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% Release:
%   PROVED OBSTRUCTED for the prescription frozen in this manuscript.
%
% The regulator is an explicit compact spacetime convolution applied
% to the switched metric arguments of an auxiliary harmonic-coordinate
% problem. It is fixed before the observable is defined.
%
% The first-order covariance test fails on the gauge algebra declared
% before the calculation. An exact positive witness is provided.
% Second-order coefficients and their retained defects are also written.
%
% No quantum admission, causal product, sharp-regulator limit,
% boundary algebra, or coupled Einstein--scalar existence theorem
% is claimed.
%
% Verification:
% No successful compilation, executed symbolic computation,
% proof-assistant verification, or independent review is claimed.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\newcommand{\Dtest}{\mathscr D}
\newcommand{\dd}{\,\mathrm d}
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\newcommand{\Boxeta}{\Box_\eta}
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\newcommand{\Seps}{S_\eps}
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\newcommand{\norm}[1]{\left\lVert #1\right\rVert}
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    \textbf{#1}\par\smallskip
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\title{\textbf{Landmark X-A}\\[0.4em]
Classical covariance of a fixed-regulator switched relational scalar\\
through order \(\kappa^2\): an explicit regulator obstruction}
\author{Ngo Cheung}
\date{}

\usepackage{tikz}
\begin{document}

\maketitle

\begin{abstract}
We freeze a compactly switched Einstein--scalar density, a
plateau-supported gauge algebra, a zero-Cauchy prescription, and one
explicit fixed regulator. The regulator convolves the switched metric
arguments before they enter an auxiliary harmonic-coordinate problem.
The resulting relational scalar is a well-defined off-shell formal
functional through order \(\kappa^2\). Its coefficient kernels are
smooth and compactly supported, and its metric dependence obeys an
explicit regulator-thickened support bound.

The regulated coordinate problem does not intertwine the frozen gauge
action. On a pure-gauge first coefficient, its coordinate response is
\(S_\eps\xi\), rather than \(\xi\). We prove an exact first-order
observable defect and exhibit one fixed observation test, one
plateau-supported gauge vector, and one free scalar for which its
smearing is strictly positive at every allowed fixed regulator.
The same gap occurs when the observable is evaluated on the scalar
solution selected by the switched action. The action itself satisfies
the retained classical plateau Noether identity; consequently the
failure is attributable to the coordinate regulator, not to omitted
switch derivatives or changed Cauchy data.

We also derive the second-order coordinate and relational coefficients,
the retained infinitesimal defects, and the complete second-order
pure-gauge test. A separate nonzero forced scalar example checks the
action-derived response. The release is a proved obstruction for this
prescription on its originally declared gauge algebra. No quantum
product, quantum Ward identity, sharp-regulator limit, or general
no-go theorem for regulated relational 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) {Convolution sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Defined coefficient};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Gauge identity};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Existence of the coefficient does not give the gauge identity.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A defined coefficient can still break the gauge identity. 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-conv}
\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) {Convolution sheet};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Gauge identity};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Existence of the coefficient does not give the gauge identity.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A defined coefficient can still break the gauge identity. 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-conv}
\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) {Convolution sheet};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Defined coefficient};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Gauge identity};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Existence of the coefficient does not give the gauge identity.};
\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-conv}
\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) {Convolution sheet};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {Gauge identity};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Defined coefficient. Existence of the coefficient does not give the gauge identity.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Convolution sheet is not Gauge identity. 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-conv}
\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: Convolution sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Defined coefficient};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Gauge identity};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Existence of the coefficient does not give the gauge identity.};
\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-conv}
\end{figure}

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


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



\scopebox{Release: proved obstructed}{%
The construction below is not advertised as a gauge-invariant
regulated observable. Its defining problems and coefficients exist,
but its first-order gauge identity fails on the gauge algebra frozen
before the calculation.

The defect is retained. The gauge algebra is not reduced after the
defect is found, and no second-order or quantum correction is used to
cancel a nonzero first-order classical coefficient.}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.52\linewidth}X@{}}
\toprule
Object & Status\\
\midrule
One switched classical density representative & Frozen explicitly\\
One fixed regulator and coordinate problem & Defined explicitly\\
Coefficientwise Cauchy existence and uniqueness & Proved\\
Relational expansion through \(\kappa^2\) & Derived\\
Fixed-regulator coefficient kernels & Smooth and compact\\
Action-level plateau Noether identity & Proved\\
Coordinate and observable covariance & Explicitly obstructed\\
Second-order pure-gauge test & Derived, with defects retained\\
Nonzero forced scalar test & Proved\\
Quantum admission and products & Not undertaken\\
Sharp-regulator limit & Not taken\\
Coupled Einstein--scalar existence & Not claimed\\
\bottomrule
\end{tabularx}
\end{center}

\tableofcontents

%=====================================================================
\section{Question, prerequisite boundary, and release discipline}
\label{sec:introduction}
%=====================================================================

The question is whether a specified regulated coordinate prescription
transforms with the scalar under the same gauge action. A regulator
must participate in this calculation. Classical covariance of an
unregulated prescription does not establish covariance after
convolution, and regularity of coefficient kernels does not establish
gauge invariance.

This paper chooses the obstruction branch of that question. The
regulator, switch, gauge algebra, and observation test are fixed in
Section~\ref{sec:sheet}. The preflight calculation in
Section~\ref{sec:preflight} already determines the release. Subsequent
sections identify the coefficients and isolate the failed identity;
they do not attempt a repair.

\subsection{What the supplied manuscripts establish}

The following statements are comparison data, not lemmas about the
regulated operator defined here.

\paragraph{VII-A.}
For its switched scalar action and its \emph{unswitched} harmonic
Cauchy problem, VII-A proves first- and second-order plateau
intertwining. Its first-order cancellation uses
\[
gK\xi=K\xi,\qquad
\phi_{1,g}[h+K\xi]-\phi_{1,g}[h]=X_\xi\phi_0,
\qquad
w[h+K\xi]-w[h]=\xi.
\]
It also retains the larger-support defect. Its named metric equations
are unswitched, and it does not construct a switched gravitational
quantum theory \cite{VIIA}.

\paragraph{VI.}
VI separates geometric solution-jet descent from a switched retarded
assignment on unchanged free inputs. It proves a first-order
compact-switch defect for that assignment and explains why a
second-order term cannot change the defective first coefficient
\cite{VI}.

\paragraph{III.}
III distinguishes an unchanged free-product quotient, free-shell
tangency, and descent on a named coupled first-order jet. Those are
different equation spaces. None is the off-shell domain used here
\cite{III}.

\paragraph{VII-B and IX-A.}
VII-B excludes its sharp zero-Cauchy insertion from its specified
off-shell polynomial microcausal domain. IX-A places that insertion
in a different classical carrier with a boundary-sensitive topology,
without constructing a quantum product. Neither defines the
regulated auxiliary metric used below \cite{VIIB,IXA}.

\paragraph{Other tracks.}
No result from IX-E-2 or IX-E-3 is used. In particular, a boundary-trace
certificate or a conditional support bridge is not a construction of
a two-insertion product. No cone model, measurement token, finite
contact matrix, or Hilbert-space model enters this paper.

Relational-coordinate constructions are comparison background
\cite{Frob}. They do not supply the covariance of the convolution
prescription frozen here.

\subsection{Three logically distinct outcomes}

For a frozen prescription, the alternatives are:
\begin{enumerate}
\item equations, data, and covariance are constructed;
\item a nonzero defect is proved on the frozen gauge algebra;
\item a named construction or proof obligation remains unresolved.
\end{enumerate}
The result below is the second alternative. It is not an unresolved
choice of regulator and is not a theorem excluding every possible
regulator.

%=====================================================================
\section{Convention sheet and frozen data}
\label{sec:sheet}
%=====================================================================

\subsection{Spacetime and configurations}

Let
\[
M=\R^4,\qquad M_T=[0,8]\times\R^3,\qquad
\Sigma=\{t=0\},\qquad
\eta=\diag(1,-1,-1,-1).
\]
Coordinates are denoted \(x=(t,\mathbf x)\), and
\[
\Boxeta=\partial_t^2-\Delta_{\mathbf x}.
\]
All indices are raised and lowered with \(\eta\) unless explicitly
stated otherwise. Parentheses include the factor \(1/2\).

The configuration space is
\[
\Conf
=
\Cinfty(M;S^2T^*M)\oplus\Cinfty(M;\R),
\]
with its compact-open smooth Fr\'echet topology. Its elements are
\((h,\phi)\). Fields are defined on all of \(M\), so convolution
requires no extension choice at the slab boundary.

For coefficient-jet calculations we use
\[
\Conf_{\mathrm{jet}}
=
\Cinfty(M;S^2T^*M)^2\oplus\Cinfty(M;\R),
\]
with elements \((h,k,\phi)\). These are smooth off-shell
configuration spaces. No equation-of-motion, gauge, or gauge-fixed
quotient is taken.

The metric is formal:
\begin{equation}
q=\eta+\kappa h,
\label{eq:baremetric}
\end{equation}
or, when evaluating a coefficient jet,
\begin{equation}
q=\eta+\kappa h+\kappa^2k\modthree.
\label{eq:jetmetric}
\end{equation}
There is no factor \(1/2\) in \(k\). No numerical smallness,
convergence, or global invertibility is asserted.

We fix a massless minimally coupled scalar. The coupling \(\kappa\)
has the sign and normalization specified by the action in
Section~\ref{sec:action}.

\subsection{An explicit switch and plateau}

Let \(x_*=(4,0,0,0)\), and use the Euclidean norm only to specify
compact supports and the regulator. Put
\[
r(x)=|x-x_*|_{\mathrm E}.
\]
Define
\[
u(s)=
\begin{cases}
e^{-1/s},&s>0,\\
0,&s\leq0,
\end{cases}
\qquad
c(s)=\frac{u(s)}{u(s)+u(1-s)}.
\]
The denominator is everywhere positive. Then \(c=0\) for \(s\leq0\)
and \(c=1\) for \(s\geq1\).

Freeze
\begin{equation}
g(x)=1-c\!\left(\frac{r(x)^2-4}{5}\right).
\label{eq:switch}
\end{equation}
Thus \(0\leq g\leq1\), \(g=1\) on \(r\leq2\), and
\[
\supp g\subset\{r\leq3\}\Subset(0,8)\times\R^3.
\]
In particular \(g=0\) near \(\Sigma\) and near \(t=8\).

Freeze the open sets
\begin{equation}
U=\{r<1\},\qquad O=\{r<1/4\}.
\label{eq:UO}
\end{equation}
The switch is background data and is not transformed.

\subsection{Gauge algebra and formal transformations}

For a vector field \(\xi\), set
\[
(K\xi)_{\mu\nu}
=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,
\qquad
X_\xi=\xi^\mu\partial_\mu.
\]
The frozen gauge algebra is
\begin{equation}
\mathfrak g_U=\Cc(U;TM).
\label{eq:gaugealg}
\end{equation}
Each member vanishes on a neighborhood of the edge of \(U\), hence
vanishes there to every order. It also vanishes near both temporal
boundaries.

The corresponding group is generated by flows of members of
\(\mathfrak g_U\). At second formal order we use
\begin{equation}
F_\kappa^*
=
\exp(\kappa\Lie_\xi+\kappa^2\Lie_\zeta)\modthree,
\qquad \xi,\zeta\in\mathfrak g_U.
\label{eq:formalgroup}
\end{equation}
This parametrization is closed under formal composition because
\(\mathfrak g_U\) is closed under Lie brackets.

For the single-field metric \eqref{eq:baremetric}, the infinitesimal
action is
\begin{equation}
\delta_\xi h=K\xi+\kappa\Lie_\xi h,
\qquad
\delta_\xi\phi=\kappa X_\xi\phi.
\label{eq:infaction}
\end{equation}
For \eqref{eq:jetmetric}, finite coefficient extraction gives
\begin{align}
h'&=h+K\xi,
\label{eq:hprime}\\
k'&=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi^2\eta.
\label{eq:kprime}
\end{align}
The nonlinear term in \eqref{eq:infaction} and the second coefficient
in \eqref{eq:kprime} are retained.

\subsection{One regulator, fixed before all products}

Define the real even compact mollifier
\[
\varrho(v)=
\begin{cases}
Z^{-1}\exp\!\left(-\dfrac1{1-|v|_{\mathrm E}^2}\right),
&|v|_{\mathrm E}<1,\\[4pt]
0,&|v|_{\mathrm E}\geq1,
\end{cases}
\qquad
Z=\int_{\R^4}
\exp\!\left(-\frac1{1-|v|_{\mathrm E}^2}\right)
\mathbf 1_{\{|v|_{\mathrm E}<1\}}\dd^4v.
\]
Freeze one number
\begin{equation}
0<\eps\leq1/8.
\label{eq:epsrange}
\end{equation}
All statements concern this fixed number. Define
\begin{equation}
\varrho_\eps(v)=\eps^{-4}\varrho(v/\eps),
\qquad
\Seps a=\varrho_\eps*a,
\qquad
\Teps a=\Seps(ga),
\label{eq:regulator}
\end{equation}
componentwise in the displayed background coordinates.

This is a spacetime convolution, not a Lorentz-invariant or
diffeomorphism-natural regulator. It regularizes the metric arguments
of the coordinate functional. It does not regulate the action, the
scalar field argument, or the Cauchy inverse itself.

For a metric jet define the auxiliary metric
\begin{equation}
\widehat q_{g,\eps}
=
\eta+\kappa H+\kappa^2J,
\qquad
H=\Teps h,\quad J=\Teps k.
\label{eq:auxmetric}
\end{equation}
The action uses \(q\), not \(\widehat q_{g,\eps}\). This distinction
is part of the prescription.

\subsection{One observation test}

Define the nonzero real bump
\[
\psi(x)=
\begin{cases}
\exp\!\left(1-\dfrac1{1-64r(x)^2}\right),&r(x)<1/8,\\
0,&r(x)\geq1/8,
\end{cases}
\qquad
N=\int_M\psi(x)\dd^4x>0.
\]
Freeze
\begin{equation}
f=\frac{\psi}{N}\in\Cc(O),\qquad
\xi_*=\psi\,\partial_1\in\mathfrak g_U.
\label{eq:fandxi}
\end{equation}
The same \(f\), \(g\), \(\xi_*\), and regulator are used throughout
the obstruction witness. In particular, the gauge algebra is not
chosen from the kernel of the regulator defect.

\scopebox{Frozen contract}{%
Signature \(+---\); mass \(m=0\); one compact background switch
\eqref{eq:switch}; gauge algebra \eqref{eq:gaugealg};
metric-argument regulator \eqref{eq:regulator}; auxiliary metric
\eqref{eq:auxmetric}; zero coordinate coefficient data at \(t=0\);
one label-space test \eqref{eq:fandxi}; formal coefficients through
\(\kappa^2\); no quantum order.}

%=====================================================================
\section{The switched action and its coefficient equations}
\label{sec:action}
%=====================================================================

\subsection{Curvature and density representative}

Fix
\[
R_{\mu\nu}
=
\partial_\rho\Gamma^\rho_{\mu\nu}
-\partial_\nu\Gamma^\rho_{\mu\rho}
+\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}.
\]
Write
\[
d(q)=\sqrt{-\det q},
\qquad A(q)^{\mu\nu}=d(q)q^{\mu\nu}.
\]
The unswitched geometric action is
\begin{equation}
S_{\mathrm{ES}}[q,\phi]
=
-\frac2{\kappa^2}\int d(q)R[q]\dd^4x
+\frac12\int A(q)^{\mu\nu}
\partial_\mu\phi\,\partial_\nu\phi\dd^4x.
\label{eq:ESaction}
\end{equation}
With
\[
T_{\mu\nu}
=
\partial_\mu\phi\partial_\nu\phi
-\tfrac12q_{\mu\nu}q^{\alpha\beta}
\partial_\alpha\phi\partial_\beta\phi,
\]
variation of \eqref{eq:ESaction} gives
\[
G_{\mu\nu}=\frac{\kappa^2}{4}T_{\mu\nu},
\qquad \Box_q\phi=0.
\]
These equations fix the normalization; their coupled solutions are
not the configuration domain of this paper.

Before switching, choose the first-derivative representative
\begin{equation}
\ell_{\mathrm{grav}}(\kappa,h)
=
-\frac2{\kappa^2}A(q)^{\mu\nu}
\left(
\Gamma^\rho_{\mu\sigma}\Gamma^\sigma_{\nu\rho}
-\Gamma^\rho_{\mu\nu}\Gamma^\sigma_{\rho\sigma}
\right),
\label{eq:gravdensity}
\end{equation}
and
\[
\ell_{\mathrm{sc}}(\kappa,h,\phi)
=
\tfrac12A(q)^{\mu\nu}\partial_\mu\phi\partial_\nu\phi.
\]
Set
\begin{equation}
\ell(\kappa,h,\phi)
=
\ell_{\mathrm{grav}}+\ell_{\mathrm{sc}}
=
\ell_0+\kappa\ell_1+\kappa^2\ell_2+O(\kappa^3).
\label{eq:ellseries}
\end{equation}
The density \eqref{eq:gravdensity}, not an unspecified equivalence
class modulo divergences, is used in this expansion.

\subsection{Explicit coefficients}

For a symmetric tensor \(a\), define
\begin{align}
b_1(a)&=\tfrac12\tr a,
&
b_2(a)&=\tfrac18(\tr a)^2-\tfrac14a_{\alpha\beta}a^{\alpha\beta},
\label{eq:bcoeff}\\
a_1(a)^{\mu\nu}
&=\tfrac12(\tr a)\eta^{\mu\nu}-a^{\mu\nu},
\label{eq:aone}\\
a_2(a)^{\mu\nu}
&=
a^\mu{}_\alpha a^{\alpha\nu}
-\tfrac12(\tr a)a^{\mu\nu}
+b_2(a)\eta^{\mu\nu}.
\label{eq:atwo}
\end{align}
Thus
\[
d(\eta+\kappa h)
=1+\kappa b_1(h)+\kappa^2b_2(h)+O(\kappa^3),
\]
\[
A(\eta+\kappa h)
=\eta+\kappa a_1(h)+\kappa^2a_2(h)+O(\kappa^3).
\]

To specify the gravitational vertices without hidden integrations by
parts, put
\[
T_{\lambda\mu\nu}(h)
=
\partial_\mu h_{\lambda\nu}
+\partial_\nu h_{\lambda\mu}
-\partial_\lambda h_{\mu\nu},
\]
\[
\gamma_r^\rho{}_{\mu\nu}
=
\tfrac12(-1)^{r-1}
(h^{r-1})^\rho{}_\sigma\eta^{\sigma\lambda}
T_{\lambda\mu\nu}(h),
\qquad r=1,2,3,
\]
where \(h^0=I\). For Christoffel-type arrays set
\[
\mathcal Q_{\mu\nu}(P,Q)
=
P^\rho{}_{\mu\sigma}Q^\sigma{}_{\nu\rho}
-P^\rho{}_{\mu\nu}Q^\sigma{}_{\rho\sigma}.
\]
Then
\begin{align}
\ell_{\mathrm{grav},0}
&=-2\eta^{\mu\nu}\mathcal Q_{\mu\nu}(\gamma_1,\gamma_1),
\label{eq:gravzero}\\
\ell_{\mathrm{grav},1}
&=-2\Big[
a_1(h)^{\mu\nu}\mathcal Q_{\mu\nu}(\gamma_1,\gamma_1)
\nonumber\\
&\hspace{3em}
+\eta^{\mu\nu}
\bigl(
\mathcal Q_{\mu\nu}(\gamma_1,\gamma_2)
+\mathcal Q_{\mu\nu}(\gamma_2,\gamma_1)
\bigr)\Big],
\label{eq:gravone}\\
\ell_{\mathrm{grav},2}
&=-2\Big[
a_2(h)^{\mu\nu}\mathcal Q_{\mu\nu}(\gamma_1,\gamma_1)
\nonumber\\
&\hspace{3em}
+a_1(h)^{\mu\nu}
\bigl(
\mathcal Q_{\mu\nu}(\gamma_1,\gamma_2)
+\mathcal Q_{\mu\nu}(\gamma_2,\gamma_1)
\bigr)
\nonumber\\
&\hspace{3em}
+\eta^{\mu\nu}
\bigl(
\mathcal Q_{\mu\nu}(\gamma_1,\gamma_3)
+\mathcal Q_{\mu\nu}(\gamma_2,\gamma_2)
+\mathcal Q_{\mu\nu}(\gamma_3,\gamma_1)
\bigr)\Big].
\label{eq:gravtwo}
\end{align}
The scalar coefficients are
\[
\ell_{\mathrm{sc},0}
=\tfrac12\eta^{\mu\nu}\partial_\mu\phi\partial_\nu\phi,
\qquad
\ell_{\mathrm{sc},j}
=\tfrac12a_j(h)^{\mu\nu}\partial_\mu\phi\partial_\nu\phi,
\quad j=1,2.
\]

\subsection{Action-level switching}

Freeze
\begin{equation}
S_g^{[2]}[h,\phi]
=
\int_M
\bigl(\ell_0+\kappa g\ell_1+\kappa^2g\ell_2\bigr)\dd^4x.
\label{eq:switchedaction}
\end{equation}
The free term is interpreted variationally with compact variations;
its numerical integral need not exist on every smooth configuration.
The interaction terms have compact coefficient support.

One switch multiplies each retained interaction density. In
particular, the second scalar vertex carries \(g\), not \(g^2\).
The regulator does not enter \eqref{eq:switchedaction}.

\begin{remark}[A density representative cannot be switched silently]
If \(\widetilde\ell_j=\ell_j+\partial_\mu B^\mu\), then
\[
\int g\widetilde\ell_j
=
\int g\ell_j-\int(\partial_\mu g)B^\mu.
\]
Consequently \eqref{eq:switchedaction} is a choice of model, not
merely a notation for an unswitched equivalence class.
\end{remark}

\subsection{All retained Euler switch terms}

Let \(\mathcal E_h\) be the ordinary first-derivative Euler operator
in the tensor variable. Since every \(\ell_j\) depends on \(h\) and
\(\partial h\), but not on higher derivatives of \(h\),
\begin{align}
\frac{\delta S_g^{[2]}}{\delta h_{\alpha\beta}}
={}&
\mathcal E_h^{\alpha\beta}\ell_0
+\kappa\left[
g\mathcal E_h^{\alpha\beta}\ell_1
-(\partial_\mu g)
\frac{\partial\ell_1}{\partial(\partial_\mu h_{\alpha\beta})}
\right]
\nonumber\\
&+\kappa^2\left[
g\mathcal E_h^{\alpha\beta}\ell_2
-(\partial_\mu g)
\frac{\partial\ell_2}{\partial(\partial_\mu h_{\alpha\beta})}
\right].
\label{eq:metricEuler}
\end{align}
Equation \eqref{eq:metricEuler}, together with
\eqref{eq:gravzero}--\eqref{eq:gravtwo}, is an explicit finite
definition of the switched metric Euler equation. No coupled
existence assertion is inferred from it.

The scalar Euler equation is \(-D_g[q]\phi=0\), where
\begin{equation}
D_g[q]\phi
=
\partial_\mu\left(
\left[
\eta^{\mu\nu}
+\kappa g a_1(h)^{\mu\nu}
+\kappa^2g a_2(h)^{\mu\nu}
\right]\partial_\nu\phi
\right).
\label{eq:Dg}
\end{equation}
Its output is density valued.

Define
\begin{equation}
V_{j,g}[h]\chi
=
\partial_\mu\bigl(g a_j(h)^{\mu\nu}\partial_\nu\chi\bigr)
=
g\partial_\mu\bigl(a_j(h)^{\mu\nu}\partial_\nu\chi\bigr)
+(\partial_\mu g)a_j(h)^{\mu\nu}\partial_\nu\chi.
\label{eq:Vjg}
\end{equation}
These are all switch-derivative terms in the displayed scalar
coefficient equations. There is no mass term because \(m=0\).

For a metric jet, \(V_{2,g}[h]\) is replaced by
\begin{equation}
W_g[h,k]\chi
=
V_{1,g}[k]\chi+V_{2,g}[h]\chi.
\label{eq:Wg}
\end{equation}

%=====================================================================
\section{Cauchy operators and the regulated coordinate problem}
\label{sec:coordinates}
%=====================================================================

\subsection{The finite-time inverse}

For a smooth source \(s\), define
\begin{equation}
(\Ezero s)(t)
=
\int_0^t
\frac{\sin((t-u)\sqrt{-\Delta})}{\sqrt{-\Delta}}s(u)\dd u,
\qquad 0\leq t\leq8.
\label{eq:Ezero}
\end{equation}
The sine quotient has the continuous value \(t-u\) at zero
spatial frequency. Sources without decay are handled locally by
finite propagation.

\begin{lemma}[Cauchy properties]
\label{lem:Cauchy}
For every smooth source on a neighborhood of \(M_T\), the
zero-data wave problem has a unique smooth solution. It depends
continuously on the source on each compact truncated causal region.
In particular,
\[
\Boxeta\Ezero s=s,\qquad
\Ezero s|_\Sigma=0,\qquad
\partial_t\Ezero s|_\Sigma=0.
\]
For smooth \(a\),
\[
\Ezero\Boxeta a=a-H_0[a],
\]
where \(H_0[a]\) is the homogeneous wave solution with the Cauchy
data of \(a\). Thus \(\Ezero\Boxeta a=a\) when those data vanish.
\end{lemma}

\begin{proof}
For spatially compact sources, differentiation of
\eqref{eq:Ezero} gives the equation and the data. On a finite time
interval, differentiated energy estimates give
\[
\norm{u(t)}_{H^{s+1}}+\norm{\partial_tu(t)}_{H^s}
\leq C_{s,T}
\left(
\norm{u(0)}_{H^{s+1}}+\norm{\partial_tu(0)}_{H^s}
+\int_0^T\norm{s(u)}_{H^s}\dd u
\right).
\]
The same estimates give uniqueness. Increasing \(s\) and using
Sobolev embedding give smooth dependence. Local energy estimates
give finite propagation, allowing compatible compact cutoffs for
general smooth sources. These arguments are the standard smooth
wave Cauchy theory \cite{BGP}.

Finally, \(a-\Ezero\Boxeta a\) solves the homogeneous equation
with the data of \(a\), proving the last assertion.
\end{proof}

\subsection{The actual regulated problem}

The coordinates are scalar functions with an external label
\(\mu\). Define them by
\begin{equation}
\partial_\alpha
\left(
A(\widehat q_{g,\eps})^{\alpha\beta}
\partial_\beta X^\mu_{g,\eps}
\right)=0\modthree,
\label{eq:regcoord}
\end{equation}
with
\begin{equation}
X^\mu_{g,\eps}|_\Sigma=x^\mu|_\Sigma,
\qquad
\partial_tX^\mu_{g,\eps}|_\Sigma=\delta^\mu_0.
\label{eq:coorddata}
\end{equation}
Equivalently, \eqref{eq:regcoord} is
\(\Box_{\widehat q_{g,\eps}}X^\mu_{g,\eps}=0\) formally.
The auxiliary metric is exactly \eqref{eq:auxmetric}.

Write
\begin{equation}
X^\mu_{g,\eps}
=
x^\mu+\kappa w^\mu+\kappa^2v^\mu\modthree.
\label{eq:Xexp}
\end{equation}
Define
\begin{equation}
C^\mu(a)
=
\partial_\nu a^{\nu\mu}
-\tfrac12\partial^\mu\tr a
=
-\partial_\nu a_1(a)^{\nu\mu}.
\label{eq:C}
\end{equation}

\begin{proposition}[Coordinate coefficients and uniqueness]
\label{prop:coordcoeff}
The unique coefficients satisfying
\eqref{eq:regcoord}--\eqref{eq:coorddata} are
\begin{align}
w^\mu[h]
&=\Ezero C^\mu(H),
\label{eq:w}\\
v^\mu[h,k]
&=
\Ezero C^\mu(J)
-\Ezero\left[
\partial_\alpha a_2(H)^{\alpha\mu}
+\partial_\alpha
\bigl(a_1(H)^{\alpha\beta}\partial_\beta w^\mu[h]\bigr)
\right].
\label{eq:v}
\end{align}
Both coefficients and their first time derivatives vanish on
\(\Sigma\).
\end{proposition}

\begin{proof}
The auxiliary density expansion is
\[
A(\widehat q_{g,\eps})
=
\eta+\kappa a_1(H)
+\kappa^2\bigl(a_1(J)+a_2(H)\bigr)\modthree.
\]
Substitution in \eqref{eq:regcoord} gives
\[
\Boxeta w^\mu=C^\mu(H),
\]
\[
\Boxeta v^\mu
=
C^\mu(J)-\partial_\alpha a_2(H)^{\alpha\mu}
-\partial_\alpha
\bigl(a_1(H)^{\alpha\beta}\partial_\beta w^\mu\bigr).
\]
The sources are smooth. Lemma~\ref{lem:Cauchy} gives the formulas
and successive uniqueness.
\end{proof}

\subsection{Switch derivatives inside this prescription}

Convolution commutes with constant-coefficient derivatives. Therefore
\begin{align}
\partial_\alpha H_{\mu\nu}
&=
\Seps\bigl(g\partial_\alpha h_{\mu\nu}
+(\partial_\alpha g)h_{\mu\nu}\bigr),
\label{eq:dH}\\
C^\mu(H)
&=
\Seps\left(
gC^\mu(h)
+(\partial_\nu g)
\left[h^{\nu\mu}-\tfrac12\eta^{\nu\mu}\tr h\right]
\right).
\label{eq:CHswitch}
\end{align}
The same identities hold with \(J,k\).

For clarity, the nonlinear divergence in \eqref{eq:v} is
\begin{align}
\partial_\alpha a_2(H)^{\alpha\mu}
={}&
(\partial_\alpha H^\alpha{}_\rho)H^{\rho\mu}
+H^\alpha{}_\rho\partial_\alpha H^{\rho\mu}
\nonumber\\
&-\tfrac12(\partial_\alpha\tr H)H^{\alpha\mu}
-\tfrac12(\tr H)\partial_\alpha H^{\alpha\mu}
\nonumber\\
&+\eta^{\alpha\mu}
\left(
\tfrac14(\tr H)\partial_\alpha\tr H
-\tfrac12H_{\rho\sigma}\partial_\alpha H^{\rho\sigma}
\right),
\label{eq:da2}
\end{align}
and
\begin{align}
\partial_\alpha
\bigl(a_1(H)^{\alpha\beta}\partial_\beta w^\mu\bigr)
={}&
\left(
\tfrac12\eta^{\alpha\beta}\partial_\alpha\tr H
-\partial_\alpha H^{\alpha\beta}
\right)\partial_\beta w^\mu
\nonumber\\
&+a_1(H)^{\alpha\beta}\partial_\alpha\partial_\beta w^\mu.
\label{eq:da1w}
\end{align}
Equations \eqref{eq:dH}--\eqref{eq:da1w} retain the actual
switch-derivative terms. No \(\partial^2g\) term occurs in these
displayed coefficient sources. Such terms would occur in some
further differentiated presentations, but no such presentation is
used here.

In particular,
\[
a_2\bigl(\Seps(gh)\bigr)
\]
is not replaced by \(g\,a_2(h)\), by
\(\Seps(g\,a_2(h))\), or by \(a_2(gh)\).
Those are different prescriptions.

%=====================================================================
\section{Regulator preflight and an exact positive defect}
\label{sec:preflight}
%=====================================================================

\begin{lemma}[Pure-gauge response of the regulated operator]
\label{lem:preflight}
For every \(\xi\in\mathfrak g_U\),
\begin{equation}
\Lg(K\xi)
:=
\Ezero C\Teps(K\xi)
=
\Seps\xi.
\label{eq:preflight}
\end{equation}
The right-hand side has zero Cauchy data near \(\Sigma\).
\end{lemma}

\begin{proof}
Since \(g=1\) on a neighborhood of \(\supp\xi\),
\[
gK\xi=K\xi.
\]
Constant-coefficient differentiation commutes with convolution, so
\[
\Teps(K\xi)=K(\Seps\xi),
\qquad
C K(\Seps\xi)=\Boxeta(\Seps\xi).
\]
The identity \(CK=\Boxeta\) follows directly from
\eqref{eq:C}: the derivative of the divergence of \(\xi\) cancels
the derivative of half the trace of \(K\xi\).

The support of \(\Seps\xi\) lies in the Euclidean
\(\eps\)-neighborhood of \(\supp\xi\), which is separated from
\(\Sigma\). Lemma~\ref{lem:Cauchy} therefore gives
\[
\Ezero\Boxeta(\Seps\xi)=\Seps\xi.
\]
\end{proof}

\begin{remark}[The necessary identity fails]
The first-order identity required for coordinate covariance is
\[
\Lg(K\xi)=\xi.
\]
Its actual value is \eqref{eq:preflight}. There is no claim that
\(\Lg\) globally factors as an operator on \(Ch\): the
switch-derivative term in \eqref{eq:CHswitch} prevents that
simplification. On the pure-gauge test, however, the failure is
exactly the failure of the regulated inverse to return \(\xi\).
\end{remark}

\begin{lemma}[Strict convolution gap]
\label{lem:positivegap}
For the fixed \(f,\xi_*\) of \eqref{eq:fandxi}, define
\begin{equation}
c_\eps
=
\int_M f(x)\bigl(\psi(x)-\Seps\psi(x)\bigr)\dd^4x.
\label{eq:ceps}
\end{equation}
Then
\begin{equation}
c_\eps
=
\frac1{2N}
\int_{\R^4}\varrho_\eps(a)
\int_{\R^4}
\bigl(\psi(x)-\psi(x-a)\bigr)^2\dd^4x\,\dd^4a
>0.
\label{eq:positiveenergy}
\end{equation}
\end{lemma}

\begin{proof}
Translation preserves \(\int\psi^2\). Expanding the square in
\eqref{eq:positiveenergy} and using
\(\int\varrho_\eps=1\) gives
\[
\frac1N\left(\int\psi^2-\int\psi\,\Seps\psi\right),
\]
which is \eqref{eq:ceps}.

For every \(a\neq0\), the inner integral is strictly positive.
Otherwise continuity would give \(\psi(x)=\psi(x-a)\) everywhere.
Iteration would make a nonzero compactly supported function invariant
under arbitrarily large translations in the direction \(a\), which
is impossible. The mollifier is positive on the open ball
\(0<|a|_{\mathrm E}<\eps\), a set of positive measure.
Thus the outer integral is strictly positive.
\end{proof}

\begin{theorem}[Preflight obstruction on the frozen algebra]
\label{thm:preflightstop}
The coordinate prescription
\eqref{eq:regcoord}--\eqref{eq:coorddata} fails first-order gauge
covariance on \(\mathfrak g_U\). For the free scalar
\begin{equation}
\phi_0(x)=x^1,
\label{eq:phiwitness}
\end{equation}
the first-order relational variation smeared with the frozen \(f\)
is \(c_\eps>0\).
\end{theorem}

\begin{proof}
The scalar variation is \(X_{\xi_*}\phi_0=\psi\). The coordinate
variation is \(\Seps\xi_*\), whose action on \(\phi_0\) is
\(\Seps\psi\). Their difference has smearing \eqref{eq:ceps},
which is positive by Lemma~\ref{lem:positivegap}.
The complete relational expansion proving the insertion sign
appears in Section~\ref{sec:observable}.
\end{proof}

\scopebox{The preflight decision}{%
The gauge algebra \(\mathfrak g_U\), the switch, the observation test,
and the regulator remain unchanged. The first-order covariance gate
is proved obstructed. The remaining calculations document this
prescription and its defects; they do not reopen a successful
covariance claim.}

%=====================================================================
\section{Relational expansion and retained infinitesimal defects}
\label{sec:observable}
%=====================================================================

\subsection{Off-shell functional}

The formal inverse of \eqref{eq:Xexp} is
\begin{equation}
Y^\mu(z)
=
z^\mu-\kappa w^\mu(z)
+\kappa^2\bigl(w^\nu\partial_\nu w^\mu-v^\mu\bigr)(z)
\modthree.
\label{eq:inverse}
\end{equation}
It is obtained by substitution in \(X(Y(z))=z\); no numerical
global inverse is required.

Define
\begin{equation}
\Phi_{g,\eps}[h,k,\phi](z)=\phi(Y(z)),
\qquad
\Ocal_{f,g,\eps}[h,k,\phi]
=
\int f(z)\Phi_{g,\eps}(z)\dd^4z.
\label{eq:observable}
\end{equation}
The test \(f\) is fixed in relational-label space. It is not
transformed as a spacetime-coordinate test.

For a vector \(w\), write
\begin{equation}
\mathcal J(w)
=
(w^\nu\partial_\nu w^\mu)\partial_\mu
+\tfrac12w^\mu w^\nu\partial_\mu\partial_\nu.
\label{eq:Jw}
\end{equation}

\begin{proposition}[Relational coefficients]
\label{prop:Ocoeff}
The coefficients of \eqref{eq:observable} are
\begin{align}
\Ocal^{(0)}&=\int f\phi,
\label{eq:Ozero}\\
\Ocal^{(1)}&=-\int fX_w\phi,
\label{eq:Oone}\\
\Ocal^{(2)}&=\int f\bigl(\mathcal J(w)\phi-X_v\phi\bigr).
\label{eq:Otwo}
\end{align}
For the bare single-field functional, set \(k=0\) in these formulas.
\end{proposition}

\begin{proof}
Taylor-expand \(\phi(Y(z))\) using \eqref{eq:inverse}.
The linear displacement gives \(-X_w\phi\). The second displacement
gives
\((w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi\),
and the quadratic Taylor term gives
\(\frac12w^\mu w^\nu\partial_\mu\partial_\nu\phi\).
\end{proof}

\subsection{A bilinear notation for the second coordinate coefficient}

Let \(a_2(H_1,H_2)\) denote the symmetric polarization of
\(a_2(H)\). Explicitly,
\begin{align}
a_2(H_1,H_2)^{\mu\nu}
={}&
\tfrac12\left(
H_1^\mu{}_\rho H_2^{\rho\nu}
+H_2^\mu{}_\rho H_1^{\rho\nu}
\right)
\nonumber\\
&-\tfrac14\left(
(\tr H_1)H_2^{\mu\nu}
+(\tr H_2)H_1^{\mu\nu}
\right)
\nonumber\\
&+\left(
\tfrac18(\tr H_1)(\tr H_2)
-\tfrac14H_{1,\rho\sigma}H_2^{\rho\sigma}
\right)\eta^{\mu\nu}.
\label{eq:a2polar}
\end{align}
Set \(W(H)=\Ezero CH\), and define
\begin{align}
\mathcal V^\mu(H_1,H_2)
=
-\Ezero\Big[
&\partial_\alpha a_2(H_1,H_2)^{\alpha\mu}
\nonumber\\
&+\tfrac12\partial_\alpha\bigl(
a_1(H_1)^{\alpha\beta}\partial_\beta W(H_2)^\mu
+a_1(H_2)^{\alpha\beta}\partial_\beta W(H_1)^\mu
\bigr)\Big].
\label{eq:Vbil}
\end{align}
Then
\begin{equation}
w=W(H),\qquad v=W(J)+\mathcal V(H,H).
\label{eq:vbil}
\end{equation}

\subsection{The defects at both retained infinitesimal orders}

For the bare functional \(k=0\), split
\[
\delta_\xi=\delta_\xi^{(0)}+\kappa\delta_\xi^{(1)},
\]
where
\[
\delta_\xi^{(0)}h=K\xi,\quad
\delta_\xi^{(0)}\phi=0,\quad
\delta_\xi^{(1)}h=\Lie_\xi h,\quad
\delta_\xi^{(1)}\phi=X_\xi\phi.
\]
Put
\[
a=\Seps\xi,\qquad
d_2=\Ezero C\Teps(\Lie_\xi h)
+2\mathcal V(H,Ka),
\]
and
\begin{equation}
\mathcal R_{2,\xi}
=
d_2-X_\xi w,
\label{eq:Rtwo}
\end{equation}
where \(X_\xi w\) means the vector with components
\(\xi^\nu\partial_\nu w^\mu\).

\begin{proposition}[Exact retained infinitesimal defects]
\label{prop:infdefects}
For every smooth \((h,\phi)\) and \(\xi\in\mathfrak g_U\),
\begin{align}
\delta_\xi\Ocal_{f,g,\eps}
={}&
\kappa\int fX_{\xi-a}\phi
\nonumber\\
&+\kappa^2\int f
\left[
X_{X_{a-\xi}w}\phi
-X_{\mathcal R_{2,\xi}}\phi
+X_wX_{a-\xi}\phi
\right]
+O(\kappa^3).
\label{eq:infdefect}
\end{align}
In particular, first-order covariance would require \(a=\xi\)
in the coordinate identity. That requirement is not satisfied.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:preflight} gives
\[
\delta_\xi^{(0)}w=a.
\]
Linearity of \(W\Teps\) gives
\[
\delta_\xi^{(1)}w=\Ezero C\Teps(\Lie_\xi h),
\]
and polarization gives
\[
\delta_\xi^{(0)}v=2\mathcal V(H,Ka).
\]
Hence
\[
\delta_\xi^{(0)}v+\delta_\xi^{(1)}w=d_2.
\]

At first order,
\[
\delta_\xi^{(0)}\Ocal^{(1)}
+\delta_\xi^{(1)}\Ocal^{(0)}
=
\int fX_{\xi-a}\phi.
\]
At second order,
\[
\delta_\xi^{(1)}(-X_w\phi)
=
-X_{\delta_\xi^{(1)}w}\phi-X_wX_\xi\phi.
\]
Direct differentiation of \eqref{eq:Jw} gives
\[
D\mathcal J(w)[a]=X_wX_a+X_{X_aw}.
\]
Therefore the second-order integrand is
\[
-X_{d_2}\phi+X_wX_a\phi+X_{X_aw}\phi-X_wX_\xi\phi.
\]
Using \eqref{eq:Rtwo} yields \eqref{eq:infdefect}.
\end{proof}

\begin{corollary}[No second-order cancellation of the witness]
\label{cor:nohigher}
No modification beginning at order \(\kappa^2\) makes the retained
functional gauge invariant on \(\mathfrak g_U\) while preserving
its first-order coefficient.
\end{corollary}

\begin{proof}
Formal invariance requires equality of each coefficient separately.
Theorem~\ref{thm:preflightstop} gives a nonzero coefficient of
\(\kappa\). A coefficient beginning at \(\kappa^2\) cannot change it.
\end{proof}

%=====================================================================
\section{Smooth coefficient kernels, support, and continuity}
\label{sec:kernels}
%=====================================================================

These are classical kernel statements. They are not a quantum-domain
or product theorem.

\subsection{First coordinate kernel}

For a tensor component label \(A=(\alpha\beta)\), let \(e_A\) denote
the constant symmetric component basis. Define
\[
H_{A,y}(u)=g(y)\varrho_\eps(u-y)e_A.
\]
Then
\begin{equation}
\Lambda_A^\mu(z,y)
=
\bigl[\Ezero C H_{A,y}\bigr]^\mu(z)
\label{eq:Lambda}
\end{equation}
is the first coordinate kernel:
\[
w^\mu[h](z)=\int\Lambda_A^\mu(z,y)h_A(y)\dd^4y.
\]

\begin{lemma}[Smooth parameter dependence]
\label{lem:smoothkernel}
For fixed \(\eps\), \(\Lambda_A^\mu(z,y)\) is jointly smooth on a
neighborhood of \(M_T\times M\). The polarized second kernel
\[
\Omega_{AB}^\mu(z;y,y')
=
\mathcal V^\mu(H_{A,y},H_{B,y'})(z)
\]
is jointly smooth as well. Both vanish in each input variable
outside \(\supp g\).
\end{lemma}

\begin{proof}
The source \(H_{A,y}(u)\) is jointly smooth and has a common compact
\(u\)-support when \(y\in\supp g\). Parameter differentiation
commutes with the wave problem by uniqueness and the differentiated
energy estimates of Lemma~\ref{lem:Cauchy}. Thus
\eqref{eq:Lambda} is jointly smooth.

Every source in \eqref{eq:Vbil} is a finite sum of products and
derivatives of such smooth parameter families. The same argument
applies to \(\Omega\). The factor \(g(y)\), respectively
\(g(y)g(y')\), proves the input support assertion.
\end{proof}

\subsection{Joint kernels of the observable}

Integration by parts in the scalar argument gives
\begin{equation}
\Ocal^{(1)}[h,\phi]
=
\int u_{1,A}(y,z)h_A(y)\phi(z)\dd^4y\,\dd^4z,
\qquad
u_{1,A}=\partial_{z^\mu}\bigl(f(z)\Lambda_A^\mu(z,y)\bigr).
\label{eq:uone}
\end{equation}

For the quadratic metric part define
\begin{align}
P_{AB}^\mu(z;y,y')
={}&
\tfrac12\left(
\Lambda_A^\nu(z,y)\partial_{z^\nu}\Lambda_B^\mu(z,y')
+\Lambda_B^\nu(z,y')\partial_{z^\nu}\Lambda_A^\mu(z,y)
\right)
-\Omega_{AB}^\mu(z;y,y'),
\label{eq:Pkernel}\\
Q_{AB}^{\mu\nu}(z;y,y')
={}&
\tfrac14\left(
\Lambda_A^\mu(z,y)\Lambda_B^\nu(z,y')
+\Lambda_B^\mu(z,y')\Lambda_A^\nu(z,y)
\right).
\label{eq:Qkernel}
\end{align}
Then
\begin{align}
\Ocal^{(2)}[h,k,\phi]
={}&
\int u_{1,A}(y,z)k_A(y)\phi(z)\dd^4y\,\dd^4z
\nonumber\\
&+\int u_{2,AB}(y,y',z)
h_A(y)h_B(y')\phi(z)\dd^4y\,\dd^4y'\,\dd^4z,
\label{eq:utwoformula}\\
u_{2,AB}
={}&
-\partial_{z^\mu}\bigl(fP_{AB}^\mu\bigr)
+\partial_{z^\mu}\partial_{z^\nu}
\bigl(fQ_{AB}^{\mu\nu}\bigr).
\label{eq:utwo}
\end{align}
The kernels are symmetric in their two metric slots.

\begin{proposition}[Classical regularity and derivative kernels]
\label{prop:regularity}
At fixed \(\eps\),
\[
u_1\in\Cc(M^2),\qquad u_2\in\Cc(M^3),
\]
with the indicated tensor components. Their supports are contained in
\[
\supp g\times\supp f,\qquad
\supp g\times\supp g\times\supp f,
\]
respectively. The coefficients are continuous polynomial functionals
on the declared smooth configuration spaces.

Every joint directional derivative kernel is obtained from
\eqref{eq:uone} and \eqref{eq:utwo} by slot contraction with the
specified smooth directions and the finite polynomial Leibniz rule.
Consequently those derivative kernels are smooth and compactly
supported in the remaining slots.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:smoothkernel} gives smoothness. The compact factors
\(g(y)\), \(g(y')\), and \(f(z)\), including their derivatives after
integration by parts, give compact support.

Pairing a smooth compact kernel against smooth fields is a
continuous multilinear map: each pairing is bounded by a finite
product of compact-open seminorms of its arguments. This proves
polynomial continuity. Differentiation of a finite polynomial
contracts slots of its fixed kernels, proving the final assertion.
\end{proof}

\subsection{The actual support bound}

Let
\[
\mathcal D_f=J_\eta^-(\supp f)\cap\{t\geq0\},
\qquad
\mathcal D_f^{(\eps)}
=
\mathcal D_f+\overline B_{\mathrm E}(0,\eps).
\]

\begin{proposition}[Regulator-thickened metric dependence]
\label{prop:support}
Through order \(\kappa^2\), the observable depends on \(h,k\) only
in
\[
\supp g\cap\mathcal D_f^{(\eps)},
\]
and on \(\phi\) only in \(\supp f\), in the usual sense that agreement
on neighborhoods of these sets gives agreement of the coefficients.
\end{proposition}

\begin{proof}
A coordinate source point \(u\) affecting an observation point
\(z\) lies in \(J_\eta^-(z)\cap\{t\geq0\}\). A convolution input
\(y\) affecting \(H(u)\) or \(J(u)\) satisfies
\[
y\in\supp g,\qquad |u-y|_{\mathrm E}\leq\eps.
\]
For a nested second-order term, its intermediate wave source points
remain in \(\mathcal D_f\) by transitivity of the causal relation.
Each metric input is within one convolution radius of such a point.
The scalar argument and its derivatives are evaluated only on
\(\supp f\).
\end{proof}

\begin{remark}[Not an exact causal support theorem]
Spacetime convolution can sample a metric input slightly to the
future of a coordinate source point. Proposition~\ref{prop:support}
is therefore an \(\eps\)-thickened support statement, not exact
causal dependence in the original field arguments. No causal
factorization theorem is inferred from it.
\end{remark}

\subsection{Fixed-regulator test continuity}

If \(f\) is allowed to vary in \(\Dtest(O)\), the maps
\[
f\longmapsto u_1,\qquad f\longmapsto u_2
\]
are continuous into the smooth compact-kernel spaces on each fixed
support stage, and hence in their test-function LF topologies.
Indeed, \eqref{eq:uone} and \eqref{eq:utwo} use only finite
differentiation and multiplication by fixed smooth functions on
common compact sets.

All estimates are allowed to depend on \(\eps\). No uniform estimate
as \(\eps\) changes is asserted.

%=====================================================================
\section{The switched scalar solution and the same obstruction}
\label{sec:scalarsolution}
%=====================================================================

\subsection{Action-derived scalar coefficients}

Let \(\rho_0\) be any smooth free scalar,
\(\Boxeta\rho_0=0\). Define
\[
\rho_g
=
\rho_0+\kappa\rho_{1,g}+\kappa^2\rho_{2,g}\modthree
\]
by the switched scalar equation and zero positive-order Cauchy data.
For the metric jet,
\begin{align}
\Boxeta\rho_{1,g}
&=-V_{1,g}[h]\rho_0,
\label{eq:rhooneeq}\\
\Boxeta\rho_{2,g}
&=-V_{1,g}[h]\rho_{1,g}-W_g[h,k]\rho_0.
\label{eq:rhotwoeq}
\end{align}
Hence
\begin{align}
\rho_{1,g}
&=-\Ezero V_{1,g}[h]\rho_0,
\label{eq:rhoone}\\
\rho_{2,g}
&=
\Ezero V_{1,g}[h]\Ezero V_{1,g}[h]\rho_0
-\Ezero W_g[h,k]\rho_0.
\label{eq:rhotwo}
\end{align}
The sources are smooth and compactly supported because their
coefficients contain \(g\) or its derivatives.

Evaluation of the relational functional on this scalar solution gives
\begin{align}
\Phi_g^{[0]}&=\rho_0,
\label{eq:evalzero}\\
\Phi_g^{[1]}&=\rho_{1,g}-X_w\rho_0,
\label{eq:evalone}\\
\Phi_g^{[2]}&=
\rho_{2,g}-X_w\rho_{1,g}
+\mathcal J(w)\rho_0-X_v\rho_0.
\label{eq:evaltwo}
\end{align}
Square brackets distinguish evaluated solution coefficients from the
bare coefficients \eqref{eq:Ozero}--\eqref{eq:Otwo}.

\subsection{Scalar covariance proved for the actual action}

For a metric jet, set
\[
A_g[q]=(1-g)\eta+gA(q).
\]
The density operator is \(D_g[q]\rho=\partial_\mu
(A_g[q]^{\mu\nu}\partial_\nu\rho)\) through the retained order.

\begin{lemma}[Scalar density covariance on the frozen plateau]
\label{lem:scalarcov}
For a diffeomorphism generated by \(\mathfrak g_U\),
\[
D_g[F^*q](F^*\rho)=F_{\mathrm{dens}}^*(D_g[q]\rho),
\]
coefficientwise through the retained order. The scalar solution with
fixed free Cauchy data therefore obeys
\[
\rho_g[F_\kappa^*q;\rho_0]
=
F_\kappa^*\rho_g[q;\rho_0]\modthree.
\]
\end{lemma}

\begin{proof}
The curved density \(A(q)\) is natural under pullback, and \(F^*g=g\).
The difference between \(F^*A_g[q]\) and \(A_g[F^*q]\) is
\[
(1-g)(F^*\eta-\eta),
\]
where \(\eta\) in this expression denotes its contravariant density.
On the transformation support \(g=1\), and outside that support \(F\)
and its derivative are the identity. The difference is zero.
Divergence of a vector density is natural, proving the operator
identity.

The pullback solution has the same Cauchy data because \(F\) is the
identity near \(\Sigma\). Successive uniqueness of
\eqref{eq:rhooneeq}--\eqref{eq:rhotwoeq} proves the solution identity.
\end{proof}

This proof concerns the switched scalar problem in
\eqref{eq:switchedaction}. It does not establish covariance of the
auxiliary metric map \(q\mapsto\widehat q_{g,\eps}\).

\begin{theorem}[The defect survives evaluation on the switched solution]
\label{thm:evaluatedgap}
Take \(\rho_0=x^1\), the fixed \(f,\xi_*\), and the pure-gauge metric
jet obtained by pulling back the flat metric. Then the difference
between the first evaluated relational coefficient and its flat
value is
\[
\int f\Phi_g^{[1]}=c_\eps>0.
\]
\end{theorem}

\begin{proof}
Lemma~\ref{lem:scalarcov} gives
\[
\rho_{1,g}=X_{\xi_*}\rho_0=\psi.
\]
Lemma~\ref{lem:preflight} gives
\[
X_w\rho_0=\Seps\psi.
\]
Equation \eqref{eq:evalone} and
Lemma~\ref{lem:positivegap} give the result.
\end{proof}

No scalar free-shell transformation is being assumed here. The scalar
is the forced solution of \eqref{eq:rhooneeq}, and the comparison is
between solutions selected by the same action and Cauchy prescription.

%=====================================================================
\section{Independent action variation and the plateau Noether identity}
\label{sec:noether}
%=====================================================================

The preceding defect was obtained from the coordinate problem.
We now vary the action independently. This calculation shows that
the plateau action identity does not remove the regulator defect.

\subsection{The fixed divergence relating the gravitational densities}

Define the formal vector density
\begin{equation}
B^\mu(\kappa,h)
=
-\frac2{\kappa^2}A(q)^{\alpha\beta}\Gamma^\mu_{\alpha\beta}
+\frac2{\kappa^2}A(q)^{\mu\alpha}\Gamma^\beta_{\alpha\beta}.
\label{eq:Bdensity}
\end{equation}
Direct integration by parts in the coordinate formula for \(R\)
gives
\begin{equation}
-\frac2{\kappa^2}d(q)R[q]
=
\ell_{\mathrm{grav}}+\partial_\mu B^\mu.
\label{eq:Hilbertsplit}
\end{equation}
To verify the identity, expand the derivative terms in
\(d(q)q^{\alpha\beta}R_{\alpha\beta}\), move the two derivatives
of \(\Gamma\) into the displayed divergence, and use metric
compatibility to differentiate \(A(q)^{\alpha\beta}\).
The remaining terms are exactly
\[
A(q)^{\alpha\beta}
\left(
\Gamma^\rho_{\alpha\sigma}\Gamma^\sigma_{\beta\rho}
-\Gamma^\rho_{\alpha\beta}\Gamma^\sigma_{\rho\sigma}
\right),
\]
with the coefficient \(-2/\kappa^2\). This proves
\eqref{eq:Hilbertsplit} without altering the switched representative.

The formal series \(B^\mu\) may begin at \(\kappa^{-1}\).
The density \(\ell\) in \eqref{eq:ellseries} begins at
\(\kappa^0\).

\subsection{Local current before switching}

For the infinitesimal action \eqref{eq:infaction}, naturality of
the Hilbert and scalar densities gives
\[
\delta_\xi(\ell+\partial_\mu B^\mu)
=
\kappa\partial_\mu
\bigl(\xi^\mu(\ell+\partial_\nu B^\nu)\bigr).
\]
Consequently
\begin{equation}
\delta_\xi\ell=\partial_\mu\mathcal N^\mu,
\qquad
\mathcal N^\mu
=
\kappa\xi^\mu\ell
+\kappa\xi^\mu\partial_\nu B^\nu
-\delta_\xi B^\mu.
\label{eq:current}
\end{equation}
This formula explicitly defines a finite-order local current
coefficient at every retained order. Negative-power current
coefficients, if present, have identically zero divergence because
the left side has no negative powers.

Write
\[
n_j^\mu=\coeff{\mathcal N^\mu}{j},
\qquad j=0,1,2.
\]
Coefficient comparison gives
\begin{align}
\delta_\xi^{(0)}\ell_0
&=\partial_\mu n_0^\mu,
\label{eq:Noetherzero}\\
\delta_\xi^{(0)}\ell_1+\delta_\xi^{(1)}\ell_0
&=\partial_\mu n_1^\mu,
\label{eq:Noetherone}\\
\delta_\xi^{(0)}\ell_2+\delta_\xi^{(1)}\ell_1
&=\partial_\mu n_2^\mu.
\label{eq:Noethertwo}
\end{align}
All \(n_j\) depend locally on the fields and a finite jet of \(\xi\).
They vanish outside the support of that jet.

\subsection{Switch and boundary terms exposed}

Applying \eqref{eq:Noetherzero}--\eqref{eq:Noethertwo} to
\eqref{eq:switchedaction} gives, modulo \(\kappa^3\),
\begin{align}
\delta_\xi S_g^{[2]}
={}&
\int\partial_\mu n_0^\mu
\nonumber\\
&+\kappa\left[
\int\partial_\mu(gn_1^\mu)
-\int(\partial_\mu g)n_1^\mu
+\int(1-g)\delta_\xi^{(1)}\ell_0
\right]
\nonumber\\
&+\kappa^2\left[
\int\partial_\mu(gn_2^\mu)
-\int(\partial_\mu g)n_2^\mu
\right].
\label{eq:switchedNoether}
\end{align}
This identity is the varied-action calculation, not an inference from
an intertwiner.

\begin{proposition}[Classical action identity on the frozen algebra]
\label{prop:actionidentity}
For every \(\xi\in\mathfrak g_U\),
\[
\delta_\xi S_g^{[2]}=O(\kappa^3)
\]
as a formal variational identity.
\end{proposition}

\begin{proof}
Every current in \eqref{eq:switchedNoether} is compactly supported
in the gauge-support region. Thus the spatial and temporal boundary
integrals vanish.

On a neighborhood of this region \(g=1\) and \(dg=0\).
Therefore
\[
(\partial_\mu g)n_j^\mu=0,
\qquad
(1-g)\delta_\xi^{(1)}\ell_0=0.
\]
All retained terms in \eqref{eq:switchedNoether} vanish.
The discarded term from varying \(\kappa^2g\ell_2\) with
\(\kappa\delta_\xi^{(1)}\) is of order \(\kappa^3\).
\end{proof}

\subsection{Scalar sign check}

At first order,
\[
\ell_{\mathrm{sc},1}
=-\tfrac12h^{\mu\nu}T_{\mu\nu}(\eta,\phi).
\]
The off-shell identity
\[
\partial_\mu T^{\mu\nu}=(\Boxeta\phi)\partial^\nu\phi
\]
gives
\[
\delta_\xi^{(0)}\int g\ell_{\mathrm{sc},1}
=
\int g(\Boxeta\phi)X_\xi\phi
+\int(\partial_\mu g)T^{\mu\nu}\xi_\nu.
\]
Meanwhile the free scalar variation is
\[
\delta_\xi^{(1)}S_{\mathrm{sc},0}
=
-\int(\Boxeta\phi)X_\xi\phi.
\]
Their sum is
\begin{equation}
\int(g-1)(\Boxeta\phi)X_\xi\phi
+\int(\partial_\mu g)T^{\mu\nu}\xi_\nu.
\label{eq:scalarNoethercheck}
\end{equation}
Both terms vanish for \(\xi\in\mathfrak g_U\). This explicitly
checks the scalar Euler and switch signs.

\begin{remark}[What this identity does not say]
The regulator does not enter the action, so there is no regulator
term in \eqref{eq:switchedNoether}. It enters the coordinate
functional and produces \eqref{eq:infdefect}. An action Noether
identity does not turn a non-natural regulated coordinate map into
a natural one.

No quantum Ward identity is contained in
Proposition~\ref{prop:actionidentity}.
\end{remark}

%=====================================================================
\section{The complete second-order pure-gauge test}
\label{sec:puregauge}
%=====================================================================

\subsection{A genuine pure-gauge jet}

For \(\xi\in\mathfrak g_U\) and a real parameter \(\lambda\), take
\begin{equation}
q_\lambda
=
\exp(\lambda\kappa\Lie_\xi)\eta\modthree,
\qquad
\phi_\lambda
=
\exp(\lambda\kappa X_\xi)\phi_0\modthree.
\label{eq:puregauge}
\end{equation}
Thus
\begin{align}
h_\lambda&=\lambda K\xi,
&
k_\lambda&=\tfrac{\lambda^2}{2}\Lie_\xi^2\eta,
\label{eq:puremetriccoeff}\\
\phi_\lambda
&=
\phi_0+\lambda\kappa X_\xi\phi_0
+\tfrac{\lambda^2\kappa^2}{2}X_\xi^2\phi_0.
\label{eq:purescalarcoeff}
\end{align}
The straight-line metric \(\eta+\lambda\kappa K\xi\) is not
substituted for \eqref{eq:puregauge}.

Let
\begin{equation}
a=\Seps\xi,\qquad
d_\xi
=
\tfrac12\Teps(\Lie_\xi^2\eta)
-\tfrac12\Lie_a^2\eta.
\label{eq:differencejet}
\end{equation}
Then the regulated auxiliary jet is
\[
\widehat q_\lambda
=
\eta+\lambda\kappa Ka
+\lambda^2\kappa^2
\left(\tfrac12\Lie_a^2\eta+d_\xi\right).
\]

\begin{lemma}[Coordinate coefficients for the pure-gauge test]
\label{lem:purecoordinates}
For \eqref{eq:puregauge},
\begin{equation}
w_\lambda=\lambda a,
\qquad
v_\lambda^\mu
=
\lambda^2\left(
\tfrac12a^\nu\partial_\nu a^\mu
+\Ezero C^\mu(d_\xi)
\right).
\label{eq:purecoordcoeff}
\end{equation}
All coefficient Cauchy data vanish.
\end{lemma}

\begin{proof}
The first assertion is Lemma~\ref{lem:preflight}.

For the second, first consider the auxiliary pure-gauge metric
\[
\widetilde q_\lambda
=\exp(\lambda\kappa\Lie_a)\eta.
\]
The functions
\[
\widetilde X_\lambda^\mu
=
\exp(\lambda\kappa X_a)x^\mu
=
x^\mu+\lambda\kappa a^\mu
+\tfrac{\lambda^2\kappa^2}{2}
a^\nu\partial_\nu a^\mu
\]
solve its harmonic equation. This follows directly from scalar
naturality of the wave operator applied to
\(\Boxeta x^\mu=0\). Since \(a\) vanishes near \(\Sigma\), their
data are the frozen coordinate data. Uniqueness therefore proves
these are its coordinate coefficients.

The actual auxiliary jet has the same first coefficient and an
additional second coefficient \(\lambda^2d_\xi\).
Equation \eqref{eq:v} is linear in that second coefficient, so it
adds \(\lambda^2\Ezero Cd_\xi\).
The data statement follows from the support of \(a,d_\xi\) and
the zero-data inverse.
\end{proof}

This proof uses naturality of an auxiliary unregulated harmonic
problem as an explicitly verified comparison. It does not assert
naturality of \(q\mapsto\widehat q_{g,\eps}\).

\subsection{The relational coefficients on that jet}

\begin{proposition}[Pure-gauge relational defect through second order]
\label{prop:pureobservable}
Substitution of \eqref{eq:puregauge} into the regulated relational
functional gives
\begin{align}
\Phi_\lambda^{[1]}
&=\lambda X_{\xi-a}\phi_0,
\label{eq:pureOone}\\
\Phi_\lambda^{[2]}
&=
\lambda^2\left[
\tfrac12X_\xi^2
-X_aX_\xi
+\tfrac12X_a^2
-X_{\Ezero Cd_\xi}
\right]\phi_0
\nonumber\\
&=
\lambda^2\left[
\tfrac12X_{\xi-a}^2
+\tfrac12X_{[\xi,a]}
-X_{\Ezero Cd_\xi}
\right]\phi_0.
\label{eq:pureOtwo}
\end{align}
For \(\xi=\xi_*\), \(\phi_0=x^1\), and the frozen \(f\),
the first smearing is \(\lambda c_\eps\).
\end{proposition}

\begin{proof}
The coefficient substitution in
\eqref{eq:evalone}--\eqref{eq:evaltwo}, with
\eqref{eq:purescalarcoeff} and \eqref{eq:purecoordcoeff}, gives
the first line and
\[
\tfrac12X_\xi^2-X_aX_\xi
+\mathcal J(a)
-X_{\frac12X_aa+\Ezero Cd_\xi}.
\]
The identity
\[
\mathcal J(a)=\tfrac12X_a^2+\tfrac12X_{X_aa}
\]
reduces this to the first expression in \eqref{eq:pureOtwo}.
Finally,
\[
X_\xi^2-2X_aX_\xi+X_a^2
=
X_{\xi-a}^2+[X_\xi,X_a]
=
X_{\xi-a}^2+X_{[\xi,a]}.
\]
The smearing statement is Lemma~\ref{lem:positivegap}.
\end{proof}

The coordinate equations and data have therefore been checked on
the full second-order pure-gauge jet. The check does not yield
cancellation: the first coefficient is already nonzero, and the
second coefficient records the additional retained regulator terms.

%=====================================================================
\section{A nonzero forced scalar example and a symbolic diagnostic}
\label{sec:sourced}
%=====================================================================

This section tests the action-derived scalar response. It does not
claim a coupled solution of the switched Einstein equation.

Take
\begin{equation}
h_{\mu\nu}=\eta_{\mu\nu}\psi,\qquad k=0,\qquad \rho_0=t,
\label{eq:conformaltest}
\end{equation}
with the fixed bump \(\psi\) from Section~\ref{sec:sheet}.
Since \(\supp\psi\Subset O\), \(g\psi=\psi\).

\begin{proposition}[A nonzero derived source]
\label{prop:sourced}
For \eqref{eq:conformaltest},
\[
a_1(h)^{\mu\nu}=\psi\eta^{\mu\nu},
\qquad a_2(h)^{\mu\nu}=0,
\]
and
\begin{equation}
V_{1,g}[h]\rho_0=\partial_t\psi\not\equiv0.
\label{eq:nonzerosource}
\end{equation}
The selected zero-data correction is
\begin{equation}
\rho_{1,g}
=
-\Ezero\partial_t\psi
=
-\partial_t\Ezero\psi,
\label{eq:sourcedresponse}
\end{equation}
and it is not identically zero. The second correction is
\begin{equation}
\rho_{2,g}
=
\Ezero V_{1,g}[h]\Ezero\partial_t\psi.
\label{eq:sourcedsecond}
\end{equation}
\end{proposition}

\begin{proof}
Here \(\tr h=4\psi\) and
\(h_{\alpha\beta}h^{\alpha\beta}=4\psi^2\).
Equations \eqref{eq:aone}--\eqref{eq:atwo} give
\(a_1(h)=\psi\eta\) and \(a_2(h)=0\).
Since \(\partial_\nu t=\delta_\nu^0\),
\[
\partial_\mu(\psi\eta^{\mu\nu}\partial_\nu t)
=\partial_t\psi.
\]
The chosen bump has nonconstant time dependence, proving
\eqref{eq:nonzerosource}.

Equation \eqref{eq:rhoone} gives the first formula in
\eqref{eq:sourcedresponse}. The source \(\psi\) vanishes near
\(\Sigma\), so \(\partial_t\Ezero\psi\) has zero value and first
time derivative there and solves
\(\Boxeta(\partial_t\Ezero\psi)=\partial_t\psi\).
Uniqueness gives the second formula. If \(\rho_{1,g}=0\), its
equation would imply \(\partial_t\psi=0\), a contradiction.
Equation \eqref{eq:rhotwo} and \(V_{2,g}[h]=0\) give
\eqref{eq:sourcedsecond}.
\end{proof}

For the auxiliary coordinates, put \(\chi=\Seps\psi\).
Then \(H=\eta\chi\), and
\begin{equation}
w^\mu=-\Ezero\partial^\mu\chi,
\qquad
v^\mu=-\Ezero\partial_\alpha
\bigl(\chi\partial^\alpha w^\mu\bigr).
\label{eq:conformalcoordinates}
\end{equation}
These are the actual regulated coordinate coefficients for this
test, not the unregulated coordinates.

\scopebox{Symbolic diagnostic only}{%
For a formal conformal metric \(q=(1+\kappa\psi)\eta\) in four
dimensions,
\[
\sqrt{-\det q}\,q^{\mu\nu}
=(1+\kappa\psi)\eta^{\mu\nu}.
\]
Thus its quadratic massless scalar density coefficient vanishes
exactly. This independently checks \(a_2(\eta\psi)=0\) and the absence
of a second local scalar vertex in
\eqref{eq:sourcedsecond}. It is a finite algebraic diagnostic, not a
proof of gauge covariance, not a numerical simulation, and not an
executed computer-algebra calculation.}

The metric profile in \eqref{eq:conformaltest} is an off-shell
configuration used in a scalar Cauchy problem. It is not declared to
satisfy \eqref{eq:metricEuler}. No externally added force has been
introduced: the nonzero source is the variation of the frozen
scalar interaction evaluated on this configuration.

%=====================================================================
\section{Controls, dependencies, and the exact release boundary}
\label{sec:release}
%=====================================================================

\subsection{Zero-switch and flat controls}

If the switch is specialized to zero in both the action and the
coordinate prescription, then
\[
\Teps h=\Teps k=0,\qquad w=v=0,
\]
and
\[
\Ocal_{f,0,\eps}[h,k,\phi]=\int f\phi\modthree.
\]
The scalar interaction vertices also vanish, so
\(\rho_{1,0}=\rho_{2,0}=0\).

This is a normalization specialization, not a gauge theorem for a
nonempty plateau at \(g=0\). The frozen gauge algebra belongs to
the nonzero switch \eqref{eq:switch}.

If \(h=k=0\) with the original frozen switch, then likewise \(w=v=0\).
These checks fix the insertion and response signs. They do not
cancel the nonzero pure-gauge witness.

\subsection{Dependency audit}

\begin{center}
\small
\renewcommand{\arraystretch}{1.15}
\begin{tabularx}{\linewidth}{@{}XXX@{}}
\toprule
Proved here & Assumed or frozen & Imported statement actually used\\
\midrule
Regulated coefficient equations and uniqueness
&
Smooth off-shell configurations; fixed convolution and data
&
Standard smooth wave Cauchy theory and finite propagation
\cite{BGP}; the estimates used are recorded in
Lemma~\ref{lem:Cauchy}.\\

Exact first-order regulator gap and positive smearing
&
The explicit \(g,U,O,\eps,f,\xi_*\)
&
None.\\

Second-order bare coefficients and retained defects
&
Formal equality modulo \(\kappa^3\)
&
None.\\

Plateau scalar solution identity
&
The action-level switching in
\eqref{eq:switchedaction}
&
VII-A is comparison only: its coordinates are unswitched.
No theorem about the present regulator is imported.\\

Varied-action Noether identity
&
The curvature and density representative fixed here
&
None of the series intertwiners is used in this proof.\\

Off-shell domain and kernel continuity
&
Compact-open smooth topology and fixed-regulator kernel topology
&
VII-B and IX-A are comparison only. Neither supplies the topology
or a product for this construction.\\

Coefficientwise non-repair statement
&
The first coefficient is held fixed
&
VI provides the comparison principle that different formal orders
must not be conflated; the argument here is proved directly.\\

No quotient or representation conclusion
&
No on-shell or gauge-fixed quotient
&
III is used only to identify a different equation-space question.\\
\bottomrule
\end{tabularx}
\end{center}

VII-C's hypotheses are not invoked or tested here. No sharp-limit
argument is attempted. IX-E-2 and IX-E-3 supply no premise.

\subsection{Decision table}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.52\linewidth}X@{}}
\toprule
Evidence & Decision\\
\midrule
Regulator, switched problem, data, and coefficients specified
&
Not unresolved at a definition or uniqueness obligation\\

\(\Lg(K\xi)=\Seps\xi\), with zero data preserved
&
The failed identity is coordinate covariance, not the data\\

Exact positive \(c_\eps\) for the frozen test and gauge vector
&
Proved obstructed on \(\mathfrak g_U\)\\

Action Noether terms vanish on the same algebra
&
Action compatibility does not repair the coordinate regulator\\

Second-order pure-gauge coefficients explicitly retained
&
No second-order cancellation of the first-order gap\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{What may and may not be handed to X-B}

The available classical material is:
\begin{enumerate}
\item the off-shell coefficients
\eqref{eq:Ozero}--\eqref{eq:Otwo};
\item the joint kernels
\eqref{eq:uone} and \eqref{eq:utwo};
\item the named configuration and test spaces;
\item the support and continuity statements of
Section~\ref{sec:kernels};
\item the action identity \eqref{eq:switchedNoether};
\item the nonzero regulator defect \eqref{eq:infdefect};
\item the single first interaction
\[
V_1(g)=\int g\ell_1,
\]
including both its gravitational and scalar terms.
\end{enumerate}

There is \emph{no cleared classical gauge-invariant handoff} for this
prescription. A later paper cannot treat the defect as absent.
A changed regulator or coordinate prescription would define a new
problem and would require a new classical proof before being used
as the invariant input of X-B.

No kernel is labeled quantum-admitted here. No product, extension,
causal factorization identity, or quantum normalization is defined.
Smooth classical kernels do not by themselves constitute the
required end-to-end quantum construction.

\scopebox{Fixed-regulator theorem boundary}{%
Every theorem in this manuscript concerns the fixed nonzero
\(\eps\) in \eqref{eq:epsrange}. No limit as \(\eps\to0\), no
topology for that limit, and no identification with a sharp Cauchy
insertion are asserted.

The result neither constructs nor obstructs a boundary algebra.
It does not alter VII-C, identify the regulated coordinates with an
IX kernel, or use a \(2+1\)-dimensional cone model.}

%=====================================================================
\section{Conclusion}
%=====================================================================

The prescribed regulator produces a complete classical coefficient
calculation, but not a covariant relational scalar.

The decisive identity is
\[
\Ezero C\Seps(gK\xi)=\Seps\xi,
\qquad \xi\in\mathfrak g_U,
\]
where covariance would require \(\xi\). The Cauchy data are preserved
and the plateau switch terms are controlled. Nevertheless, for the
one observation test and gauge vector frozen in the convention sheet,
\[
\int f(\xi_*-\Seps\xi_*)^\mu\partial_\mu x^1
=c_\eps>0.
\]
The same positive coefficient occurs after evaluation on the scalar
solution of the switched action.

The second-order expansion includes the nonlinear metric gauge
action, the genuine second metric coefficient in a pure-gauge jet,
the inverse-coordinate derivative term, and the quadratic Taylor
term. None changes the defective first coefficient.

Thus the precise closing statement is:
\begin{quote}
For the frozen switched action, spacetime-convolution regulator,
Cauchy prescription, smooth off-shell configuration domain, and
plateau-supported gauge algebra, the declared relational scalar
fails gauge invariance already at order \(\kappa\), and therefore
fails gauge invariance through order \(\kappa^2\), at fixed
\(\eps>0\).
\end{quote}
This is an obstruction to this prescription, not to every regulated
relational construction.

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 and regulator audit}
%=====================================================================

\subsection{The three signs that enter the witness}

With signature \(+---\),
\[
\Boxeta=\partial_t^2-\Delta,
\qquad
\Ezero s
=
+\int_0^t
\frac{\sin((t-u)|D|)}{|D|}s(u)\dd u.
\]
The scalar action has a positive kinetic density, so its Euler
derivative is \(-D_g[q]\phi\). The first scalar coefficient satisfies
\[
\Boxeta\rho_{1,g}=-V_{1,g}[h]\rho_0,
\qquad
\rho_{1,g}=-\Ezero V_{1,g}[h]\rho_0.
\]
The inverse coordinate map begins with \(-w\), so the first bare
relational insertion is
\[
-\int fX_w\phi.
\]
Consequently the scalar pullback and coordinate variation combine as
\[
+\int fX_\xi\phi-\int fX_{\Seps\xi}\phi,
\]
which has the positive witness \eqref{eq:positiveenergy}.

\subsection{Why the regulator is nontrivial}

The regulator smooths every original metric input slot before the
harmonic problem is solved. Its joint kernels are the smooth
parameter families of Section~\ref{sec:kernels}. It does not merely
move an initial surface and does not average identical solutions.

The strict identity \eqref{eq:positiveenergy} proves that this
regularization is nontrivial on an admitted gauge direction.
That nontrivial action is exactly what breaks the required
right-inverse identity.

\subsection{Why the edge condition is not changed}

The gauge vector \(\xi_*=\psi\partial_1\) is supported in the smaller
observation region \(O\Subset U\). Thus it satisfies the original
edge and initial-data conditions with room to spare. The witness
does not exploit a gauge vector meeting \(dg\), \(\Sigma\), or the
temporal cutoff.

Removing \(\xi_*\) after the calculation would change the frozen
problem. No such removal is made.

\subsection{Why classical smoothness is not a quantum conclusion}

The coefficient kernels proved smooth here concern one explicitly
modified off-shell classical functional. Their existence does not
identify that functional with the sharp insertion studied in VII-B
or IX-A. It also does not construct products with the selected
interaction, prove a quantum gauge identity, or supply a causal
algebra.

In addition, the regulator-thickened support bound is not exact
causal dependence in the original metric arguments. A later causal
construction would have to confront this fact explicitly.

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

\paragraph{Nature of the release.}
This manuscript gives the proved-obstructed branch for one explicitly
frozen classical prescription. It does not present an obstructed
construction as a gauge-invariant observable.

\paragraph{Relation to the supplied series.}
The supplied manuscripts are cited only for the specific comparison
statements identified in Section~\ref{sec:introduction}.
No series theorem about a sharp insertion, boundary topology,
free product, or support certificate is used as a theorem about the
regulated coordinate map defined here.

\paragraph{Classical and quantum scope.}
All products used in the proofs are products of smooth classical
functions or pairings with smooth compact kernels. No quantum
contraction or renormalization operation is performed.

\paragraph{Coupled equations.}
The action and its retained Euler equations are specified. The
sourced diagnostic solves the scalar equation for an off-shell
metric configuration. It is not a coupled Einstein--scalar
existence result.

\paragraph{Computation and verification.}
The positive witness is analytic. No numerical calculation or
executed symbolic computation is used. No successful
\texttt{pdflatex} compilation, proof-assistant verification, or
independent referee review is claimed.

%=====================================================================
\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 only for standard smooth wave Cauchy theory and
finite propagation.

\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 Relational-coordinate comparison background only;
no quantum result is transferred to the present regulator.

\bibitem{III}
\emph{Descent of first-order relational scalars, and the free-product
obstruction}.
\newblock Landmark III, unpublished manuscript supplied with this
project, version as supplied in \texttt{III.tex}.
\newblock Used only to distinguish its equation-space and quotient
questions from the off-shell problem studied here.

\bibitem{VI}
\emph{Relational Einstein--scalar observables at second order:
classical descent and a compact-switch obstruction}.
\newblock Landmark VI, unpublished manuscript supplied with this
project, version as supplied in \texttt{VI.tex}.
\newblock Comparison for the separation of geometric descent,
switched assignments, and coefficientwise obstructions.

\bibitem{VIIA}
\emph{Plateau-compatible relational scalars:
a second-order classical intertwiner}.
\newblock Landmark VII-A, unpublished manuscript supplied with this
project, version as supplied in \texttt{VII-A.tex}.
\newblock Comparison only; its harmonic-coordinate equation is
unswitched and unregulated.

\bibitem{VIIB}
\emph{Cauchy harmonic insertions and second-order retarded products:
a sharp-boundary obstruction in an off-shell polynomial domain}.
\newblock Landmark VII-B, unpublished manuscript supplied with this
project, version as supplied in \texttt{VII-B.tex}.
\newblock Comparison only; its sharp insertion and quantum domain
are not the definitions of the present classical prescription.

\bibitem{IXA}
\emph{An off-shell admission domain and boundary-sensitive topology
for sharp Cauchy relational insertions}.
\newblock Landmark IX-A, unpublished manuscript supplied with this
project, version as supplied in \texttt{IX-A.tex}.
\newblock Comparison only; its carrier and topology are not imported,
and no quantum product is inferred.

\end{thebibliography}

\end{document}