
There are no independent quantum fields $\widehat h,\widehat k$.
Write $\tau=\eta^{\mu\nu}\gamma_{\mu\nu}$ and set
\begin{align}
 b_1[\gamma]&=\tfrac12\tau,
 &
 b_2[\gamma]&=\tfrac18\tau^2
             -\tfrac14\gamma_{\mu\nu}\gamma^{\mu\nu},
 \label{eq:b}\\
 a_1^{\mu\nu}[\gamma]&=\tfrac12\tau\eta^{\mu\nu}
                         -\gamma^{\mu\nu},
 \label{eq:a1}\\
 a_2^{\mu\nu}[\gamma]&=
 \gamma^\mu{}_\rho\gamma^{\rho\nu}
 -\tfrac12\tau\gamma^{\mu\nu}+b_2[\gamma]\eta^{\mu\nu}.
 \label{eq:a2}
\end{align}
Thus
\[
 \sqrt{-q}=1+\kappa b_1+\kappa^2b_2+O(\kappa^3),
 \qquad
 \Adens_q^{\mu\nu}
 =\eta^{\mu\nu}+\kappa a_1^{\mu\nu}
       +\kappa^2a_2^{\mu\nu}+O(\kappa^3).
\]
These symbols are not coefficients of $\sqrt{-q}R[q]$.

The action is defined as a full formal generating expression before
its Euler derivative is projected through $\kappa^2$. In particular,
$\gamma$ may later be evaluated on a formal solution family.
This does not introduce another fundamental metric field.

\subsection{Inherited relational prescription}

X-A supplies the coordinate prescription
\[
 \partial_\alpha\left(
  \bigl[\eta^{\alpha\beta}
  +\sig(\Adens_q^{\alpha\beta}-\eta^{\alpha\beta})\bigr]
       \partial_\beta\Coord_q^\mu\right)=0,
\]
with the fixed data
$\Coord_q^\mu|_\Sigma=x^\mu|_\Sigma$ and
$\partial_t\Coord_q^\mu|_\Sigma=\delta^\mu_0$.
The observable is the fixed label-space smearing of
$\phi\circ\Coord_q^{-1}$.

Its plateau covariance and X-A's Route II are premises
\cite{XA}. They are not reproved here. The ramp does not enter the
gravitational action.

X-B, X-C, and X-D are not inputs. VI and VII-A are comparison
manuscripts with different signature and metric prescriptions.
VII-C, X-Enter, and the cone papers supply no premise. No quantum
cancellation from another relational prescription is imported
\cite{Frob}.

\subsection{What coupling order requires}

The factor $\kappa\Coord_1[\gamma]$ has one metric leg.
This observation alone does not eliminate a cubic Einstein vertex:
that vertex has coefficient $\kappa$, so the two factors have total
coupling degree two before any additional graph requirements are
specified. No rooted-tree topology is selected or calculated here.

More importantly, an action identity is not a graph selection rule.
The nonlinear classical identities require the relevant Einstein
coefficients. The Fierz--Pauli-only alternative is explicitly audited
in Section~\ref{sec:quadraticfailure}.

%=====================================================================
\section{One action, one gauge-fixing term, and two sign checks}
\label{sec:action}
%=====================================================================

\subsection{Curvature and action}

Fix the curvature convention
\begin{equation}
 R_{\mu\nu}
 =\partial_\rho\Gamma^\rho_{\mu\nu}
  -\partial_\nu\Gamma^\rho_{\mu\rho}
  +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
  -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho},
 \label{eq:ricci}
\end{equation}
with the Levi-Civita Christoffel symbols and
$G_{\mu\nu}=R_{\mu\nu}-\frac12q_{\mu\nu}R$.
Use
\begin{equation}
 S_{\EH}[q]=-\frac{2}{\kappa^2}
                 \int_{\MT}\sqrt{-q}\,R[q]\dd^4x.
 \label{eq:EH}
\end{equation}
There is no cosmological constant and no gravitational switch.

The unchanged X-A scalar density is
\begin{align}
 \Lag_q[\phi]&=\tfrac12\sqrt{-q}
       \left(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
                    -m^2\phi^2\right),\\
 \Lag_\eta[\phi]&=\tfrac12
       \left(\partial_\mu\phi\partial^\mu\phi-m^2\phi^2\right),\\
 S_g^{\mathrm{sc}}[q,\phi]
 &=\int_{\MT}\bigl((1-g)\Lag_\eta[\phi]+g\Lag_q[\phi]\bigr)\dd^4x.
 \label{eq:scalar-action}
\end{align}
Actions are interpreted through compact interior variations and compact
differences. Numerical finiteness on every smooth configuration is not
assumed. No metric boundary variation is taken.

Define
\[
 \bar\gamma_{\mu\nu}
 =\gamma_{\mu\nu}-\tfrac12\eta_{\mu\nu}\tau,
 \qquad
 D^\nu(\gamma)=\partial_\mu\bar\gamma^{\mu\nu}.
\]
The single frozen gauge-fixing term is
\begin{equation}
 S_{\gf}[\gamma]=\int_{\MT}D_\nu(\gamma)D^\nu(\gamma)\dd^4x.
 \label{eq:gf}
\end{equation}
Its coefficient has been fixed; no gauge-parameter family is used.

\subsection{Explicit quadratic action}

\begin{proposition}[Quadratic gravitational coefficient]
\label{prop:FP}
Modulo a total divergence, the quadratic bulk coefficient of
\eqref{eq:EH} is
\begin{align}
 S_{\FP}[\gamma]=\frac12\int_{\MT}\Bigl(
 &\partial_\lambda\gamma_{\mu\nu}
          \partial^\lambda\gamma^{\mu\nu}
 -2(\partial_\mu\gamma^{\mu\nu})
          (\partial^\rho\gamma_{\rho\nu})
 \nonumber\\
 &+2(\partial_\mu\gamma^{\mu\nu})\partial_\nu\tau
 -\partial_\lambda\tau\,\partial^\lambda\tau
 \Bigr)\dd^4x.
 \label{eq:FP}
\end{align}
Its Euler derivative is $2G_{\mathrm{lin}}^{\mu\nu}(\gamma)$, where
\begin{equation}
 G_{\mathrm{lin}}^{\mu\nu}(\gamma)
 =-\tfrac12\Box\bar\gamma^{\mu\nu}
  +\partial^{(\mu}D^{\nu)}
  -\tfrac12\eta^{\mu\nu}\partial_\rho D^\rho.
 \label{eq:Glin}
\end{equation}
\end{proposition}

\begin{proof}
Expanding \eqref{eq:ricci} to first order gives
\eqref{eq:Glin}. Variation with respect to the covariant metric gives
\[
 \delta S_{\EH}
 =\frac{2}{\kappa^2}\int\sqrt{-q}\,
             G^{\mu\nu}(q)\delta q_{\mu\nu}\dd^4x.
\]
Consequently the quadratic coefficient has derivative
$2G_{\mathrm{lin}}(\gamma)$. Integration by parts in \eqref{eq:FP}
gives that same derivative. Both quadratic functionals vanish at
$\gamma=0$; integrating their equal directional derivatives along
$s\gamma$, $0\leq s\leq1$, proves equality as bulk variational
functionals. The linear Einstein--Hilbert coefficient is a
divergence and has no bulk Euler derivative.
\end{proof}

\begin{lemma}[Kinetic sign]
\label{lem:TT}
For $\gamma_{0\mu}=0$, $\partial_i\gamma_{ij}=0$, and
$\delta^{ij}\gamma_{ij}=0$, the quadratic density becomes
\[
 \Lag_{\FP}^{\TT}
 =\tfrac12\dot\gamma_{ij}\dot\gamma_{ij}
  -\tfrac12\partial_k\gamma_{ij}\partial_k\gamma_{ij}.
\]
Its time-derivative term has the same sign as
$\frac12\dot\phi^2$.
\end{lemma}

\begin{proof}
The trace and divergence terms in \eqref{eq:FP} vanish.
Raising both spatial indices contributes two minus signs, so
$\gamma^{ij}=\gamma_{ij}$. The stated expression follows.
\end{proof}

\begin{lemma}[Newtonian sign comparison]
\label{lem:Newton}
Let $H=q-\eta$ denote the physical metric perturbation. For an
auxiliary static conserved source
$T_{00}=\rho\geq0$, $T_{0i}=T_{ij}=0$, $\rho$ smooth compactly
supported and nonzero, the linear unswitched equation is solved by
\[
 H_{00}=2\Phi_N,\qquad
 H_{ij}=2\Phi_N\delta_{ij},\qquad H_{0i}=0,
\]
where
\begin{equation}
 \Delta\Phi_N=\frac{\kappa^2}{8}\rho,\qquad
 \Phi_N(x)=-\frac{\kappa^2}{32\pi}
       \int_{\R^3}\frac{\rho(y)}{|x-y|}\dd^3y<0.
 \label{eq:Newton}
\end{equation}
Thus $\kappa^2=32\pi G_N$ gives the attractive Newtonian sign.
\end{lemma}

\begin{proof}
The scalar metric variation computed below is negative, so the
unswitched metric equation is $G_{\mu\nu}=\kappa^2T_{\mu\nu}/4$.
For the displayed perturbation,
$\bar H_{00}=4\Phi_N$, $\bar H_{ij}=0$, and $D(H)=0$.
Equation \eqref{eq:Glin} gives
$G_{\mathrm{lin},00}(H)=2\Delta\Phi_N$ and zero spatial components.
The Newton inverse of $\Delta$ gives \eqref{eq:Newton}.
\end{proof}

This static calculation is a sign diagnostic, not a solution of the
compact-time switched model and not an adiabatic-limit construction.
The canonically normalized field is $\gamma=H/\kappa$; confusing
$\gamma_{00}$ with $H_{00}$ would change the normalization check.

%=====================================================================
\section{Metric Euler derivative through coupling order two}
\label{sec:euler}
%=====================================================================

\subsection{Stress tensor and the first vertex}

Put $p_\mu=\partial_\mu\phi$ and define
\[
 \ell_q=\tfrac12(q^{\alpha\beta}p_\alpha p_\beta-m^2\phi^2),
 \qquad
 T_q^{\mu\nu}
 =q^{\mu\alpha}q^{\nu\beta}p_\alpha p_\beta-q^{\mu\nu}\ell_q.
\]
Write $T_0=T_\eta$ and $\ell_0=\Lag_\eta$.

\begin{lemma}[Stress sign and inherited vertex]
\label{lem:vertex}
For a covariant metric variation,
\[
 \delta_q\Lag_q=-\tfrac12\sqrt{-q}\,
                    T_q^{\mu\nu}\delta q_{\mu\nu}.
\]
In particular,
\begin{equation}
 V_{1,g}
 =\tfrac12\int g(a_1^{\mu\nu}p_\mu p_\nu-m^2b_1\phi^2)\dd^4x
 =-\tfrac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}\dd^4x.
 \label{eq:vertex}
\end{equation}
\end{lemma}

\begin{proof}
Use
$\delta\sqrt{-q}=\frac12\sqrt{-q}\,q^{\mu\nu}\delta q_{\mu\nu}$
and
$\delta q^{\alpha\beta}=-q^{\alpha\mu}q^{\beta\nu}\delta q_{\mu\nu}$.
For the second assertion,
\[
 a_1^{\mu\nu}p_\mu p_\nu-m^2b_1\phi^2
 =-\gamma^{\mu\nu}p_\mu p_\nu+\tau\ell_0
 =-\gamma_{\mu\nu}T_0^{\mu\nu}.
\]
\end{proof}

The second matter coefficient is fixed, not guessed:
\begin{equation}
 V_{2,g}=\tfrac12\int
     g(a_2^{\mu\nu}p_\mu p_\nu-m^2b_2\phi^2)\dd^4x.
 \label{eq:V2}
\end{equation}
One switch multiplies this coefficient; replacing it by $g^2$
would change X-A's prescription.

\subsection{Exact metric derivative and its expansion}

Define the Einstein tensor-density coefficients by
\begin{equation}
 \sqrt{-q}\,G^{\mu\nu}(q)
 =\kappa\Hcal_1^{\mu\nu}
  +\kappa^2\Hcal_2^{\mu\nu}
  +\kappa^3\Hcal_3^{\mu\nu}+O(\kappa^4).
 \label{eq:Hdef}
\end{equation}
Here $\Hcal_1=G_{\mathrm{lin}}(\gamma)$.
Appendix~\ref{app:coefficients} gives finite explicit formulas for
all three coefficients, without an external curvature convention.

Let
\[
 \sqrt{-q}\,T_q^{\mu\nu}
 =T_0^{\mu\nu}+\kappa J_1^{\mu\nu}+O(\kappa^2).
\]
Direct expansion gives
\begin{align}
 J_1^{\mu\nu}[\gamma,\phi]
 ={}&b_1T_0^{\mu\nu}
 -\gamma^{\mu\alpha}p_\alpha p^\nu
 -\gamma^{\nu\alpha}p_\alpha p^\mu
 \nonumber\\
 &+\gamma^{\mu\nu}\ell_0
 +\tfrac12\eta^{\mu\nu}\gamma^{\alpha\beta}p_\alpha p_\beta.
 \label{eq:J1}
\end{align}

\begin{theorem}[Metric Euler coefficients]
\label{thm:euler}
With
$\Egamma^{\mu\nu}=\delta S_{\inv}/\delta\gamma_{\mu\nu}$,
the exact formal derivative and its order-two expansion are
\begin{align}
 \Egamma^{\mu\nu}
 &=\frac{2}{\kappa}\sqrt{-q}\,G^{\mu\nu}(q)
       -\frac{\kappa}{2}g\sqrt{-q}\,T_q^{\mu\nu},
 \label{eq:eexact}\\
 &=2\Hcal_1^{\mu\nu}
   +\kappa\left(2\Hcal_2^{\mu\nu}
                         -\tfrac12gT_0^{\mu\nu}\right)
   +\kappa^2\left(2\Hcal_3^{\mu\nu}
                         -\tfrac12gJ_1^{\mu\nu}\right)
   +O(\kappa^3).
 \label{eq:eexp}
\end{align}
The gauge-fixing contribution is separately
\begin{equation}
 \Egamma_{\gf}^{\mu\nu}
 =-2\partial^{(\mu}D^{\nu)}
       +\eta^{\mu\nu}\partial_\rho D^\rho,
 \qquad
 \partial_\mu\Egamma_{\gf}^{\mu\nu}=-\Box D^\nu.
 \label{eq:egf}
\end{equation}
Thus $\delta S_{\mathrm{tot}}/\delta\gamma=\Egamma+\Egamma_{\gf}$.
There are no $\partial g$ terms in \eqref{eq:eexact} or
\eqref{eq:eexp}.
\end{theorem}

\begin{proof}
Combine the Einstein variation in Proposition~\ref{prop:FP},
Lemma~\ref{lem:vertex}, and $\delta q=\kappa\delta\gamma$.
The flat scalar density is independent of $\gamma$.
Equations \eqref{eq:Hdef} and \eqref{eq:J1} give the expansion.
Varying $D_\nu D^\nu$ and integrating by parts gives
\eqref{eq:egf}; taking its divergence gives the last identity.
No integration by parts in a metric variation of the scalar density
is needed because that density has no metric derivatives.
\end{proof}

For $\phi=\phi_0+\kappa\phi_1+\cdots$, the order-one scalar source
is exactly $-\frac12gT_0[\phi_0]$. At order two, replace
$J_1[\gamma,\phi]$ in \eqref{eq:eexp} by
\[
 J_1[\gamma,\phi_0]+\dot T_0[\phi_0;\phi_1],
\]
where
\[
 \dot T_0^{\mu\nu}
 =p_0^\mu p_1^\nu+p_1^\mu p_0^\nu
  -\eta^{\mu\nu}
       (p_{0\alpha}p_1^\alpha-m^2\phi_0\phi_1).
\]
This distinguishes the metric variation of the stress density from
the correction due to the scalar solution.

\subsection{Where switch derivatives actually occur}

The scalar Euler derivative is
\begin{equation}
 \Ephi=\frac{\delta S_{\inv}}{\delta\phi}
 =-\left(\Pfree+\kappa\mathsf V_{1,g}
                  +\kappa^2\mathsf V_{2,g}\right)\phi+O(\kappa^3),
 \label{eq:Ephi}
\end{equation}
with
\begin{align}
 \mathsf V_{1,g}\phi
 &=g\left[\partial_\mu(a_1^{\mu\nu}p_\nu)+m^2b_1\phi\right]
       +(\partial_\mu g)a_1^{\mu\nu}p_\nu,\\
 \mathsf V_{2,g}\phi
 &=g\left[\partial_\mu(a_2^{\mu\nu}p_\nu)+m^2b_2\phi\right]
       +(\partial_\mu g)a_2^{\mu\nu}p_\nu.
 \label{eq:scalar-switch}
\end{align}
Also,
\[
 \partial_\mu(gT_0^{\mu\nu})
 =g(\Pfree\phi)\partial^\nu\phi
        +(\partial_\mu g)T_0^{\mu\nu}.
\]
These are the switch-derivative terms required by the stated model.
They must not be moved into its metric Euler derivative.

%=====================================================================
\section{Noether identity and the inherited Route II}
\label{sec:noether}
%=====================================================================

\subsection{Imported finite varied-action formula}

For a compactly supported diffeomorphism $F$ in the interior,
X-A's Route II gives
\begin{align}
 \delta_F S_g^{\mathrm{sc}}
 ={}&\int(1-g)
          \bigl(\Lag_\eta[F^*\phi]-\Lag_\eta[\phi]\bigr)\dd^4x
 \nonumber\\
 &-\int(\partial_\mu g)
                Z_F^\mu[\Lag_q[\phi]]\dd^4x,
 \label{eq:RouteII}
\end{align}
where
$F_{\dens}^*r-r=\partial_\mu Z_F^\mu[r]$.
For the flow of $\kappa\xi+\kappa^2\zeta$,
\[
 Z_F^\mu[r]
 =\kappa\xi^\mu r
   +\kappa^2\left(\zeta^\mu r
          +\tfrac12\xi^\mu\partial_\rho(\xi^\rho r)\right)
   +O(\kappa^3).
\]
The proof of \eqref{eq:RouteII} is not repeated.

The unswitched Einstein--Hilbert density is natural. The same
divergence lemma therefore shows
$\delta_FS_{\EH}=0$ for compact interior $F$.
Consequently \eqref{eq:RouteII} is also the finite variation of
$S_{\inv}$. There is no analogous assertion for $S_{\gf}$.

\subsection{The correct adjoint gauge operator}

For a symmetric contravariant density $u^{\mu\nu}$ define
\begin{equation}
 \Ncal_\nu(u)
 =2\partial_\mu(u^{\mu\rho}q_{\rho\nu})
            -u^{\mu\rho}\partial_\nu q_{\mu\rho}.
 \label{eq:N}
\end{equation}
For an arbitrary interior compact vector field $\xi$, the scaled
infinitesimal transformation is
\[
 \delta_\xi q=\kappa\Lie_\xi q,\qquad
 \delta_\xi\gamma=\Lie_\xi q,\qquad
 \delta_\xi\phi=\kappa X_\xi\phi.
\]
Thus $\delta_\xi\gamma=K\xi+\kappa\Lie_\xi\gamma$.

Introduce the explicit background-breaking density
\begin{equation}
 \Bcal_\nu
 =(\partial_\mu g)T_0^\mu{}_\nu
       -(1-g)(\Pfree\phi)\partial_\nu\phi
       +(\partial_\nu g)(\Lag_\eta[\phi]-\Lag_q[\phi]).
 \label{eq:B}
\end{equation}

\begin{theorem}[Fixed-background Noether identity]
\label{thm:noether}
Off shell, the invariant action satisfies
\begin{equation}
 \boxed{\quad
 \Ncal_\nu(\Egamma)
 =\kappa\left(\Ephi\,\partial_\nu\phi-\Bcal_\nu\right).
 \quad}
 \label{eq:noether}
\end{equation}
The gauge-fixed action instead satisfies
\begin{equation}
 \Ncal_\nu(\Egamma+\Egamma_{\gf})
 =\kappa\left(\Ephi\,\partial_\nu\phi-\Bcal_\nu\right)
       +\Ncal_\nu(\Egamma_{\gf}).
 \label{eq:noether-gf}
\end{equation}
On $U$, $\Bcal_\nu=0$. This removes the switching breaking,
not the gauge-fixing contribution.
\end{theorem}

\begin{proof}
By symmetry of $\Egamma$ and integration by parts,
\[
 \delta_\xi S_{\inv}
 =\int\xi^\nu
       \left[-\Ncal_\nu(\Egamma)
                    +\kappa\Ephi\,\partial_\nu\phi\right]\dd^4x.
\]
Take the infinitesimal coefficient of the imported
\eqref{eq:RouteII}. The flat scalar variation is
\[
 \delta_\xi\Lag_\eta
 =\kappa\left(
   T_0^\mu{}_\nu\partial_\mu\xi^\nu
       +\partial_\mu(\xi^\mu\Lag_\eta)\right).
\]
After integrating these two terms, the coefficient of
$\kappa\xi^\nu$ is exactly \eqref{eq:B}.
The fundamental lemma for arbitrary compact interior $\xi$ proves
\eqref{eq:noether}. Adding the separate gauge-fixing variation proves
\eqref{eq:noether-gf}. On $U$, both $1-g$ and $\partial g$ vanish.
\end{proof}

\begin{corollary}[Plateau contraction and Route II]
For $\xi\in\ga$,
\[
 \int\xi^\nu
       \left[-\Ncal_\nu(\Egamma)
                  +\kappa\Ephi\,\partial_\nu\phi\right]\dd^4x=0.
\]
This is the contraction of \eqref{eq:noether} and the infinitesimal
form of X-A's Route II. It is not a new covariance theorem for the
ramp observable.
\end{corollary}

\subsection{Coefficient and sign audit}

For a $\kappa$-independent scalar write
$\Egamma=e_0+\kappa e_1+\kappa^2e_2+\cdots$ and
\[
 Q_\nu(u)
 =2\partial_\mu(u^{\mu\rho}\gamma_{\rho\nu})
                    -u^{\mu\rho}\partial_\nu\gamma_{\mu\rho}.
\]
Then \eqref{eq:noether} gives
\begin{align}
 2\partial_\mu(e_0^{\mu\rho}\eta_{\rho\nu})&=0,\\
 2\partial_\mu(e_1^{\mu\rho}\eta_{\rho\nu})+Q_\nu(e_0)
 &=-g(\Pfree\phi)\partial_\nu\phi
                    -(\partial_\mu g)T_0^\mu{}_\nu,
 \label{eq:N1}\\
 2\partial_\mu(e_2^{\mu\rho}\eta_{\rho\nu})+Q_\nu(e_1)
 &=-(\mathsf V_{1,g}\phi)\partial_\nu\phi
                    +(\partial_\nu g)\Lag^{(1)}[\gamma,\phi],
 \label{eq:N2}
\end{align}
where
$\Lag^{(1)}
 =\frac12(a_1^{\alpha\beta}p_\alpha p_\beta-m^2b_1\phi^2)$.
These formulas expose the missing coupling and nonlinear metric terms
in a bare partial-divergence identity. The source sign in
\eqref{eq:N1} agrees with $-\frac12gT_0$ in the metric derivative.

\subsection{Why the quadratic and gauge-fixed alternatives fail}
\label{sec:quadraticfailure}

\begin{proposition}[Fierz--Pauli alone is insufficient]
Replacing $S_{\EH}$ by $S_{\FP}$ does not preserve the nonlinear
Noether identity even on the plateau.
\end{proposition}

\begin{proof}
Set $\phi=0$. For the complete Einstein action,
$\Ncal_\nu(\Egamma_{\EH})=0$.
Its order-one identity requires
\[
 2\partial_\mu(2\Hcal_2^{\mu\rho}\eta_{\rho\nu})
                  +Q_\nu(2\Hcal_1)=0.
\]
Deleting the cubic action deletes $2\Hcal_2$ but does not delete
$Q_\nu(2\Hcal_1)$. The latter is not identically zero.
For example, where $\gamma=\psi(t)\eta$,
\[
 Q_0(2G_{\mathrm{lin}}(\gamma))
                  =-6\dot\psi\,\ddot\psi.
\]
Choose a smooth compact tensor supported in $U$ which has this form
near a point where $\dot\psi\ddot\psi\neq0$.
The proposed quadratic nonlinear identity fails there.
\end{proof}

\begin{proposition}[Gauge fixing is not plateau invariant]
\label{prop:gfbad}
The fixed term \eqref{eq:gf} is not invariant under $\Gp$.
At the linear level its breaking is $-\Box D^\nu$, as in
\eqref{eq:egf}.
\end{proposition}

\begin{proof}
Take $\xi=f\partial_1$, with nonzero $f\in C_c^\infty(U)$.
Directly,
\[
 D(K\xi)^\nu=\Box\xi^\nu,
 \qquad
 S_{\gf}[K\xi]=-\int(\Box f)^2\dd^4x\neq0.
\]
A nonzero compact interior $f$ cannot satisfy $\Box f=0$, by
Cauchy uniqueness with zero past data.
Starting from $\gamma=0$, a genuine pure-gauge metric has
$\gamma'=K\xi+O(\kappa)$, so the nonzero displayed term is the
leading gauge-fixing action. The Einstein action on that flat
pullback is unchanged. Hence the discrepancy is gauge fixing,
not a switching defect.
\end{proof}

No compact plateau gauge algebra is recovered by simply demanding
$\Box\xi=0$: compact support to the future of $\Sigma$ and Cauchy
uniqueness then force $\xi=0$.

%=====================================================================
\section{Covariance of the invariant metric Euler operator}
\label{sec:covariance}
%=====================================================================

Let $F^*_{(2,0),\dens}$ denote pullback of a contravariant
rank-two density of weight one:
\begin{align}
 (F^*_{(2,0),\dens}u)^{\mu\nu}(x)
 ={}&\det DF(x)\,
 (DF(x)^{-1})^\mu{}_\alpha
 (DF(x)^{-1})^\nu{}_\beta
 \nonumber\\
 &\hspace{18mm}\times u^{\alpha\beta}(F(x)).
 \label{eq:tensorpull}
\end{align}
All admitted flows preserve orientation.

\begin{theorem}[Plateau metric Euler covariance]
\label{thm:covariance}
For $F\in\Gp$, put
$q'=F^*q$, $\phi'=F^*\phi$, and
$\gamma'=(q'-\eta)/\kappa$. Then
\begin{equation}
 \Egamma^{\mu\nu}[q',\phi']
   =(F^*_{(2,0),\dens}\Egamma[q,\phi])^{\mu\nu}.
 \label{eq:covariance}
\end{equation}
The identity is formal and holds after expansion through
$\kappa^2$, provided all coefficients needed before projection are
retained. It does not hold with $\Egamma$ replaced by
$\Egamma+\Egamma_{\gf}$ in general.
\end{theorem}

\begin{proof}
The Einstein tensor-density in \eqref{eq:eexact} is natural.
The scalar stress density $\sqrt{-q}T_q^{\mu\nu}$ is natural as
well. Since $F^*g=g$, its switched multiple transforms by
\eqref{eq:tensorpull}. This proves \eqref{eq:covariance}.
The counterexample in Proposition~\ref{prop:gfbad}, or the explicit
formula \eqref{eq:egf}, excludes the gauge-fixed assertion.
\end{proof}

If covariant indices are preferred, define
\[
 \mathsf E_{\mu\nu}^{\mathrm{gr}}
       =q_{\mu\alpha}q_{\nu\beta}\Egamma^{\alpha\beta}.
\]
It transforms as a \emph{covariant rank-two density}; its two indices
also acquire Jacobian factors. Treating it as four-by-four scalar
densities would be incorrect.

For a general compact $F$ which need not preserve $g$, the exact
metric covariance defect is
\begin{align}
 &\Egamma[q',\phi']-F^*_{(2,0),\dens}\Egamma[q,\phi]
 \nonumber\\
 &\qquad
 =-\frac{\kappa}{2}(g-F^*g)\,
                   \sqrt{-q'}\,T_{q'}[\phi'].
 \label{eq:covariance-defect}
\end{align}
Its infinitesimal form involves $X_\xi g$.
The divergence defect involves $(\partial_\mu g)T^{\mu\nu}$,
as in \eqref{eq:N1}; these two defects are related but not identical
formulas.

In particular, $\partial g$ is a source-conservation issue, not an
extra term in the minimally coupled metric Euler derivative.
The restriction to $\Gp$ removes the covariance defect. It does
not imply global solvability of the metric equation in the
switching region.

%=====================================================================
\section{The empty fixed-input Einstein jet}
\label{sec:jet}
%=====================================================================

\subsection{Definition and necessary compatibility}

\begin{definition}[Fixed-input Einstein jet]
\label{def:jet}
For a fixed free scalar $\phi_0$, let $\Jcal_g(\phi_0)$ consist of
smooth formal coefficient families
\[
 \gamma(\kappa)=\gamma_0+\kappa\gamma_1+\kappa^2\gamma_2,
 \qquad
 \phi(\kappa)=\phi_0+\kappa\phi_1+\kappa^2\phi_2,
\]
such that the coefficients through $\kappa^2$ of
$\Egamma=0$ and $\Ephi=0$ vanish. Positive-order scalar
coefficients have the X-A zero Cauchy data.
No condition is imposed on metric Cauchy data.

This is the jet of the invariant Einstein equation, not of the
gauge-fixed Euler equation. Requiring $D=0$ in addition is a
possible further restriction, not a replacement of that equation.
\end{definition}

\begin{lemma}[Necessary source conservation]
\label{lem:compatibility}
If $\Jcal_g(\phi_0)$ is nonempty, then
\begin{equation}
 (\partial_\mu g)T_0^{\mu\nu}[\phi_0]=0.
 \label{eq:compatibility}
\end{equation}
For the source-response subclass $\gamma_0=0$, this is already the
compatibility condition for
\begin{equation}
 G_{\mathrm{lin}}^{\mu\nu}(\gamma_1)
                       =\tfrac14gT_0^{\mu\nu}[\phi_0].
 \label{eq:source-jet}
\end{equation}
\end{lemma}

\begin{proof}
The linearized Bianchi identity
$\partial_\mu G_{\mathrm{lin}}^{\mu\nu}=0$ follows directly from
\eqref{eq:Glin}. Also
$\partial_\mu T_0^{\mu\nu}
  =(\Pfree\phi_0)\partial^\nu\phi_0=0$.
Taking the divergence of \eqref{eq:source-jet} proves the assertion
in that subclass.

For arbitrary $\gamma_0$, evaluate the order-one coefficient of
the exact Noether identity on a purported jet.
Since the metric Euler coefficients vanish, its left side
vanishes. Since $\Pfree\phi_0=0$, the order-zero scalar Euler
term vanishes. The order-zero value of $\Bcal_\nu$ is precisely
$(\partial_\mu g)T_0^\mu{}_\nu[\phi_0]$.
Equation \eqref{eq:noether} therefore forces
\eqref{eq:compatibility}. This argument permits arbitrary vacuum
leading metric data and does not presume a harmonic section.
\end{proof}

\subsection{A fixed input for which the jet is empty}

For the chosen mass define
\begin{equation}
 \phi_m(t,x)=
 \begin{cases}
  \sin(mt)/m,&m>0,\\
  t,&m=0.
 \end{cases}
 \label{eq:input}
\end{equation}
It is a smooth free solution for every $m\geq0$.

\begin{theorem}[Empty fixed-input Einstein jet]
\label{thm:empty}
For every nonzero compact switch of the frozen class,
\begin{equation}
 \boxed{\qquad \Jcal_g(\phi_m)=\varnothing. \qquad}
 \label{eq:empty}
\end{equation}
The obstruction occurs in the first sourced metric coefficient,
before the second-order scalar or metric coefficients can repair it.
\end{theorem}

\begin{proof}
The input has no spatial derivative. A direct computation gives
\[
 T_0^{00}[\phi_m]=\tfrac12,\qquad
 T_0^{0i}[\phi_m]=0,
\]
and
\[
 T_0^{ij}[\phi_m]=
 \begin{cases}
 \tfrac12\cos(2mt)\delta^{ij},&m>0,\\
 \tfrac12\delta^{ij},&m=0.
 \end{cases}
\]
Consequently the $\nu=0$ component of
\eqref{eq:compatibility} is
\[
 \tfrac12\partial_tg=0.
\]
A nonzero function compactly supported in the slab interior cannot
be independent of time: for each fixed spatial point it vanishes
near both time boundaries. Since the switch has a nonempty plateau,
$\partial_tg$ is not identically zero. The necessary compatibility
condition fails.
\end{proof}

The same obstruction can use compact scalar Cauchy data.
Choose a spatial cutoff equal to one on a ball containing the
initial causal shadows of $\supp g$, and prescribe
$\phi_0(0)=0$, $\dot\phi_0(0)$ equal to that cutoff.
Finite propagation and uniqueness imply that this compact-data
solution equals \eqref{eq:input} on a neighbourhood of $\supp g$.
Thus its stress gives the same witness. This uses the classical
Cauchy theory already admitted in X-A, not a gravitational propagator.

\subsection{A quantitative bump diagnostic}

Because $g$ rises from zero to one somewhere, there is an interior
open bump region on which $\partial_tg\geq c>0$.
For nonnegative $f$ supported there with $\int f\dd^4x=1$,
\begin{equation}
 \int f\,(\partial_\mu g)T_0^{\mu0}[\phi_m]\dd^4x
                  \geq\frac c2>0.
 \label{eq:bump}
\end{equation}
This is an evaluated conservation obstruction, not an Einstein
solution.

In the massless diagnostic $\phi_0=t$, take also
$\gamma=\psi\eta$, $\psi$ smooth. Then
\[
 T_0^{00}=\tfrac12,\qquad
 (\partial_\mu g)T_0^{\mu0}=\tfrac12\partial_tg,
 \qquad
 G_{\mathrm{lin}}[\psi\eta]_{\mu\nu}
 =-\partial_\mu\partial_\nu\psi+\eta_{\mu\nu}\Box\psi.
\]
Choosing a conformal test tensor does not alter the nonzero number
\eqref{eq:bump} and does not solve the coupled metric equation.
This is a check of the compatibility gate only.

\subsection{Pure gauge and limits of the emptiness statement}

For every smooth vector field,
\[
 G_{\mathrm{lin}}[K\xi]=0,\qquad D(K\xi)=\Box\xi.
\]
Substitution into \eqref{eq:Glin} verifies the first identity.
For $\xi\in\ga$ this is the required pure-gauge diagnostic.
It checks the linearization, not the existence of a sourced jet.

The unrestricted family of Einstein jets is not empty:
$\phi_0=0$ and all metric and scalar coefficients zero solve the
equations. Other inputs satisfying \eqref{eq:compatibility} are not
classified here. Necessity is not being promoted to a general
sufficiency theorem.

No second-order correction can cancel the nonzero order-one
compatibility coefficient. The stress tensor is not modified, and
the Einstein--Hilbert density is not switched. A possible removal
or controlled comparison of compact cutoffs belongs to a successor
local-coupling or adiabatic problem; no such limit is taken.

%=====================================================================
\section{Why gauge-fixed wave solvability is a different question}
\label{sec:gaugefixed}
%=====================================================================

At quadratic order the sum of \eqref{eq:FP} and \eqref{eq:gf} is
\[
 S_{\FP}+S_{\gf}
 =\int\left(
 \tfrac12\partial_\lambda\gamma_{\mu\nu}
               \partial^\lambda\gamma^{\mu\nu}
 -\tfrac14\partial_\lambda\tau\,\partial^\lambda\tau
 \right)\dd^4x,
\]
with Euler operator
\begin{equation}
 2G_{\mathrm{lin}}^{\mu\nu}+\Egamma_{\gf}^{\mu\nu}
                    =-\Box\bar\gamma^{\mu\nu}.
 \label{eq:quadraticoperator}
\end{equation}

In the source-response expansion, the gauge-fixed equation is
\[
 -\Box\bar\gamma_1^{\mu\nu}
                     =\tfrac12gT_0^{\mu\nu}[\phi_0].
\]
Componentwise smooth Cauchy theory permits such an equation with a
nonconserved smooth compact source. Taking its divergence gives
\begin{equation}
 \Box D^\nu(\gamma_1)
             =-\tfrac12(\partial_\mu g)T_0^{\mu\nu}[\phi_0].
 \label{eq:constraintbreaking}
\end{equation}
For the witness input, $D$ therefore cannot vanish identically.

Thus a gauge-fixed solution is not an Einstein solution in de Donder
gauge unless the source compatibility is restored. Gauge fixing
does not remove the Bianchi obstruction; it changes the equation
being solved. Equation \eqref{eq:quadraticoperator} specifies an
operator only. No graviton Green function, two-point function,
propagator normalization, or state is constructed.

%=====================================================================
\section{Decision table and handoff}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.54\linewidth}X@{}}
\toprule
Gate & Result\\
\midrule
Signature, curvature, coupling normalization & Frozen\\
Positive TT kinetic sign & Proved\\
Attractive Newtonian comparison & Proved\\
Quadratic action and de Donder term & Explicit\\
Metric Euler derivative through $\kappa^2$ & Proved; no
$\partial g$ in the minimal metric variation\\
Scalar and source-divergence switch derivatives & Displayed\\
Invariant nonlinear Noether identity & Proved with the correct
adjoint gauge operator and coupling factors\\
Gauge-fixed unbroken Noether identity & Proved false; breaking
term retained\\
Invariant metric Euler covariance on $\ga$ & Proved as
tensor-density covariance\\
Nonlinear Fierz--Pauli-only replacement & Proved insufficient\\
Fixed-input Einstein jet $\Jcal_g(\phi_m)$ & Empty\\
Universal emptiness for all inputs & False; vacuum counterexample\\
Gauge-fixed wave equation & Different equation; not an
Einstein-jet repair\\
Quantum metric, propagator, tree, BRST class & Not constructed\\
\bottomrule
\end{tabularx}
\end{center}

The corrected sheet reaches \textbf{Constructed}. The specified
on-shell Einstein reading reaches \textbf{Proved obstructed}.
The literal demand for a nonlinear covariant, gauge-fixed
Fierz--Pauli-only sheet cannot receive a construction theorem.

XI-B receives:
\begin{enumerate}
\item the signed quadratic operator
$2G_{\mathrm{lin}}+\Egamma_{\gf}=-\Box\bar\gamma$;
\item the explicit de Donder term and its breaking;
\item the stress-tensor vertex \eqref{eq:vertex};
\item the unswitched Einstein action and the order bookkeeping
needed for its nonlinear identities;
\item the Noether identity \eqref{eq:noether}, including its
background-switch defect;
\item the empty fixed-input Einstein fiber and its distinction
from gauge-fixed wave solvability.
\end{enumerate}

It receives no propagator, ghost, antifield, tree, or quantum state.
The algebra $\ga$ acts on classical configurations; it has not
been shown to act on a quantum graviton. Establishing a BRST
description of gauge fixing would be a later construction, not a
consequence of naming the de Donder term \cite{BFR}.

\subsection*{Conclusion}

A coherent classical sheet has been frozen without putting the
metric on shell. Its invariant and gauge-fixed derivatives are
distinguished, and its source obstruction has a named input:
\[
 \boxed{
 \text{signed classical action and identities constructed},
 \qquad
 \Jcal_g(\phi_m)=\varnothing.
 }
\]
The ramp relational prescription remains the inherited X-A object.
No quantum-gravitational observable has been constructed.

%=====================================================================
\appendix
\section{Finite formulas for the gravitational Euler coefficients}
\label{app:coefficients}
%=====================================================================

This appendix makes \eqref{eq:Hdef} an explicit coefficient
prescription. All contractions use $\eta$ unless a coefficient
$u_i$ is written.

Set
\[
 u_0^{\mu\nu}=\eta^{\mu\nu},\qquad
 u_1^{\mu\nu}=-\gamma^{\mu\nu},\qquad
 u_2^{\mu\nu}=\gamma^\mu{}_\rho\gamma^{\rho\nu}.
\]
For $n=1,2,3$ define
\begin{equation}
 \Gamma_n^\rho{}_{\mu\nu}
 =\tfrac12u_{n-1}^{\rho\sigma}
    \left(\partial_\mu\gamma_{\nu\sigma}
         +\partial_\nu\gamma_{\mu\sigma}
         -\partial_\sigma\gamma_{\mu\nu}\right).
 \label{eq:Gamma-n}
\end{equation}
These are precisely the first three coefficients of the
Christoffel symbols of $q=\eta+\kappa\gamma$.

Define, for $1\leq n\leq3$,
\begin{align}
 R_{n,\mu\nu}
 ={}&\partial_\rho\Gamma_n^\rho{}_{\mu\nu}
       -\partial_\nu\Gamma_n^\rho{}_{\mu\rho}
 \nonumber\\
 &+\sum_{\substack{a+b=n\\a,b\geq1}}
 \left(
 \Gamma_a^\rho{}_{\rho\lambda}\Gamma_b^\lambda{}_{\mu\nu}
 -\Gamma_a^\rho{}_{\nu\lambda}\Gamma_b^\lambda{}_{\mu\rho}
 \right),
 \label{eq:R-n}\\
 r_n&=\sum_{\substack{i+j=n\\i\geq0,\ j\geq1}}
                       u_i^{\mu\nu}R_{j,\mu\nu},
 \qquad r_0=0,
 \label{eq:r-n}\\
 C_{n,\mu\nu}
 &=R_{n,\mu\nu}
       -\tfrac12\eta_{\mu\nu}r_n
       -\tfrac12\gamma_{\mu\nu}r_{n-1}.
 \label{eq:C-n}
\end{align}
Only $u_0,u_1,u_2$ occur in these sums. Let $b_0=1$ and use
$b_1,b_2$ from \eqref{eq:b}. Then
\begin{equation}
 \Hcal_n^{\mu\nu}
 =
 \sum_{\substack{i+j+\ell+s=n\\
                  i,j,\ell\geq0,\ s\geq1}}
 b_i\,u_j^{\mu\alpha}u_\ell^{\nu\beta}C_{s,\alpha\beta},
 \qquad 1\leq n\leq3.
 \label{eq:H-n}
\end{equation}
Again, no index above two is required for $b_i,u_j,u_\ell$.
Equations \eqref{eq:Gamma-n}--\eqref{eq:H-n} are finite local
differential formulas, not placeholders for unknown vertices.

\begin{lemma}
The formulas in this appendix give the coefficients
$\Hcal_1,\Hcal_2,\Hcal_3$ of \eqref{eq:Hdef}.
\end{lemma}

\begin{proof}
Expand
$q^{-1}=u_0+\kappa u_1+\kappa^2u_2+\cdots$
in the exact Christoffel formula. Since $\partial q=\kappa\partial
\gamma$, the order-$n$ Christoffel coefficient is
\eqref{eq:Gamma-n}. Substitution into \eqref{eq:ricci} yields
\eqref{eq:R-n}. Contracting the Ricci expansion with the inverse
metric yields \eqref{eq:r-n}; multiplying $q_{\mu\nu}R$ gives
\eqref{eq:C-n}. Finally, multiplying by $\sqrt{-q}$ and by two
inverse metrics gives \eqref{eq:H-n}.
\end{proof}

The Einstein action has bulk expansion
\[
 S_{\EH}=S_{\FP}+\kappa S_{\EH,3}
                    +\kappa^2S_{\EH,4}+O(\kappa^3),
\]
where the higher coefficients are fixed by \eqref{eq:EH}.
Their Euler derivatives are $2\Hcal_2$ and $2\Hcal_3$.
This identifies their classical role without constructing a tree.

Because the Einstein Euler derivative contains $1/\kappa$,
its order-two coefficient involves the order-three metric curvature
coefficient. A pullback covariance check must likewise retain
the corresponding metric Taylor coefficients before projection.
Truncating the metric pullback prematurely at second order would
lose required information. Genuine compact flows avoid that
bookkeeping ambiguity.

%=====================================================================
\section{A compact audit of dependencies and non-identifications}
\label{app:audit}
%=====================================================================

The proofs use:
\begin{enumerate}
\item the explicit curvature convention and action;
\item algebraic density expansions and compact interior variations;
\item X-A's finite varied-action formula and divergence lemma;
\item the ordinary tensorial naturality of curvature and scalar stress;
\item the displayed linearized Bianchi identity;
\item classical Cauchy uniqueness and finite propagation for the
optional compact-data witness.
\end{enumerate}

They do not use an external graviton propagator, quantum Ward
identity, counterterm catalogue, or response matrix.

The following objects remain distinct:
\[
 \frac{\delta S_{\inv}}{\delta\gamma},
 \qquad
 \frac{\delta S_{\mathrm{tot}}}{\delta\gamma},
 \qquad
 G_{\mathrm{lin}}(\gamma),
 \qquad
 -\Box\bar\gamma.
\]
The first obeys the nonlinear fixed-background identity.
The second includes gauge-fixing breaking.
The third has the linearized Bianchi identity.
The fourth is the gauge-fixed quadratic operator.

Similarly, a conserved Einstein source, a switched nonconserved
source, and a source admitted by a gauge-fixed wave equation are not
interchangeable. The witness distinguishes them at a lower order
than any radiative or renormalization question.

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

\paragraph{Classical scope.}
All fields and products in this manuscript are smooth classical
coefficient fields. The metric is an independent classical
configuration variable, not a quantized field.

\paragraph{Scope correction.}
The title refers to an empty \emph{fixed-input} Einstein jet.
Gauge fixing is not claimed to preserve the invariant action's
Noether identity. The full Einstein action is specified because
nonlinear classical identities cannot be certified from the
quadratic coefficient alone.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, independently checked symbolic calculation, or
referee review is asserted. No empirical data or numerical graph
calculation is used.

\paragraph{Successor work.}
A free multiplet, gravitational propagator, ghost and antifield
sector, BRST identity, quantum state, relational product
renormalization, tree calculation, and sharp-ramp or adiabatic
limit remain separate tasks.

%=====================================================================
\begin{thebibliography}{9}
\addcontentsline{toc}{section}{References}

\bibitem{XA}
N.~Cheung,
\emph{Landmark X-A: Classical covariance of one fixed-regulator
switched relational scalar through order $\kappa^2$}.
Supplied companion manuscript.
Used only for the declared classical prescription, its Cauchy
facts, and the varied-action formula identified in the text.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
``Quantum gravity from the point of view of locally covariant
quantum field theory,''
\emph{Communications in Mathematical Physics}
\textbf{345} (2016), 741--779.
\href{https://arxiv.org/abs/1306.1058}{arXiv:1306.1058}.
Comparison framework only; no quantum construction is imported.

\bibitem{Donoghue}
J.~F.~Donoghue,
``General relativity as an effective field theory:
The leading quantum corrections,''
\emph{Physical Review D}
\textbf{50} (1994), 3874--3888.
\href{https://arxiv.org/abs/gr-qc/9405057}{arXiv:gr-qc/9405057}.
Comparison background only; the sign checks in this manuscript
are derived in the declared signature.

\bibitem{Frob}
M.~B.~Fr\"ob,
``Gauge-invariant quantum gravitational corrections to
correlation functions,''
\emph{Classical and Quantum Gravity}
\textbf{35} (2018), 055006.
\href{https://arxiv.org/abs/1710.00839}{arXiv:1710.00839}.
Comparison background only; no quantum cancellation or
coordinate Green-function prescription is used as a premise.

\end{thebibliography}

\end{document}