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% X-A only. One frozen switched classical prescription, one explicit
% regulator, one Cauchy prescription, one plateau-supported gauge
% algebra. The relational scalar is expanded through kappa^2 and its
% classical gauge covariance is proved at fixed regulator width.
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% Ending reached: (1) Constructed.
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% Not constructed: any quantum product, quantum Ward identity,
% sharp-regulator limit, boundary algebra, admission topology, or
% identification with the VII--IX or cone constructions.
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\newcommand{\Adens}{\mathfrak A}
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  \textbf{#1}\par\smallskip #2
  \end{minipage}}
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}

\title{\textbf{Landmark X-A}\\[0.4em]
Classical covariance of one fixed-regulator switched\\
relational scalar through order \(\kappa^2\)}
\author{Ngo Cheung}
\date{}

\usepackage{tikz}
\begin{document}

\maketitle

\begin{abstract}
We freeze one classical prescription. A minimally coupled scalar lives
on the formal metric jet \(q=\eta+\kappa h+\kappa^2k\); the
curved-minus-flat scalar density is multiplied by one compact switch
\(g\) before variation. Four coordinate functions are defined by the
divergence-form equation of an edge-regulated auxiliary density
\(\Aeps[q]=\eta^{-1}+\sig(\Adens_q-\eta^{-1})\), where \(\sig\) is an
explicit smooth time ramp of width \(\eps\) and \(\Adens_q\) is the
metric density of \(q\), with fixed Cauchy data on the initial surface.
The gauge algebra consists of vector fields supported in a plateau
\(U\) whose closure lies inside \(\{g=1\}\cap\{t>\eps\}\).

The regulator preflight is passed exactly: the regulated first
coordinate coefficient returns \(\xi\) on every pure gauge \(K\xi\) of
the frozen algebra. An explicit witness shows that the same identity
fails for gauge parameters entering the edge zone \(0<t<\eps\), so the
plateau condition is part of the freeze and not removable.

The relational scalar \(\Phi=\phi\circ\Co_q^{-1}\), smeared by one
fixed label-space test, is proved gauge invariant through \(\kappa^2\)
at fixed \(\eps>0\), off shell in the metric and in the scalar, by two
independent routes: covariance of the regulated coordinate operator and
of the switched Euler operator under plateau-supported transformations
together with coefficientwise Cauchy uniqueness; and an exact
varied-action identity in which every switch term is displayed and
cancels only through \(g=1\) and \(\dd g=0\) on the gauge support. The
switched scalar solution with fixed free input intertwines the same
action, so the on-shell evaluation is invariant as well. The
pure-gauge two-jet and a sourced diagnostic are checked. The linear
coordinate kernel is smooth: the regulator replaces the hypersurface
delta of the sharp prescription by a ramp term.

No quantum product, quantum Ward identity, sharp-regulator limit,
boundary algebra, admission topology, or identification with the
VII--IX or cone constructions is claimed.
\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) {Switched scalar density};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Edge-regulated coordinates};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A second prescription};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The release is this sheet only. It is not the convolution sheet.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{One frozen classical 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-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) {Switched scalar density};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A second prescription};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The release is this sheet only. It is not the convolution sheet.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{One frozen classical 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-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) {Switched scalar density};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Edge-regulated coordinates};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A second prescription};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The release is this sheet only. It is not the convolution 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-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) {Switched scalar density};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A second prescription};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Edge-regulated coordinates. The release is this sheet only. It is not the convolution sheet.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Switched scalar density is not A second prescription. 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-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: Switched scalar density};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Edge-regulated coordinates};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A second prescription};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The release is this sheet only. It is not the convolution 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-A}
\end{figure}

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


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



\scopebox{Release claim: X-A only, ending (1) Constructed}{%
For the frozen switched action, regulator, Cauchy prescription,
configuration domain, and plateau-supported gauge algebra of
Section~\ref{sec:sheet}, the declared relational scalar is gauge
invariant through order \(\kappa^2\) at fixed \(\eps>0\)
(Theorem~\ref{thm:main}). No quantum product, quantum Ward identity,
sharp-regulator limit, boundary algebra, or identification with the
VII--IX or cone constructions is claimed.}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.50\linewidth}X@{}}
\toprule
Object & Status here\\
\midrule
Convention sheet (action, switch, regulator, data, algebra) & Frozen before any calculation\\
Regulator preflight \(\Co_1[K\xi]=\xi\) on the frozen algebra & Proved exactly\\
Edge-zone failure of the same identity & Proved by an explicit witness\\
Switched scalar coefficient equations with switch derivatives & Derived and displayed\\
Regulated coordinate coefficient equations with regulator derivatives & Derived and displayed\\
Relational expansion and insertion signs & Fixed\\
Coordinate intertwiner through \(\kappa^2\) (Route~I) & Proved\\
Off-shell observable invariance through \(\kappa^2\) & Proved\\
On-shell invariance with fixed free input & Proved\\
Varied-action identity with all switch terms (Route~II) & Proved\\
Pure-gauge two-jet and sourced diagnostic & Checked\\
Smooth first-derivative kernels at fixed \(\eps\) & Proved; not labelled admitted\\
Quantum product, \(\mathsf T_2\), \(R_1\), quantum Ward identity & Not constructed\\
Sharp-regulator limit & Not taken\\
Admission topology or boundary algebra & Not constructed\\
\bottomrule
\end{tabularx}
\end{center}

\tableofcontents

%=====================================================================
\section{The question and the ending reached}
\label{sec:introduction}
%=====================================================================

A relational scalar built from Cauchy-prescribed harmonic coordinates
is a classical object before it is a quantum insertion. The IX track
admitted the sharp zero-Cauchy insertion to a boundary-sensitive
carrier; VII-B excluded the same insertion from an off-shell
microcausal polynomial domain. Neither manuscript constructed a
regulated classical coordinate problem, and neither proved covariance
of such a problem. That identity has to be proved, or obstructed,
here.

X-A therefore opens a new classical line. One switched classical
prescription, one explicit regulator, one Cauchy prescription, and one
plateau-supported gauge algebra are frozen in Section~\ref{sec:sheet}.
The relational scalar is expanded through \(\kappa^2\), and its gauge
covariance, including regulator and Cauchy data, is decided on the
frozen algebra.

\subsection{Three legal endings}

\begin{enumerate}
 \item \textbf{Constructed.} Equations, regulator, data, and
 observable identities are proved through \(\kappa^2\) at fixed
 \(\eps>0\).
 \item \textbf{Proved obstructed.} An explicit nonzero switch or
 regulator defect holds on the frozen group.
 \item \textbf{Unresolved.} The regulator is unspecified, uniqueness is
 missing, or the second-order cancellation is unfinished.
\end{enumerate}

The ending reached is (1). The preflight of
Section~\ref{sec:preflight} decides it: the frozen regulator is exact
on every pure gauge of the frozen algebra, and the full second-order
mechanism is then available. A proved defect appears only for gauge
parameters outside the frozen algebra, in the edge zone of the
regulator; it is recorded as the boundary of the freeze, not as a
theorem on the frozen group.

\subsection{Two issues settled before the long proof}

\paragraph{Linearized metric does not mean linear gauge action.}
The metric jet \(\eta+\kappa h+\kappa^2k\) is retained with its
complete second-order pullback law. The second metric coefficient
\(k\) transforms by \(\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi^2\eta\), and
this term enters the second-order coordinate coefficient. No
second-order relational expansion is combined with only the first
term of the metric transformation.

\paragraph{The regulator passes its own covariance test.}
Smoothing the dressing is not assumed to preserve the gauge identity.
The frozen regulator multiplies the deviation of the metric density
from flat by a smooth time ramp \(\sig\) inside the divergence-form
coordinate operator. For a pure gauge supported where \(\sig=1\) and
\(\dd\sig=0\), the regulated first coefficient equals the exact
first coefficient and returns \(\xi\). For a pure gauge supported in
the ramp zone it does not. Both statements are proved in
Section~\ref{sec:preflight} before any second-order formula is
written.

\subsection{The main theorem}

\begin{theorem}[X-A classical covariance]
\label{thm:main}
Fix the data of Section~\ref{sec:sheet}: the slab \(\MT\), the formal
metric jet \(q=\eta+\kappa h+\kappa^2k\), the switch \(g\), the
regulator \(\sig\), the plateau \(U\) with
\(\overline U\subset P_g\cap\{\eps<t<T\}\), the gauge group \(\Gp\),
the switched scalar action \(S_g^{\mathrm{sc}}\), the regulated
coordinate prescription of Definition~\ref{def:coordinates}, and one
label-space test \(f\in C_c^\infty(\MTo)\). Then:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item the regulated coordinate problem has a unique formal solution
 \(\Co_q=x+\kappa\Co_1+\kappa^2\Co_2\pmod{\kappa^3}\), with explicit
 zero-data Cauchy coefficients;
 \item \(\Co_1[K\xi]=\xi\) for every \(\xi\in\ga\);
 \item for every \(F\in\Gp\),
 \(\Co_{F^*q}=F^*\Co_q\pmod{\kappa^3}\), with coefficient laws
 \eqref{eq:X1law}--\eqref{eq:X2law};
 \item for every configuration \((h,k,\phi)\) and every \(F\in\Gp\),
 \[
  \Obs_{f,g,\eps}[F^*q,F^*\phi]
  =
  \Obs_{f,g,\eps}[q,\phi]\pmod{\kappa^3};
 \]
 \item the switched scalar solution with fixed free input satisfies
 \(\rho_g[F^*q;\phi_0]=F^*\rho_g[q;\phi_0]\pmod{\kappa^3}\), and
 \(\Obs_{f,g,\eps}[q,\rho_g[q;\phi_0]]\) is invariant under \(\Gp\);
 \item the varied switched action is an explicit sum of terms
 carrying a factor \(1-g\) or \(\partial g\), and vanishes
 identically on \(\Gp\);
 \item the coefficients depend on the configuration only in
 \(\mathcal D_f=J^-(\supp f)\cap\MT\), are continuous in the smooth
 topologies at fixed \(\eps\), and their first functional derivatives
 at fixed configuration are smooth compactly supported kernels.
\end{enumerate}
No quantum object is part of the statement.
\end{theorem}

The proof occupies
Sections~\ref{sec:preflight}--\ref{sec:tests}; the handoff
statements of (vii) are in Section~\ref{sec:decision}.

%=====================================================================
\section{Prerequisite sheet}
\label{sec:prereq}
%=====================================================================

These are the statements actually delivered by the earlier
manuscripts, copied in their own conventions. None is a theorem about
the regulated operator introduced below. Shared vocabulary is not a
transfer.

\paragraph{VII-A, copied.}
Signature \((-,+,+,+)\). Minimally coupled scalar on the formal metric
\(\mathbf g=\eta+\kappa h+\kappa^2k\), compactly switched scalar
action, unswitched harmonic coordinates with fixed Cauchy data.
Admitted transformations are supported strictly inside a region where
the interaction switch equals one and are the identity near the
initial surface. On that restricted algebra: exact first-order
cancellation and a second-order classical intertwiner, with all switch
derivatives retained. The proof uses covariance of the switched
density-valued Euler operator and uniqueness of the coefficientwise
Cauchy problems. The first-order plateau identity is
\[
 A_1^{(g)}[h+K\xi;\phi_0]=A_1^{(g)}[h;\phi_0],
 \qquad
 gK\xi=K\xi,\quad
 \phi_{1,g}[h+K\xi]-\phi_{1,g}[h]=X_\xi\phi_0,\quad
 w[h+K\xi]-w[h]=\xi,
\]
for \(\xi\) supported in the plateau. Outside that support,
\[
 A_1^{(g)}[h+K\xi]-A_1^{(g)}[h]
 =\mathsf E_m\mathsf Q^{(m)}_{(1-g)K\xi}\phi_0,
\]
and a higher-order coefficient cannot repair this first-order defect.
The metric equations in the named Einstein--scalar jet are
unswitched; the scalar equation is switched. A causal support bound
and a classical comparison map between switches are proved. Equality
for switches sharing only an observation plateau is not asserted. No
renormalized product, quantum Ward identity, or counterterm
separation.

\paragraph{VI, copied.}
Signature \((-,+,+,+)\). Second-order classical descent for a
specified relational Einstein--scalar jet, and a separate exact
first-order compact-switch defect for the identification
\((h,\phi_0)\sim(h+K\xi,\phi_0)\):
\[
 \mathcal A_{f,g}^{(1)}(h+K\xi,\phi_0)-\mathcal A_{f,g}^{(1)}(h,\phi_0)
 =\int f\Bigl[(g-1)X_\xi\phi_0
 +\mathsf G_{\mathrm R}\bigl((\partial_\mu g)\overline{K\xi}^{\mu\nu}
 \partial_\nu\phi_0-[P,g]X_\xi\phi_0\bigr)\Bigr]\dd^4x,
\]
with \([P,g]u=(\Box_\eta g)u+2(\partial^\mu g)\partial_\mu u\). No
correction beginning at second order removes it while the
identification and the switch remain as stated. No renormalized
response and no quantum Ward identity.

\paragraph{III, copied; a different equation space.}
Signature \((-,+,+,+)\). An unchanged free-product quotient does not
receive the Poisson-dressed symbol. Moving the scalar does not preserve
the massive free shell for general compact gauges. On a specified
coupled first-order jet with forced scalar correction the functional
descends, with a unique retarded harmonic section. A compact coupled
Ward pair is nonzero. No quantum representation.

\paragraph{VII-B and IX-A, non-transfer.}
VII-B (signature \((+,-,-,-)\)) excludes the sharp zero-Cauchy
insertion from its stated off-shell microcausal polynomial domain:
the adjoint \(L^*a=\theta C^*b+\delta(t)\mathsf B(b|_{\Sigma_0})\)
has timelike conormal wavefront covectors. IX-A (signature
\((-,+,+,+)\)) admits that same insertion in a different carrier with
boundary-trace topology \(\tau_\Sigma\), and proves that unchanged
regular approximants do not converge to it in that topology. Neither
defines a regulated classical coordinate problem, and neither
constructs a quantum product. The regulated kernel of
Lemma~\ref{lem:kernel} below is not identified with either sharp
object.

\paragraph{Not prerequisites.}
V's external \(R_{1,1}\), oscillator coefficient, and Newton--wave
cubic; II's finite contact matrix; IV's three Hilbert models;
VII-E/F and VIII-A--C conical certificates and token promises;
IX-Free-2's partial free product; IX-D-v3's conditional subtraction;
IX-E-2's \(\operatorname{Im}K\) formula; IX-E-3's conditional support
bridge. No arrow from those manuscripts into X-A is a theorem.

The relational-coordinate literature \cite{BFR,Frob} is comparison
background only. It does not supply covariance of this switched,
regulated prescription.

%=====================================================================
\section{Convention sheet}
\label{sec:sheet}
%=====================================================================

The first deliverable is a sheet. Every later identity refers to it.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.20\linewidth}X@{}}
\toprule
Item & Declaration\\
\midrule
Spacetime & \(\MT=[0,T]\times\R^3\), \(\Sigma=\{t=0\}\),
\(\MTo=(0,T)\times\R^3\); \(\eta=\diag(1,-1,-1,-1)\),
\(\Box=\partial_t^2-\Delta\)\\
Fields & Smooth real symmetric \(h_{\mu\nu},k_{\mu\nu}\) on a
neighbourhood of \(\MT\); formal real scalar
\(\phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2\); no quotient, no decay,
no field equation\\
Metric & \(q=\eta+\kappa h+\kappa^2k\) in
\(C^\infty[\kappa]/(\kappa^3)\); formal only\\
Action & Switched minimally coupled scalar
\eqref{eq:action}, mass \(m\geq0\); no gravitational action is used,
the metric is off shell\\
Switching & \(g\in C_c^\infty(\MTo;[0,1])\) multiplies the
curved-minus-flat scalar density before variation
\eqref{eq:Ag}; \(g\) does not enter the coordinate prescription\\
Regulator & \(\sig(t,\mathbf x)=\chi(t/\eps)\), \(\chi\) explicit
\eqref{eq:chi}, \(0<\eps<T\); enters only the coordinate prescription,
through the auxiliary density \eqref{eq:Aeps}\\
Gauge algebra & \(\ga=C_c^\infty(U;TM)\), \(U\) open,
relatively compact, \(\overline U\subset P_g\cap\{\eps<t<T\}\),
\(P_g=\Int\{g=1\}\); all parameters and all derivatives vanish
outside \(U\), hence near \(\Sigma\) and in the edge zone
\(\{0\leq t\leq\eps\}\)\\
Cauchy data & \(\Co^\mu|_\Sigma=x^\mu\),
\(\partial_t\Co^\mu|_\Sigma=\delta^\mu_0\);
positive-order scalar coefficients have zero data;
gauge transformations are the identity near \(\Sigma\)\\
Observable & \(\Phi_{g,\eps}=\phi\circ\Co_q^{-1}\) smeared by one
\(f\in C_c^\infty(\MTo;\R)\) in label space\\
Orders & Through \(\kappa^2\); no quantum coefficient\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Spacetime, formal ring, configurations}

Indices are raised and lowered with \(\eta\); parentheses on index
pairs include the factor \(\tfrac12\). Put
\[
 \mathbb D_2=\R[\kappa]/(\kappa^3).
\]
Equality modulo \(\kappa^3\) means coefficient equality; no numerical
remainder estimate is implied. For a covector field \(\xi_\mu\) and
vector field \(\xi^\mu=\eta^{\mu\nu}\xi_\nu\),
\[
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu
 =(\Lie_\xi\eta)_{\mu\nu},
 \qquad
 X_\xi=\xi^\mu\partial_\mu,
 \qquad
 D_\xi r=\partial_\mu(\xi^\mu r).
\]
Here \(D_\xi\) is the Lie derivative of a scalar density represented
relative to \(\dd^4x\).

The configuration space is
\[
 \Conf=\bigl\{(h,k,\phi)\bigr\},
 \qquad
 h,k\in C^\infty(S^2T^*M)\ \text{real},
 \qquad
 \phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2,\
 \phi_j\in C^\infty(M;\R).
\]
All fields are smooth on a neighbourhood of \(\MT\) and are used on
\(\MT\). The case of a \(\kappa\)-independent scalar is
\(\phi_1=\phi_2=0\).

\subsection{Metric jet and density coefficients}

Write \(h=\eta^{\mu\nu}h_{\mu\nu}\) for the trace where no confusion
can arise, and set
\begin{align}
 b_1[h]&=\tfrac12h,
 &
 b_2[h]&=\tfrac18h^2-\tfrac14h_{\alpha\beta}h^{\alpha\beta},
 \label{eq:b}\\
 a_1^{\mu\nu}[h]&=\tfrac12h\,\eta^{\mu\nu}-h^{\mu\nu},
 &
 a_2^{\mu\nu}[h]&=h^\mu{}_\alpha h^{\alpha\nu}
 -\tfrac12h\,h^{\mu\nu}+b_2[h]\eta^{\mu\nu}.
 \label{eq:a}
\end{align}
The symmetric bilinear polarizations \(a_2[h,h']\) and \(b_2[h,h']\)
are defined by \(a_2[h,h]=a_2[h]\), \(b_2[h,h]=b_2[h]\).

\begin{lemma}[Density expansion]
\label{lem:density}
With \(\sqrt{-q}=\sqrt{-\det q}\) and
\(\Adens_q^{\mu\nu}=\sqrt{-q}\,q^{\mu\nu}\),
\begin{align}
 \sqrt{-q}
 &=1+\kappa b_1[h]+\kappa^2\bigl(b_1[k]+b_2[h]\bigr)\formal,
 \label{eq:volexp}\\
 \Adens_q^{\mu\nu}
 &=\eta^{\mu\nu}+\kappa a_1^{\mu\nu}[h]
 +\kappa^2\bigl(a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h]\bigr)\formal.
 \label{eq:Aexp}
\end{align}
\end{lemma}

\begin{proof}
The inverse metric is
\(q^{\mu\nu}=\eta^{\mu\nu}-\kappa h^{\mu\nu}
+\kappa^2(h^\mu{}_\alpha h^{\alpha\nu}-k^{\mu\nu})\).
Expanding
\(\exp\bigl(\tfrac12\tr\log(\id+\kappa\eta^{-1}h+\kappa^2\eta^{-1}k)\bigr)\)
gives \eqref{eq:volexp}. Multiplication gives \eqref{eq:Aexp}.
\end{proof}

Two identities used repeatedly follow from \eqref{eq:a}:
\begin{equation}
 C^\mu[h]:=\partial_\nu h^{\nu\mu}-\tfrac12\partial^\mu h
 =-\partial_\nu a_1^{\nu\mu}[h],
 \qquad
 C^\mu[K\xi]=\Box\xi^\mu.
 \label{eq:C}
\end{equation}
For the second, \(\tr K\xi=2\partial_\rho\xi^\rho\) and
\(\partial_\nu(K\xi)^{\nu\mu}=\Box\xi^\mu+\partial^\mu\partial_\rho\xi^\rho\).

\subsection{Switch, regulator, plateau, gauge group}

Fix
\begin{equation}
 g\in C_c^\infty(\MTo;[0,1]),
 \qquad
 P_g=\Int\{x:g(x)=1\}\neq\varnothing.
 \label{eq:switch}
\end{equation}

Fix the ramp
\begin{equation}
 \chi(s)=
 \begin{cases}
  0,& s\leq0,\\[2pt]
  \dfrac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},& 0<s<1,\\[8pt]
  1,& s\geq1,
 \end{cases}
 \label{eq:chi}
\end{equation}
which is smooth, nondecreasing, and satisfies \(\chi'(s)>0\) for
\(0<s<1\): with \(a=e^{-1/s}\), \(b=e^{-1/(1-s)}\),
\(\chi'=(a'b-ab')/(a+b)^2\), \(a'=a/s^2>0\), \(b'=-b/(1-s)^2<0\).
Fix \(0<\eps<T\) and
\begin{equation}
 \sig(t,\mathbf x)=\chi(t/\eps).
 \label{eq:sigma}
\end{equation}
Then \(\sig=0\) on \(\{t\leq0\}\), \(\sig=1\) on \(\{t\geq\eps\}\),
\(\partial_\mu\sig=\delta^0_\mu\sig'\) with \(\sig'(t)=\eps^{-1}\chi'(t/\eps)\),
and \(\sig'>0\) exactly on \(\{0<t<\eps\}\).

Fix a nonempty open, relatively compact \(U\) with
\begin{equation}
 \overline U\subset P_g\cap\{\eps<t<T\},
 \label{eq:U}
\end{equation}
and set
\begin{equation}
 \ga=C_c^\infty(U;TM).
 \label{eq:ga}
\end{equation}
It is a Lie algebra under the bracket of vector fields. For
\(\xi,\zeta\in\ga\) let \(V_\kappa=\kappa\xi+\kappa^2\zeta\) and let
\(F_\kappa=\Fl^1_{V_\kappa}\) be its time-one flow, a diffeomorphism of
\(M\) for every real \(\kappa\).

\begin{lemma}[Admitted transformations]
\label{lem:group}
\begin{enumerate}[label=\textup{(\alph*)}]
 \item \(F_\kappa\) is the identity outside a compact subset of
 \(U\); in particular on a neighbourhood of \(\{t\leq\eps\}\) and of
 \(\{t\geq T\}\). It maps \(U\) onto \(U\) and \(\MT\) onto \(\MT\).
 \item \(F_\kappa^*g=g\) and \(F_\kappa^*\sig=\sig\).
 \item For every smooth tensor field or tensor density \(\mathsf T\),
 \[
  F_\kappa^*\mathsf T
  =\mathsf T+\kappa\Lie_\xi\mathsf T
  +\kappa^2\bigl(\Lie_\zeta+\tfrac12\Lie_\xi^2\bigr)\mathsf T
  +O(\kappa^3),
 \]
 with the Lie derivative of the appropriate type and the remainder
 \(O(\kappa^3)\) in the smooth topology on compact sets.
 \item The formal operators
 \(\exp(\kappa\Lie_\xi+\kappa^2\Lie_\zeta)\pmod{\kappa^3}\),
 \(\xi,\zeta\in\ga\), form a group \(\Gp\) under composition.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) The flow of a vector field supported in a compact
\(K_0\subset U\) is the identity outside \(K_0\) and preserves every
set containing \(K_0\) whose complement it fixes. (b) On \(U\) both
\(g\) and \(\sig\) equal one by \eqref{eq:U}; outside \(U\),
\(F_\kappa=\id\). (c) Let \(\Psi(s)=(\Fl^s_{V_\kappa})^*\mathsf T\).
Then \(\partial_s\Psi=(\Fl^s_{V_\kappa})^*\Lie_{V_\kappa}\mathsf T\),
so
\[
 \Psi(1)=\mathsf T+\Lie_{V_\kappa}\mathsf T
 +\int_0^1\!\!\int_0^s(\Fl^r_{V_\kappa})^*\Lie_{V_\kappa}^2\mathsf T\dd r\dd s
 =\mathsf T+\Lie_{V_\kappa}\mathsf T+\tfrac12\Lie_{V_\kappa}^2\mathsf T
 +O(\kappa^3),
\]
because \(\Lie_{V_\kappa}^3=O(\kappa^3)\). Insert
\(\Lie_{V_\kappa}=\kappa\Lie_\xi+\kappa^2\Lie_\zeta\). (d) By the
Baker--Campbell--Hausdorff formula modulo \(\kappa^3\),
\[
 \exp(\kappa\Lie_\xi+\kappa^2\Lie_\zeta)
 \exp(\kappa\Lie_{\xi'}+\kappa^2\Lie_{\zeta'})
 =\exp\bigl(\kappa\Lie_{\xi+\xi'}
 +\kappa^2(\Lie_{\zeta+\zeta'}+\tfrac12\Lie_{[\xi,\xi']})\bigr),
\]
and \([\xi,\xi']\in\ga\). Inverses are obtained with
\((\xi,\zeta)\mapsto(-\xi,-\zeta)\).
\end{proof}

For \(\kappa\)-dependent objects \(\mathsf T=\sum_j\kappa^j\mathsf T_j\)
the formal pullback is applied coefficientwise and truncated; this is
the Taylor expansion of the genuine pullback of the genuine
\(\kappa\)-dependent field by \(F_\kappa\). Explicitly,
\begin{align}
 F_\kappa^*\psi
 &=\psi+\kappa X_\xi\psi+\kappa^2\bigl(X_\zeta+\tfrac12X_\xi^2\bigr)\psi,
 \label{eq:scalarpull}\\
 F_{\kappa,\mathrm{dens}}^*r
 &=r+\kappa D_\xi r+\kappa^2\bigl(D_\zeta+\tfrac12D_\xi^2\bigr)r,
 \label{eq:densitypull}\\
 F_\kappa^*q&=\eta+\kappa h'+\kappa^2k',
 \qquad
 h'=h+K\xi,
 \qquad
 k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi,
 \label{eq:metricpull}
\end{align}
all modulo \(\kappa^3\). Equation \eqref{eq:metricpull} is the
complete second-order metric law. Setting \(\zeta=0\) and reading the
\(\lambda\)-linear part of the one-parameter family
\(\exp(\lambda\kappa\Lie_\xi)\) gives the infinitesimal law
\(\delta_\xi h=K\xi\), \(\delta_\xi k=\Lie_\xi h\), equivalently, in
the notation \(q=\eta+\kappa h\) with \(\kappa\)-dependent \(h\),
\(\delta_\xi h=K\xi+\kappa\Lie_\xi h\).

\subsection{Cauchy operators}

\begin{lemma}[Smooth Cauchy problem on the slab]
\label{lem:cauchy}
Let \(\mu\geq0\), \(s\in C^\infty(\MT)\), \(u_0,u_1\in C^\infty(\R^3)\).
There is a unique \(u\in C^\infty(\MT)\) with
\[
 (\Box+\mu^2)u=s,\qquad u|_\Sigma=u_0,\qquad \partial_tu|_\Sigma=u_1.
\]
For \(y\in\MT\), \(u(y)\) depends only on the restrictions of
\(s,u_0,u_1\) to \(J^-(y)\cap\MT\). If \(s,u_0,u_1\) vanish outside a
compact set \(K_0\) and its causal future, then
\(\supp u\subset J^+(K_0)\cap\MT\). For every compact \(K\subset\MT\)
and every \(N\) there are \(N'\) and \(C\) with
\[
 \|u\|_{C^N(K)}\leq C\bigl(
 \|s\|_{C^{N'}(J^-(K)\cap\MT)}
 +\|u_0\|_{C^{N'}(J^-(K)\cap\Sigma)}
 +\|u_1\|_{C^{N'}(J^-(K)\cap\Sigma)}\bigr).
\]
\end{lemma}

\begin{proof}
For compactly supported smooth data and source, existence,
uniqueness, the support bound, and continuous dependence are standard
\cite[Chapter~3]{BGP}. Finite propagation for the constant-coefficient
operator \(\Box+\mu^2\) is the energy argument of
\cite[\S2.4.3]{Evans}, with \(\mu^2u^2\) included in the energy density:
a smooth solution of the homogeneous equation whose data vanish on
\(J^-(y)\cap\Sigma\) and whose source vanishes on \(J^-(y)\cap\MT\)
vanishes at \(y\).

For general data, fix a compact \(K\subset\MT\). The set
\(\mathcal D_K=J^-(K)\cap\MT\) is compact. Choose
\(\varrho\in C_c^\infty(M)\) equal to one on a neighbourhood of
\(\mathcal D_K\), and let \(u_K\) be the compact-data solution for
\((\varrho s,\varrho u_0,\varrho u_1)\). If \(\varrho'\) is another
such cutoff, the difference \(u_K-u_{K}'\) has source and data
vanishing on \(\mathcal D_K\), hence vanishes on \(K\) by finite
propagation. Define \(u=u_K\) on \(K\). Compatibility over different
\(K\) follows in the same way, so \(u\) is a well-defined smooth
solution. Uniqueness and the dependence statement are finite
propagation again. The estimate is the compact-case estimate applied
to \((\varrho s,\varrho u_0,\varrho u_1)\).
\end{proof}

Write \(\Emu s\) for the solution with zero data; \(\Ez\) is the
wave case and \(\Em\) the Klein--Gordon case. In spatial Fourier
variables, with \(\Omega_\mu=\sqrt{-\Delta+\mu^2}\),
\begin{equation}
 (\Emu s)(t)=\int_0^t\frac{\sin((t-s')\Omega_\mu)}{\Omega_\mu}s(s')\dd s',
 \label{eq:Duhamel}
\end{equation}
interpreted locally by finite propagation; the sine quotient has its
continuous value \(t-s'\) at zero frequency when \(\mu=0\). For a
compactly supported smooth \(a\) on \(M\), \(\EA a\) denotes the
advanced solution of \((\Box+\mu^2)b=a\), smooth and supported in
\(J^-(\supp a)\) \cite[Chapter~3]{BGP}.

\begin{lemma}[Transposition on the slab]
\label{lem:transpose}
Let \(a\in C_c^\infty(\MTo)\) and \(s\in C^\infty(\MT)\), scalar or
componentwise. Then
\[
 \int_{\MT}a\,\Emu s\dd^4x=\int_{\MT}(\EA a)\,s\dd^4x.
\]
\end{lemma}

\begin{proof}
Put \(u=\Emu s\) and \(b=\EA a\). The set \(J^-(\supp a)\cap\MT\) is
compact and \(b\) vanishes on a neighbourhood of \(\{t\geq T\}\) in
\(\MT\), since \(\supp a\subset\{t\leq T-\delta\}\) for some
\(\delta>0\). Green's identity on \(\MT\) gives
\[
 \int_{\MT}\bigl((\Box+\mu^2)b\,u-b\,(\Box+\mu^2)u\bigr)
 =\Bigl[\int_{\R^3}(\partial_tb\,u-b\,\partial_tu)\dd^3x\Bigr]_{t=0}^{t=T}.
\]
At \(t=T\) the integrand vanishes because \(b\) does; at \(t=0\) it
vanishes because \(u\) has zero data. There is no spatial boundary term
because \(b\) is compactly supported in each slice.
\end{proof}

\subsection{Observation test}

Fix \(f\in C_c^\infty(\MTo;\R)\). It is a function of relational
labels. It is never transformed, and no spacetime volume density is
inserted when it is paired with the relational scalar.

\subsection{Two cautions recorded on the sheet}

\begin{enumerate}
 \item The density representative is frozen before multiplication by
 \(g\). The switched scalar density is
 \(\Lag_g=(1-g)\Lag_\eta+g\Lag_q\) with \(\Lag_q\) the natural
 curved density \eqref{eq:Lq}. An integration by parts performed
 before switching need not equal one performed afterwards; the
 multiplied-equation prescription \(g\cdot(\text{curved operator})\)
 is not the Euler operator of the switched action
 (Appendix~\ref{app:audit}).
 \item The convention is not \(q=\eta+\kappa gh\). That substitution
 produces different powers of \(g\) and different switch-derivative
 terms. It is not used anywhere below.
\end{enumerate}

The coordinate prescription uses an auxiliary density different from
the density of the switched action: the edge-regulated density
\(\Aeps[q]\) of \eqref{eq:Aeps}, regulated by \(\sig\) and not by
\(g\). This split is declared here. It is the analogue of VII-A's
split between switched scalar action and unswitched harmonic equation,
with the unswitched equation replaced by the edge-regulated one.

%=====================================================================
\section{Regulator preflight}
\label{sec:preflight}
%=====================================================================

\subsection{The regulated coordinate problem}

Let \(\eta^{-1}\) denote the constant tensor density
\(\Adens_\eta^{\alpha\beta}=\eta^{\alpha\beta}\) (note \(\sqrt{-\eta}=1\)).

\begin{definition}[Edge-regulated auxiliary density and coordinate operator]
\label{def:Aeps}
For the formal metric \(q=\eta+\kappa h+\kappa^2k\), set
\begin{equation}
 \Aeps[q]^{\alpha\beta}
 =\eta^{\alpha\beta}+\sig\bigl(\Adens_q^{\alpha\beta}-\eta^{\alpha\beta}\bigr),
 \qquad
 \Dco[q]\psi=\partial_\alpha\bigl(\Aeps[q]^{\alpha\beta}\partial_\beta\psi\bigr).
 \label{eq:Aeps}
\end{equation}
The output of \(\Dco[q]\) is a scalar density relative to \(\dd^4x\).
\end{definition}

\begin{definition}[Regulated coordinate prescription]
\label{def:coordinates}
The coordinate functions
\(\Co_q^\mu=x^\mu+\kappa\Co_1^\mu+\kappa^2\Co_2^\mu\),
\(\mu=0,1,2,3\), are the formal solutions of
\begin{equation}
 \Dco[q]\Co_q^\mu=0\formal,
 \qquad
 \Co_q^\mu|_\Sigma=x^\mu|_\Sigma,
 \qquad
 \partial_t\Co_q^\mu|_\Sigma=\delta^\mu_0.
 \label{eq:coordproblem}
\end{equation}
In the notation of the production plan, the operator is
\(\mathcal H_{g,\eps}[h]=\Dco[q]\), the boundary operator is Cauchy
restriction on \(\Sigma\), and the data
\(d^\mu_{g,\eps}=(x^\mu|_\Sigma,\delta^\mu_0)\) are independent of
the configuration and of \(g\).
\end{definition}

The fixed background objects are: \(\eta\), \(\Sigma\), \(\sig\), and
the data. The switch \(g\) does not enter. The index \(\mu\) labels four
scalar functions; it does not indicate a vector field.

\begin{lemma}[Operator expansion]
\label{lem:Dexp}
\(\Dco[q]=\Box+\kappa\Vc[h]+\kappa^2\Wc[h,k]\pmod{\kappa^3}\), with
\begin{align}
 \Vc[h]\psi
 &=\partial_\alpha\bigl(\sig a_1^{\alpha\beta}[h]\partial_\beta\psi\bigr)
 =\sig\,\partial_\alpha\bigl(a_1^{\alpha\beta}[h]\partial_\beta\psi\bigr)
 +\sig'\,a_1^{0\beta}[h]\partial_\beta\psi,
 \label{eq:V}\\
 \Wc[h,k]\psi
 &=\partial_\alpha\bigl(\sig(a_1^{\alpha\beta}[k]+a_2^{\alpha\beta}[h])
 \partial_\beta\psi\bigr)
 \nonumber\\
 &=\sig\,\partial_\alpha\bigl((a_1^{\alpha\beta}[k]+a_2^{\alpha\beta}[h])
 \partial_\beta\psi\bigr)
 +\sig'\,(a_1^{0\beta}[k]+a_2^{0\beta}[h])\partial_\beta\psi.
 \label{eq:W}
\end{align}
The regulator enters with at most one derivative; there are no
\(\sig''\) terms and no \(g\) terms.
\end{lemma}

\begin{proof}
Insert \eqref{eq:Aexp} into \eqref{eq:Aeps} and expand the
divergence; \(\partial_\alpha\sig=\delta^0_\alpha\sig'\).
\end{proof}

\begin{remark}[Action-level regulation]
Each \(\Co_q^\mu\) is a critical point of
\(\tfrac12\int\Aeps[q]^{\alpha\beta}\partial_\alpha\Co^\mu\partial_\beta\Co^\mu\)
under compactly supported variations. The regulated prescription is
therefore the action-level switching of VII-A with \(g\) replaced by
\(\sig\), not the multiplied equation \(\Box\Co_1=\sig C[h]\). The two
differ by the \(\sig'\) term in \eqref{eq:V}; see
Lemma~\ref{lem:kernel}. The density \(\Aeps[q]\) is the density of a
formal auxiliary metric, since in four dimensions
\(\det\Adens=\det q\) and \(q^{\mu\nu}=\Adens^{\mu\nu}/\sqrt{-\det\Adens}\);
this fact is not used.
\end{remark}

\subsection{What the regulator alters}

For \(t\leq0\), \(\Aeps[q]=\eta^{-1}\); for \(t\geq\eps\),
\(\Aeps[q]=\Adens_q\). The sharp onset \(\theta(t)\) of the zero-Cauchy
coupling in the VII-B and IX-A prescriptions is replaced by the ramp
\(\sig\). The Cauchy data remain on \(\Sigma=\{t=0\}\); the initial
surface is not moved. What changes is the singular structure of the
linear coordinate kernel.

\begin{lemma}[The regulated linear kernel is smooth]
\label{lem:kernel}
For \(a\in C_c^\infty(\MTo;T^*M)\) define
\(\Jc(a)[h]=\int_{\MT}a_\mu\Co_1^\mu[h]\dd^4x\). Then
\begin{equation}
 \Jc(a)[h]=\int_{\MT}h_{\rho\sigma}\,q_{a,\eps}^{\rho\sigma}\dd^4x,
 \qquad
 q_{a,\eps}=\sig\,C^*(\EA a),
 \qquad
 (C^*b)^{\rho\sigma}=-\partial^{(\rho}b^{\sigma)}
 +\tfrac12\eta^{\rho\sigma}\partial_\mu b^\mu.
 \label{eq:qaeps}
\end{equation}
The kernel \(q_{a,\eps}\) is smooth, vanishes on \(\{t\leq0\}\), and is
supported in \(J^-(\supp a)\cap\{t\geq0\}\).

By contrast, the multiplied-equation prescription
\(\widetilde\Co_1=\Ez(\sig C[h])\) has kernel
\(C^*(\sig\EA a)=\sig C^*(\EA a)+\sig'\,\mathsf B(\EA a)\), with
\(\mathsf B^{\rho\sigma}(v)=-n^{(\rho}v^{\sigma)}+\tfrac12\eta^{\rho\sigma}n_\mu v^\mu\),
\(n_\mu=\partial_\mu t\). The two prescriptions differ by the
switch-derivative term \(\sig'\mathsf B(\EA a)\). Neither is identified
with the sharp kernel \(\theta C^*b+\delta(t)\mathsf B(b|_\Sigma)\) of
VII-B, and no \(\eps\to0\) limit is taken here.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:Dexp} the first coefficient equation is
\(\Box\Co_1^\mu=-\partial_\alpha(\sig a_1^{\alpha\mu}[h])\) with zero
data, so \(\Co_1^\mu=-\Ez\partial_\alpha(\sig a_1^{\alpha\mu}[h])\).
Lemma~\ref{lem:transpose} with \(b=\EA a\) gives
\[
 \Jc(a)[h]=-\int_{\MT}b_\mu\,\partial_\alpha(\sig a_1^{\alpha\mu}[h])
 =\int_{\MT}\partial_\alpha b_\mu\,\sig a_1^{\alpha\mu}[h].
\]
The boundary term at \(t=0\) vanishes because \(\sig(0)=0\); at
\(t=T\) because \(b\) vanishes there; spatially because \(b\) is
compactly supported in each slice. Inserting \eqref{eq:a},
\[
 \partial_\alpha b_\mu a_1^{\alpha\mu}[h]
 =\tfrac12(\partial_\mu b^\mu)h-\partial_\alpha b_\mu h^{\alpha\mu}
 =h_{\rho\sigma}\bigl(\tfrac12\eta^{\rho\sigma}\partial_\mu b^\mu
 -\partial^{(\rho}b^{\sigma)}\bigr).
\]
Smoothness and support follow from those of \(b\) and \(\sig\). For
the multiplied-equation prescription,
\(\int a\,\Ez(\sig Ch)=\int\sig b\,Ch=\int C^*(\sig b)\,h\), and the
product rule with \(\partial_\mu\sig=n_\mu\sig'\) gives the displayed
decomposition, exactly as in the VII-B adjoint computation with
\(\theta\) replaced by \(\sig\).
\end{proof}

\subsection{Pure-gauge exactness on the frozen algebra}

\begin{proposition}[Preflight identity]
\label{prop:preflight}
Let \(\xi\in C_c^\infty(\MTo;TM)\) with \(\supp\xi\subset\{t\geq\eps\}\).
Then
\begin{equation}
 \Co_1[K\xi]=\xi.
 \label{eq:preflight}
\end{equation}
In particular \eqref{eq:preflight} holds for every \(\xi\in\ga\). In
the notation of the plan, \(\mathcal E_{g,\eps}\Box\xi=\xi\) with the
prescribed data.
\end{proposition}

\begin{proof}
By \eqref{eq:a}, \(a_1[K\xi]=(\partial_\rho\xi^\rho)\eta-K\xi^{\cdot\cdot}\)
is supported in \(\supp\xi\subset\{t\geq\eps\}\), where \(\sig=1\)
and \(\sig'=0\). Hence, by \eqref{eq:V} and \eqref{eq:C},
\[
 \Box\Co_1^\mu[K\xi]
 =-\partial_\alpha(\sig a_1^{\alpha\mu}[K\xi])
 =-\partial_\alpha a_1^{\alpha\mu}[K\xi]
 =C^\mu[K\xi]=\Box\xi^\mu.
\]
Both \(\Co_1[K\xi]\) and \(\xi\) have zero Cauchy data on \(\Sigma\),
since \(\xi\) vanishes on \(\{t<\eps\}\). Uniqueness in
Lemma~\ref{lem:cauchy} gives \eqref{eq:preflight}.
\end{proof}

\subsection{Edge-zone failure: the boundary of the freeze}

The condition \(\overline U\subset\{t>\eps\}\) was declared in
Section~\ref{sec:sheet}. The following computation shows it is not
removable. It concerns gauge parameters outside \(\ga\) and is not a
defect on the frozen group.

\begin{proposition}[Explicit edge-zone defect]
\label{prop:edge}
Let \(\alpha\in C_c^\infty((0,\eps))\) be nonnegative and not
identically zero, let \(\beta\in C_c^\infty(\R^3)\) with
\(\int\beta\dd^3x=1\), and let \(\xi=\alpha(t)\beta(\mathbf x)\,\partial_1\),
that is \(\xi^1=\alpha\beta\) and \(\xi^0=\xi^2=\xi^3=0\). Then for
every \(t\in[\eps,T]\),
\begin{equation}
 \int_{\R^3}\bigl(\Co_1^1[K\xi]-\xi^1\bigr)(t,\mathbf x)\dd^3x
 =-\int_0^\eps\sig'(s)\alpha(s)\dd s<0.
 \label{eq:edgegap}
\end{equation}
Consequently \(\Co_1[K\xi]\neq\xi\). Moreover, with
\(\Delta^{(1)}_{\eps,\xi}:=(\xi-\Co_1[K\xi])^\mu\partial_\mu\phi_0\),
there are a smooth \(\phi_0\) and an \(f\in C_c^\infty(\MTo)\) with
\(\int f\Delta^{(1)}_{\eps,\xi}\dd^4x\neq0\). This \(\xi\) is not in
\(\ga\).
\end{proposition}

\begin{proof}
The source \(-\partial_\alpha(\sig a_1^{\alpha\mu}[K\xi])\) is smooth
and compactly supported, so \(u^\mu:=\Co_1^\mu[K\xi]\) has compact
spatial support on each slice and the spatial integrals
\(\mathcal X^\mu(t)=\int u^\mu(t,\mathbf x)\dd^3x\),
\(\Xi^\mu(t)=\int\xi^\mu(t,\mathbf x)\dd^3x\) are smooth in \(t\).
Integrating the coefficient equation over space, the Laplacian term
and spatial divergences drop out:
\[
 \ddot{\mathcal X}^\mu
 =-\frac{\dd}{\dd t}\Bigl(\sig\int_{\R^3}a_1^{0\mu}[K\xi]\dd^3x\Bigr).
\]
From \(a_1^{0\mu}[K\xi]=(\partial_\rho\xi^\rho)\eta^{0\mu}-\partial^0\xi^\mu-\partial^\mu\xi^0\),
using \(\eta^{0\mu}=\delta^\mu_0\), \(\partial^0=\partial_0\), and the
vanishing of integrated spatial derivatives,
\[
 \int_{\R^3}a_1^{0\mu}[K\xi]\dd^3x
 =\delta^\mu_0\dot\Xi^0-\dot\Xi^\mu-\delta^\mu_0\dot\Xi^0
 =-\dot\Xi^\mu.
\]
Hence \(\ddot{\mathcal X}^\mu=(\sig\dot\Xi^\mu)^{\boldsymbol\cdot}\). Both
\(\dot{\mathcal X}^\mu\) and \(\sig\dot\Xi^\mu\) vanish at \(t=0\), so
\(\dot{\mathcal X}^\mu=\sig\dot\Xi^\mu\) and
\(\mathcal X^\mu(t)=\int_0^t\sig\dot\Xi^\mu\dd s\).
For \(\mu=1\), \(\Xi^1=\alpha\), and for \(t\geq\eps\),
\[
 \mathcal X^1(t)=\int_0^\eps\sig\alpha'\dd s
 =\bigl[\sig\alpha\bigr]_0^\eps-\int_0^\eps\sig'\alpha\dd s
 =-\int_0^\eps\sig'\alpha\dd s,
\]
while \(\xi^1(t,\cdot)=0\) for \(t\geq\eps\). The integral is strictly
positive because \(\sig'>0\) on \((0,\eps)\) and \(\alpha\geq0\) is not
identically zero. This proves \eqref{eq:edgegap}.

On \(\{t\geq\eps\}\) the function \(u^1\) solves the homogeneous wave
equation and has nonzero spatial integral, so \(u^1(t_1,\mathbf x_1)\neq0\)
at some \((t_1,\mathbf x_1)\in\MTo\). Take \(\phi_0=x^1\); then
\(\Delta^{(1)}_{\eps,\xi}=(\xi-u)^1=-u^1\) near that point, and a
nonnegative bump \(f\) supported where \(u^1\) has fixed sign gives a
nonzero smeared value. A free Klein--Gordon solution with
\(\partial_1\phi_0\neq0\) at a point where \(u^1\neq0\) works as well.
Finally \(\supp\xi\subset\{0<t<\eps\}\) is disjoint from \(U\).
\end{proof}

\begin{remark}[Reading of the defect]
For a gauge parameter in the edge zone the first-order observable
changes, under \((h,\phi_0)\mapsto(h+K\xi,\phi_0+\kappa X_\xi\phi_0)\),
by \(\kappa\int f\Delta^{(1)}_{\eps,\xi}\) at order \(\kappa\)
(Section~\ref{sec:expansion}). By the order argument of VI and VII-A, no
coefficient beginning at \(\kappa^2\) can cancel it. The regulator
preserves exactly the gauge directions supported where it equals one;
it does not preserve directions supported where it ramps. The frozen
algebra \eqref{eq:ga} was chosen accordingly before any calculation.
\end{remark}

\subsection{Preflight verdict}

The identity \(\mathcal E_{g,\eps}\Box\xi=\xi\) holds on the frozen
algebra (Proposition~\ref{prop:preflight}), and the second-order
mechanism is not merely plausible but available: the regulated
operator is in divergence form with an auxiliary density that is
natural under plateau transformations
(Lemma~\ref{lem:plateaucoord}). The second-order manuscript is opened.

%=====================================================================
\section{Switched scalar action and coefficient equations}
\label{sec:action}
%=====================================================================

\subsection{The frozen action}

Let
\begin{equation}
 \Lag_q[\phi]=\tfrac12\bigl(\Adens_q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
 -m^2\sqrt{-q}\,\phi^2\bigr),
 \qquad
 \Lag_\eta[\phi]=\tfrac12\bigl(\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
 -m^2\phi^2\bigr),
 \label{eq:Lq}
\end{equation}
both scalar densities relative to \(\dd^4x\). Define the switched
density coefficients
\begin{equation}
 \Ag[q]^{\mu\nu}=\eta^{\mu\nu}+g\bigl(\Adens_q^{\mu\nu}-\eta^{\mu\nu}\bigr),
 \qquad
 \varrho_g[q]=1+g\bigl(\sqrt{-q}-1\bigr),
 \label{eq:Ag}
\end{equation}
and the switched density
\begin{equation}
 \Lag_g[q,\phi]
 =\tfrac12\bigl(\Ag[q]^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
 -m^2\varrho_g[q]\phi^2\bigr)
 =(1-g)\Lag_\eta[\phi]+g\Lag_q[\phi].
 \label{eq:Lg}
\end{equation}
The switched scalar action is
\begin{equation}
 S_g^{\mathrm{sc}}[q,\phi]=\int_{\MT}\Lag_g[q,\phi]\dd^4x,
 \label{eq:action}
\end{equation}
understood variationally: through its Euler--Lagrange density and
through differences of integrands with compact support. Its numerical
value need not be finite. This is exactly the prescription ``one
switch multiplies the complete curved-minus-flat density before
variation''.

The Euler operator, with the sign convention
\(\delta S_g^{\mathrm{sc}}/\delta\phi=-\Eul[q]\phi\), is
\begin{equation}
 \Eul[q]\phi=\partial_\mu\bigl(\Ag[q]^{\mu\nu}\partial_\nu\phi\bigr)
 +m^2\varrho_g[q]\phi.
 \label{eq:Euler}
\end{equation}
Its output is a scalar density.

\subsection{Coefficient operators with switch derivatives displayed}

\begin{lemma}[Expansion of the switched Euler operator]
\label{lem:Eexp}
\(\Eul[q]=(\Box+m^2)+\kappa\mathsf V_{1,g}[h]+\kappa^2\mathsf V_{2,g}[h,k]\pmod{\kappa^3}\),
where
\begin{align}
 \mathsf V_{1,g}[h]\psi
 &=\partial_\mu\bigl(g\,a_1^{\mu\nu}[h]\partial_\nu\psi\bigr)+g\,m^2b_1[h]\psi
 \nonumber\\
 &=g\bigl[\partial_\mu(a_1^{\mu\nu}[h]\partial_\nu\psi)+m^2b_1[h]\psi\bigr]
 +(\partial_\mu g)\,a_1^{\mu\nu}[h]\partial_\nu\psi,
 \label{eq:V1g}\\
 \mathsf V_{2,g}[h,k]\psi
 &=\partial_\mu\bigl(g\,(a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h])\partial_\nu\psi\bigr)
 +g\,m^2\bigl(b_1[k]+b_2[h]\bigr)\psi
 \nonumber\\
 &=g\bigl[\partial_\mu((a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h])\partial_\nu\psi)
 +m^2(b_1[k]+b_2[h])\psi\bigr]
 +(\partial_\mu g)\,(a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h])\partial_\nu\psi.
 \label{eq:V2g}
\end{align}
In particular \(\mathsf V_{1,g}[h]=\mathsf Q^{(m)}_{gh}\), where
\(\mathsf Q^{(m)}_h\psi=\partial_\mu(a_1^{\mu\nu}[h]\partial_\nu\psi)+m^2b_1[h]\psi\)
is the unswitched first variation.
\end{lemma}

\begin{proof}
Insert Lemma~\ref{lem:density} into \eqref{eq:Ag}--\eqref{eq:Euler}
and expand. Linearity of \(a_1,b_1\) gives the last statement.
\end{proof}

\begin{lemma}[Pure-gauge variation of the scalar operator]
\label{lem:QK}
For every smooth \(\psi\) and every smooth vector field \(\xi\),
\begin{equation}
 \mathsf Q^{(m)}_{K\xi}\psi=D_\xi(\Box+m^2)\psi-(\Box+m^2)X_\xi\psi.
 \label{eq:QK}
\end{equation}
\end{lemma}

\begin{proof}
Direct expansion. With \(a_1[K\xi]=(\partial_\rho\xi^\rho)\eta-\partial^\mu\xi^\nu-\partial^\nu\xi^\mu\)
and \(b_1[K\xi]=\partial_\rho\xi^\rho\),
\[
 \mathsf Q^{(m)}_{K\xi}\psi
 =(\partial_\rho\xi^\rho)(\Box+m^2)\psi-(\Box\xi^\nu)\partial_\nu\psi
 -2\partial^\mu\xi^\nu\partial_\mu\partial_\nu\psi,
\]
because \(\partial_\mu\partial^\nu\xi^\mu\partial_\nu\psi\) cancels
against \(\partial_\mu(\partial_\rho\xi^\rho)\partial^\mu\psi\). The
right-hand side of \eqref{eq:QK} expands to the same expression, since
\(X_\xi\) commutes with the constant mass term.
\end{proof}

\subsection{The switched solution with fixed free input}

\begin{definition}[Switched scalar jet]
\label{def:rho}
Fix a smooth real \(\phi_0\) with \((\Box+m^2)\phi_0=0\) on \(\MT\).
The switched solution \(\rho_g[q;\phi_0]=\rho_0+\kappa\rho_{1,g}+\kappa^2\rho_{2,g}\)
is defined by \(\rho_0=\phi_0\), \(\Eul[q]\rho_g=0\pmod{\kappa^3}\),
and zero Cauchy data for \(\rho_{1,g},\rho_{2,g}\). Explicitly,
\begin{align}
 (\Box+m^2)\rho_{1,g}&=-\mathsf V_{1,g}[h]\rho_0,
 &\rho_{1,g}&=-\Em\mathsf V_{1,g}[h]\phi_0,
 \label{eq:rho1}\\
 (\Box+m^2)\rho_{2,g}&=-\mathsf V_{1,g}[h]\rho_{1,g}-\mathsf V_{2,g}[h,k]\rho_0,
 &\rho_{2,g}&=\Em\mathsf V_{1,g}[h]\Em\mathsf V_{1,g}[h]\phi_0
 -\Em\mathsf V_{2,g}[h,k]\phi_0.
 \label{eq:rho2}
\end{align}
\end{definition}

\begin{lemma}[Uniqueness and support of the scalar jet]
\label{lem:rhounique}
Equations \eqref{eq:rho1}--\eqref{eq:rho2} give the unique formal
solution with the stated data. The sources are smooth and compactly
supported in \(\supp g\), and
\(\supp\rho_{j,g}\subset J^+(\supp g)\cap\MT\) for \(j=1,2\).
\end{lemma}

\begin{proof}
Coefficient comparison and Lemma~\ref{lem:cauchy}, applied first at
order \(\kappa\) and then at order \(\kappa^2\). The coefficients of
\(\mathsf V_{1,g}\) and \(\mathsf V_{2,g}\) carry the factor \(g\) or
\(\partial g\).
\end{proof}

\begin{remark}[No quotient is defined by these equations]
Using \eqref{eq:rho1}--\eqref{eq:rho2} to evaluate the observable on a
perturbative solution does not define an on-shell quotient of the
configuration domain. The descent of III is on a named coupled jet
with a different gauge group; it is not this evaluation.
\end{remark}

%=====================================================================
\section{Regulated coordinate coefficient equations and data}
\label{sec:coordinates}
%=====================================================================

\begin{proposition}[Coordinate coefficient equations]
\label{prop:coordeq}
The problem \eqref{eq:coordproblem} has a unique formal solution. Its
coefficients are the zero-data solutions of
\begin{align}
 \Box\Co_1^\mu
 &=-\Vc[h]x^\mu
 =\sig\,C^\mu[h]-\sig'\,a_1^{0\mu}[h],
 \label{eq:X1eq}\\
 \Box\Co_2^\mu
 &=-\Vc[h]\Co_1^\mu-\Wc[h,k]x^\mu
 \nonumber\\
 &=\sig\Bigl(C^\mu[k]-\partial_\alpha a_2^{\alpha\mu}[h]
 -\partial_\alpha\bigl(a_1^{\alpha\beta}[h]\partial_\beta\Co_1^\mu\bigr)\Bigr)
 -\sig'\Bigl(a_1^{0\mu}[k]+a_2^{0\mu}[h]
 +a_1^{0\beta}[h]\partial_\beta\Co_1^\mu\Bigr),
 \label{eq:X2eq}
\end{align}
that is,
\begin{equation}
 \Co_1^\mu=-\Ez\,\partial_\alpha\bigl(\sig a_1^{\alpha\mu}[h]\bigr),
 \qquad
 \Co_2^\mu=-\Ez\bigl(\Vc[h]\Co_1^\mu+\Wc[h,k]x^\mu\bigr).
 \label{eq:Xsol}
\end{equation}
Both coefficients and their first time derivatives vanish on
\(\Sigma\). The value \(\Co_j^\mu(y)\) depends on \((h,k)\) only in
\(J^-(y)\cap\MT\). The coefficient \(\Co_2\) is affine in \(k\):
\begin{equation}
 \Co_2[h,k]=\Co_1[k]+\Co_2[h,0].
 \label{eq:X2affine}
\end{equation}
\end{proposition}

\begin{proof}
Since \(\Box x^\mu=0\), inserting Lemma~\ref{lem:Dexp} into
\eqref{eq:coordproblem} gives at order \(\kappa\) the equation
\(\Box\Co_1^\mu+\Vc[h]x^\mu=0\) and at order \(\kappa^2\) the equation
\(\Box\Co_2^\mu+\Vc[h]\Co_1^\mu+\Wc[h,k]x^\mu=0\). The displayed
right-hand sides follow from \eqref{eq:V}, \eqref{eq:W},
\(\partial_\beta x^\mu=\delta^\mu_\beta\), and \eqref{eq:C}. Existence,
uniqueness, and dependence are Lemma~\ref{lem:cauchy}. Equation
\eqref{eq:X2affine} holds because the \(k\)-dependent part of the
second source is \(-\partial_\alpha(\sig a_1^{\alpha\mu}[k])\), the
first source for \(k\).
\end{proof}

\paragraph{Term ledger for \eqref{eq:X1eq}--\eqref{eq:X2eq}.}
\begin{itemize}
 \item Ordinary metric terms: \(\sig C^\mu[h]\); \(\sig C^\mu[k]\);
 \(-\sig\partial_\alpha a_2^{\alpha\mu}[h]\);
 \(-\sig\partial_\alpha(a_1^{\alpha\beta}[h]\partial_\beta\Co_1^\mu)\).
 \item Regulator-derivative terms actually present: those with one
 factor \(\sig'\). No \(\sig''\) term occurs.
 \item Switch terms: none; \(g\) does not enter the coordinate
 prescription.
 \item Data: zero Cauchy data for \(\Co_1,\Co_2\); the data of
 \(\Co_q\) are \((x^\mu,\delta^\mu_0)\).
\end{itemize}
The second coefficient is derived from the frozen regulated problem,
not obtained by attaching \(\sig\) to the unswitched equation.

\begin{lemma}[Kernel of the first coefficient]
\label{lem:Lambda}
Let \(G_{\mathrm R}\) be the retarded fundamental solution of \(\Box\)
on \(M\). Then
\begin{equation}
 \Co_1^\mu[h](x)=\int\Lambda_\eps^{\mu,\rho\sigma}(x,z)h_{\rho\sigma}(z)\dd^4z,
 \qquad
 \Lambda_\eps^{\mu,\rho\sigma}(x,z)
 =\sig(z)\,P^{\alpha\mu,\rho\sigma}\,\partial^z_\alpha G_{\mathrm R}(x-z),
 \label{eq:Lambda}
\end{equation}
with
\(P^{\alpha\mu,\rho\sigma}=\tfrac12\eta^{\alpha\mu}\eta^{\rho\sigma}
-\tfrac12(\eta^{\alpha\rho}\eta^{\mu\sigma}+\eta^{\alpha\sigma}\eta^{\mu\rho})\).
As a distribution in \((x,z)\),
\(\WF(\Lambda_\eps)\subset\WF\bigl(G_{\mathrm R}(x-z)\bigr)\).
\end{lemma}

\begin{proof}
The source \(-\partial_\alpha(\sig a_1^{\alpha\mu}[h])\) vanishes on
\(\{t\leq0\}\), so \(\Ez\) acts on it as the retarded operator.
Integrating by parts in \(z\) over the compact set
\(J^-(x)\cap\{z^0\geq0\}\), with no boundary contribution because
\(\sig(z)=0\) at \(z^0=0\), gives \eqref{eq:Lambda}. Multiplication by
the smooth function \(\sig(z)\) and differentiation do not enlarge
wavefront sets \cite[Section~8.1]{Hormander}.
\end{proof}

The factor \(\theta(z^0)\) in the VII-B kernel \(\Lambda\) is replaced
here by \(\sig(z)\). This is a kernel-level inclusion, not an
admission statement; joint wavefront sets and boundary traces for the
observable kernels are X-B obligations.

%=====================================================================
\section{Relational expansion and insertion signs}
\label{sec:expansion}
%=====================================================================

\begin{lemma}[Formal inverse]
\label{lem:inverse}
The map \(\Co_q\) has the unique formal inverse
\begin{equation}
 \Co_q^{-1}(z)=z-\kappa\Co_1(z)
 +\kappa^2\bigl(\Co_1^\nu\partial_\nu\Co_1-\Co_2\bigr)(z)\formal.
 \label{eq:inverse}
\end{equation}
\end{lemma}

\begin{proof}
Substitute into \(\Co_q(\Co_q^{-1}(z))\): the first-order terms cancel,
and at second order the Taylor term \(-\Co_1^\nu\partial_\nu\Co_1\) is
cancelled by the displayed coefficient. Uniqueness is coefficientwise.
No numerical nonperturbative inverse is asserted.
\end{proof}

\begin{definition}[Relational scalar and observable]
\label{def:observable}
For \((h,k,\phi)\in\Conf\),
\begin{equation}
 \Phi_{g,\eps}[q,\phi](z)=\phi\bigl(\Co_q^{-1}(z)\bigr),
 \qquad
 \Obs_{f,g,\eps}[q,\phi]=\int f(z)\,\Phi_{g,\eps}[q,\phi](z)\dd^4z
 \quad\in\mathbb D_2.
 \label{eq:observable}
\end{equation}
The subscript \(g\) records that the on-shell evaluation of
Definition~\ref{def:rho} uses \(g\); the functional itself depends on
\(g\) only through that evaluation.
\end{definition}

\begin{lemma}[Expansion and insertion signs]
\label{lem:Phiexp}
\(\Phi_{g,\eps}=\Phi^{(0)}+\kappa\Phi^{(1)}+\kappa^2\Phi^{(2)}\pmod{\kappa^3}\), with
\begin{align}
 \Phi^{(0)}&=\phi_0,
 \label{eq:Phi0}\\
 \Phi^{(1)}&=\phi_1-\Co_1^\mu\partial_\mu\phi_0,
 \label{eq:Phi1}\\
 \Phi^{(2)}&=\phi_2-\Co_1^\mu\partial_\mu\phi_1
 +\bigl(\Co_1^\nu\partial_\nu\Co_1^\mu-\Co_2^\mu\bigr)\partial_\mu\phi_0
 +\tfrac12\Co_1^\mu\Co_1^\nu\partial_\mu\partial_\nu\phi_0.
 \label{eq:Phi2}
\end{align}
For a \(\kappa\)-independent scalar, \(\phi_1=\phi_2=0\). For the
on-shell evaluation \(\phi=\rho_g[q;\phi_0]\), the coefficients are
\begin{align}
 A_1^{(g,\eps)}&=\rho_{1,g}-\Co_1^\mu\partial_\mu\phi_0,
 \label{eq:A1}\\
 A_2^{(g,\eps)}&=\rho_{2,g}-\Co_1^\mu\partial_\mu\rho_{1,g}
 +\bigl(\Co_1^\nu\partial_\nu\Co_1^\mu-\Co_2^\mu\bigr)\partial_\mu\phi_0
 +\tfrac12\Co_1^\mu\Co_1^\nu\partial_\mu\partial_\nu\phi_0.
 \label{eq:A2}
\end{align}
\end{lemma}

\begin{proof}
Taylor expansion of \(\phi(\Co_q^{-1}(z))\) with \eqref{eq:inverse}.
\end{proof}

The insertion sign is fixed: the first-order dressing is
\(-\Co_1^\mu\partial_\mu\phi_0\). The five contributions to
\eqref{eq:A2} are iterated scalar propagation, the second-order scalar
vertex, the mixed term, the second-order inverse-coordinate term, and
the quadratic Taylor term. None may be omitted. The off-shell
expansion \eqref{eq:Phi0}--\eqref{eq:Phi2} and the on-shell evaluation
\eqref{eq:A1}--\eqref{eq:A2} are different objects: the first is a
functional on \(\Conf\), the second its value on the switched solution.

%=====================================================================
\section{Covariance by equations and data (Route I)}
\label{sec:routeI}
%=====================================================================

\subsection{Formal naturality}

\begin{lemma}[Formal naturality]
\label{lem:natural}
For \(F=F_\kappa\) as in Lemma~\ref{lem:group} and every formal scalar
\(\psi\), the following hold modulo \(\kappa^3\):
\begin{align}
 \Adens_{F^*q}&=F^*\Adens_q,
 \qquad
 \sqrt{-F^*q}=F^*_{\mathrm{dens}}\sqrt{-q},
 \qquad
 \Lag_{F^*q}[F^*\psi]=F^*_{\mathrm{dens}}\Lag_q[\psi],
 \label{eq:nat1}\\
 \partial_\alpha\bigl((F^*\Adens)^{\alpha\beta}\partial_\beta F^*\psi\bigr)
 &=F^*_{\mathrm{dens}}\,\partial_\alpha\bigl(\Adens^{\alpha\beta}\partial_\beta\psi\bigr)
 \quad\text{for every tensor density }\Adens.
 \label{eq:nat2}
\end{align}
Here \(F^*\) on a contravariant tensor density is
\((F^*\Adens)^{\alpha\beta}(x)=\det DF(x)\,(DF(x)^{-1})^\alpha{}_\gamma
(DF(x)^{-1})^\beta{}_\delta\Adens^{\gamma\delta}(F(x))\).
\end{lemma}

\begin{proof}
For each fixed point \(x\), choose a compact neighbourhood \(K\) of
\(\{x\}\cup F_\kappa(x)\) for \(|\kappa|\) small. On \(K\) and for
\(|\kappa|\) small, \(q(\kappa)=\eta+\kappa h+\kappa^2k\) is a
nondegenerate Lorentzian metric, and the identities hold as genuine
tensor identities for the genuine diffeomorphism \(F_\kappa\): the
density of a pulled-back metric is the pulled-back density, the
pulled-back volume factor is the density pullback, the divergence of a
vector density and the contraction of a density with a covector are
natural. Both sides are smooth in \(\kappa\) and their Taylor
coefficients through order two are polynomial differential expressions
in \((h,k,\xi,\zeta,\psi_j)\); equality for all small \(\kappa\)
gives equality of the coefficients.
\end{proof}

\subsection{Plateau covariance of the two operators}

\begin{lemma}[Plateau covariance of the regulated coordinate operator]
\label{lem:plateaucoord}
For \(F\in\Gp\) and every formal scalar \(\psi\),
\begin{equation}
 \Dco[F^*q](F^*\psi)=F^*_{\mathrm{dens}}\bigl(\Dco[q]\psi\bigr)\formal.
 \label{eq:plateaucoord}
\end{equation}
Only the condition \(\overline U\subset\{\eps<t<T\}\) is used.
\end{lemma}

\begin{proof}
Write \(\Aeps[q]=(1-\sig)\eta^{-1}+\sig\Adens_q\). By
Lemma~\ref{lem:group}(b) and \eqref{eq:nat1},
\[
 F^*\Aeps[q]=(1-\sig)F^*\eta^{-1}+\sig\Adens_{F^*q},
\]
so \(F^*\Aeps[q]-\Aeps[F^*q]=(1-\sig)(F^*\eta^{-1}-\eta^{-1})\). On
\(U\) the factor \(1-\sig\) vanishes by \eqref{eq:U}; outside \(U\),
\(F=\id\) and \(DF=\id\), so \(F^*\eta^{-1}=\eta^{-1}\). Hence
\(F^*\Aeps[q]=\Aeps[F^*q]\), and \eqref{eq:nat2} gives
\eqref{eq:plateaucoord}.
\end{proof}

\begin{lemma}[Plateau covariance of the switched Euler operator]
\label{lem:plateaueuler}
For \(F\in\Gp\) and every formal scalar \(\psi\),
\begin{equation}
 \Eul[F^*q](F^*\psi)=F^*_{\mathrm{dens}}\bigl(\Eul[q]\psi\bigr)\formal,
 \qquad
 \Lag_g[F^*q,F^*\psi]=F^*_{\mathrm{dens}}\Lag_g[q,\psi]\formal.
 \label{eq:plateaueuler}
\end{equation}
Here the condition \(\overline U\subset P_g\) is used.
\end{lemma}

\begin{proof}
Identical to Lemma~\ref{lem:plateaucoord} with \(\sig\) replaced by
\(g\), for both \(\Ag[q]=(1-g)\eta^{-1}+g\Adens_q\) and
\(\varrho_g[q]=(1-g)\cdot1+g\sqrt{-q}\); the density pullback of the
constant density \(1\) is \(\det DF=1\) outside \(U\). The identity
for \(\Lag_g\) follows from \eqref{eq:Lg} and \eqref{eq:nat1} in the
same way.
\end{proof}

\begin{remark}[Switch derivatives have not disappeared]
Both lemmas are identities for the complete divergence-form operators.
They do not replace \(\partial_\mu(g a_1^{\mu\nu}\partial_\nu)\) by
\(g\partial_\mu(a_1^{\mu\nu}\partial_\nu)\), nor
\(\partial_\alpha(\sig a_1^{\alpha\beta}\partial_\beta)\) by
\(\sig\partial_\alpha(a_1^{\alpha\beta}\partial_\beta)\). The terms
with \(\partial g\) in \eqref{eq:V1g}--\eqref{eq:V2g} and with
\(\sig'\) in \eqref{eq:X1eq}--\eqref{eq:X2eq} remain part of the
operators; they are covariant because they vanish where the
transformation acts.
\end{remark}

\subsection{The coordinate intertwiner}

\begin{theorem}[Coordinate intertwiner through second order]
\label{thm:coordinate}
For every \((h,k)\) and every \(F\in\Gp\),
\begin{equation}
 \Co_{F^*q}=F^*\Co_q=\Co_q\circ F\formal.
 \label{eq:intertwiner}
\end{equation}
Equivalently, with \(h',k'\) as in \eqref{eq:metricpull},
\begin{align}
 \Co_1[h']&=\Co_1[h]+\xi,
 \label{eq:X1law}\\
 \Co_2[h',k']&=\Co_2[h,k]+\xi^\nu\partial_\nu\Co_1[h]
 +\zeta+\tfrac12\xi^\nu\partial_\nu\xi.
 \label{eq:X2law}
\end{align}
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:plateaucoord},
\(\Dco[F^*q](F^*\Co_q)=F^*_{\mathrm{dens}}(\Dco[q]\Co_q)=0\pmod{\kappa^3}\).
Since \(F\) is the identity on a neighbourhood of \(\{t\leq\eps\}\),
\(F^*\Co_q=\Co_q\circ F\) has the Cauchy data of \(\Co_q\), namely
\((x^\mu,\delta^\mu_0)\). Uniqueness in Proposition~\ref{prop:coordeq}
gives \eqref{eq:intertwiner}. Expanding with \eqref{eq:scalarpull},
\[
 F^*\Co_q^\mu
 =x^\mu+\kappa\bigl(\Co_1^\mu+X_\xi x^\mu\bigr)
 +\kappa^2\bigl(\Co_2^\mu+X_\xi\Co_1^\mu+X_\zeta x^\mu+\tfrac12X_\xi^2x^\mu\bigr),
\]
and \(X_\xi x^\mu=\xi^\mu\), \(X_\xi^2x^\mu=\xi^\nu\partial_\nu\xi^\mu\).
\end{proof}

\begin{remark}[Infinitesimal form]
Setting \(\zeta=0\) and taking the \(\lambda\)-linear part of the
family \(\exp(\lambda\kappa\Lie_\xi)\), so that
\(\delta^{(0)}h=K\xi\) and \(\delta^{(1)}h=\Lie_\xi h\) in the
notation \(q=\eta+\kappa h\) with \(\kappa\)-dependent \(h\),
\eqref{eq:X1law}--\eqref{eq:X2law} and \eqref{eq:X2affine} read
\[
 \delta^{(0)}_\xi\Co_1=\xi,
 \qquad
 \delta^{(0)}_\xi\Co_2+\delta^{(1)}_\xi\Co_1=\xi^\nu\partial_\nu\Co_1,
\]
with \(\delta^{(1)}_\xi\Co_1=\Co_1[\Lie_\xi h]\) and
\(\delta^{(0)}_\xi\Co_2=\Co_2'_h[K\xi]\). These are the two coefficient
identities named in the plan. The second is the main calculation; it
is proved by Theorem~\ref{thm:coordinate} and verified directly in
Proposition~\ref{prop:audit}.
\end{remark}

\subsection{The scalar intertwiner}

\begin{proposition}[Switched scalar intertwiner]
\label{prop:scalar}
For every \((h,k)\), every free \(\phi_0\), and every \(F\in\Gp\),
\begin{equation}
 \rho_g[F^*q;\phi_0]=F^*\rho_g[q;\phi_0]\formal.
 \label{eq:scalarintertwiner}
\end{equation}
Coefficientwise,
\begin{equation}
 \rho_{1,g}'=\rho_{1,g}+X_\xi\phi_0,
 \qquad
 \rho_{2,g}'=\rho_{2,g}+X_\xi\rho_{1,g}+X_\zeta\phi_0+\tfrac12X_\xi^2\phi_0.
 \label{eq:rholaw}
\end{equation}
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:plateaueuler}, \(F^*\rho_g[q;\phi_0]\) solves the
switched equation for \(F^*q\). Its zeroth coefficient is \(\phi_0\),
and its positive-order coefficients
\(\rho_{1,g}+X_\xi\phi_0\), \(\rho_{2,g}+X_\xi\rho_{1,g}+X_\zeta\phi_0+\tfrac12X_\xi^2\phi_0\)
have zero Cauchy data because \(\xi,\zeta\) vanish near \(\Sigma\).
Lemma~\ref{lem:rhounique} gives \eqref{eq:scalarintertwiner}.
\end{proof}

\subsection{Proof of the main covariance statements}

\begin{proof}[Proof of Theorem~\ref{thm:main}, items (i)--(v)]
Item (i) is Proposition~\ref{prop:coordeq}; (ii) is
Proposition~\ref{prop:preflight}; (iii) is
Theorem~\ref{thm:coordinate}.

For (iv), formal inversion of \eqref{eq:intertwiner} gives
\(\Co_{F^*q}^{-1}=F^{-1}\circ\Co_q^{-1}\pmod{\kappa^3}\). With
\(F^*\phi=\phi\circ F\),
\[
 \Phi_{g,\eps}[F^*q,F^*\phi]
 =(\phi\circ F)\circ(F^{-1}\circ\Co_q^{-1})
 =\phi\circ\Co_q^{-1}
 =\Phi_{g,\eps}[q,\phi]\formal,
\]
pointwise in the label \(z\). Smearing with the fixed \(f\) gives
(iv). A direct coefficient verification of this cancellation, with all
quadratic gauge terms retained, is in Appendix~\ref{app:direct}.

For (v), Proposition~\ref{prop:scalar} and (iv) give
\[
 \Obs_{f,g,\eps}\bigl[F^*q,\rho_g[F^*q;\phi_0]\bigr]
 =\Obs_{f,g,\eps}\bigl[F^*q,F^*\rho_g[q;\phi_0]\bigr]
 =\Obs_{f,g,\eps}\bigl[q,\rho_g[q;\phi_0]\bigr]\formal.
\]
\end{proof}

The off-shell statement (iv) uses only coordinate covariance and the
fixed coordinate data; it is valid for the larger algebra of fields
supported in any \(U'\) with \(\overline{U'}\subset\{\eps<t<T\}\). The
on-shell statement (v) additionally uses \(\overline U\subset P_g\).
The theorem restricts to any \(\Gp\)-invariant subset of
configurations; invariance of a given subset, such as the
Einstein--scalar jet of VII-A, is a separate statement about the metric
equations and is not reproved here.

\subsection{Operator-level second-order audit}

\begin{proposition}[Direct verification of \eqref{eq:X2law}]
\label{prop:audit}
Write \(\Vc=\Vc[h]\), \(\Wc=\Wc[h,k]\), \(\Vc'=\Vc[h']\),
\(\Wc'=\Wc[h',k']\), \(X=X_\xi\), \(T_2=X_\zeta+\tfrac12X_\xi^2\),
\(D=D_\xi\), \(\Delta_2=D_\zeta+\tfrac12D_\xi^2\). Expanding
Lemma~\ref{lem:plateaucoord} gives the operator identities
\begin{align}
 \Vc'&=\Vc+D\Box-\Box X,
 \label{eq:Vid}\\
 \Wc'+\Vc'X+\Box T_2&=\Wc+D\Vc+\Delta_2\Box.
 \label{eq:Wid}
\end{align}
The candidate
\(\Co_2^{\mathrm c}=\Co_2+X\Co_1+T_2x\) satisfies
\(\Box\Co_2^{\mathrm c}=-\Vc'(\Co_1+Xx)-\Wc'x\) and has zero data;
hence \(\Co_2^{\mathrm c}=\Co_2[h',k']\).
\end{proposition}

\begin{proof}
Equations \eqref{eq:Vid}--\eqref{eq:Wid} are the \(\kappa\) and
\(\kappa^2\) coefficients of
\((\Box+\kappa\Vc'+\kappa^2\Wc')(1+\kappa X+\kappa^2T_2)
=(1+\kappa D+\kappa^2\Delta_2)(\Box+\kappa\Vc+\kappa^2\Wc)\).
Using \(\Box\Co_1=-\Vc x\), \(\Box\Co_2=-\Vc\Co_1-\Wc x\), and
\(\Box x=0\),
\begin{align*}
 \Box\Co_2^{\mathrm c}
 &=-\Vc\Co_1-\Wc x+\Box X\Co_1+\Box T_2x\\
 &=-\Vc\Co_1-\Wc x+(D\Box+\Vc-\Vc')\Co_1
 +(\Wc+D\Vc+\Delta_2\Box-\Wc'-\Vc'X)x\\
 &=-\Vc\Co_1-\Wc x-D\Vc x+\Vc\Co_1-\Vc'\Co_1
 +\Wc x+D\Vc x-\Wc'x-\Vc'Xx\\
 &=-\Vc'(\Co_1+Xx)-\Wc'x.
\end{align*}
Since \(Xx^\mu=\xi^\mu\) and \(T_2x^\mu=\zeta^\mu+\tfrac12\xi^\nu\partial_\nu\xi^\mu\),
the candidate is the right-hand side of \eqref{eq:X2law}; it vanishes
with its time derivative on \(\Sigma\). Uniqueness concludes.
\end{proof}

This audit retains the terms proportional to \(\Box\) in
\eqref{eq:Vid}--\eqref{eq:Wid}; they are operator identities, not
identities on solutions. It does not use any informal cancellation of
regulator terms.

%=====================================================================
\section{The varied action (Route II)}
\label{sec:routeII}
%=====================================================================

Route II computes the variation of the frozen switched action
independently of the coordinate problem, exposes every switch term,
and records the classical Noether identity a later construction must
respect. The gauge parameter is allowed to be an arbitrary compactly
supported vector field in \(\MTo\), so that the switch terms are
visible before the plateau condition is imposed.

For a compactly supported \(\xi,\zeta\in C_c^\infty(\MTo;TM)\), let
\(F=F_\kappa\) be the genuine time-one flow of \(\kappa\xi+\kappa^2\zeta\)
and define the finite variation
\begin{equation}
 \delta_FS_g^{\mathrm{sc}}[q,\phi]
 :=\int_{\MT}\bigl(\Lag_g[F^*q,F^*\phi]-\Lag_g[q,\phi]\bigr)\dd^4x.
 \label{eq:deltaS}
\end{equation}
The integrand is supported in the compact set where \(F\neq\id\), so
the integral is finite even though \(S_g^{\mathrm{sc}}\) itself need
not be.

\begin{lemma}[Divergence form of density pullback]
\label{lem:Z}
For every smooth scalar density \(r\),
\[
 F^*_{\mathrm{dens}}r-r=\partial_\mu Z_F^\mu[r],
 \qquad
 Z_F^\mu[r]=\kappa\xi^\mu r+\kappa^2\bigl(\zeta^\mu r+\tfrac12\xi^\mu D_\xi r\bigr)
 +O(\kappa^3),
\]
with \(Z_F[r]\) supported in \(\supp\xi\cup\supp\zeta\).
\end{lemma}

\begin{proof}
From \eqref{eq:densitypull}, \(D_\xi r=\partial_\mu(\xi^\mu r)\),
\(D_\zeta r=\partial_\mu(\zeta^\mu r)\), and
\(D_\xi^2r=\partial_\mu(\xi^\mu\partial_\nu(\xi^\nu r))=\partial_\mu(\xi^\mu D_\xi r)\).
\end{proof}

\begin{proposition}[Exact varied-action identity with all switch terms]
\label{prop:varied}
For every configuration and every compactly supported \(F\) as above,
\begin{equation}
 \delta_FS_g^{\mathrm{sc}}[q,\phi]
 =\int_{\MT}(1-g)\bigl(\Lag_\eta[F^*\phi]-\Lag_\eta[\phi]\bigr)\dd^4x
 -\int_{\MT}(\partial_\mu g)\,Z_F^\mu\bigl[\Lag_q[\phi]\bigr]\dd^4x
 \formal.
 \label{eq:varied}
\end{equation}
Through second order,
\begin{align}
 \Lag_\eta[F^*\phi]-\Lag_\eta[\phi]
 &=\kappa\bigl(\partial^\mu\phi\,\partial_\mu(X_\xi\phi)-m^2\phi X_\xi\phi\bigr)
 \nonumber\\
 &\quad+\kappa^2\Bigl(\partial^\mu\phi\,\partial_\mu(T_2\phi)-m^2\phi T_2\phi
 +\tfrac12\partial^\mu(X_\xi\phi)\partial_\mu(X_\xi\phi)
 -\tfrac12m^2(X_\xi\phi)^2\Bigr),
 \label{eq:Letavar}\\
 Z_F^\mu[\Lag_q[\phi]]
 &=\kappa\,\xi^\mu\Lag_\eta[\phi]
 +\kappa^2\Bigl(\xi^\mu\Lag^{(1)}[h,\phi]+\zeta^\mu\Lag_\eta[\phi]
 +\tfrac12\xi^\mu D_\xi\Lag_\eta[\phi]\Bigr),
 \label{eq:Zexp}
\end{align}
where \(\Lag^{(1)}[h,\phi]=\tfrac12(a_1^{\mu\nu}[h]\partial_\mu\phi\partial_\nu\phi-m^2b_1[h]\phi^2)\)
and, for a \(\kappa\)-dependent \(\phi\), the displayed products are
expanded coefficientwise.
\end{proposition}

\begin{proof}
By \eqref{eq:Lg} and the naturality \eqref{eq:nat1} of \(\Lag_q\),
\[
 \Lag_g[F^*q,F^*\phi]-\Lag_g[q,\phi]
 =(1-g)\bigl(\Lag_\eta[F^*\phi]-\Lag_\eta[\phi]\bigr)
 +g\bigl(F^*_{\mathrm{dens}}\Lag_q[\phi]-\Lag_q[\phi]\bigr).
\]
Lemma~\ref{lem:Z} turns the second term into
\(g\,\partial_\mu Z_F^\mu=\partial_\mu(gZ_F^\mu)-(\partial_\mu g)Z_F^\mu\).
The divergence of the compactly supported \(gZ_F\) integrates to zero.
The expansions are \eqref{eq:scalarpull} and Lemma~\ref{lem:Z} with
\(\Lag_q=\Lag_\eta+\kappa\Lag^{(1)}+O(\kappa^2)\).
\end{proof}

Every term of \eqref{eq:varied} carries a factor \(1-g\) or
\(\partial_\mu g\). The identities \(g=1\) and \(\dd g=0\) on the gauge
support cancel them, and only them, after they have been written.

\begin{proposition}[First-order display for a general compact gauge parameter]
\label{prop:firstorder}
Let \(\phi\) be \(\kappa\)-independent and let
\(T^{\mu\nu}[\phi]=\partial^\mu\phi\partial^\nu\phi-\eta^{\mu\nu}\Lag_\eta[\phi]\).
Then
\begin{equation}
 \bigl[\delta_FS_g^{\mathrm{sc}}\bigr]_{\kappa^1}
 =\int_{\MT}(1-g)\,T^{\mu\nu}[\phi]\,\partial_\mu\xi_\nu\dd^4x
 =\int_{\MT}(\partial_\mu g)\,T^{\mu\nu}[\phi]\,\xi_\nu\dd^4x
 -\int_{\MT}(1-g)\,\xi_\nu\,(\Box+m^2)\phi\,\partial^\nu\phi\dd^4x.
 \label{eq:firstorder}
\end{equation}
\end{proposition}

\begin{proof}
The \(\kappa^1\) coefficient of \eqref{eq:varied} is
\[
 \int(1-g)\bigl(\partial^\mu\xi^\nu\partial_\mu\phi\partial_\nu\phi
 +\xi^\nu\partial_\nu\Lag_\eta\bigr)
 -\int(\partial_\mu g)\xi^\mu\Lag_\eta.
\]
Since
\(\partial_\mu\bigl((1-g)\xi^\mu\Lag_\eta\bigr)
=-(\partial_\mu g)\xi^\mu\Lag_\eta+(1-g)(\partial_\mu\xi^\mu)\Lag_\eta+(1-g)\xi^\mu\partial_\mu\Lag_\eta\)
integrates to zero, the expression equals
\(\int(1-g)\bigl(\partial^\mu\xi^\nu\partial_\mu\phi\partial_\nu\phi-(\partial_\mu\xi^\mu)\Lag_\eta\bigr)
=\int(1-g)T^{\mu\nu}\partial_\mu\xi_\nu\). Integration by parts and
\(\partial_\mu T^{\mu\nu}=(\Box+m^2)\phi\,\partial^\nu\phi\) give the
second form.
\end{proof}

\begin{corollary}[Vanishing on the frozen group; the Noether identity]
\label{cor:noether}
For \(F\in\Gp\), \(\delta_FS_g^{\mathrm{sc}}[q,\phi]=0\pmod{\kappa^3}\)
identically on \(\Conf\), off shell. Differentiating this identity in
\(\phi\) recovers \eqref{eq:plateaueuler}. In particular the
first-order switch defect \eqref{eq:firstorder} vanishes because
\(1-g\) and \(\partial g\) vanish on \(\supp\xi\subset U\).
\end{corollary}

\begin{proof}
On \(U\supset\supp(F-\id)\), \(1-g=0\) and \(\partial g=0\), so both
integrands in \eqref{eq:varied} vanish. The Euler identity follows by
varying \(\phi\) with compact support and comparing Euler--Lagrange
densities, which is the content of Lemma~\ref{lem:plateaueuler}.
\end{proof}

This is the classical Noether identity of the frozen prescription. It
is not a quantum Ward identity. Its first-order defect
\eqref{eq:firstorder} for parameters leaving the plateau is the
same structure as the switched Noether defect of VI, in the present
signature and switching convention.

The coordinate prescription is not action-derived in the sense of
\eqref{eq:action}; its covariance was proved in
Lemma~\ref{lem:plateaucoord}. The two routes are independent: Route I
proves invariance of the observable through the defining equations and
data; Route II proves that the dynamics selecting the on-shell
evaluation is itself invariant, with all switch terms displayed.

%=====================================================================
\section{Two mandatory tests and the zero-switch check}
\label{sec:tests}
%=====================================================================

\subsection{The pure-gauge two-jet}

\begin{proposition}[Pure-gauge jet through second order]
\label{prop:puregauge}
Let \(\xi\in\ga\), \(\zeta=0\), and take the pure-gauge jet of the
flat configuration,
\[
 h=K\xi,\qquad k=\tfrac12\Lie_\xi K\xi=\tfrac12\Lie_\xi^2\eta,
 \qquad
 \phi=F_\kappa^*\phi_0=\phi_0+\kappa X_\xi\phi_0+\tfrac12\kappa^2X_\xi^2\phi_0.
\]
Then
\begin{equation}
 \Co_1=\xi,\qquad
 \Co_2=\tfrac12\xi^\nu\partial_\nu\xi^\mu,\qquad
 \Phi^{(1)}=0,\qquad \Phi^{(2)}=0,
 \label{eq:puregaugejet}
\end{equation}
and \(\Co_1,\Co_2\) are independent of \(\eps\) for every
\(\eps\) with \(\overline U\subset\{t>\eps\}\).
\end{proposition}

\begin{proof}
The jet is \(F_\kappa^*(\eta,\phi_0)\) with \(\zeta=0\). Since
\(\Co_\eta=x\), Theorem~\ref{thm:coordinate} gives
\(\Co_q=F_\kappa^*x=x+\kappa\xi+\kappa^2\tfrac12\xi^\nu\partial_\nu\xi\).
The first coordinate equation was verified directly in
Proposition~\ref{prop:preflight}. For the second, the identities
\eqref{eq:Vid}--\eqref{eq:Wid} at \(q=\eta\) read
\(\Vc[K\xi]=D\Box-\Box X\) and
\(\Wc[K\xi,\tfrac12\Lie_\xi K\xi]+\Vc[K\xi]X+\tfrac12\Box X^2=\tfrac12D^2\Box\),
so that
\[
 \Box\Co_2=-\Vc[K\xi]\xi-\Wc x
 =-\Vc[K\xi]\xi+\Vc[K\xi]\xi+\tfrac12\Box X^2x
 =\Box\bigl(\tfrac12\xi^\nu\partial_\nu\xi\bigr),
\]
with zero data. Then \(\Phi^{(1)}=X_\xi\phi_0-\xi^\mu\partial_\mu\phi_0=0\),
and
\[
 \Phi^{(2)}=\tfrac12X_\xi^2\phi_0-X_\xi(X_\xi\phi_0)
 +\tfrac12\xi^\nu\partial_\nu\xi^\mu\partial_\mu\phi_0
 +\tfrac12\xi^\mu\xi^\nu\partial_\mu\partial_\nu\phi_0=0,
\]
because \(X_\xi^2\phi_0=\xi^\nu\partial_\nu\xi^\mu\partial_\mu\phi_0
+\xi^\mu\xi^\nu\partial_\mu\partial_\nu\phi_0\). Regulator
independence holds because \(\sig=1\) and \(\sig'=0\) on the supports
of all sources.
\end{proof}

Testing only \(h=K\xi\) with \(k=0\) is not a second-order pure-gauge
jet: the straight-line family \(\eta+\kappa K\xi\) is not pure gauge
through second order, and \(\Co_2[K\xi,0]=\tfrac12\xi\cdot\partial\xi-\Co_1[\tfrac12\Lie_\xi K\xi]\)
by \eqref{eq:X2affine}.

\subsection{A sourced diagnostic}

\begin{proposition}[Sourced first coefficient, diagnostic only]
\label{prop:sourced}
Let \(m=0\), \(\phi_0=t\), \(h_{\mu\nu}=\psi\,\eta_{\mu\nu}\) with
real \(\psi\in C_c^\infty(\MTo)\), and \(k=0\). Then
\(a_1^{\mu\nu}[h]=\psi\eta^{\mu\nu}\), \(b_1[h]=2\psi\),
\[
 \mathsf V_{1,g}[h]\phi_0=\partial_t(g\psi),
 \qquad
 \rho_{1,g}=-\Ez\partial_t(g\psi),
 \qquad
 \Co_1^0=-\Ez\partial_t(\sig\psi),
 \qquad
 \Co_1^i=\Ez(\sig\partial_i\psi),
\]
and the first relational coefficient \eqref{eq:A1} is
\begin{equation}
 A_1^{(g,\eps)}=-\Ez\,\partial_t\bigl((g-\sig)\psi\bigr).
 \label{eq:A1diag}
\end{equation}
If \(\supp\psi\subset\{t\geq\eps\}\setminus\supp g\) and
\(\partial_t\psi\not\equiv0\), then \(A_1^{(g,\eps)}=\Ez\partial_t\psi\neq0\)
and there is \(f\) with \(\int fA_1^{(g,\eps)}\dd^4x\neq0\). If
\(\supp\psi\subset P_g\cap\{t\geq\eps\}\), then \(A_1^{(g,\eps)}=0\)
although \(\rho_{1,g}\) and \(\Co_1\) are individually nonzero whenever
\(\partial_t\psi\not\equiv0\).
\end{proposition}

\begin{proof}
\(\Box t=0\), so \(\phi_0\) is free. From \eqref{eq:V1g} with
\(\partial_\nu t=\delta^0_\nu\) and \(\eta^{\mu0}=\delta^\mu_0\),
\(\mathsf V_{1,g}[h]\phi_0=\partial_\mu(g\psi\eta^{\mu0})=\partial_t(g\psi)\).
From \eqref{eq:X1eq}, \(C^\mu[\psi\eta]=\partial^\mu\psi-2\partial^\mu\psi=-\partial^\mu\psi\)
and \(a_1^{0\mu}=\psi\delta^\mu_0\), so
\(\Box\Co_1^0=-\sig\partial_t\psi-\sig'\psi=-\partial_t(\sig\psi)\) and
\(\Box\Co_1^i=-\sig\partial^i\psi=\sig\partial_i\psi\). Then
\(A_1=\rho_{1,g}-\Co_1^\mu\partial_\mu t=\rho_{1,g}-\Co_1^0\), which is
\eqref{eq:A1diag}. In the first case \(g=0\) and \(\sig=1\) on
\(\supp\psi\), so the source is \(\partial_t\psi\neq0\) and
\(A_1\neq0\) because \(\Box A_1\neq0\); a bump \(f\) where \(A_1\) has
fixed sign gives a nonzero smearing. In the second case
\(g=\sig=1\) on \(\supp\psi\).
\end{proof}

This diagnostic checks the sign bookkeeping of \eqref{eq:A1} on a
sourced coefficient without adding an external source to the action.
It is not a coupled existence theorem, the metric is not put on shell,
and the vanishing in the second case is a property of the particular
pair \((\psi\eta,t)\), not a general identity.

\subsection{Zero-switch specialization}

\begin{proposition}[The \(g=0\) prescription]
\label{prop:zeroswitch}
If \(g=0\), then \(\mathsf V_{1,g}=\mathsf V_{2,g}=0\) and
\(\rho_g[q;\phi_0]=\phi_0\), while \(\Co_q\) is unchanged. Hence
\[
 \Obs_{f,0,\eps}[q,\phi_0]
 =\int f\bigl(\phi_0-\kappa\Co_1^\mu\partial_\mu\phi_0
 +\kappa^2\bigl[(\Co_1\cdot\partial\Co_1-\Co_2)^\mu\partial_\mu\phi_0
 +\tfrac12\Co_1^\mu\Co_1^\nu\partial_\mu\partial_\nu\phi_0\bigr]\bigr)\dd^4x,
\]
which differs from \(\int f\phi_0\) in general: in the setting of
Proposition~\ref{prop:sourced} the first coefficient is
\(\Ez\partial_t(\sig\psi)\neq0\) for suitable \(\psi\). The dressing
survives at \(g=0\); \(\Phi\neq\phi\). The declared free prescription
is: harmonic-type dressing with respect to the edge-regulated density,
no scalar correction. At \(g=0\) the set \(P_g\) is empty, so no
plateau \(U\) satisfies \eqref{eq:U} and the on-shell statement (v) is
vacuous; the off-shell statements (iii)--(iv) remain valid for fields
supported in any \(U'\) with \(\overline{U'}\subset\{\eps<t<T\}\).
\end{proposition}

\begin{proof}
Immediate from Lemma~\ref{lem:Eexp}, Definition~\ref{def:rho},
Proposition~\ref{prop:sourced}, and the proof of
Theorem~\ref{thm:main}(iv).
\end{proof}

%=====================================================================
\section{Decision, X-B handoff, and what not to add}
\label{sec:decision}
%=====================================================================

\subsection{Decision table}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.56\linewidth}X@{}}
\toprule
Evidence & Release\\
\midrule
Equations, regulator, data, and observable identities proved through
\(\kappa^2\) (Sections~\ref{sec:preflight}--\ref{sec:tests}) &
\textbf{Classical covariance at fixed \(\eps\): reached}\\
Explicit nonzero switch or regulator defect on the frozen group &
Not the case; the only proved defect
(Proposition~\ref{prop:edge}) is outside the frozen algebra\\
Regulator unspecified, uniqueness missing, or second-order
cancellation unfinished & Not the case\\
\bottomrule
\end{tabularx}
\end{center}

The gauge algebra was not shrunk after a defect appeared: the
condition \(\overline U\subset\{t>\eps\}\) is in the sheet of
Section~\ref{sec:sheet}, and Proposition~\ref{prop:edge} shows it is
not removable.

\subsection{Support and fixed-\(\eps\) continuity}

\begin{proposition}[Support and continuity]
\label{prop:continuity}
Let \(\mathcal D_f=J^-(\supp f)\cap\MT\), a compact set.
\begin{enumerate}[label=\textup{(\alph*)}]
 \item \(\Obs^{(j)}_{f,g,\eps}[h,k,\phi]\), \(j=0,1,2\), depends on
 \((h,k,\phi)\) only through their restrictions to every
 neighbourhood of \(\mathcal D_f\); the same holds for the on-shell
 evaluation, with \(\phi_0\) in place of \(\phi\).
 \item For fixed \(\eps\), \(h\mapsto\Co_1[h]\) is a continuous linear
 map and \((h,k)\mapsto\Co_2[h,k]\) a continuous polynomial map of
 \(C^\infty\) into \(C^\infty\), with respect to the smooth
 topologies on compact sets; \((h,k,\phi)\mapsto\Obs^{(j)}_{f,g,\eps}\)
 is a continuous polynomial functional, and
 \(f\mapsto\Obs^{(j)}_{f,g,\eps}[h,k,\phi]\) is a continuous linear
 functional on \(C_c^\infty(\MTo)\).
\end{enumerate}
\end{proposition}

\begin{proof}
(a) By Lemma~\ref{lem:cauchy}, \(\Co_j(z)\) depends on \((h,k)\) in
\(J^-(z)\cap\MT\); the local differential operators in
\eqref{eq:X1eq}--\eqref{eq:X2eq} do not enlarge support, and nested
terms obey \(J^-(z')\cap\MT\subset J^-(z)\cap\MT\) for
\(z'\in J^-(z)\). The relational coefficients at \(z\) involve
\(\phi\) and its derivatives at \(z\). Taking the union over
\(z\in\supp f\) gives \(\mathcal D_f\). The same argument applies to
\(\rho_{j,g}\). (b) The estimate in Lemma~\ref{lem:cauchy} and the
continuity of products and derivatives in \(C^\infty\) give the
continuity of the coefficient maps; pairing with \(f\) is continuous.
\end{proof}

\subsection{Derivative kernels at fixed configuration}

\begin{proposition}[Smooth first-derivative kernels]
\label{prop:kernels}
Fix a configuration \((h,k,\phi)\). The first functional derivatives
of the observable coefficients are:
\begin{align}
 \bigl(\Obs^{(1)}\bigr)'_h&=-q_{f\dd\phi_0,\eps}=-\sig\,C^*\bigl(\EA(f\dd\phi_0)\bigr),
 &
 \bigl(\Obs^{(1)}\bigr)'_{\phi_0}&=\partial_\mu\bigl(f\Co_1^\mu[h]\bigr),
 &
 \bigl(\Obs^{(1)}\bigr)'_{\phi_1}&=f,
 \label{eq:kern1}\\
 \bigl(\Obs^{(2)}\bigr)'_k&=-\sig\,C^*\bigl(\EA(f\dd\phi_0)\bigr),
 &
 \bigl(\Obs^{(2)}\bigr)'_{\phi_1}&=\partial_\mu\bigl(f\Co_1^\mu[h]\bigr),
 &
 \bigl(\Obs^{(2)}\bigr)'_{\phi_2}&=f,
 \label{eq:kern2}\\
 \bigl(\Obs^{(2)}\bigr)'_{\phi_0}
 &=-\partial_\mu\bigl(f(\Co_1\cdot\partial\Co_1-\Co_2)^\mu\bigr)
 +\tfrac12\partial_\mu\partial_\nu\bigl(f\Co_1^\mu\Co_1^\nu\bigr),
 \label{eq:kern3}
\end{align}
where \((f\dd\phi_0)_\mu=f\partial_\mu\phi_0\), and, for a smooth
symmetric direction \(\tilde h\),
\begin{equation}
 \bigl(\Obs^{(2)}\bigr)'_h[\tilde h]
 =\int f\Bigl[-\Co_1[\tilde h]\cdot\partial\phi_1
 +\bigl(\Co_1[\tilde h]\cdot\partial\Co_1+\Co_1\cdot\partial\Co_1[\tilde h]
 -(\Co_2)'_h[\tilde h]\bigr)\cdot\partial\phi_0
 +\Co_1[\tilde h]^\mu\Co_1^\nu\partial_\mu\partial_\nu\phi_0\Bigr]\dd^4x,
 \label{eq:kern4}
\end{equation}
with
\begin{equation}
 (\Co_2)'_h[\tilde h]
 =-\Ez\Bigl(\Vc[\tilde h]\Co_1[h]+\Vc[h]\Co_1[\tilde h]
 +2\,\partial_\alpha\bigl(\sig a_2^{\alpha\mu}[h,\tilde h]\bigr)\Bigr),
 \qquad
 (\Co_2)'_k[\tilde k]=\Co_1[\tilde k].
 \label{eq:X2deriv}
\end{equation}
Every one of these derivatives is represented by a smooth compactly
supported function or tensor on \(\MT\), supported in
\(\mathcal D_f\). In particular the metric derivatives have no
hypersurface delta term.
\end{proposition}

\begin{proof}
Equations \eqref{eq:kern1}--\eqref{eq:kern4} follow from
\eqref{eq:Phi1}--\eqref{eq:Phi2} by differentiation and integration
by parts in the \(\phi\) slots; \eqref{eq:X2deriv} follows from
\eqref{eq:Xsol} and the polarization of \(a_2\). The \(h\)-derivative
of \(\Obs^{(1)}\) and the \(k\)-derivative of \(\Obs^{(2)}\) are
\(-\Jc(f\dd\phi_0)\), whose kernel is given by Lemma~\ref{lem:kernel}.
For \eqref{eq:kern4}, each term is of the form
\(\Jc(c)[\tilde h]\) for a smooth compactly supported covector
\(c\) built from \(f\), \(\phi\), \(\Co_1[h]\), and their derivatives,
or of the form
\(\int(\EA c)\cdot\bigl(\Vc[\tilde h]\Co_1[h]+2\partial_\alpha(\sig a_2^{\alpha\mu}[h,\tilde h])\bigr)\)
by Lemma~\ref{lem:transpose} with \(c=f\dd\phi_0\); after
integrating by parts, the latter is the pairing of \(\tilde h\) with
a smooth compactly supported tensor built from \(\sig\), \(\EA c\),
\(\Co_1[h]\), \(h\), and derivatives. The term
\(\int(\EA c)\cdot\Vc[h]\Co_1[\tilde h]\) equals
\(\Jc(c')[\tilde h]\) with
\(c'_\mu=\partial_\beta\bigl(\partial_\alpha(\EA c)_\mu\,\sig a_1^{\alpha\beta}[h]\bigr)\)
smooth and compactly supported, hence has kernel
\(\sig C^*(\EA c')\). Supports follow from
Proposition~\ref{prop:continuity}(a).
\end{proof}

These are classical kernel facts at fixed \(\eps\). They are not
labelled quantum-admitted. Joint wavefront sets of the mixed kernels,
boundary traces, continuity in any admission topology, and a causal
product are X-B obligations. The topology \(\tau_\Sigma\) of IX-A is
not asserted to be the topology of this regulated observable.

\subsection{Handoff list}

If X-B is opened, it receives only:
\begin{itemize}
 \item the coefficients \(\Obs^{(0)},\Obs^{(1)},\Obs^{(2)}\) of
 Definition~\ref{def:observable} with Lemma~\ref{lem:Phiexp}, and the
 derivative kernels of Proposition~\ref{prop:kernels};
 \item the configuration space \(\Conf\) and the test space
 \(C_c^\infty(\MTo;\R)\) in label space;
 \item the support statement \(\mathcal D_f=J^-(\supp f)\cap\MT\)
 of Proposition~\ref{prop:continuity}(a);
 \item the fixed-\(\eps\) continuity statements of
 Proposition~\ref{prop:continuity}(b);
 \item the classical gauge identity of Theorem~\ref{thm:main}(iv)--(vi),
 together with the retained defect terms \eqref{eq:varied} and
 \eqref{eq:firstorder} for parameters leaving the plateau, and the
 edge-zone defect \eqref{eq:edgegap};
 \item the single interaction selected for a later retarded product:
 the first-order switched scalar vertex
 \[
  V_{1,g}[h,\phi]=\int_{\MT}g\,\Lag^{(1)}[h,\phi]\dd^4x
  =\tfrac12\int_{\MT}g\bigl(a_1^{\mu\nu}[h]\partial_\mu\phi\partial_\nu\phi
  -m^2b_1[h]\phi^2\bigr)\dd^4x,
 \]
 whose scalar Hessian is \(\mathsf V_{1,g}[h]\) of \eqref{eq:V1g}.
\end{itemize}

\subsection{What not to add}

No \(\mathsf T_2\), no \(R_1\), and no quantum Ward identity is
constructed. The sharp-regulator limit \(\eps\to0\) is not taken; the
regulated kernel of Lemma~\ref{lem:kernel} is not identified with
VII-B's sharp insertion, with IX-A's admitted element, or with IX-E-2's
kernel \(K\). VII-C is not repaired by this classical identity. VIII's
token and the \(2+1\) cone are not imported. III's coupled-jet descent,
VI's defect, and VII-A's plateau intertwiner are three statements on
three different equation spaces; the present theorem is a fourth, on
the frozen regulated prescription.

\subsection{Conclusion}

For the frozen switched action, regulator, Cauchy prescription,
configuration domain, and plateau-supported gauge algebra, the
declared relational scalar is gauge invariant through order
\(\kappa^2\) at fixed \(\eps>0\).

No quantum product, quantum Ward identity, sharp-regulator limit,
boundary algebra, or identification with the VII--IX or cone
constructions is claimed.

%=====================================================================
\appendix
\section{Sign, density, switch, and regulator audit}
\label{app:audit}
%=====================================================================

\subsection{Euler sign}

With signature \((+,-,-,-)\) and the density \eqref{eq:Lg},
\[
 \frac{\delta S_g^{\mathrm{sc}}}{\delta\phi}
 =-\partial_\mu\bigl(\Ag^{\mu\nu}\partial_\nu\phi\bigr)-m^2\varrho_g\phi
 =-\Eul[q]\phi.
\]
The leading part of \(\Eul[q]\) is \(\Box+m^2=\partial_t^2+\Omega_m^2\),
and the first-order equation is \((\Box+m^2)\rho_{1,g}=-\mathsf V_{1,g}\phi_0\),
hence \(\rho_{1,g}=-\Em\mathsf V_{1,g}\phi_0\) with the positive
Duhamel kernel \eqref{eq:Duhamel}. The retarded operator for sources
supported in \(\{t\geq0\}\) agrees with \(\Em\). A convention with the
opposite sign for the perturbation operator must change the response
sign as well; the two cannot be mixed.

\subsection{The divergence term in a density transformation}

For an orientation-preserving diffeomorphism,
\(F^*_{\mathrm{dens}}r=(\det DF)\,(r\circ F)\). Along the flow of \(\xi\),
\(\frac{\dd}{\dd s}\big|_{s=0}F^*_{s,\mathrm{dens}}r=\xi^\mu\partial_\mu r+(\partial_\mu\xi^\mu)r=D_\xi r\).
Omitting the divergence term would replace the off-shell identity
\eqref{eq:QK} by an incorrect commutator; the term disappears on a
free scalar only after imposing \((\Box+m^2)\phi_0=0\).

\subsection{Switching before variation}

For any smooth symmetric coefficient \(A^{\mu\nu}\) and any smooth
switch \(s\),
\[
 \partial_\mu\bigl(sA^{\mu\nu}\partial_\nu\psi\bigr)
 =s\,\partial_\mu\bigl(A^{\mu\nu}\partial_\nu\psi\bigr)
 +(\partial_\mu s)A^{\mu\nu}\partial_\nu\psi.
\]
Multiplying an unswitched equation by \(s\) is therefore not the Euler
equation of the switched density. This applies twice here: to the
interaction switch \(g\) in \eqref{eq:V1g}--\eqref{eq:V2g}, and to the
regulator \(\sig\) in \eqref{eq:V}--\eqref{eq:W}. The frozen
prescriptions are the divergence-form ones.

\subsection{Action-level versus multiplied-equation regulation}

The two regulated first coefficients
\[
 \Co_1=-\Ez\partial_\alpha(\sig a_1^{\alpha\mu}[h]),
 \qquad
 \widetilde\Co_1=\Ez(\sig C^\mu[h]),
\]
differ by \(\Co_1-\widetilde\Co_1=-\Ez(\sig'a_1^{0\mu}[h])\), and
their adjoint kernels differ by \(\sig'\mathsf B(\EA a)\)
(Lemma~\ref{lem:kernel}). Both agree with the unregulated first
coefficient on pure gauges supported in \(\{t\geq\eps\}\). Only the
first is in divergence form with a natural auxiliary density, which is
what Lemma~\ref{lem:plateaucoord} uses at second order. The frozen
prescription is the first.

\subsection{Zero-data identities on the plateau}

For \(\xi,\zeta\in\ga\), the fields \(\xi\), \(\zeta\), \(X_\xi\phi_0\),
\(X_\zeta\phi_0\), and \(X_\xi^2\phi_0\) vanish on a neighbourhood of
\(\Sigma\). Accordingly \(\Ez\Box\xi=\xi\) and
\(\Em(\Box+m^2)(X_\xi\phi_0)=X_\xi\phi_0\). The relevant fact is the
vanishing Cauchy data; compact support in \(U\) is the sufficient
condition declared on the sheet. For a parameter not vanishing near
\(\Sigma\), \(\Ez\Box\xi=\xi-H_0[\xi]\), where \(H_0[\xi]\) is the
homogeneous solution with the data of \(\xi\); such terms would have
to be included in the transformation law and are excluded here by the
support condition, not by a universal identity.

\subsection{The regulator does not move the initial surface}

The data of \(\Co_q\) are prescribed on \(\Sigma=\{t=0\}\) for every
\(\eps\). What \(\sig\) changes is the coefficient of the coordinate
operator on \(\{0<t<\eps\}\). Lemma~\ref{lem:kernel} shows the
consequence for the linear kernel: the hypersurface delta of the sharp
prescription is replaced by the smooth term \(\sig C^*b\), and in the
multiplied-equation variant by \(\sig C^*b+\sig'\mathsf B(b)\). No
statement about \(\eps\to0\) is made.

%=====================================================================
\section{Direct coefficient verification of observable invariance}
\label{app:direct}
%=====================================================================

Using the coefficient laws \eqref{eq:X1law}--\eqref{eq:X2law} and
\eqref{eq:scalarpull}, write
\(\phi_1'=\phi_1+X_\xi\phi_0\),
\(\phi_2'=\phi_2+X_\xi\phi_1+X_\zeta\phi_0+\tfrac12X_\xi^2\phi_0\),
\(\Co_1'=\Co_1+\xi\), \(\Co_2'=\Co_2+\xi\cdot\partial\Co_1+\zeta+\tfrac12\xi\cdot\partial\xi\).
Then
\[
 \Phi^{(1)\prime}=\phi_1+X_\xi\phi_0-(\Co_1+\xi)\cdot\partial\phi_0=\Phi^{(1)}.
\]
For the second coefficient,
\begin{align*}
 \Phi^{(2)\prime}-\Phi^{(2)}
 ={}&\bigl(X_\xi\phi_1-\xi\cdot\partial\phi_1\bigr)
 +\bigl(X_\zeta\phi_0-\zeta\cdot\partial\phi_0\bigr)\\
 &+\tfrac12X_\xi^2\phi_0
 -\Co_1\cdot\partial(X_\xi\phi_0)-\xi\cdot\partial(X_\xi\phi_0)\\
 &+\bigl(\Co_1\cdot\partial\xi+\xi\cdot\partial\Co_1+\xi\cdot\partial\xi
 -\xi\cdot\partial\Co_1-\tfrac12\xi\cdot\partial\xi\bigr)\cdot\partial\phi_0\\
 &+\tfrac12\bigl(\Co_1^\mu\xi^\nu+\xi^\mu\Co_1^\nu+\xi^\mu\xi^\nu\bigr)
 \partial_\mu\partial_\nu\phi_0.
\end{align*}
The first two brackets vanish. With
\(X_\xi^2\phi_0=\xi\cdot\partial\xi\cdot\partial\phi_0+\xi^\mu\xi^\nu\partial_\mu\partial_\nu\phi_0\)
and
\(\Co_1\cdot\partial(X_\xi\phi_0)=\Co_1^\nu\partial_\nu\xi^\mu\partial_\mu\phi_0+\Co_1^\nu\xi^\mu\partial_\nu\partial_\mu\phi_0\),
the coefficients of
\(\xi\cdot\partial\xi\cdot\partial\phi_0\),
\(\xi\xi\partial\partial\phi_0\),
\(\Co_1\cdot\partial\xi\cdot\partial\phi_0\), and
\(\Co_1\xi\partial\partial\phi_0\) are respectively
\(\tfrac12-1+\tfrac12\), \(\tfrac12-1+\tfrac12\), \(-1+1\), \(-1+1\),
all zero. This verifies Theorem~\ref{thm:main}(iv) coefficientwise
without suppressing the quadratic terms in the gauge parameter.

%=====================================================================
\section{Proved dependencies}
\label{app:deps}
%=====================================================================

The main theorem uses only:
\begin{enumerate}
 \item the explicit switched density \eqref{eq:Lg} and auxiliary
 density \eqref{eq:Aeps}, with the explicit ramp \eqref{eq:chi};
 \item coefficientwise Cauchy existence, uniqueness, finite
 propagation, and continuity on the slab (Lemma~\ref{lem:cauchy});
 \item the support conditions \(\overline U\subset P_g\) and
 \(\overline U\subset\{\eps<t<T\}\);
 \item naturality of metric densities and of divergence-form scalar
 density operators under genuine diffeomorphisms, transferred to the
 formal group by Taylor expansion (Lemmas~\ref{lem:group}
 and~\ref{lem:natural});
 \item formal inversion of a map with leading term the identity;
 \item the complete second-order metric pullback \eqref{eq:metricpull}.
\end{enumerate}
No quantum product, counterterm classification, earlier-series theorem,
or finite response matrix is used. The companion manuscripts are
quoted in Section~\ref{sec:prereq} for comparison and non-transfer
only; none of their theorems is a premise.

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

\paragraph{Scope of the result.}
This manuscript establishes the X-A classical gate for one frozen
regulated switched prescription. It does not begin the admission,
product, or causal-product problems of a later X-B.

\paragraph{Dependence on prior manuscripts.}
The statements of VII-A, VI, III, VII-B, and IX-A quoted in
Section~\ref{sec:prereq} are comparison background. No numbered result
from them is used as a premise. In particular VII-A's identity
\(w[h+K\xi]-w[h]=\xi\) for the unregulated unswitched inverse is not
used as a lemma about the regulated operator; the corresponding
statement is proved here as Proposition~\ref{prop:preflight}.

\paragraph{Verification status.}
No successful \texttt{pdflatex} execution, proof-assistant check,
numerical residual computation, or independent review is claimed. The
source uses no external figures, bibliography database, or executable
code. The nonzero witnesses are analytic.

\paragraph{Quantum boundary.}
The classical identities proved here are not advertised as a quantum
Ward identity, a renormalized product, or a sharp-limit statement.
The derivative kernels of Proposition~\ref{prop:kernels} are smooth
classical kernels at fixed \(\eps\); they are not labelled admitted to
any quantum domain.

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

\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.
\newblock arXiv:0806.1036.
\newblock Used for the smooth Cauchy problem, Green operators, and
finite propagation.

\bibitem{Evans}
L.~C.~Evans.
\newblock \emph{Partial Differential Equations}.
\newblock Graduate Studies in Mathematics 19, second edition.
American Mathematical Society, 2010.
\newblock Used for the energy argument giving the domain of dependence
of the constant-coefficient wave and Klein--Gordon operators.

\bibitem{Hormander}
L.~H\"ormander.
\newblock \emph{The Analysis of Linear Partial Differential Operators I}.
\newblock Grundlehren der mathematischen Wissenschaften 256, second
edition. Springer, 1990.
\newblock Used only for the rule that smooth multiplication and
differentiation do not enlarge wavefront sets.

\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{Communications in Mathematical Physics}
\textbf{345} (2016), 741--779.
\newblock arXiv:1306.1058.
\newblock Comparison background for relational observables; no theorem
is used.

\bibitem{Frob}
M.~B.~Fr\"ob.
\newblock Gauge-invariant quantum gravitational corrections to
correlation functions.
\newblock \emph{Classical and Quantum Gravity} \textbf{35} (2018), 055006.
\newblock arXiv:1710.00839.
\newblock Comparison background; its Feynman-prescribed coordinate
Green function is not the present Cauchy prescription.

\end{thebibliography}

\end{document}