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\title{
Relational Einstein--scalar observables at second order:\\
classical descent and a compact-switch obstruction
}
\author{}
\date{}

\begin{document}

\maketitle

\begin{abstract}
We examine the compatibility of a finite-time harmonic relational
prescription with compactly switched Einstein--scalar perturbation theory.
For a specified second-order Einstein--scalar solution-jet space, we
construct harmonic coordinate coefficients with fixed Cauchy data and
prove that the resulting scalar functional descends under formal
diffeomorphisms that are trivial near the initial surface. The metric is
an unknown of the coefficient equations; an explicit sourced family
establishes nonvacuity.

The same geometric dressing does not automatically descend after the
scalar correction is replaced by the retarded response of a compactly
switched vertex. We derive the exact first-order switch defect and give
two representatives with a strictly nonzero smeared gap, even though the
switch equals one near the observation smearing. No correction beginning
at second order can remove this first-order defect while that
representative identification and lower-order prescription are retained.

We distinguish the classical second-order coefficient from the
four-term expansion of a renormalized interacting composite. We also
state and prove the finite-dimensional certificate criterion, including
its catalogue-completeness and finite-sampling requirements.
No renormalized second-order relational product, complete
finite-renormalization response space, or radiative response matrix is
constructed here. The quantum certification problem is therefore
\textnormal{RG-open}. The closed result is the classical descent theorem
together with the compact-switch obstruction, not a quantum
renormalization separator.
\end{abstract}

\noindent
\fbox{%
\begin{minipage}{0.945\textwidth}
\textbf{Scope.}
This paper proves a second-order classical descent statement and a
first-order obstruction to a specified switched realization of that
statement. It does not prove a quantum Ward identity or a
scheme-independent radiative correction.

\smallskip
\textbf{Excluded endpoints.}
No ultraviolet completion, adiabatic limit, interacting Hilbert-space
construction, localized probe, dense transverse-traceless
representability theorem, or global multi-graviton contact
classification is asserted.
\end{minipage}}

\tableofcontents

\section{The problem and the corrected logical order}

\subsection{One proposed construction, two different maps}

A relational scalar functional can be defined geometrically before an
interaction switch is introduced. A retarded interacting field can also
be defined perturbatively after an interaction is multiplied by a
compact test function. These operations do not commute merely because
both use retarded inverses.

The distinction is between
\begin{equation}
 \text{solution jet}
 \longmapsto
 \text{relational scalar}
 \label{eq:geometric-arrow}
\end{equation}
and
\begin{equation}
 \text{free input}
 \longmapsto
 \text{switched retarded correction}
 \longmapsto
 \text{dressed expression}.
 \label{eq:switched-arrow}
\end{equation}
The first map has a natural coefficient gauge action inherited from
diffeomorphisms. The second requires an intertwining statement between
that action and the switched dynamics. We prove the first statement
through second order and exhibit failure of the second for a specified
naive identification.

This obstruction must be resolved before a finite-renormalization
separator is interpreted as a gauge-invariant response.

Figure~\ref{fig:VI-two-maps} separates the geometric map from
the proposed switched realization and records the required
logical order.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[sbox, text width=11.6cm] (top) {classical descent through order $\kappa^2$: proved};
\node[mbox, below=of top] (j) {$\mathcal J_T$};
\node[mbox, right=of j] (x) {$X_\gamma$};
\node[mbox, right=of x] (y) {$Y_\gamma$};
\node[sbox, right=of y] (phi) {$\phi\circ Y_\gamma$};
\node[sbox, right=of phi] (o) {$\mathcal O_f$};
\draw[arr] (j)--(x); \draw[arr] (x)--(y); \draw[arr] (y)--(phi); \draw[arr] (phi)--(o);
\node[cbox, text width=11.6cm, below=7mm of j] (low) {specified unchanged-free-input identification};
\node[sbox, below=of low] (h) {$(h,\phi_0)$};
\node[sbox, right=of h] (f1) {$-G_{\mathrm R}\mathsf Q_{gh}\phi_0$};
\node[cbox, right=of f1] (f2) {$-G_{\mathrm R}\mathsf Q_{gh}\phi_0-X_{w[h]}\phi_0$};
\draw[arr] (h)--(f1); \draw[arr] (f1)--(f2);
\draw[arrg] (phi.south)--(f1.north) node[cbox, font=\scriptsize, midway, fill=white, align=center] {not supplied automatically\\ by retarded inverses};
\node[gbox, text width=11.6cm, below=5mm of h] (seq) {switch-compatible gauge realization $\longrightarrow$ renormalized products $\longrightarrow$ complete response catalogue $\longrightarrow$ certificate};
\node[cbox, font=\scriptsize, below=of seq] {uncompleted};
\end{tikzpicture}
\caption{Two maps and their compatibility problem. The
geometric construction acts on solution jets and descends
under the named formal gauge group. The specified switched
assignment acts on unchanged free scalar inputs and does
not automatically implement that coefficient action.
Gauge compatibility must be addressed before a quantum
response separator is interpreted as gauge invariant.}
\label{fig:VI-two-maps}
\end{figure}


\subsection{Relation to existing constructions}

Relational constructions and second-order expansions are already part
of perturbative gravitational field theory
\cite{BFHPR,Frob}. In particular, Fr\"ob computed gauge-invariant
one-loop scalar correlation functions and exhibited cancellation between
gauge-dependent self-energy contributions and relational corrections
\cite{Frob}. Neither the existence of relational observables nor the
existence of such a cancellation is claimed as a new result here.

The locally covariant, renormalized Batalin--Vilkovisky treatment provides
a framework for perturbative effective gravity
\cite{BFR}. It does not, by citation alone, establish the Ward identity
of a newly specified boundary prescription or its compatibility with a
fixed compact switch.

Hack treats the linearized Einstein--Klein--Gordon system on on-shell
backgrounds \cite{Hack}. Our background specialization is
\((\eta,0)\). The nonlinear coefficient problem studied below is not a
linearization about a fixed nonzero scalar on Minkowski spacetime.

Retarded products and the Action Ward Identity are developed in
\cite{DF}. The Action Ward Identity concerns the dependence on local
functionals rather than their density representatives. It must not be
confused with a proved anomaly-free diffeomorphism Ward identity for the
composite considered here.

\subsection{Five corrections that determine the scope}

The following distinctions are needed before formulating the theorems.

\begin{enumerate}
\item
The pair \((\eta,0)\), with zero cosmological constant, \emph{is} an exact
Einstein--scalar background solution. A nonzero free scalar on a fixed
Minkowski metric is generally not an exact background solution at
nonzero gravitational coupling.

\item
The bare dressing coefficient is not the first interacting coefficient.
If both constructions exist, the first interacting coefficient is
\begin{equation}
 \OO_f^{(1)}+R_1(V_1;\OO_f^{(0)}).
 \label{eq:first-interacting}
\end{equation}
An external-metric scalar response supplies only the retarded term.

\item
Turning on a second-order source does not retrospectively alter a
previously fixed first-order coefficient. It changes the equations and
data needed at the next order.

\item
A second-order interacting coefficient requires products involving two
first-order vertices. Consequently, the required
\(T_2(V_1,V_1)\) and three-insertion constructions cannot all be postponed
until after that coefficient has supposedly been proved.

\item
A fit on finitely many smearings does not prove a global counterterm
identity. Conversely, a separator is valid only against the
\emph{complete} allowed response space.
\end{enumerate}

\subsection{Closed and unclosed statements}

\begin{center}
\begin{tabular}{p{0.49\textwidth}p{0.41\textwidth}}
\toprule
Object & Status here\\
\midrule
Second-order Einstein--scalar coefficient equations
& Defined with action normalization\\
Finite-time harmonic relational functional
& Constructed classically\\
Descent under the named formal gauge group
& Proved\\
Sourced nonzero second-order example
& Constructed classically\\
Naively switched first-order realization
& Explicit gauge gap\\
Four-term interacting expansion
& Algebraic identity, conditional on products\\
Quantum switched Ward identity
& Not constructed\\
Complete action and composite counterterm catalogue
& Not constructed\\
Evaluated radiative vector and response matrix
& Not constructed\\
Quantum finite-renormalization certificate
& RG-open\\
\bottomrule
\end{tabular}
\end{center}

The unresolved entries are not filled by assigning them theorem names.
The obstruction branch gives a definite result without claiming the
unperformed radiative calculation.

\section{Action, signs, background, and coefficient domain}

\subsection{Conventions}

Let
\[
 M=\R^{1+3},\qquad
 \eta=\operatorname{diag}(-1,1,1,1),\qquad
 \Boxeta=-\partial_t^2+\Delta,
 \qquad P=\Boxeta-m^2,
 \qquad m>0.
\]
Fix a finite time \(T>0\), an initial surface \(t=0\), and the slab
\[
 M_T=[0,T]\times\R^3.
\]
All fields are smooth on a neighborhood of the slab. Compact test
functions used for observations and interactions have support in its
interior.

The metric is denoted by \(\gamma\); the letter \(g\) denotes only an
interaction switch.

For a symmetric tensor \(a\), write
\begin{align}
 \bar a_{\mu\nu}
 &=a_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta a,
 &
 D(a)^\nu&=\partial_\mu\bar a^{\mu\nu},\\
 (K\xi)_{\mu\nu}
 &=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,
 &
 X_\xi&=\xi^\mu\partial_\mu.
\end{align}
Then
\begin{equation}
 D(K\xi)^\mu=\Boxeta\xi^\mu.
 \label{eq:DK}
\end{equation}
The scalar first-variation operator is
\begin{equation}
 L_a=-a^{\mu\nu}\partial_\mu\partial_\nu
       -D(a)^\nu\partial_\nu.
 \label{eq:L}
\end{equation}
In particular,
\begin{equation}
 L_{K\xi}=-[P,X_\xi].
 \label{eq:pure-gauge-operator}
\end{equation}
These identities follow by expanding the derivatives.

\subsection{The action and the source normalization}

The action is
\begin{equation}
 S[\gamma,\phi]
 =
 \frac{2}{\kappa^2}\int\sqrt{-\gamma}\,R[\gamma]\dd^4x
 -
 \frac12\int\sqrt{-\gamma}
 \left(\gamma^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
       +m^2\phi^2\right)\dd^4x.
 \label{eq:action}
\end{equation}
The curvature convention is fixed by requiring
\begin{equation}
 G^{(1)}_{\mu\nu}(a)
 =
 -\frac12\Boxeta\bar a_{\mu\nu}
 +\partial_{(\mu}D(a)_{\nu)}
 -\frac12\eta_{\mu\nu}\partial^\alpha D(a)_\alpha.
 \label{eq:G1}
\end{equation}
Thus
\[
 G^{(1)}K=0,\qquad
 \partial^\mu G^{(1)}_{\mu\nu}(a)=0.
\]

For the matter action in \eqref{eq:action},
\[
 T_{\mu\nu}(\gamma,\phi)
 =
 \partial_\mu\phi\partial_\nu\phi
 -
 \frac12\gamma_{\mu\nu}
 \left(\gamma^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi
       +m^2\phi^2\right).
\]
Variation with respect to the inverse metric gives
\begin{equation}
 G_{\mu\nu}(\gamma)
 =
 \frac{\kappa^2}{4}T_{\mu\nu}(\gamma,\phi),
 \qquad
 (\Box_\gamma-m^2)\phi=0.
 \label{eq:full-equations}
\end{equation}
The factor \(1/4\) belongs to the normalization \(2/\kappa^2\).
It cannot be removed from a coefficient equation without redefining the
coupling or the source.

\subsection{Formal solution jets}

Work coefficientwise modulo \(\kappa^3\), and write
\begin{equation}
 \gamma=\eta+\kappa h+\kappa^2j,
 \qquad
 \phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2.
 \label{eq:jets}
\end{equation}
Define \(G^{(2)}[h,h]\) by
\[
 G(\eta+\kappa h+\kappa^2j)
 =
 \kappa G^{(1)}(h)
 +\kappa^2\bigl(G^{(1)}(j)+G^{(2)}[h,h]\bigr)
 \formal.
\]
Similarly, define \(\cM_h\) by
\begin{equation}
 \Box_{\eta+\kappa h+\kappa^2j}-m^2
 =
 P+\kappa L_h+\kappa^2(L_j+\cM_h)
 \formal.
 \label{eq:wave-expansion}
\end{equation}
An explicit formula for \(\cM_h\) is given in
Section~\ref{sec:scalar-expansion}.

\begin{definition}[Named solution-jet space]
\label{def:jet-space}
The space \(\cJ_T\) consists of real smooth tuples
\((h,j,\phi_0,\phi_1,\phi_2)\) on \(M_T\) satisfying
\begin{align}
 G^{(1)}(h)&=0,
 \label{eq:jet-gravity-one}\\
 G^{(1)}(j)+G^{(2)}[h,h]
 &=\frac14T(\eta,\phi_0),
 \label{eq:jet-gravity-two}\\
 P\phi_0&=0,
 \label{eq:jet-scalar-zero}\\
 P\phi_1&=-L_h\phi_0,
 \label{eq:jet-scalar-one}\\
 P\phi_2&=-L_h\phi_1-(L_j+\cM_h)\phi_0.
 \label{eq:jet-scalar-two}
\end{align}
No assertion that arbitrary unconstrained metric data extend to this
space is included in the definition.
\end{definition}

The coefficient \(\phi_0\) is a free field configuration in a
coupling expansion. It is not being declared the nonzero scalar of an
exact Minkowski Einstein--scalar background.

Figure~\ref{fig:VI-coefficient-domain} organizes the named
coefficient equations and their action-dependent source
normalization.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[mbox, text width=11cm] (bg) {$(\eta,0)$ is the exact background,\quad $G(\gamma)=\kappa^2 T(\gamma,\phi)/4$};
\node[sbox, text width=11cm, below=of bg] (exp) {$\gamma=\eta+\kappa h+\kappa^2 j$,\quad $\phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2$};
\node[gbox, below=of exp] (k0) {$\kappa^0$};
\node[gbox, right=8mm of k0] (k1) {$\kappa^1$};
\node[gbox, right=8mm of k1] (k2) {$\kappa^2$};
\node[mbox, text width=5.2cm, below left=4mm and 0mm of k1] (met) {$G^{(1)}(h)=0$\\ $G^{(1)}(j)+G^{(2)}[h,h]=\frac14 T(\eta,\phi_0)$};
\node[sbox, text width=5.2cm, below right=4mm and 0mm of k1] (sc) {$P\phi_0=0$\\ $P\phi_1=-L_h\phi_0$\\ $P\phi_2=-L_h\phi_1-(L_j+\mathsf M_h)\phi_0$};
\node[cbox, text width=11cm, below=4mm of met] {source factor $1/4$\\ $\phi_0$ is a coupling-expansion coefficient, not an exact nonzero Minkowski scalar background};
\end{tikzpicture}
\caption{Coefficient equations on the named jet space.
The action normalization fixes the second-order metric
source to \(T(\eta,\phi_0)/4\). The free scalar coefficient
belongs to the coupling expansion and is not being treated
as a nonzero exact scalar background on Minkowski spacetime.
The displayed equations define the admissible jet domain.}
\label{fig:VI-coefficient-domain}
\end{figure}


\subsection{The named gauge group}

Let
\[
 \mathfrak g_T=C_c^\infty((0,T)\times\R^3;TM).
\]
Gauge transformations are formal pullbacks
\begin{equation}
 F_{\xi,\zeta}^*
 =
 \exp\bigl(\kappa\Lie_\xi+\kappa^2\Lie_\zeta\bigr)
 \formal,
 \qquad \xi,\zeta\in\mathfrak g_T.
 \label{eq:formal-pullback}
\end{equation}
The corresponding transformations are the identity near the initial
surface. They form a group of formal second-order transformations under
composition. Closure follows from the Baker--Campbell--Hausdorff formula
and closure of \(\mathfrak g_T\) under the Lie bracket.

The use of compact support is a convenient sufficient condition.
No residual-wave gauge group is substituted for this group.

\begin{proposition}[Coefficient gauge action]
\label{prop:gauge-action}
The action \eqref{eq:formal-pullback} induces
\begin{align}
 h'&=h+K\xi,
 \label{eq:h-transform}\\
 j'&=j+\Lie_\xi h+K\zeta+\frac12\Lie_\xi^2\eta,
 \label{eq:j-transform}\\
 \phi_0'&=\phi_0,
 \label{eq:phi0-transform}\\
 \phi_1'&=\phi_1+X_\xi\phi_0,
 \label{eq:phi1-transform}\\
 \phi_2'&=\phi_2+X_\xi\phi_1+X_\zeta\phi_0
                         +\frac12X_\xi^2\phi_0.
 \label{eq:phi2-transform}
\end{align}
It preserves \(\cJ_T\) and all coefficient Cauchy data at \(t=0\).
\end{proposition}

\begin{proof}
Expand the exponential in \eqref{eq:formal-pullback} and collect powers
of \(\kappa\) after substituting \eqref{eq:jets}. This gives the displayed
formulas.

The tensor identity
\[
 G(F^*\gamma)=F^*G(\gamma)
\]
and the scalar identity
\[
 (\Box_{F^*\gamma}-m^2)(F^*\phi)
 =
 F^*((\Box_\gamma-m^2)\phi)
\]
preserve the coefficient equations. These identities follow from the
tensorial definitions of the Einstein tensor and the scalar wave
operator. Equivalently, they may be verified by substituting the
pullback formulas into their coordinate definitions and applying the
chain rule.

Since \(\xi,\zeta\) vanish on a neighborhood of \(t=0\), all displayed
changes and their time derivatives vanish there.
\end{proof}

Figure~\ref{fig:VI-gauge-action} displays the full coefficient
action and the initial-surface support restriction.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[font=\scriptsize, sgray] (sch) {schematic slab $0\le t\le T$};
\draw[sgray] (0,-1.15) -- (4.8,-1.15) node[right, font=\scriptsize] {$t=0$};
\draw[sgray] (0,-1.15) -- (0,1.05) node[above, font=\scriptsize] {$t=T$};
\node[cbox, font=\scriptsize] at (1.5,-0.15) {$\operatorname{supp}\xi$};
\node[cbox, font=\scriptsize] at (3.3,-0.15) {$\operatorname{supp}\zeta$};
\node[mbox, text width=6.8cm, right=6mm of sch] (act) {$F_{\xi,\zeta}^*=\exp(\kappa\mathcal L_\xi+\kappa^2\mathcal L_\zeta)\ \mathrm{mod}\,\kappa^3$\\ $h'=h+K\xi$\\ $j'=j+\mathcal L_\xi h+K\zeta+\frac12\mathcal L_\xi^2\eta$\\ $\phi_0'=\phi_0$,\ $\phi_1'=\phi_1+X_\xi\phi_0$\\ $\phi_2'=\phi_2+X_\xi\phi_1+X_\zeta\phi_0+\frac12 X_\xi^2\phi_0$};
\node[gbox, text width=6.8cm, below=of act] {preserves $\mathcal J_T$; preserves coefficient Cauchy data; not a substituted residual-wave gauge group};
\end{tikzpicture}
\caption{Named formal gauge action. Compact generators
supported away from the initial surface induce simultaneous
metric and scalar coefficient transformations. Covariance
preserves the solution-jet equations, while the support
restriction preserves all coefficient Cauchy data. The
scalar transformations are essential parts of the action.}
\label{fig:VI-gauge-action}
\end{figure}


\section{The linearized interface and a sourced nonempty domain}

\subsection{What linearization at the chosen background gives}

At \((\eta,0)\), the ordinary linearized scalar gauge variation vanishes,
because the background scalar has zero derivative. With canonical metric
normalization, the free Hessian and gauge map are
\begin{equation}
 \cP_0(a,u)=(-2G^{(1)}(a),Pu),
 \qquad
 \cK_0\xi=(K\xi,0).
 \label{eq:free-operators}
\end{equation}
Metric indices in the first entry are raised when it is paired with a
covariant variation.

For compact tests \((J,c)\), use
\[
 \pair{(J,c)}{(a,u)}
 =
 \int J^{\mu\nu}a_{\mu\nu}\dd^4x
 +
 \int cu\dd^4x.
\]
The formal adjoint gauge map is
\begin{equation}
 (\cK_0^*(J,c))_\nu
 =
 -2\eta_{\nu\beta}\partial_\mu J^{\mu\beta}.
 \label{eq:free-Kstar}
\end{equation}

\begin{proposition}[Algebraic observable quotient]
\label{prop:linear-quotient}
The quotient
\begin{equation}
 \cE_{\mathrm{lin}}
 =
 \frac{
 \{(J,c)\in C_c^\infty(S^2TM)\oplus C_c^\infty(M):
                    \partial_\mu J^{\mu\nu}=0\}
 }{
 \cP_0\bigl(C_c^\infty(S^2T^*M)\oplus C_c^\infty(M)\bigr)
 }
 \label{eq:linear-quotient}
\end{equation}
is well defined. Its elements induce gauge-invariant linear
functionals on free solutions.
\end{proposition}

\begin{proof}
The linearized Bianchi identity implies
\(\cK_0^*\cP_0=0\), so the denominator is contained in the numerator.
The free Hessian is formally self-adjoint: it is the second variation of
the quadratic action, and its explicit expression also verifies this by
integration by parts. Consequently, a test in the denominator pairs to
zero with every solution of \(\cP_0(a,u)=0\).

A numerator test annihilates every smooth pure gauge:
\[
 \int J^{\mu\nu}(K\xi)_{\mu\nu}\dd^4x
 =
 -2\int \xi_\nu\partial_\mu J^{\mu\nu}\dd^4x=0.
\]
All integrations by parts are justified by compactness of the tests.
\end{proof}

We do not claim that \eqref{eq:linear-quotient} is a complete
classification of all continuous observables, that its induced pairing
has a specified radical, or that it already carries the nonlinear
relational observable.

Figure~\ref{fig:VI-linear-quotient} summarizes the algebraic
linear-observable quotient and its limited claim.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[gbox] (g) {$\mathfrak g$};
\node[mbox, right=10mm of g] (au) {$(a,u)$};
\node[sbox, right=10mm of au] (eq) {$(-2G^{(1)}(a),Pu)$};
\draw[arr] (g)--(au) node[font=\scriptsize, midway, above] {$\mathcal K_0$};
\draw[arr] (au)--(eq) node[font=\scriptsize, midway, above] {$\mathcal P_0$};
\node[mbox, below=of au] {$\mathcal K_0\xi=(K\xi,0)$};
\node[sbox, text width=9cm, below=of eq] (star) {$\mathcal K_0^*(J,c)=0\Longleftrightarrow\partial_\mu J^{\mu\nu}=0$\\ quotient by $\operatorname{im}\mathcal P_0$;\ $\mathcal K_0^*\mathcal P_0=0$};
\node[gbox, below=of star] (scope) {algebraic gauge-invariant linear functionals only};
\node[cbox, text width=3.4cm, below left=3mm and -6mm of scope] {not a complete continuous classification};
\node[cbox, text width=3.2cm, below=of scope] {not a specified radical};
\node[cbox, text width=3.4cm, below right=3mm and -6mm of scope] {not a nonlinear relational observable};
\end{tikzpicture}
\caption{Linearized observable interface at \((\eta,0)\).
The scalar entry of the linearized gauge map vanishes for
the chosen zero-scalar background. Compact conserved tensor
tests annihilate pure gauges, and field-equation tests
annihilate free solutions. The quotient supplies linear
functionals, not a classification of the nonlinear
relational observable or its quantum realization.}
\label{fig:VI-linear-quotient}
\end{figure}


The decoupling in \eqref{eq:free-operators} is a consequence of the
chosen zero-scalar background. It is not a prescription for assigning a
direct-product symplectic form to a different triangular system with a
fixed nonzero scalar. The general coupled-background construction is
treated in \cite{Hack}.

\subsection{A sourced family in the coefficient space}

The metric in Definition~\ref{def:jet-space} is an unknown, not an
external coefficient. We next exhibit solutions in which it is
nontrivially sourced.

For a spatial Schwartz source define the Newton inverse
\begin{equation}
 (\Delta^{-1}b)(x)
 =
 -\frac1{4\pi}\int_{\R^3}\frac{b(y)}{|x-y|}\dd^3y.
 \label{eq:newton}
\end{equation}
Only this inverse branch is used below.

\begin{proposition}[Nonempty sourced second-order family]
\label{prop:sourced-family}
Let \(\phi_0\) be any real Klein--Gordon solution with compact smooth
Cauchy data. There is a tuple in \(\cJ_T\) with
\[
 h=0,\qquad \phi_1=0,
\]
and with \(j\) satisfying
\[
 D(j)=0,\qquad G^{(1)}(j)=\frac14T(\eta,\phi_0).
\]
For nonzero scalar data the source is nonzero.
\end{proposition}

\begin{proof}
Put
\[
 Q_{\mu\nu}=\frac14T_{\mu\nu}(\eta,\phi_0).
\]
The identity
\[
 \partial^\mu T_{\mu\nu}(\eta,\phi_0)
 =
 (P\phi_0)\partial_\nu\phi_0
\]
shows that \(Q\) is conserved. Finite propagation makes \(Q\) spatially
compact on every bounded time interval.

At \(t=0\), set
\[
 \rho=Q_{00}(0),\qquad J_i=Q_{0i}(0),\qquad
 a=-2\Delta^{-1}\rho,\qquad V_i=2\Delta^{-1}J_i,
\]
and
\[
 B_{ij}=\partial_iV_j+\partial_jV_i-\delta_{ij}\partial_\ell V_\ell.
\]
Prescribe trace-reversed metric data by
\begin{align}
 \bar j_{00}(0)&=a,&
 \bar j_{0i}(0)&=0,&
 \bar j_{ij}(0)&=0,\\
 \partial_t\bar j_{00}(0)&=0,&
 \partial_t\bar j_{0i}(0)&=0,&
 \partial_t\bar j_{ij}(0)&=B_{ij}.
\end{align}
Solve
\begin{equation}
 \Boxeta\bar j=-2Q.
 \label{eq:sourced-reduced}
\end{equation}
The Newton data are smooth, and the finite-time wave Cauchy problem
therefore gives a smooth solution. Existence can also be obtained
directly from the cosine and sine propagators. The multipliers
\(|p|^{-2}\) in the position data are locally integrable in three
spatial dimensions.

Write \(d_\nu=-\partial_t\bar j_{0\nu}+\partial_i\bar j_{i\nu}\).
The data give \(d_\nu(0)=0\). Since
\[
 \Delta a=-2\rho,\qquad \partial_i B_{ij}=2J_j,
\]
equation \eqref{eq:sourced-reduced} gives
\(\partial_t d_\nu(0)=0\). Conservation of \(Q\) implies
\(\Boxeta d_\nu=0\). Wave uniqueness gives \(D(j)=0\), and
\eqref{eq:G1} then gives \(G^{(1)}(j)=Q\).

Finally, solve
\[
 P\phi_2=-L_j\phi_0
\]
with zero Cauchy data. The forcing is smooth and spatially compact on
finite slabs because it contains derivatives of \(\phi_0\).
The resulting tuple satisfies all equations of
Definition~\ref{def:jet-space}.

For nonzero scalar data,
\[
 \int_{\R^3}T_{00}(\eta,\phi_0)(0,x)\dd^3x
 =
 \frac12\int_{\R^3}
 \bigl(|\dot\phi_0|^2+|\nabla\phi_0|^2+m^2|\phi_0|^2\bigr)(0,x)
 \dd^3x>0.
\]
Thus \(Q\neq0\).
\end{proof}

This proposition supplies a family, not a general constrained
Einstein--scalar Cauchy theorem. It permits the Newton tails required
by its chosen data.

Figure~\ref{fig:VI-sourced-family} shows the explicit sourced
family and the propagation of its harmonic constraints.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[sbox] (phi) {$\phi_0$};
\node[sbox, right=of phi] (q) {$Q=T(\eta,\phi_0)/4$};
\node[sbox, right=of q] (rj) {$(\rho,J_i)$};
\draw[arr] (phi)--(q); \draw[arr] (q)--(rj);
\node[mbox, text width=5cm, below left=5mm and -4mm of q] (nv) {$a=-2\Delta^{-1}\rho$\\ $V_i=2\Delta^{-1}J_i$\\ $B_{ij}=\partial_iV_j+\partial_jV_i-\delta_{ij}\partial_\ell V_\ell$};
\node[mbox, text width=4.4cm, right=3mm of nv] (wave) {trace-reversed data\\ $\Box_\eta\bar j=-2Q$};
\node[mbox, text width=9.8cm, below=of nv] (con) {$D(j)(0)=\partial_t D(j)(0)=0$ and $\Box_\eta D(j)=0$ imply $D(j)=0$};
\node[sbox, below=of con] (end) {$G^{(1)}(j)=Q$,\ $P\phi_2=-L_j\phi_0$,\ $h=\phi_1=0$};
\node[gbox, font=\scriptsize, below=of end] {schematic Newton tails, not compact metric support; explicit family, not a general constrained Cauchy theorem};
\end{tikzpicture}
\caption{A nonempty sourced solution-jet family. Conserved
scalar stress determines Newton initial data and the
second-order metric forcing. The harmonic constraint has
zero Cauchy data and propagates homogeneously, giving
\(G^{(1)}(j)=T(\eta,\phi_0)/4\). The construction permits
Newton tails and establishes a family rather than a
general constrained existence theorem.}
\label{fig:VI-sourced-family}
\end{figure}


\section{Finite-time harmonic coordinates and second-order descent}

\subsection{Explicit scalar operator expansion}
\label{sec:scalar-expansion}

For \(h\), define
\begin{align}
 d_h&=\frac12\tr_\eta h,\\
 e_h&=\frac18(\tr_\eta h)^2-\frac14h_{\alpha\beta}h^{\alpha\beta},\\
 A_h^{\mu\nu}&=d_h\eta^{\mu\nu}-h^{\mu\nu},\\
 C_h^{\mu\nu}
 &=h^\mu{}_\rho h^{\rho\nu}
   -d_h h^{\mu\nu}+e_h\eta^{\mu\nu}.
 \label{eq:C-h}
\end{align}

\begin{lemma}[Second variation of the scalar wave operator]
\label{lem:second-wave}
The operator in \eqref{eq:wave-expansion} is
\begin{equation}
 \cM_h u
 =
 \partial_\mu(C_h^{\mu\nu}\partial_\nu u)
 -d_hL_hu-e_h\Boxeta u.
 \label{eq:M}
\end{equation}
\end{lemma}

\begin{proof}
The inverse metric and volume density are
\begin{align}
 \gamma^{\mu\nu}
 &=
 \eta^{\mu\nu}-\kappa h^{\mu\nu}
 +\kappa^2(h^\mu{}_\rho h^{\rho\nu}-j^{\mu\nu})
 \formal,\\
 \sqrt{-\gamma}
 &=
 1+\kappa d_h+\kappa^2(d_j+e_h)
 \formal.
\end{align}
Hence
\[
 \sqrt{-\gamma}\,\gamma^{\mu\nu}
 =
 \eta^{\mu\nu}
 +\kappa A_h^{\mu\nu}
 +\kappa^2(A_j^{\mu\nu}+C_h^{\mu\nu})
 \formal.
\]
Substitute this and
\[
 (\sqrt{-\gamma})^{-1}
 =
 1-\kappa d_h+\kappa^2(d_h^2-d_j-e_h)
 \formal
\]
into the divergence formula for \(\Box_\gamma\).
The first coefficient is
\[
 \partial_\mu(A_h^{\mu\nu}\partial_\nu)-d_h\Boxeta=L_h.
\]
The part linear in \(j\) is \(L_j\). The remaining part is
\[
 \partial_\mu(C_h^{\mu\nu}\partial_\nu)
 -d_h\partial_\mu(A_h^{\mu\nu}\partial_\nu)
 +(d_h^2-e_h)\Boxeta,
\]
which equals \eqref{eq:M}.
\end{proof}

\subsection{The finite-time inverse}

For a smooth source \(b\), let \(\Gzero b\) denote the solution of
\[
 \Boxeta u=b,\qquad u(0)=\partial_tu(0)=0.
\]
On \(0\leq t\leq T\),
\begin{equation}
 (\Gzero b)(t)
 =
 -\int_0^t
 \frac{\sin((t-s)|D|)}{|D|}b(s)\dd s.
 \label{eq:Gzero}
\end{equation}
For sources without spatial decay, the formula is interpreted locally
using finite propagation and compatible compact cutoffs. Values on a
compact target set depend only on a compact truncated past region.
Ordinary wave Cauchy theory justifies this construction
\cite{BGP}.

No inverse from an infinite past is needed.

\subsection{Coordinate coefficients}

Define four scalar coordinate functions, with an external coordinate
label \(\mu\), by
\begin{equation}
 X_\gamma^\mu(x)
 =
 x^\mu+\kappa w^\mu(x)+\kappa^2v^\mu(x)
 \formal.
 \label{eq:coordinates}
\end{equation}
They are required to satisfy
\begin{equation}
 \Box_\gamma X_\gamma^\mu=0\formal,
 \qquad
 X_\gamma^\mu|_{t=0}=x^\mu|_{t=0},
 \qquad
 \partial_tX_\gamma^\mu|_{t=0}=\delta^\mu_0.
 \label{eq:harmonic-coordinate-data}
\end{equation}

\begin{proposition}[Existence and uniqueness of the coordinate jet]
\label{prop:coordinates}
For every smooth \(h,j\), the conditions
\eqref{eq:harmonic-coordinate-data} have the unique coefficient solution
\begin{align}
 w^\mu[h]&=\Gzero D(h)^\mu,
 \label{eq:w}\\
 v^\mu[h,j]
 &=
 \Gzero\bigl(
 D(j)^\mu-\cM_hx^\mu-L_hw^\mu[h]
 \bigr).
 \label{eq:v}
\end{align}
Both coefficients and their first time derivatives vanish at \(t=0\).
\end{proposition}

\begin{proof}
Since \(\Boxeta x^\mu=0\) and
\(L_hx^\mu=-D(h)^\mu\), the first coefficient of
\(\Box_\gamma X^\mu\) is
\[
 \Boxeta w^\mu-D(h)^\mu.
\]
The second coefficient is
\[
 \Boxeta v^\mu+L_hw^\mu-D(j)^\mu+\cM_hx^\mu.
\]
The zero-data wave problem gives the formulas and uniqueness.
\end{proof}

The functions \(X^\mu\) are scalar functions under spacetime pullback.
Their index \(\mu\) labels the four coordinate functions; it does not
mean that \(X^\mu\) transforms as the components of a vector field.

Figure~\ref{fig:VI-harmonic-coordinates} summarizes the
finite-time coordinate equations and their fixed Cauchy data.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[mbox, text width=11cm] (x) {$\Box_\gamma X_\gamma^\mu=0$,\ $X_\gamma^\mu=x^\mu+\kappa w^\mu+\kappa^2 v^\mu$\\ $X^\mu|_0=x^\mu|_0$,\ $\partial_t X^\mu|_0=\delta^\mu_0$};
\node[mbox, text width=5.2cm, below left=4mm and 0mm of x] (w) {$\Box_\eta w^\mu=D(h)^\mu$\\ $w^\mu=G_{\Box,0}D(h)^\mu$};
\node[mbox, text width=5.2cm, below right=4mm and 0mm of x] (v) {$\Box_\eta v^\mu=D(j)^\mu-\mathsf M_h x^\mu-L_h w^\mu$};
\draw[arr] (x)--(w); \draw[arr] (x)--(v);
\node[sbox, text width=11cm, below=4mm of w] (g0) {$G_{\Box,0}b=-\int_0^t\sin((t-s)|D|)/|D|\,b(s)\,\mathrm ds$};
\node[gbox, font=\scriptsize, below=of g0] {schematic compact target and truncated causal past. Finite-time local dependence; no infinite-past inverse. $\mu$ labels four scalar coordinate functions.};
\end{tikzpicture}
\caption{Finite-time harmonic coordinate jet. Zero-data
wave problems determine the first and second coordinate
coefficients uniquely. Their values on a compact target
depend on a truncated past region. The index labels four
scalar coordinate functions, and the construction requires
neither an infinite-past inverse nor a global numerical
coordinate map.}
\label{fig:VI-harmonic-coordinates}
\end{figure}


\subsection{Coordinate covariance}

\begin{lemma}[Transformation of the coordinate coefficients]
\label{lem:coordinate-transform}
For the transformations in Proposition~\ref{prop:gauge-action},
\begin{equation}
 X_{\gamma'}=F_{\xi,\zeta}^*X_\gamma\formal.
 \label{eq:coordinate-covariance}
\end{equation}
Consequently,
\begin{align}
 w'^\mu&=w^\mu+\xi^\mu,
 \label{eq:w-transform}\\
 v'^\mu
 &=v^\mu+\xi^\nu\partial_\nu w^\mu+\zeta^\mu
                       +\frac12\xi^\nu\partial_\nu\xi^\mu.
 \label{eq:v-transform}
\end{align}
\end{lemma}

\begin{proof}
Scalar covariance gives
\[
 \Box_{F^*\gamma}(F^*X_\gamma^\mu)
 =
 F^*(\Box_\gamma X_\gamma^\mu)=0\formal.
\]
Because \(F\) is the identity on a neighborhood of the initial surface,
\(F^*X_\gamma^\mu\) has exactly the data in
\eqref{eq:harmonic-coordinate-data}. Uniqueness in
Proposition~\ref{prop:coordinates} proves
\eqref{eq:coordinate-covariance}. Expanding the pullback gives
\eqref{eq:w-transform}--\eqref{eq:v-transform}.
\end{proof}

\subsection{The observable}

The formal inverse coordinate map is
\begin{equation}
 Y_\gamma^\mu(X)
 =
 X^\mu-\kappa w^\mu(X)
 +\kappa^2\bigl(w^\nu\partial_\nu w^\mu-v^\mu\bigr)(X)
 \formal.
 \label{eq:inverse-coordinate}
\end{equation}
The formula follows by substituting it into \(X_\gamma(Y_\gamma(X))\).

For \(f\in C_c^\infty((0,T)\times\R^3;\R)\), define
\begin{equation}
 \OO_f[\gamma,\phi]
 =
 \int f(X)\,\phi(Y_\gamma(X))\dd^4X
 \formal.
 \label{eq:observable}
\end{equation}
This is a formal coordinate-space smearing. Its definition does not
assume that a numerical nonlinear coordinate map exists globally.

\begin{theorem}[Second-order classical descent]
\label{thm:descent}
On the jet space \(\cJ_T\), the functional \eqref{eq:observable} has
coefficients
\begin{align}
 \OO_f^{[0]}&=\int f\phi_0\dd^4x,
 \label{eq:O0-jet}\\
 \OO_f^{[1]}&=\int f\psi_1\dd^4x,
 &
 \psi_1&=\phi_1-w^\mu\partial_\mu\phi_0,
 \label{eq:O1-jet}\\
 \OO_f^{[2]}&=\int f\psi_2\dd^4x,
 \label{eq:O2-jet}\\
 \psi_2
 &=
 \phi_2-w^\mu\partial_\mu\phi_1
 +(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0
 +\frac12w^\mu w^\nu\partial_\mu\partial_\nu\phi_0.
 \label{eq:psi2}
\end{align}
Each coefficient is invariant under the formal gauge group
\eqref{eq:formal-pullback}. Hence \(\OO_f\) descends to the orbit space
of \(\cJ_T\).
\end{theorem}

\begin{proof}
Taylor expansion of \(\phi(Y_\gamma(X))\), using
\eqref{eq:jets} and \eqref{eq:inverse-coordinate}, gives
\eqref{eq:O0-jet}--\eqref{eq:psi2}.

For invariance, equation \eqref{eq:coordinate-covariance} says
\(X_{\gamma'}=X_\gamma\circ F\). Therefore
\[
 Y_{\gamma'}=F^{-1}\circ Y_\gamma\formal.
\]
Since \(\phi'=F^*\phi=\phi\circ F\),
\[
 \phi'\circ Y_{\gamma'}
 =
 \phi\circ F\circ F^{-1}\circ Y_\gamma
 =
 \phi\circ Y_\gamma
 \formal.
\]
Smearing with the same fixed function of the relational labels proves
coefficientwise invariance.

All coefficients are integrals of smooth functions against compact
\(f\). The coordinate coefficients at points of \(\supp f\) depend on
finite-time wave problems, so the definition is meaningful on the
declared smooth domain.
\end{proof}

\begin{remark}[Off-shell strength and on-shell restriction]
The invariance proof uses coordinate covariance and fixed coordinate
Cauchy data, not the Einstein or scalar equations. It therefore also
gives an off-shell formal invariance statement. Restriction to
\(\cJ_T\) makes it an observable on the named solution-jet quotient.
This does not establish quantum renormalizability of the functional.
\end{remark}

Figure~\ref{fig:VI-classical-descent} illustrates the
composition identity underlying coefficientwise classical
descent.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[mbox] (xlab) {physical labels $x$};
\node[mbox, right=of xlab] (Xlab) {relational labels $X$};
\node[sbox, text width=11cm, below=of xlab] (id) {$X_{\gamma'}=X_\gamma\circ F$,\ $Y_{\gamma'}=F^{-1}\circ Y_\gamma$,\ $\phi'=\phi\circ F$\\ $\phi'\circ Y_{\gamma'}=\phi\circ Y_\gamma$};
\node[sbox, below=of id] (o) {$\mathcal O_f=\int f(X)\,\phi(Y_\gamma(X))\,\mathrm d^4X$};
\node[sbox, text width=11cm, below=of o] (psi) {$\psi_1=\phi_1-X_w\phi_0$\\ $\psi_2=\phi_2-X_w\phi_1+(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0+\frac12 w^\mu w^\nu\partial_\mu\partial_\nu\phi_0$};
\node[gbox, below=of psi] {formal through order two; fixed relational-label smearing; classical covariance, not quantum renormalizability};
\end{tikzpicture}
\caption{Classical descent by coordinate covariance.
The inverse coordinate map compensates the pullback of
the scalar, leaving the relational scalar unchanged through
second order. The proof applies formally off shell and
restricts to the named solution-jet quotient. It does not
establish the quantum products or renormalizability of the
composite.}
\label{fig:VI-classical-descent}
\end{figure}


\subsection{Dependence on the relational prescription}

Changing coordinate Cauchy data is not a gauge transformation in the
group used above. For example, if \(w\) is replaced by \(w+a\), where
\(\Boxeta a=0\), then at first order
\begin{equation}
 \widetilde\OO_f^{[1]}-\OO_f^{[1]}
 =
 -\int f\,a^\mu\partial_\mu\phi_0\dd^4x.
 \label{eq:dressing-change}
\end{equation}
This need not vanish.

The following operations must therefore remain separate:
\begin{enumerate}
\item applying a diffeomorphism in the named gauge group;
\item changing a gauge-fixing term used for a perturbative construction;
\item changing coordinate initial data or the relational prescription;
\item changing solution Cauchy data.
\end{enumerate}
Theorem~\ref{thm:descent} concerns only the first operation with the
other choices fixed.

Figure~\ref{fig:VI-prescription-choices} separates gauge
descent from changes of the relational or dynamical data.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[sbox, text width=5.4cm] (one) {1. named diffeomorphism with fixed coordinate data};
\node[sbox, right=of one] (thm) {classical descent};
\draw[arr] (one)--(thm);
\node[cbox, text width=5.4cm, below=of one] (two) {2. change of perturbative gauge-fixing term};
\node[cbox, text width=5.4cm, below=of two] (three) {3. change of relational coordinate Cauchy data};
\node[cbox, text width=5.4cm, below=of three] (four) {4. change of solution Cauchy data};
\node[mbox, text width=5.6cm, right=of three] (chg) {$w\mapsto w+a$,\ $\Box_\eta a=0$\\ $\widetilde{\mathcal O}_f^{[1]}-\mathcal O_f^{[1]}=-\int f\,a^\mu\partial_\mu\phi_0$\\ generally changes the observable};
\node[gbox, text width=11.4cm, below=of four] {no equality arrows among the four operations; all other choices held fixed};
\end{tikzpicture}
\caption{Four distinct changes. The descent theorem concerns
the named diffeomorphism action with the relational
prescription and Cauchy data fixed. Changing coordinate
initial data can change the observable already at first
order. Gauge fixing, relational prescription, and solution
data are therefore not interchangeable choices.}
\label{fig:VI-prescription-choices}
\end{figure}


\section{Bare dressing, solution coefficients, and external response}

\subsection{Bare functional coefficients are not solution-jet coefficients}

For perturbative quantization one usually expands a functional of
independent fields \((h,\varphi)\), with
\(\gamma=\eta+\kappa h\), before substituting interacting solutions.
Let
\[
 v_0[h]=v[h,0].
\]
The bare geometric coefficients are
\begin{align}
 \OO_f^{(0)}[h,\varphi]
 &=\int f\varphi\dd^4x,\\
 \OO_f^{(1)}[h,\varphi]
 &=-\int f\,w^\mu[h]\partial_\mu\varphi\dd^4x,
 \label{eq:bare-O1}\\
 \OO_f^{(2)}[h,\varphi]
 &=
 \int f\left[
 (w^\nu\partial_\nu w^\mu-v_0^\mu)\partial_\mu\varphi
 +\frac12w^\mu w^\nu\partial_\mu\partial_\nu\varphi
 \right]\dd^4x.
 \label{eq:bare-O2}
\end{align}
Round parentheses label bare functional coefficients. Square brackets
in Theorem~\ref{thm:descent} label coefficients after substitution of a
solution jet.

The relation between the two expansions is explicit:
\[
 v[h,j]=w[j]+v_0[h].
\]
Substituting
\[
 h\longmapsto h+\kappa j,\qquad
 \varphi\longmapsto\phi_0+\kappa\phi_1+\kappa^2\phi_2
\]
into the bare expansion gives exactly
\eqref{eq:O0-jet}--\eqref{eq:psi2}. In particular, one must not insert a
sourced \(j\) into a bare second-order term and then count the same
dynamical correction again through an interacting-field map.

\subsection{The compactly switched minimal scalar vertex}

Fix
\[
 g\in C_c^\infty((0,T)\times\R^3;\R).
\]
For a smooth external tensor \(h\), the matter vertex is
\begin{equation}
 V_1^{\mathrm m}(g;h,\varphi)
 =
 \frac12\int gh^{\mu\nu}T_{\mu\nu}(\eta,\varphi)\dd^4x.
 \label{eq:matter-vertex}
\end{equation}
For \(H=gh\), define
\begin{equation}
 \cQ_H
 =
 -\partial_\mu(\bar H^{\mu\nu}\partial_\nu)
 -\frac12m^2\tr_\eta H.
 \label{eq:Q}
\end{equation}
It is the scalar Hessian contribution of
\eqref{eq:matter-vertex}, and it is formally self-adjoint. Direct
expansion gives
\begin{equation}
 \cQ_H=L_H+\frac12(\tr_\eta H)P.
 \label{eq:QL}
\end{equation}
Hence, on a free scalar,
\begin{equation}
 \cQ_{gh}\phi_0
 =
 gL_h\phi_0
 -(\partial_\mu g)\bar h^{\mu\nu}\partial_\nu\phi_0.
 \label{eq:Q-switch}
\end{equation}

Let \(\GR\) be the retarded inverse of \(P\), with time kernel
\[
 \GR(t)=-\Theta(t)\frac{\sin(t\Omega_m)}{\Omega_m},
 \qquad \Omega_m=\sqrt{-\Delta+m^2}.
\]
Since \(g\) is compactly supported after \(t=0\), the first scalar
response with zero past data is
\begin{equation}
 \phi_{1,g}[h,\phi_0]
 =
 -\GR\cQ_{gh}\phi_0.
 \label{eq:external-response}
\end{equation}

\begin{proposition}[Exact external-response comparison]
\label{prop:external-comparison}
For \(P\phi_0=0\),
\begin{equation}
 \phi_{1,g}[h,\phi_0]=\Szero(gh)\phi_0,
 \label{eq:external-duhamel}
\end{equation}
where
\begin{equation}
 (\Szero(a)\phi_0)(t)
 =
 \int_0^t
 \frac{\sin((t-s)\Omega_m)}{\Omega_m}L_a\phi_0(s)\dd s.
 \label{eq:Szero}
\end{equation}
For \(u_f=-\GA f\), its smearing is
\begin{equation}
 \int f\phi_{1,g}\dd^4x
 =
 \int (\cQ_{gh}u_f)\phi_0\dd^4x.
 \label{eq:external-smearing}
\end{equation}
\end{proposition}

\begin{proof}
Equation \eqref{eq:QL} gives
\(\cQ_{gh}\phi_0=L_{gh}\phi_0\).
Both sides of \eqref{eq:external-duhamel} have zero Cauchy data at
\(t=0\) and solve \(P\psi=-L_{gh}\phi_0\).
Klein--Gordon uniqueness proves equality.

For the smeared identity, use transposition
\(\GR^*=\GA\), then formal self-adjointness of \(\cQ_{gh}\):
\[
 -\int f\GR\cQ_{gh}\phi_0
 =
 -\int(\GA f)\cQ_{gh}\phi_0
 =
 \int(\cQ_{gh}u_f)\phi_0.
\]
All vertex integrations involve compact coefficients.
\end{proof}

This proposition concerns the forced scalar term. It does not identify
that term with the complete relational coefficient
\(\phi_1-X_w\phi_0\).

At the quantum level, the corresponding first external-metric
retarded coefficient is represented by the same linear smearing when
the two-insertion normalization is fixed accordingly. That external
calculation neither constructs a local graviton product nor supplies
the second-order relational composite.

\subsection{Why the first-order matching claim needs qualification}

Three expressions must be distinguished:
\begin{align}
 \text{bare dressing:}\quad&
 -\int fX_{w[h]}\phi_0,\\
 \text{external forced response:}\quad&
 \int f\phi_{1,g}[h,\phi_0],\\
 \text{candidate dressed response:}\quad&
 \int f\bigl(\phi_{1,g}[h,\phi_0]-X_{w[h]}\phi_0\bigr).
 \label{eq:candidate-dressed}
\end{align}
The first two are not equal in general. The third is the object whose
switch compatibility must be checked.

Even when a special harmonic restriction makes \(w[h]=0\), equality
of two displayed expressions on that restriction is not a proof that
their constructions intertwine the gauge actions on a larger domain.

Figure~\ref{fig:VI-three-coefficients} distinguishes bare,
solution-jet, and switched-response coefficients.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[gbox, text width=3.6cm] (bare) {bare parentheses\\ $\mathcal O_f^{(n)}$\\ independent $(h,\varphi)$};
\node[sbox, text width=3.8cm, right=3mm of bare] (jet) {solution-jet brackets\\ $\mathcal O_f^{[n]}$\\ after $h+\kappa j$ and the scalar jet};
\node[cbox, text width=3.8cm, right=3mm of jet] (sw) {switched candidate\\ $\int f\phi_{1,g}-\int f X_{w[h]}\phi_0$};
\node[mbox, below=of jet] (v) {$v[h,j]=w[j]+v_0[h]$};
\node[cbox, text width=11.6cm, below=of v] {forced and dressing terms are distinct; no double counting of sourced $j$; the external response alone is not the complete relational coefficient};
\end{tikzpicture}
\caption{Three coefficient constructions. Bare dressing
coefficients are functionals of independent fields;
solution-jet coefficients include the dynamical scalar
and metric corrections. The proposed switched first
coefficient combines an external forced response with a
geometric dressing. Identifying only one contribution with
the complete coefficient omits terms or double counts
dynamical corrections.}
\label{fig:VI-three-coefficients}
\end{figure}


\section{An exact compact-switch obstruction}

\subsection{The candidate and the tested identification}

Define
\begin{equation}
 \cA_{f,g}^{(1)}(h,\phi_0)
 =
 \int f\bigl(
 -\GR\cQ_{gh}\phi_0-X_{w[h]}\phi_0
 \bigr)\dd^4x,
 \qquad P\phi_0=0.
 \label{eq:A1g}
\end{equation}
The proposed free-input representative identification is
\begin{equation}
 (h,\phi_0)\sim(h+K\xi,\phi_0),
 \qquad \xi\in\mathfrak g_T.
 \label{eq:free-representative-identification}
\end{equation}
This is the identification that would have to be respected by an
equivariant realization of the geometric observable through this
particular switched retarded assignment.

It is not the full solution-jet action
\eqref{eq:phi1-transform}. The question is whether the retarded
assignment supplies that missing scalar transformation automatically.

\begin{theorem}[Exact first-order switch defect]
\label{thm:switch-defect}
For every smooth free scalar \(\phi_0\) and every
\(\xi\in\mathfrak g_T\),
\begin{align}
 &\cA_{f,g}^{(1)}(h+K\xi,\phi_0)
       -\cA_{f,g}^{(1)}(h,\phi_0)
 \nonumber\\
 &\quad=
 \int f\left[
 (g-1)X_\xi\phi_0
 +
 \GR\left(
   (\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi_0
   -[P,g]X_\xi\phi_0
 \right)\right]\dd^4x,
 \label{eq:switch-defect}
\end{align}
where
\begin{equation}
 [P,g]u=(\Boxeta g)u+2(\partial^\mu g)\partial_\mu u.
 \label{eq:P-g}
\end{equation}
\end{theorem}

\begin{proof}
Since \(\xi\) has zero Cauchy data at \(t=0\), equations
\eqref{eq:DK} and \eqref{eq:w} imply
\[
 w[h+K\xi]-w[h]=\Gzero\Boxeta\xi=\xi.
\]
The dressing variation is therefore \(-X_\xi\phi_0\).

Equations \eqref{eq:pure-gauge-operator} and \eqref{eq:Q-switch} give
\[
 \cQ_{gK\xi}\phi_0
 =
 -gP(X_\xi\phi_0)
 -(\partial_\mu g)\overline{K\xi}^{\mu\nu}
                       \partial_\nu\phi_0.
\]
Consequently,
\[
 \delta_\xi\phi_{1,g}
 =
 \GR\left(
 gP(X_\xi\phi_0)
 +(\partial_\mu g)\overline{K\xi}^{\mu\nu}
                       \partial_\nu\phi_0
 \right).
\]
Use
\[
 gP(X_\xi\phi_0)=P(gX_\xi\phi_0)-[P,g]X_\xi\phi_0.
\]
The function \(gX_\xi\phi_0\) is compactly supported, so
\(\GR P(gX_\xi\phi_0)=gX_\xi\phi_0\).
Subtracting the dressing variation gives
\eqref{eq:switch-defect}.
\end{proof}

Equation \eqref{eq:switch-defect} is a classical retarded identity. It is
not an anomalous quantum Master Ward Identity. In particular, its
nonzero right-hand side is not a loop anomaly.

Figure~\ref{fig:VI-switch-defect} separates the dressing
variation from the switched dynamical variation.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[gbox] (rep) {$(h,\phi_0)\mapsto(h+K\xi,\phi_0)$};
\node[mbox, text width=5.1cm, below left=6mm and 0mm of rep] (co) {coordinate branch\\ $\delta w=\xi$\\ $\delta(-X_w\phi_0)=-X_\xi\phi_0$};
\node[sbox, text width=5.3cm, below right=6mm and 0mm of rep] (dyn) {switched branch\\ $\delta\phi_{1,g}=gX_\xi\phi_0$ plus retarded derivatives of $g$};
\draw[arr] (rep)--(co); \draw[arr] (rep)--(dyn);
\node[cbox, text width=10.8cm, below=4mm of co] (gap) {$\delta\mathcal A_{f,g}^{(1)}=\int f[(g-1)X_\xi\phi_0+G_{\mathrm R}(\mathrm{switching\ terms})]$\\ $[P,g]u=(\Box_\eta g)u+2(\partial^\mu g)\partial_\mu u$};
\node[gbox, below=of gap] {exact classical retarded identity; not a loop anomaly or quantum Master Ward Identity};
\end{tikzpicture}
\caption{Exact first-order switch defect. The geometric
compensator changes by \(\xi\), while the compactly switched
scalar response supplies \(gX_\xi\phi_0\) and additional
retarded switching terms. Their difference gives the
displayed classical defect. A nonzero defect here is not
a computed quantum anomaly.}
\label{fig:VI-switch-defect}
\end{figure}


\subsection{A two-representative witness with a plateau switch}

\begin{theorem}[Plateau-switch obstruction]
\label{thm:plateau-obstruction}
There exist
\[
 f,g\in C_c^\infty((0,T)\times\R^3;\R),\qquad
 \xi\in\mathfrak g_T,
\]
and a real compact-data free scalar \(\phi_0\), such that
\begin{enumerate}
\item \(f\geq0\) and \(\int f\dd^4x=1\);
\item \(g=1\) on a neighborhood of \(\supp f\);
\item \(gK\xi=0\) everywhere;
\item both \(0\) and \(K\xi\) satisfy the vacuum linearized Einstein
equation;
\item
\begin{equation}
 \cA_{f,g}^{(1)}(K\xi,\phi_0)
 -
 \cA_{f,g}^{(1)}(0,\phi_0)
 \leq-\frac12.
 \label{eq:gap}
\end{equation}
\end{enumerate}
Thus the candidate \eqref{eq:A1g} does not descend through
\eqref{eq:free-representative-identification}, even when the switch is
constant near the observation region.
\end{theorem}

\begin{proof}
Choose \(t_*\in(0,T)\). At that time prescribe compact smooth scalar
data
\[
 \phi_0(t_*,x)=x^1\chi(x),\qquad
 \partial_t\phi_0(t_*,x)=0,
\]
where \(\chi=1\) near \(x=0\).
The Klein--Gordon solution has compact smooth Cauchy data also at
\(t=0\), by finite propagation. At \(x_*=(t_*,0)\),
\[
 \partial_1\phi_0(x_*)=1.
\]
Choose a relatively compact neighborhood
\(U\Subset(0,T)\times\R^3\) on which
\(\partial_1\phi_0\geq1/2\).
Choose a nonnegative unit-integral test \(f\) supported in \(U\).

Choose a compact smooth switch \(g\) equal to one near \(\supp f\).
Next choose
\(\chi_\xi\in C_c^\infty((0,T)\times\R^3)\)
equal to one on a neighborhood of \(\supp g\), and set
\[
 \xi=\chi_\xi\,\partial_1.
\]
All derivatives of \(\xi\) vanish near \(\supp g\).
Therefore \(gK\xi=0\), and hence
\[
 \phi_{1,g}[K\xi,\phi_0]=0.
\]
The zero metric has zero response as well.

On the other hand, \(w[K\xi]=\xi\), and
\(\xi=\partial_1\) on \(\supp f\). Thus
\[
 \cA_{f,g}^{(1)}(K\xi,\phi_0)
 -
 \cA_{f,g}^{(1)}(0,\phi_0)
 =
 -\int f\partial_1\phi_0\dd^4x
 \leq-\frac12.
\]
Finally, \(G^{(1)}K=0\) proves the vacuum statement.
\end{proof}

The witness does not exploit a change of the switch between
representatives. The same \(f,g,\phi_0\) are used twice.

Figure~\ref{fig:VI-plateau-witness} displays the nested support
construction yielding the strictly nonzero representative gap.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[font=\scriptsize, sgray] (sch) {schematic nested supports};
\node[gbox, text width=4.2cm, below=of sch] (out) {$\operatorname{supp}\chi_\xi$\\ $\chi_\xi=1$ near $\operatorname{supp} g$};
\node[cbox, text width=3.4cm, below=of out] (gsup) {$\operatorname{supp} g$\\ $g=1$ near $\operatorname{supp} f$};
\node[sbox, below=of gsup] (fsup) {$\operatorname{supp} f$};
\node[cbox, text width=6.2cm, right=6mm of out] (xi) {$\xi=\chi_\xi\partial_1$\\ derivatives of $\chi_\xi$ may leave $\operatorname{supp} g$\\ $gK\xi=0$,\ $\phi_{1,g}[K\xi,\phi_0]=0$,\ $w[K\xi]=\xi$};
\node[sbox, text width=10.8cm, below=4mm of fsup] {observation: $\partial_1\phi_0\ge 1/2$,\ $f\ge 0$,\ $\int f=1$\\ $\mathcal A^{(1)}(K\xi)-\mathcal A^{(1)}(0)=-\int f\partial_1\phi_0\le -1/2$\\ same $f,g,\phi_0$ for both representatives};
\end{tikzpicture}
\caption{Plateau-switch obstruction. The gauge parameter is
a constant translation near the entire switch support,
so \(gK\xi=0\) and both switched forced responses vanish.
The unswitched coordinate compensator nevertheless changes
by \(\xi\). A unit-integral smearing where
\(\partial_1\phi_0\geq 1/2\) gives a gap at most \(-1/2\),
despite \(g=1\) near the observation.}
\label{fig:VI-plateau-witness}
\end{figure}


\subsection{Why this does not contradict geometric descent}

For the geometric action, the transformation of the first scalar
coefficient is
\[
 \phi_1\longmapsto\phi_1+X_\xi\phi_0.
\]
In the witness, the switched retarded assignment instead gives
\[
 \phi_{1,g}[K\xi,\phi_0]
 =
 \phi_{1,g}[0,\phi_0]=0.
\]
It therefore fails to implement the scalar part of that action.

Theorem~\ref{thm:descent} says that the complete solution jet transforms
correctly. Theorem~\ref{thm:plateau-obstruction} says that a particular
switched assignment does not furnish such a transformation from the
unchanged free input. These are compatible statements.

The obstruction is not a theorem excluding relational observables in
perturbative gravity. It excludes the stated combination of:
\begin{enumerate}
\item the fixed unswitched coordinate dressing \(w[h]\);
\item the compactly switched response \(-\GR\cQ_{gh}\phi_0\);
\item all gauge parameters in \(\mathfrak g_T\);
\item the representative identification
\eqref{eq:free-representative-identification}.
\end{enumerate}

Changing the admissible gauge support, treating the switch as an
additional transforming background, or modifying the interacting
relational prescription changes the problem and requires a new proof.

\subsection{A second-order term cannot cancel the first-order gap}

\begin{corollary}[Order obstruction]
\label{cor:order-obstruction}
Let
\[
 \cA_{f,g}
 =
 \int f\phi_0
 +\kappa\cA_{f,g}^{(1)}
 +\kappa^2\cA_{f,g}^{(2)}
 \formal,
\]
where the first coefficient is fixed by \eqref{eq:A1g}.
No choice of \(\cA_{f,g}^{(2)}\), including any finite modification
beginning at order \(\kappa^2\), makes this expression invariant under
\eqref{eq:free-representative-identification} on the entire stated
domain.
\end{corollary}

\begin{proof}
Formal equality requires equality of the coefficient of \(\kappa\).
Theorem~\ref{thm:plateau-obstruction} supplies a nonzero difference in
that coefficient. Terms beginning at order \(\kappa^2\) do not alter it.
\end{proof}

The same argument distinguishes loop order: a positive power of
\(\hbar\) cannot cancel a fixed nonzero classical coefficient at
\(\hbar^0\) in a coefficientwise formal identity.

Figure~\ref{fig:VI-compatible-order-obstruction} explains the
compatible conclusions and the coefficientwise order
obstruction.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[sbox, text width=5.4cm] (geo) {geometric jet action\\ $\delta\phi_1=X_\xi\phi_0$,\ $\delta w=\xi$\\ $\delta(\phi_1-X_w\phi_0)=0$};
\node[cbox, text width=5.4cm, right=4mm of geo] (sw) {plateau switched assignment\\ $\delta\phi_{1,g}=0$,\ $\delta w=\xi$\\ $\delta(\phi_{1,g}-X_w\phi_0)=-X_\xi\phi_0$};
\node[gbox, text width=11.2cm, below=4mm of geo] (ord) {$\delta\mathcal A=\kappa\Delta_1+\kappa^2\Delta_2\ \mathrm{mod}\,\kappa^3$,\ $\Delta_1\neq 0$};
\node[cbox, below=of ord] (bad) {$\Delta_2$ cancels $\Delta_1$};
\draw[corange, thick] (bad.north west)--(bad.south east);
\draw[corange, thick] (bad.north east)--(bad.south west);
\node[gbox, font=\scriptsize, below=of bad] {formal equality is coefficientwise; positive $\hbar$-order cannot cancel a fixed $\hbar^0$ coefficient};
\end{tikzpicture}
\caption{Compatible conclusions and order obstruction.
Classical descent uses the full scalar coefficient action,
which cancels the coordinate variation. The specified
switched assignment fails to provide that scalar action
in the witness. Its nonzero first-order gap cannot be
removed by a modification beginning at second order while
the lower-order prescription and identification are retained.}
\label{fig:VI-compatible-order-obstruction}
\end{figure}


\section{A nonzero sourced classical second-order response}

The obstruction concerns a switched realization. It does not render the
classical descended functional vacuous.

\begin{theorem}[Nonzero second-order sourced coefficient]
\label{thm:sourced-nonzero}
There is a tuple in \(\cJ_T\) and a nonnegative compact smearing \(f\)
such that
\[
 h=0,\qquad \phi_1=0,\qquad
 \OO_f^{[2]}>0.
\]
The metric coefficient \(j\) is sourced by
\(T(\eta,\phi_0)/4\), not prescribed independently.
\end{theorem}

\begin{proof}
Choose a real nonnegative \(F\in C_c^\infty(\R^3)\) equal to a positive
constant \(A\) near \(x=0\). Take
\[
 \phi_0(0,x)=F(x),\qquad \dot\phi_0(0,x)=0.
\]
Use the construction of Proposition~\ref{prop:sourced-family}.

Here \(J_i=Q_{0i}(0)=0\), so \(V_i=0\) and \(B_{ij}=0\).
Also
\[
 \rho=Q_{00}(0)
 =
 \frac18\bigl(|\nabla F|^2+m^2F^2\bigr).
\]
It is nonnegative and nonzero. Therefore
\[
 a(0)
 =
 -2(\Delta^{-1}\rho)(0)>0.
\]
Trace reversal of the prescribed initial data gives
\[
 j_{00}(0,x)=\frac12a(x),\qquad
 j_{ij}(0,x)=\frac12a(x)\delta_{ij},\qquad
 j_{0i}(0,x)=0.
\]

Because \(D(j)=0\), \(h=0\), and \(\phi_1=0\), the coordinate
coefficients satisfy
\[
 w=0,\qquad v=\Gzero D(j)=0.
\]
Thus the relational second coefficient is simply
\[
 \OO_f^{[2]}=\int f\phi_2\dd^4x
\]
on this particular family.

Near the spatial origin, the initial spatial Hessian of \(\phi_0\)
vanishes and
\[
 \partial_t^2\phi_0(0,0)=-m^2A.
\]
Using \(D(j)=0\),
\[
 L_j\phi_0(0,0)
 =
 -j^{00}(0,0)\partial_t^2\phi_0(0,0)
 =
 \frac12a(0)m^2A>0.
\]
The equation \(P\phi_2=-L_j\phi_0\) and the zero Cauchy data of
\(\phi_2\) imply
\[
 \partial_t^2\phi_2(0,0)=\frac12a(0)m^2A.
\]
Consequently,
\[
 \phi_2(t,0)=\frac14a(0)m^2A\,t^2+O(t^3).
\]
For sufficiently small positive \(t\), the solution is positive in a
spacetime neighborhood. A nonzero nonnegative \(f\) supported there gives
the assertion.
\end{proof}

\subsection{Meaning of the example}

On this specially chosen harmonic family, the classical coefficient
reduces to a cubic source-response expression:
\[
 \phi_0
 \longmapsto
 T(\phi_0)
 \longmapsto
 j
 \longmapsto
 \Szero(j)\phi_0.
\]
This restricted reduction is a consequence of the explicit vanishing of
\(w,v,h,\phi_1\) in the calculation. It is not a universal
identification of a relational second-order observable with a
Newton-inverse cubic.

In particular, it does not identify:
\begin{enumerate}
\item a classical coefficient with a Wick-ordered composite;
\item a scalar-source reconstruction with an independent graviton field;
\item the displayed coefficient with a complete interacting
second-order quantum field;
\item a nonzero classical value with a scheme-independent radiative
response.
\end{enumerate}

If the source is replaced by zero while the prescribed homogeneous
metric data and scalar correction data are also set to zero, then
\(j=\phi_2=0\). This is the exact source-off control for the family.
Removing an interaction switch in another model is not this operation.

Figure~\ref{fig:VI-sourced-positive} displays the nonzero
classical coefficient and its exact source-off control.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[sbox, text width=5.6cm] (data) {$\phi_0(0)=F$,\ $\dot\phi_0(0)=0$,\ $F=A>0$ near $0$};
\node[sbox, below=of data] (src) {$\rho=\frac18(|\nabla F|^2+m^2F^2)$,\ $a=-2\Delta^{-1}\rho$,\ $a(0)>0$};
\node[mbox, below=of src] (z) {$h=\phi_1=w=v=0$,\ $P\phi_2=-L_j\phi_0$};
\node[sbox, below=of z] (lead) {$\phi_2(t,0)=\frac14 a(0)m^2 A\,t^2+O(t^3)$};
\node[gbox, font=\scriptsize, below=of lead] (sch) {schematic positive future neighborhood};
\node[sbox, below=of sch] {$f\ge 0\Rightarrow \mathcal O_f^{[2]}>0$};
\node[cbox, text width=5.4cm, right=5mm of src] (off) {source-off branch\\ zero source and zero homogeneous metric and scalar-correction data $\Rightarrow j=\phi_2=0$};
\node[gbox, font=\scriptsize, below=of off] {special classical family, not a universal quantum cubic identity};
\end{tikzpicture}
\caption{Nonzero sourced classical response. For the chosen
harmonic family, the relational coefficient reduces to
\(\int f\phi_2\), with a positive short-time leading term.
The metric is determined by scalar stress rather than
prescribed independently. Setting the source and the
specified homogeneous data to zero gives the exact
source-off control; no quantum or radiative identification
is implied.}
\label{fig:VI-sourced-positive}
\end{figure}


\section{Second-order retarded bookkeeping and the missing quantum step}

\subsection{The algebraic four-term identity}

Let \(R_n\) denote retarded products with the convention
\begin{equation}
 R_V(F)
 =
 \sum_{n=0}^\infty\frac1{n!}R_n(V^{\otimes n};F),
 \qquad R_0(F)=F.
 \label{eq:retarded-series}
\end{equation}
Suppose
\[
 V=\kappa V_1+\kappa^2V_2,\qquad
 F=\OO_f^{(0)}+\kappa\OO_f^{(1)}+\kappa^2\OO_f^{(2)}.
\]

\begin{proposition}[Conditional coefficient identity]
\label{prop:four-terms}
If the retarded products in \eqref{eq:retarded-series} exist on the
listed inputs and have the stated multilinearity, then
\begin{align}
 [R_V(F)]_{\kappa^2}
 ={}&
 \OO_f^{(2)}
 +R_1(V_1;\OO_f^{(1)})
 +R_1(V_2;\OO_f^{(0)})
 \nonumber\\
 &+\frac12R_2(V_1,V_1;\OO_f^{(0)}).
 \label{eq:four-terms}
\end{align}
\end{proposition}

\begin{proof}
The zeroth retarded term contributes \(\OO_f^{(2)}\).
The first retarded term contributes
\[
 R_1(V_1;\OO_f^{(1)})+R_1(V_2;\OO_f^{(0)}).
\]
The second retarded term contributes
\(\frac12R_2(V_1,V_1;\OO_f^{(0)})\).
Terms with three or more copies of \(V\) have order at least
\(\kappa^3\).
\end{proof}

The proposition is an algebraic expansion. It is not an existence,
renormalization, or Ward-identity theorem.

\subsection{What belongs to the interactions}

With \(\gamma=\eta+\kappa h\), the classical interaction contains
\[
 V_1=V_1^{\mathrm{grav}}+V_1^{\mathrm m},
 \qquad
 V_2=V_2^{\mathrm{grav}}+V_2^{\mathrm m}.
\]
The gravity terms are the cubic and quartic coefficients of the
Einstein--Hilbert density. The matter terms are
\begin{align}
 V_1^{\mathrm m}
 &=\frac12\int h^{\mu\nu}T_{\mu\nu}(\eta,\varphi)\dd^4x,\\
 V_2^{\mathrm m}
 &=-\frac12\int
 \left(C_h^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi
       +e_hm^2\varphi^2\right)\dd^4x.
 \label{eq:V2matter}
\end{align}
A gauge-fixed quantum construction additionally requires its ghost and
gauge-fixing interactions.

Multiplying a chosen local density by \(g\), or assigning powers or
independent copies of switches to different orders, is additional
data. These choices cannot be left implicit in a finite counterterm
classification.

\subsection{A free gauge sector does not complete the interacting one}

For orientation, a free BRST complex can be written with ghost \(c\),
antighost \(\bar c\), and auxiliary field \(b\):
\[
 s_0h=Kc,\qquad s_0\varphi=0,\qquad
 s_0c=0,\qquad s_0\bar c=b,\qquad s_0b=0.
\]
A harmonic gauge fermion is
\[
 \Psi_\alpha
 =
 \int \bar c_\nu\left(D(h)^\nu+\frac\alpha2b^\nu\right)\dd^4x,
 \qquad \alpha\neq0.
\]
Its variation gives
\[
 s_0\Psi_\alpha
 =
 \int\left(
 b_\nu D(h)^\nu+\frac\alpha2b_\nu b^\nu
 -\bar c_\nu\Boxeta c^\nu
 \right)\dd^4x.
\]
Eliminating \(b\) gives
\(-\frac1{2\alpha}\int D(h)_\nu D(h)^\nu\).

These formulas identify a free gauge sector. They do not supply the
interacting BRST differential, its switched identity, or the
renormalized products of the relational functional. No quantum theorem
below relies on pretending otherwise.

\subsection{The two-vertex products are prerequisites}

The term
\[
 R_2(V_1,V_1;\OO_f^{(0)})
\]
involves a construction with two interaction insertions and one
observable insertion. In a time-ordered formulation its definition
requires, among other ingredients, products containing
\(T_2(V_1,V_1)\) and the relevant three-insertion time-ordered product.

The term \(R_1(V_1;\OO_f^{(1)})\) creates a different problem:
\(\OO_f^{(1)}\) contains the nonlocal coordinate kernel
\(\Gzero D(h)\). The domain and extension analysis for that kernel
cannot be inferred from a calculation with an external smooth metric.

The extension theory of \cite{BF} supplies criteria for numerical
distributions once those distributions and their scaling degrees have
been determined. It does not identify them for a new composite by
notation alone.

\subsection{Coupling order is not loop order}

A declaration of ``order \(\kappa^2\)'' does not specify a loop order.
Classical source responses, tree contributions, and radiative
contributions can occur at that coupling order. A quantum statement must
also specify its \(\hbar\)-degree or graph-loop convention.

For example, suppose fields
\[
 A_f=A_f^{(0)}+\kappa A_f^{(1)}+\kappa^2 A_f^{(2)}
\]
have already been constructed in an algebra with a fixed state
\(\omega\). Then the coefficient of their connected two-point function
is
\begin{align}
 [\omega(A_fA_\ell)_{\mathrm c}]_{\kappa^2}
 ={}&
 \omega(A_f^{(2)}A_\ell^{(0)})_{\mathrm c}
 +\omega(A_f^{(0)}A_\ell^{(2)})_{\mathrm c}
 \nonumber\\
 &+\omega(A_f^{(1)}A_\ell^{(1)})_{\mathrm c}.
 \label{eq:two-point-expansion}
\end{align}
If the state itself is varied perturbatively, its expansion contributes
additional terms. Equation \eqref{eq:four-terms} for one insertion
therefore does not by itself specify a two-point response vector.

Figure~\ref{fig:VI-four-term-prerequisites} separates the
conditional coefficient identity from the product and
state data required for a quantum response.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[pbox] (mid) {$[R_V(F)]_{\kappa^2}$};
\node[pbox, text width=2.7cm, below left=8mm and 10mm of mid] (a) {$\mathcal O_f^{(2)}$};
\node[pbox, text width=2.9cm, right=2mm of a] (b) {$R_1(V_1;\mathcal O_f^{(1)})$};
\node[pbox, text width=2.9cm, right=2mm of b] (c) {$R_1(V_2;\mathcal O_f^{(0)})$};
\node[pbox, text width=3.1cm, right=2mm of c] (d) {$\frac12 R_2(V_1,V_1;\mathcal O_f^{(0)})$};
\draw[arrg] (a)--(mid); \draw[arrg] (b)--(mid); \draw[arrg] (c)--(mid); \draw[arrg] (d)--(mid);
\node[gbox, font=\scriptsize, below=of b] (ker) {nonlocal kernel $G_{\Box,0}D(h)$};
\node[gbox, font=\scriptsize, below=of d] (pre) {$T_2(V_1,V_1)$ and the three-insertion product};
\node[pbox, text width=5.6cm, below=of ker] (need) {conditional on product existence; gravity, matter, ghost, gauge fixing};
\node[cbox, text width=5.4cm, right=3mm of need] {$\kappa^2$ does not specify loop order\\ three connected two-point terms; state variations add more};
\end{tikzpicture}
\caption{Second-order interacting bookkeeping. Four terms
contribute to the one-insertion coefficient, conditional
on the required retarded products. Two-vertex products
and products involving coordinate kernels are prerequisites,
not consequences of the algebraic expansion. Coupling order
does not determine loop order, and a two-point response
requires additional product and state specifications.}
\label{fig:VI-four-term-prerequisites}
\end{figure}

The retarded-product framework is described in \cite{DF}, and
extension criteria in \cite{BF}. Neither citation constructs the
products drawn in Figure~\ref{fig:VI-four-term-prerequisites}.


\section{Finite-renormalization spaces and certificate requirements}

\subsection{What has to be fixed before a catalogue is complete}

A finite list is meaningful only relative to a declared class.
For the proposed quantum problem this class must fix, at minimum:
\begin{enumerate}
\item coupling order and loop order;
\item field content, ghost number, and discrete symmetries;
\item derivative and engineering-dimension bounds;
\item the allowed dependence on \(m\) and any renormalization scale;
\item the number and treatment of switch insertions;
\item action counterterms and composite-observable counterterms;
\item equivalences from integrations by parts, field redefinitions,
and equations of motion, with their domain of validity;
\item initial-surface terms or restrictions excluding them;
\item the Ward identity, including its permitted breaking terms;
\item lower-order normalization conditions that are held fixed.
\end{enumerate}

In particular, freezing lower-order normalizations matters for the
affine form \(r=q+Nc\). If lower-order redefinitions are allowed to vary,
their products may enter the next coefficient. An affine response space
then requires a separate parametrization argument.

\subsection{A finite-jet counting lemma}

\begin{lemma}[Finiteness under explicit polynomial bounds]
\label{lem:finite-jet}
Suppose a local counterterm class is generated by a finite set of field,
ghost, switch, and background components and their derivatives up to a
fixed order. Suppose also that:
\begin{enumerate}
\item the total degree in these generators is bounded;
\item the allowed mass and scale dependence lies in a fixed
finite-dimensional coefficient space.
\end{enumerate}
Then the resulting vector space of local polynomial densities is
finite-dimensional. Imposing linear symmetry conditions and quotienting
by linear relations preserves finite-dimensionality.
\end{lemma}

\begin{proof}
The component set and derivative bound provide finitely many jet
variables, say \(z_1,\ldots,z_n\). A total polynomial degree bound \(p\)
allows only finitely many monomials
\(z_1^{a_1}\cdots z_n^{a_n}\), with
\(\sum_i a_i\leq p\).
Tensor contractions select a subspace of this finite-dimensional space.
Multiplying by a finite basis of the allowed mass and scale dependence
still gives a finite list.

Linear covariance, parity, or Ward constraints define a subspace.
Linear identities and integrations by parts define a quotient.
Subspaces and quotients of finite-dimensional spaces are
finite-dimensional.
\end{proof}

This lemma is a counting statement, not a proof that a proposed list
spans the renormalization freedom of a nonlocal composite.

\subsection{Derivative counting alone is insufficient}

\begin{proposition}[Unrestricted switch dependence]
\label{prop:switch-infinite}
If arbitrary polynomial dependence on a dimensionless switch is
permitted without a bound on switch degree, a derivative-order bound
alone does not give a finite-dimensional local functional catalogue.

For example, at formal order \(\kappa^2\), the functionals
\begin{equation}
 C_n[g,\varphi]
 =
 \kappa^2m^4\int g(x)^n\varphi(x)^2\dd^4x,
 \qquad n=1,2,\ldots,
 \label{eq:infinite-switch}
\end{equation}
are linearly independent as functionals of the pair
\((g,\varphi)\).
\end{proposition}

\begin{proof}
Suppose a finite linear combination vanishes for all compact smooth
\(g,\varphi\):
\[
 \sum_{n=1}^N a_n C_n[g,\varphi]=0.
\]
Choose \(\varphi\not\equiv0\) supported in a ball, and choose \(g\) equal
to an arbitrary constant \(s\) on that ball. Then
\[
 m^4\left(\sum_{n=1}^N a_ns^n\right)
 \int\varphi^2\dd^4x=0.
\]
Varying \(s\) over an interval shows that the polynomial is zero.
Therefore every \(a_n=0\).
\end{proof}

These functionals have no field derivatives, are even in \(\varphi\),
and have the indicated engineering dimension when
\([\kappa]=-1\), \([m]=[\varphi]=1\), and \([g]=0\).

The proposition does \emph{not} claim that all \(C_n\) occur as finite
renormalizations of a fixed two-insertion construction.
Multilinearity and perturbative homogeneity in local coupling
insertions may exclude them. Its point is that those restrictions must
be stated and used, rather than replaced by the phrase ``fixed
derivative order.''

Nor does an infinite functional catalogue imply an infinite-dimensional
image on a fixed finite sample: every such image is a subspace of a
finite-dimensional row space.

Figure~\ref{fig:VI-catalogue-bounds} distinguishes polynomial
finiteness from coverage of the allowed response class.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[pbox, text width=5.5cm] (fin) {finite test\\ $z_1,\ldots,z_n$ and $\sum_i a_i\le p$\\ finitely many monomials, then constraints and quotients};
\node[cbox, text width=5.5cm, right=4mm of fin] (bad) {counterexample\\ bounded derivatives, unbounded switch degree\\ $C_n=\kappa^2 m^4\int g^n\varphi^2$\\ linearly independent in $(g,\varphi)$};
\node[gbox, font=\scriptsize, below=of bad] (not) {not asserted in a fixed two-insertion class};
\node[pbox, text width=11.4cm, below=4mm of fin] (comp) {complete test: action, composite, switch, boundary, Ward, and lower-order normalization};
\node[cbox, text width=5.4cm, below left=3mm and 0mm of comp] {finiteness does not prove coverage};
\node[cbox, text width=5.4cm, below right=3mm and 0mm of comp] {a finite sample still has a finite-dimensional image};
\end{tikzpicture}
\caption{Finiteness and completeness are different
requirements. Explicit bounds on jet variables, polynomial
degree, and coefficient dependence yield a finite-dimensional
local class. A derivative bound alone does not control
arbitrary powers of a dimensionless switch. Even a finite
list needs a separate proof that it covers the allowed
renormalization responses of the specified composite.}
\label{fig:VI-catalogue-bounds}
\end{figure}


\subsection{A conditional response-space definition}

\begin{definition}[Complete response catalogue]
\label{def:catalogue}
Fix a response domain \(\mathcal B\), a reference-normalized response
function \(q:\mathcal B\to\R\), and all lower-order choices.
A list \(n_1,\ldots,n_L:\mathcal B\to\R\) is a complete
finite-renormalization response catalogue if:
\begin{enumerate}
\item every allowed finite change of normalization changes the response
by \(\sum_a c_an_a\), with coefficients constant across \(\mathcal B\);
\item every listed direction is an allowed response change, or the
allowed coefficient subspace is explicitly supplied;
\item all responses, including composite and switch-dependent
contributions, have been evaluated using the same observable,
dynamics, and boundary prescription.
\end{enumerate}
\end{definition}

If the allowed coefficient subspace is proper, choose a basis of that
subspace before applying the matrix statements below.

\subsection{Finite membership and separation}

For a finite ordered family
\(S=(\beta_1,\ldots,\beta_s)\subset\mathcal B\), set
\begin{equation}
 q_S=(q(\beta_i))_{i=1}^s,\qquad
 N_S=(n_a(\beta_i))_{i,a}.
 \label{eq:response-matrix}
\end{equation}

\begin{theorem}[Finite certificate alternative]
\label{thm:finite-certificate}
Let \(N\in\mathbb K^{s\times L}\) and \(q\in\mathbb K^s\), where
\(\mathbb K\) is a field. Exactly one of the following holds:
\begin{enumerate}
\item \(q=Nc\) for some \(c\in\mathbb K^L\);
\item there is \(z\in\mathbb K^{1\times s}\) such that
\[
 zN=0,\qquad zq=1.
\]
\end{enumerate}
The second alternative is equivalent to
\[
 \rank[N\mid q]=\rank N+1.
\]
\end{theorem}

\begin{proof}
If \(q=Nc\), every functional annihilating \(\im N\) annihilates \(q\).
Thus the alternatives are disjoint.

If \(q\notin\im N\), append \(q\) to a basis of \(\im N\), extend to a
basis of \(\mathbb K^s\), and define a linear functional equal to zero
on the contact basis and all additional basis elements except \(q\),
where it equals one. Its coordinate row is \(z\).
The rank characterization is the elementary criterion for appending
an independent column.
\end{proof}

\begin{corollary}[Interpretation with a complete catalogue]
\label{cor:certificate-meaning}
Under Definition~\ref{def:catalogue}, a row \(z\) satisfying
\(zN_S=0\) gives
\[
 zr_S=zq_S
\]
for every allowed finite normalization, where
\(r_S=q_S+N_Sc\).

If also \(zq_S\neq0\), no global identity
\(q=\sum_a c_an_a\) holds on \(\mathcal B\).
A finite fit \(q_S=N_Sc\), however, proves only the fit on \(S\).
\end{corollary}

\begin{proof}
Multiply \(r_S=q_S+N_Sc\) by \(z\).
A global identity would restrict to the same identity on \(S\), which
contradicts a separator. The converse requires additional information
and is not an implication of finite linear algebra.
\end{proof}

\begin{proposition}[When a finite fit is global]
\label{prop:injective-evaluation}
Let
\[
 U=\Span_\R\{q,n_1,\ldots,n_L\}\subset\R^{\mathcal B}.
\]
If evaluation on \(S\) is injective on \(U\), then
\(q_S=N_Sc\) implies
\(q=\sum_a c_an_a\) on \(\mathcal B\).

There exists some injective evaluation family of size at most
\(\dim U\), but this existence statement supplies neither explicit
smearings nor rational evaluations.
\end{proposition}

\begin{proof}
The difference \(q-\sum_a c_an_a\) belongs to \(U\) and vanishes on
\(S\). Injectivity makes it zero.

For existence, start with no evaluation rows. If the common evaluation
kernel in \(U\) is nonzero, select a nonzero function in that kernel
and a label where it is nonzero. Adding that row strictly decreases the
kernel dimension. At most \(\dim U\) steps give zero kernel.
\end{proof}

Figure~\ref{fig:VI-certificate-alternative} illustrates the
finite certificate alternative and the additional condition
needed to promote a fit to a global identity.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[font=\scriptsize, sgray] (sch) {schematic; not an evaluated matrix};
\node[pbox, below=of sch] (w) {$W=\operatorname{im} N_S$};
\node[sbox, text width=4.8cm, below left=5mm and 0mm of w] (inn) {$q_S\in W$ gives $q_S=N_Sc$};
\node[cbox, text width=4.8cm, below right=5mm and 0mm of w] (out) {$q_S\notin W$ gives $zN_S=0$, $zq_S=1$\\ $\ker z$ contains $W$, not $q_S$};
\draw[arr] (w)--(inn); \draw[arrg] (w)--(out);
\node[pbox, text width=10.4cm, below=4mm of inn] (ev) {$U=\operatorname{Span}\{q,n_1,\ldots,n_L\}\to\mathbb R^s$ by evaluation on $S$};
\node[gbox, text width=10.4cm, below=of ev] {separator excludes a global constant-coefficient identity given coverage; a finite fit is global only if evaluation is injective on $U$; no $q_S$ or $N_S$ is evaluated here};
\end{tikzpicture}
\caption{Conditional finite-response certification.
A target vector either belongs to the counterterm response
image or admits a separating row. Catalogue completeness
is necessary for normalization-invariant interpretation.
A fit on finitely many labels is global only when evaluation
is injective on the relevant function space. The diagram
contains no evaluated quantum response data.}
\label{fig:VI-certificate-alternative}
\end{figure}


\subsection{Certificate taxonomy and the actual status}

\begin{center}
\begin{tabular}{p{0.20\textwidth}p{0.70\textwidth}}
\toprule
Status & Required result\\
\midrule
RG-separated
& An evaluated finite separator against a complete declared response
catalogue\\
RG-finite
& An exact finite-family result without a proved global identity\\
RG-no
& A global constant-coefficient counterterm identity, proved directly or
through injective evaluation\\
RG-open
& Required products, catalogue coverage, or response evaluations are not
established\\
\bottomrule
\end{tabular}
\end{center}

The radiative problem discussed in this paper is \textbf{RG-open}.
No \(q_S\) or \(N_S\) is printed because neither has been constructed for
the proposed quantum relational response.

The two-representative table in
Appendix~\ref{app:witness} is an obstruction certificate for descent of
\eqref{eq:A1g}. It is not a substitute counterterm-response matrix.

\section{Non-identifications and the precise successor problem}

\subsection{Compact smearing and locality}

The support of \(f\) is the support of a function of relational
coordinate labels. The metric dependence of the observable includes
the coordinate inverses and their initial-data prescription. Compact
support of \(f\) is therefore not, by itself, a localization theorem for
an apparatus.

Donnelly and Giddings analyze gravitational dressing and its
implications for locality under their asymptotic and charge assumptions
\cite{DG}. The present finite-time construction uses a smaller,
initial-data-preserving gauge group. It neither proves nor bypasses
their conclusions.

A localized measurement construction requires a system--probe coupling
and a suitable scattering map. In the Fewster--Verch framework, causal
factorization is a hypothesis supporting the composition theorem for
instruments \cite{FV}. No such coupling or instrument is constructed
here.

\subsection{The earlier model interfaces}

The following distinctions are mathematical, not terminological.

\begin{enumerate}
\item
A first-order compensated scalar symbol does not automatically identify
the scalar correction selected by a switched dynamical prescription.

\item
An external-metric coefficient linear in the scalar field does not
construct products of a local quantum graviton field.

\item
A finite collection of graviton modes, even with a well-defined
quadratic operator, does not by itself give a local covariant metric
field.

\item
A prescribed metric response to scalar stress does not create an
independent graviton algebra.

\item
Representability of a pairing on a fixed two-dimensional TT plane does
not decide representability of a second-order pairing on a dense
infinite-dimensional TT space.

\item
A finite operational measurement characterization supplies no
Einstein--scalar interaction, relational renormalization, or localized
gravitational apparatus.

\item
Finite contact separation for a different tree action does not classify
the finite renormalizations of this observable.
\end{enumerate}

No theorem in this paper uses any of those identifications.

\subsection{The next necessary construction}

For the proposed quantum response, the next step is not to search for a
separator among guessed columns. It is to choose and justify a switched
gauge-compatible realization of the observable.

A possible completion must specify one of the following, or another
equally explicit alternative:
\begin{enumerate}
\item a restricted gauge-support class with a proved intertwining
identity;
\item an enlarged background space in which the switch transforms,
together with the resulting Ward identity;
\item a switch-dependent relational prescription whose complete
coefficient action is proved;
\item a local-coupling construction with a precisely controlled
switching-region breaking term and a demonstrated way of extracting the
desired physical response.
\end{enumerate}
These possibilities are not asserted to be equivalent.

After that choice, the required two-vertex and composite products must
be constructed, including their coincident-point and boundary
extensions. Only then can a complete finite-renormalization response
space and its matrix be evaluated.

The adiabatic limit, a localized probe, higher radiative orders, and
dense-TT representability remain separate successor problems.

\subsection{Conclusion}

The geometric construction closes:
\[
 \text{named Einstein--scalar jet}
 \longrightarrow
 \text{finite-time harmonic coordinate jet}
 \longrightarrow
 \text{second-order descended scalar}.
\]
The naive switched realization does not:
\[
 \text{unchanged free representative}
 \longrightarrow
 -\GR\cQ_{gh}\phi_0-X_{w[h]}\phi_0
\]
has the explicit nonzero gap of
Theorem~\ref{thm:plateau-obstruction}.

That gap occurs before a one-loop calculation or counterterm matrix.
It cannot be repaired by an order-two term while the order-one
prescription is retained. The quantum certificate problem consequently
remains RG-open, with a specific obstruction identified rather than an
unproved gauge identity assumed.

Figure~\ref{fig:VI-status-successors} summarizes the closed
results and the ordered requirements of the quantum
successor problem.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[sbox, text width=3.5cm] (a) {coefficient domain and source factor};
\node[mbox, text width=3.2cm, right=2mm of a] (b) {harmonic coordinate jet};
\node[sbox, text width=3.2cm, right=2mm of b] (c) {second-order classical descent};
\node[sbox, text width=3.5cm, below=of a] (d) {sourced nonzero family};
\node[cbox, text width=3.2cm, right=2mm of d] (e) {exact switch defect};
\node[cbox, text width=3.2cm, right=2mm of e] (f) {plateau and order obstruction};
\node[pbox, text width=10.4cm, below=of e] (g) {conditional finite-dimensional certificate theorem};
\node[gbox, text width=11.2cm, below=of g] (seq) {gauge-compatible realization $\longrightarrow$ products $\longrightarrow$ quantum identities $\longrightarrow$ complete catalogue $\longrightarrow$ evaluate $q_S,N_S$ $\longrightarrow$ certificate};
\node[cbox, below=of seq] (open) {RG-open};
\node[gbox, below left=4mm and -8mm of open] (s1) {adiabatic limit};
\node[gbox, right=2mm of s1] (s2) {localized probe};
\node[gbox, right=2mm of s2] (s3) {higher radiative orders};
\node[gbox, right=2mm of s3] {dense-TT representability};
\node[cbox, text width=11.2cm, below=of s2] {compact relational-label smearing is not an apparatus-localization theorem};
\end{tikzpicture}
\caption{Closed results and quantum successor problem.
Classical descent and the specified compact-switch
obstruction are established. A quantum certificate requires
a justified switched realization, renormalized products,
the relevant identities, catalogue coverage, and evaluated
responses. These steps remain uncompleted. Adiabatic,
localization, and dense-TT questions are separate successor
problems.}
\label{fig:VI-status-successors}
\end{figure}


\appendix

\section{Sign, coefficient, and switching audit}
\label{app:signs}

\subsection{The Duhamel sign}

With
\[
 P=-(\partial_t^2+\Omega_m^2),
\]
the equation
\[
 P\psi=-L_a\phi_0,\qquad \psi(0)=\dot\psi(0)=0
\]
has solution
\[
 \psi(t)
 =
 \int_0^t\frac{\sin((t-s)\Omega_m)}{\Omega_m}
               L_a\phi_0(s)\dd s.
\]
For a source supported after the initial surface,
\[
 \psi=-\GR L_a\phi_0.
\]
Thus the plus sign in the finite-time sine formula and the minus sign
in \(-\GR L_a\phi_0\) are the same convention.

\subsection{The metric source factor}

Variation of the gravitational and matter actions with respect to
\(\gamma^{\mu\nu}\) gives
\[
 \delta S_{\mathrm{grav}}
 =
 \frac2{\kappa^2}\int\sqrt{-\gamma}\,
 G_{\mu\nu}\delta\gamma^{\mu\nu},
\qquad
 \delta S_{\mathrm m}
 =
 -\frac12\int\sqrt{-\gamma}\,
 T_{\mu\nu}\delta\gamma^{\mu\nu}.
\]
Therefore
\[
 G_{\mu\nu}=\frac{\kappa^2}{4}T_{\mu\nu}.
\]
The second-order coefficient equation is consequently
\[
 G^{(1)}(j)+G^{(2)}[h,h]=\frac14T(\eta,\phi_0).
\]
A convention with right-hand side \(T\) uses a different normalization.

The normalization displayed in
Figure~\ref{fig:VI-coefficient-domain} is used throughout.


\subsection{Two new second-order identities}

The scalar equation is
\[
 P\phi_2=-L_h\phi_1-(L_j+\cM_h)\phi_0.
\]
The second relational coefficient is
\[
 \psi_2
 =
 \phi_2-X_w\phi_1
 +(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0
 +\frac12w^\mu w^\nu\partial_\mu\partial_\nu\phi_0.
\]
Both terms involving derivatives of \(w\) are necessary when this is
rewritten using iterated directional derivatives.

Indeed, define
\[
 b^\mu=v^\mu-\frac12w^\nu\partial_\nu w^\mu.
\]
Then
\begin{equation}
 \psi_2=\phi_2-X_w\phi_1-X_b\phi_0+\frac12X_w^2\phi_0.
 \label{eq:psi2-exponential}
\end{equation}
Expanding \(X_w^2\) verifies
\eqref{eq:psi2-exponential}. Replacing \(b\) by \(v\) in this last
formula would lose half of the derivative-of-\(w\) contribution.

Figure~\ref{fig:VI-classical-descent} displays the complete
coordinate-composition coefficient.


\subsection{The local switched Noether defect}

For a not necessarily on-shell scalar \(\varphi\),
\[
 \partial_\mu T^{\mu\nu}=(P\varphi)\partial^\nu\varphi.
\]
Under \(\delta_0h=K\xi\),
\begin{align}
 \delta_0V_1^{\mathrm m}(g;h,\varphi)
 &=
 \int gT^{\mu\nu}\partial_\mu\xi_\nu\dd^4x\\
 &=
 -\int g(P\varphi)X_\xi\varphi\dd^4x
 -\int(\partial_\mu g)T^{\mu\nu}\xi_\nu\dd^4x.
 \label{eq:local-noether-defect}
\end{align}
If the first scalar variation is chosen as
\(\delta_1^g\varphi=gX_\xi\varphi\), then
\[
 \delta_1^gS_0^{\mathrm m}
 =
 \int(P\varphi)gX_\xi\varphi\dd^4x,
\]
and
\begin{equation}
 \delta_0V_1^{\mathrm m}
 +\delta_1^gS_0^{\mathrm m}
 =
 -\int(\partial_\mu g)T^{\mu\nu}\xi_\nu\dd^4x.
 \label{eq:switched-noether}
\end{equation}
This is an explicit switching-region term. Its derivation does not
authorize its omission from a quantum identity.

\subsection{The plateau witness retains the switching terms}

For the witness of Theorem~\ref{thm:plateau-obstruction},
\(gK\xi=0\), so the entire forced response vanishes.
Nevertheless,
\[
 w[K\xi]=\xi.
\]
Near \(\supp f\), \(g=1\), so the pointwise term
\((g-1)X_\xi\phi_0\) in \eqref{eq:switch-defect} vanishes there.
The retarded switching contribution must therefore equal
\(-X_\xi\phi_0\) on the observation region.

This is not a paradox. Constancy of the switch on the observation
region does not make its derivatives vanish throughout the causal
region sampled by the retarded inverse.

The nested support construction is shown in
Figure~\ref{fig:VI-plateau-witness}.


\subsection{Integration by parts changes a switched density}

Suppose two local densities obey
\[
 \widetilde{\mathcal L}=\mathcal L+\partial_\mu B^\mu.
\]
Then
\begin{equation}
 \int g\widetilde{\mathcal L}\dd^4x
 =
 \int g\mathcal L\dd^4x
 -
 \int(\partial_\mu g)B^\mu\dd^4x.
 \label{eq:switched-IBP}
\end{equation}
Thus equality of unswitched actions modulo a total derivative does not
identify two arbitrarily switched density prescriptions unless the
switch-dependent term is also included.

The Action Ward Identity is relevant to formulating this dependence
correctly \cite{DF}; it is not permission to discard the last term in
\eqref{eq:switched-IBP}.

Figure~\ref{fig:VI-switching-audit} collects the switching terms
that must be retained in the local and retarded identities.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[cbox, text width=10.8cm] (n) {$\delta_0 V_1^{\mathrm m}+\delta_1^g S_0^{\mathrm m}=-\int(\partial_\mu g)T^{\mu\nu}\xi_\nu$\\ switching-region term};
\node[cbox, text width=10.8cm, below=of n] (ibp) {$\int g\widetilde{\mathcal L}=\int g\mathcal L-\int(\partial_\mu g)B^\mu$};
\node[cbox, below=of ibp] (drop) {discard a total derivative without the switch correction};
\draw[corange, thick] (drop.north west)--(drop.south east);
\draw[corange, thick] (drop.north east)--(drop.south west);
\node[sbox, text width=10.8cm, below=of drop] (plat) {$(g-1)X_\xi\phi_0=0$ near $\operatorname{supp} f$, yet the retarded switching term is nonzero};
\node[gbox, text width=10.8cm, below=of plat] {an observation plateau is not causal-region constancy; the Action Ward Identity is not a proved diffeomorphism Ward identity for this composite};
\end{tikzpicture}
\caption{Switching-region audit. Compact switching produces
explicit terms in the local Noether identity and when
densities are changed by total derivatives. The plateau
witness retains a retarded switching contribution even
though the pointwise plateau term vanishes near the
observation. These identities do not authorize dropping
switching terms from a proposed quantum Ward identity.}
\label{fig:VI-switching-audit}
\end{figure}


\section{The finite obstruction certificate}
\label{app:witness}

\subsection{A concrete compact test family}

The proof of Theorem~\ref{thm:plateau-obstruction} can be implemented
with standard bump functions. Let
\[
 b(s)=
 \begin{cases}
 \exp(-1/(1-s^2)),& |s|<1,\\
 0,& |s|\geq1.
 \end{cases}
\]
Let \(B\in C_c^\infty(\R^4)\) be a nonnegative product of rescaled
copies of \(b\), supported in the unit coordinate cube and normalized
by
\[
 \int_{\R^4}B(y)\dd^4y=1.
\]
For a point \(x_*=(t_*,0)\), put
\[
 f_\epsilon(x)=\epsilon^{-4}B((x-x_*)/\epsilon).
\]
Choose \(\epsilon>0\) small enough that
\[
 \supp f_\epsilon\subset U,
 \qquad
 \partial_1\phi_0\geq\frac12
 \quad\hbox{on }U.
\]
Smooth plateau cutoffs give \(g\) and \(\chi_\xi\) with
\[
 g=1\ \hbox{near }\supp f_\epsilon,
 \qquad
 \chi_\xi=1\ \hbox{near }\supp g.
\]
All supports can be kept inside \((0,T)\times\R^3\).

This construction is exact. It does not claim rational values for the
bump integrals.

\subsection{The evaluated two-representative table}

Set
\[
 a_f=\int f\partial_1\phi_0\dd^4x\geq\frac12.
\]
For the same \(f,g,\phi_0\), the two rows are
\begin{center}
\begin{tabular}{ccccc}
\toprule
Metric input & \(gh\) & \(\int f\phi_{1,g}\) &
\(\int fX_{w[h]}\phi_0\) & \(\cA_{f,g}^{(1)}\)\\
\midrule
\(0\) & \(0\) & \(0\) & \(0\) & \(0\)\\
\(K\xi\) & \(0\) & \(0\) & \(a_f\) & \(-a_f\)\\
\bottomrule
\end{tabular}
\end{center}
The free-input metric quotient identifies the rows, while the candidate
functional does not.

\begin{proposition}[Quantitative failure of quotient approximation]
For any function \(F\) of the common quotient label,
\[
 \max\left\{
 |F-\cA_{f,g}^{(1)}(0,\phi_0)|,\,
 |F-\cA_{f,g}^{(1)}(K\xi,\phi_0)|
 \right\}
 \geq\frac{a_f}{2}\geq\frac14.
\]
\end{proposition}

\begin{proof}
The two candidate values differ by \(a_f\). Insert the common value
\(F\) between them and apply the triangle inequality.
\end{proof}

This bound is a finite obstruction to descent of a named classical
assignment. It says nothing about a left kernel of an unevaluated
quantum counterterm matrix.

Figure~\ref{fig:VI-two-certificates} distinguishes the
evaluated classical obstruction from the unconstructed
quantum certificate.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[sbox, text width=5.6cm] (cl) {classical certificate\\ $h=0:\ \mathcal A^{(1)}=0$\\ $h=K\xi:\ \mathcal A^{(1)}=-a_f$,\ $a_f\ge 1/2$\\ both have $gh=0$ and zero forced response};
\node[cbox, text width=5.6cm, below=of cl] (gap) {common value $F$\\ $\max\{|F|,|F+a_f|\}\ge a_f/2\ge 1/4$\\ analytic inequality, not a simulation};
\node[pbox, text width=5.2cm, right=4mm of cl] (q) {unconstructed quantum certificate\\ empty $q_S$, $N_S$, catalogue, and Ward-compatible products};
\node[cbox, below=of q] {RG-open};
\node[gbox, text width=11.2cm, below=4mm of gap] {the two-representative table is not a counterterm-response matrix};
\end{tikzpicture}
\caption{Different certificates. The two classical
representatives share a proposed quotient label but have
candidate values separated by \(a_f\geq 1/2\). Any common
quotient value therefore incurs error at least \(1/4\) on
one representative. This obstruction is established without
constructing a radiative target vector or a counterterm
response matrix; the quantum problem remains RG-open.}
\label{fig:VI-two-certificates}
\end{figure}


\section{Static audit of the proposed quantum certificate}
\label{app:quantum-audit}

\subsection{Required data and present status}

\begin{longtable}{p{0.36\textwidth}p{0.55\textwidth}}
\toprule
Required datum & Status and consequence\\
\midrule
\endfirsthead
\toprule
Required datum & Status and consequence\\
\midrule
\endhead
Classical action and normalization
&
Specified in \eqref{eq:action}; source factor fixed in
\eqref{eq:full-equations}.\\[3pt]
Classical relational prescription
&
Specified by the harmonic coordinate equation and its Cauchy data.\\[3pt]
Classical gauge group
&
Formal pullbacks generated by compact vector fields supported away
from the initial surface.\\[3pt]
Classical second-order descent
&
Proved in Theorem~\ref{thm:descent}.\\[3pt]
Switched realization on unchanged free inputs
&
The candidate \eqref{eq:A1g} fails by
Theorem~\ref{thm:plateau-obstruction}.\\[3pt]
Interacting BRST/BV prescription
&
Not supplied. A free gauge complex is not its replacement.\\[3pt]
Renormalized products with the coordinate kernels
&
Not constructed. Classical smoothness is insufficient to establish
their quantum distributional products.\\[3pt]
Quantum switch identity
&
Not proved. The classical defect is explicit, but no quantum
anomaly analysis has been performed.\\[3pt]
Action counterterms
&
No complete list for the intended quantum order is claimed.\\[3pt]
Composite counterterms
&
No complete list for the relational insertion is claimed.\\[3pt]
Boundary and switch counterterms
&
Their admissibility and bounds have not been fixed sufficiently to
supply a complete response catalogue.\\[3pt]
Chosen radiative response and state
&
Not specified to the point of an evaluated quantum coefficient.\\[3pt]
Finite family of radiative smearings
&
Not evaluated. The classical obstruction family is a different
certificate.\\[3pt]
Matrix \(N_S\) and target \(q_S\)
&
Not constructed.\\[3pt]
Rank, minor, fit, or separator
&
None claimed for the quantum problem.\\
\bottomrule
\end{longtable}

Figure~\ref{fig:VI-status-successors} records the uncompleted
quantum construction sequence, while
Figure~\ref{fig:VI-two-certificates} distinguishes it from
the evaluated classical obstruction.


\subsection{Why a selected subcatalogue cannot establish robustness}

Suppose a proposed calculation lists response columns
\(n_1,\ldots,n_L\) and produces \(zN=0\). If an allowed omitted
counterterm has response \(n_*\) with
\[
 zn_*\neq0,
\]
then \(z\) is not invariant under the complete normalization class.
The equality \(zN=0\) remains a true statement about the printed
submatrix, but its physical interpretation changes.

A complete catalogue may be redundant. Independence is not required.
Coverage is required.

\subsection{Interval evaluations}

Exact rational entries are not necessary if rigorous enclosures are
available. However, a statement that an entry is ``numerically small''
is not a zero certificate.

For example, if a candidate row \(z\) and reference vector \(q\) are
fixed exactly, and enclosures prove
\[
 |zq|\geq\delta>0,
\]
this proves only nonvanishing of that contraction. To obtain a
separator one must also prove exact annihilation of every allowed
counterterm response, or give a separately stated approximate-robustness
theorem with bounded counterterm coefficients.

Without a bound on the coefficients \(c\), a merely small
\(\|zN\|\) gives no uniform bound on \(|zNc|\).

\begin{proposition}[Approximate annihilation without coefficient bounds]
Let \(zN\neq0\). Then
\[
 \sup_{c\in\R^L}|zNc|=\infty.
\]
\end{proposition}

\begin{proof}
Choose \(c_0\) with \(zNc_0\neq0\), and replace \(c_0\) by \(tc_0\).
The absolute value grows linearly with \(|t|\).
\end{proof}

This is why a remainder smaller than a selected minor does not, by
itself, replace exact coverage and annihilation.

Figure~\ref{fig:VI-certificate-audit} illustrates why catalogue
coverage and exact annihilation cannot be replaced by a
selected submatrix or an uncontrolled small residual.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[node distance=2.6mm and 4mm]
\node[cbox, text width=5.5cm] (om) {$zN=0$ on a selected subcatalogue\\ extra allowed column $n_*$ with $zn_*\neq 0$\\ true submatrix identity; invalid complete separator\\ redundancy allowed; coverage required};
\node[cbox, text width=5.5cm, right=4mm of om] (ap) {nonzero row $zN$\\ $c=tc_0$ gives $zNc=t\,zNc_0$\\ $\sup_c|zNc|=\infty$\\ a small residual is not uniform robustness};
\node[gbox, text width=11.4cm, below=4mm of om] {a rigorous nonzero enclosure of $zq$ is not by itself exact counterterm annihilation; no tolerances or computed minors are shown};
\end{tikzpicture}
\caption{Certificate robustness requirements. An omitted
allowed response direction can invalidate the interpretation
of a left-kernel row from a selected subcatalogue. Likewise,
a nonzero annihilation residual can be amplified without
bound when counterterm coefficients are unrestricted.
Rigorous target nonvanishing and complete response
annihilation are separate requirements.}
\label{fig:VI-certificate-audit}
\end{figure}


\subsection{No submission-level quantum claim from a production plan}

The four-term expansion, a framework citation, and a proposed list of
theorem headings do not jointly constitute a radiative calculation.
The present manuscript deliberately gives the narrower theorem
supported by its calculations.

A subsequent paper claiming a quantum finite-renormalization
certificate must replace the unconstructed entries of the audit by
definitions, proofs, and evaluated responses. It must also address the
specific lower-order switching obstruction or explicitly use a
different, proved gauge-compatible prescription.

\section*{Declarations}

\paragraph{Mathematical scope.}
The established field-theoretic statements are classical formal
coefficient statements on the declared slab and gauge domain.
The retarded-product coefficient identity and matrix theorems state
their hypotheses explicitly. No quantum Ward identity, radiative
counterterm catalogue, or quantum separator is claimed.

\paragraph{Relation to the series.}
The manuscript is independent of measurement-characterization
theorems, finite-mode instruments, and finite multi-graviton
contact evaluations. Its first-order response and gauge identities are
written and proved here because the switch obstruction depends on
their precise signs and domains.

\paragraph{Data and computation.}
No empirical data, numerical loop integrations, or executed symbolic
calculations are used. The nonzero witnesses are analytic.

\paragraph{Verification.}
No proof-assistant verification, independently checked loop calculation,
successful \texttt{pdflatex} execution, or independent referee review is
asserted.

\paragraph{Publication scope.}
The title and abstract describe the obstruction result actually proved.
They do not present an unfinished quantum certification programme as a
completed contribution with a radiative prediction.

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\end{document}