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\hypersetup{
  pdftitle={XIX-B: A Witness Mismatch Between Two Coordinate Cauchy Prescriptions},
  pdfauthor={Ngo Cheung},
  pdfsubject={Classical harmonic coordinates and prepared witness completion}
}

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\title{\textbf{XIX-B}\\[3pt]
A Witness Mismatch Between Two Coordinate Cauchy Prescriptions\\
\large Unswitched Harmonic Coordinates and Prepared Coordinate Completion}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We compare the finite-time harmonic coordinate prescription of
Landmark VI with the prepared coordinate candidate recorded in
XVI-D. We first prove the scalar-wave/density identity needed to
relate VI's equation to XIX-A's displayed comparison equation,
including the conversion between their second-order coefficient
conventions. This authenticates the coordinate equation on their
common domain, not the entire predecessor observable.

For the explicit counter-propagating witness source, let
\(V_\lambda\) be its zero-data time integral and retain the response
sign \(R_\lambda=-\mathcal P_+V_\lambda\). After a declared overall
metric-sign transport, VI's first coordinate coefficient has
normalized forcing \(-\partial_tV_\lambda^{0\mu}\), whereas the
prepared candidate has forcing
\(-\partial_t(\sigma V_\lambda^{0\mu})\).
Their difference is the nonzero smooth residual
\(\partial_t((\sigma-1)V_\lambda^{0\mu})\).
An explicit compact-test construction detects that residual.
The coordinate fields consequently differ on the common slab.
The same response, placed as VI's first metric coefficient, also
fails VI's vacuum first-metric equation; this domain failure is
calculated separately.

The result is a first-order mismatch for the specified direct
comparison. It is not VI's scalar-switch defect, a universal
obstruction to another connecting construction, or a quantum
anomaly. Smooth no-decay uniqueness, operator-domain extension,
observable matching, and full-kernel determination retain their
separate roles. No second-order or positive-formal-quantum-order
term repairs the fixed first-order mismatch.
\end{abstract}

\noindent\textbf{Keywords:}
harmonic coordinates; Cauchy prescription; prepared response;
trace reversal; relational observables; no-decay uniqueness.

\tableofcontents

%=====================================================================
\section{Question and exact scope}
\label{sec:scope}
%=====================================================================

XIX-A separated time-related constructions and left the
VI--XVI-D coordinate identification undecided
\cite{XIXA}. Its displayed density equation was a comparison
prescription. It was not, merely by its label, the registered
VI coordinate construction.

The present paper makes one specified comparison and reaches
the \emph{mismatch} alternative. The comparison is:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item retain XVI-D's response tensor
      \(R_\lambda=-\mathcal P_+V_\lambda\);
\item regard that smooth response as a metric argument of the
      first harmonic-coordinate map;
\item transport the metric signature explicitly;
\item compare the resulting first harmonic coefficient with
      XVI-D's \(X_{\mathrm{cand}}\), on the same slab and with
      the same ordinary coordinate-time zero data.
\end{enumerate}
All mathematical definitions used in this comparison are printed
below. Neither a separation edge nor an unavailable companion
proof is a premise.

\paragraph{The decisive distinction.}
The supplied VI prescription is unswitched:
\[
 \Box_\gamma X_\gamma^\mu=0.
\]
The supplied XVI-D candidate uses a prepared density forcing:
\[
 \Boxp X_{\mathrm{cand}}^\mu
 =-\partial_t(\sigma V_\lambda^{0\mu}).
\]
Equality must be tested after linearization and sign normalization,
not inferred from the words ``harmonic'', ``retarded'', or
``Cauchy''.

\paragraph{The result is coefficientwise.}
The comparison parameter below probes the coordinate prescription.
It does not regrade the native gravitational coupling order of
the response \(R_\lambda\), which is bilinear in its two tensor
inputs. If a different comparison places that response in VI's
second metric coefficient, the additional second-order terms must
be included. That different comparison is not decided here.

\paragraph{Literature context.}
Relational-coordinate constructions and perturbative invariant
observables are established subjects
\cite{BFHPR,Frob}. Their existence does not identify two particular
boundary or preparation prescriptions. Likewise, Cheung's
operational characterization of measurement concerns measurement
strategies, not the Cauchy forcing studied here
\cite{CheungEnter}.
% Literature context is supported by the primary records.
% ([arxiv.org](https://arxiv.org/abs/1605.02573))

\paragraph{No false identification of the obstruction.}
VI's compact-switch defect concerns a switched scalar assignment.
It is not an obstruction to existence or uniqueness of VI's
coordinate jet. Section~\ref{sec:switch} proves the corresponding
classical identity separately; no interaction switch \(g\) enters
the coordinate mismatch theorem.

%=====================================================================
\section{The common slab and the native prescriptions}
\label{sec:native}
%=====================================================================

\subsection{Signs, tensor positions, and initial data}

Fix \(T>0\) and \(0<\varepsilon<T\). Set
\[
 M_T=[0,T]\times\R^3,\qquad
 M_T^\circ=(0,T)\times\R^3.
\]
All coefficient fields are smooth on a neighborhood of the slab.
No spatial decay is imposed unless it is explicitly stated.

Keep both background signatures visible:
\begin{equation}
\label{eq:signs}
 \eta_-=\operatorname{diag}(-1,1,1,1),\qquad
 \eta_+=-\eta_-,
 \qquad
 \Boxm=-\partial_t^2+\Delta,\quad
 \Boxp=\partial_t^2-\Delta.
\end{equation}
For a covariant symmetric tensor \(a\), define
\[
 a_\pm^{\mu\nu}
 =\eta_\pm^{\mu\alpha}\eta_\pm^{\nu\beta}a_{\alpha\beta},
 \qquad
 \tr_\pm a=\eta_\pm^{\alpha\beta}a_{\alpha\beta},
\]
and
\begin{equation}
\label{eq:trace-reverse-cov}
 \overline a_\pm^{\mu\nu}
 =a_\pm^{\mu\nu}
      -\frac12\eta_\pm^{\mu\nu}\tr_\pm a,
 \qquad
 D_\pm(a)^\mu=\partial_\alpha
                    \overline a_\pm^{\alpha\mu}.
\end{equation}
For a contravariant tensor \(B\), trace reversal means
\begin{equation}
\label{eq:Pplus}
 (\mathcal P_\pm B)^{\mu\nu}
 =B^{\mu\nu}
      -\frac12\eta_\pm^{\mu\nu}
                 \eta_{\pm,\alpha\beta}B^{\alpha\beta}.
\end{equation}
In four dimensions, \(\mathcal P_\pm^2=\id\).

The superscript \(\mu\) on a coordinate function labels one of
four scalar functions. It does not turn that function into a
vector component under a change of metric-sign convention.

\subsection{VI's coordinate equation}

The registered VI coefficient convention is
\[
 \gamma_\kappa=\eta_-+\kappa h+\kappa^2j,
 \qquad
 X_\gamma^\mu=x^\mu+\kappa w^\mu+\kappa^2v^\mu
       \pmod{\kappa^3},
\]
without factorials at order two. Its coordinate conditions are
\begin{equation}
\label{eq:VI-data}
 \Box_{\gamma_\kappa}X_\gamma^\mu=0
       \pmod{\kappa^3},\qquad
 X_\gamma^\mu|_{t=0}=x^\mu|_{t=0},\qquad
 \partial_tX_\gamma^\mu|_{t=0}=\delta^\mu_0.
\end{equation}
Thus \(w,v,\partial_tw,\partial_tv\) vanish at \(t=0\).
The first equation is
\begin{equation}
\label{eq:VI-first}
 \Boxm w^\mu=D_-(h)^\mu.
\end{equation}
These are ordinary coordinate-time data, not metric-dependent
unit-normal data. VI's coordinate proposition is stated for every
smooth \(h,j\), whereas its descended solution-jet observable has
the additional equation-space restriction
\cite[\sid{prop:coordinates}, \sid{eq:DK},
\sid{thm:descent}]{CheungVI}.
% Native coordinate equation, data, and first coefficient.
% ([enter.grok.me](https://enter.grok.me/passage/VI-proposition-prop-coordinates))

\subsection{Coordinate domain versus solution-jet domain}

The necessary first metric condition in VI's named
\(\mathcal J_T\) is
\begin{equation}
\label{eq:VI-vacuum}
 \mathcal G_-(h)=0,
\end{equation}
where its fixed linearized Einstein tensor is
\begin{equation}
\label{eq:Einstein}
 \mathcal G_{-,\mu\nu}(h)
 =-\frac12\Boxm\overline h_{-,\mu\nu}
   +\partial_{(\mu}D_-(h)_{\nu)}
   -\frac12\eta_{-,\mu\nu}\partial_\rho D_-(h)^\rho.
\end{equation}
Here lowering in the latter formula uses \(\eta_-\), and
parentheses denote symmetrization with factor \(1/2\).
This normalization is sufficient for our domain test.

The higher metric coefficient, scalar coefficients, and their
equations impose further conditions in \(\mathcal J_T\).
We do not assert that an arbitrary smooth metric argument extends
to that solution-jet space
\cite[\sid{def:jet-space}]{CheungVI}.
% The source explicitly distinguishes smooth metric arguments from J_T.
% ([enter.grok.me](https://enter.grok.me/passage/VI-definition-def-jet-space))

\paragraph{Logical consequence.}
Failure of \eqref{eq:VI-vacuum} prevents applying VI's
solution-jet descent theorem to that input. It does \emph{not}
prevent solving the independent smooth coordinate Cauchy problem
\eqref{eq:VI-data}.

%=====================================================================
\section{Lemma A: the density identity and the XIX-A comparison}
\label{sec:lemmaA}
%=====================================================================

For a formal metric \(q\) with invertible zeroth-order coefficient,
write
\[
 \rho_q=\sqrt{-\det q},\qquad
 \mathfrak a_q^{\alpha\beta}=\rho_q q^{\alpha\beta}.
\]
For a scalar \(u\), the scalar-wave definition gives
\begin{equation}
\label{eq:density-identity}
 \Box_qu
 =\rho_q^{-1}\partial_\alpha
       \left(\mathfrak a_q^{\alpha\beta}\partial_\beta u\right).
\end{equation}

\begin{lemma}[Lemma A: equation identity on a matched jet]
\label{lem:A}
On \(M_T\), use the same metric jet, the same initial surface,
and the same ordinary coordinate-time data.

Then the density-coordinate equation of XIX-A and the
scalar-wave coordinate equation of VI are equivalent
coefficientwise through order two. For the conventions used in
XIX-A,
\[
 e=\kappa,\qquad k_{\mathrm{XIX-A}}=2j,\qquad
 W_1=w,\qquad W_2=2v.
\]
The equivalence applies on the shared domain, including XIX-A's
spatially compact metric subclass. It does not identify the
complete scalar observables, enlarge XIX-A's original domain,
or turn its label \(\mathsf C_{\mathrm{VI}}\) into an authentication
of every VI construction.
\end{lemma}

\begin{proof}
The zeroth coefficient of \(\rho_q\) is \(1\), so \(\rho_q\)
has an invertible formal power series modulo \(\kappa^3\).
Multiplication by it preserves vanishing modulo \(\kappa^3\).
Equation \eqref{eq:density-identity} therefore proves equivalence
of the two coordinate equations.

For completeness, in the convention
\(q=\eta_-+\kappa h+\kappa^2j\), put
\[
 \mathfrak a_q=\eta_-^{-1}+\kappa a_1+\kappa^2a_2.
\]
Matrix inversion and determinant expansion give
\begin{align}
\label{eq:a1}
 a_1^{\alpha\beta}
 &=\frac12(\tr_-h)\eta_-^{\alpha\beta}
      -h_-^{\alpha\beta}
   =-\overline h_-^{\alpha\beta},\\
\label{eq:a2}
 a_2^{\alpha\beta}
 &=-\overline j_-^{\alpha\beta}
   +h^\alpha{}_\rho h_-^{\rho\beta}
   -\frac12(\tr_-h)h_-^{\alpha\beta}
   \nonumber\\
 &\quad+
  \left(\frac18(\tr_-h)^2
        -\frac14h_{\rho\sigma}h_-^{\rho\sigma}\right)
                     \eta_-^{\alpha\beta}.
\end{align}
All mixed indices in \eqref{eq:a2} use \(\eta_-\).
The density equation for
\(X=x+\kappa w+\kappa^2v\) has coefficients
\begin{align}
\label{eq:density-first}
 \Boxm w^\mu&=-\partial_\alpha a_1^{\alpha\mu}
                  =D_-(h)^\mu,\\
\label{eq:density-second}
 \Boxm v^\mu
 &=-\partial_\alpha a_2^{\alpha\mu}
   -\partial_\alpha
             \left(a_1^{\alpha\beta}\partial_\beta w^\mu\right).
\end{align}
The initial coefficient data are unchanged.

XIX-A wrote the second metric and coordinate coefficients with
a factor \(1/2\). Substitution of \(k=2j\) and \(W_2=2v\)
therefore yields the displayed conversion. The argument uses
equations and data only; it does not compare scalar dynamics
or observable domains.
\end{proof}

The density method is also the method recorded for VII-A, with
no interaction switch in its coordinate equation
\cite{CheungVIIA,CheungXA}. VII-A is a method comparison here,
not an existence theorem imported into VI.

\begin{remark}[The domain qualification matters]
The witness below is spatially homogeneous and noncompact.
It does not belong to the spatially compact metric subclass
originally printed in XIX-A. We consequently evaluate VI's
smooth coordinate prescription directly and independently
construct its no-decay solution. Lemma~\ref{lem:A} alone is not
used to extend the old XIX-A domain.
\end{remark}

%=====================================================================
\section{The explicit prepared witness}
\label{sec:witness}
%=====================================================================

\subsection{Frozen source data}

Let \(\sigma\in C^\infty(\R;\R)\) satisfy
\begin{equation}
\label{eq:sigma}
 \sigma(t)=0\quad(t\le0),\qquad
 \sigma(t)=1\quad(t\ge\varepsilon).
\end{equation}
No limit in \(\varepsilon\) is taken. Continuity guarantees a
nonempty open interval on which \(0<\sigma<1\).

Fix \(\lambda>0\) and a Euclidean unit vector
\(\mathbf n\in\R^3\). The native plus-signature tensor inputs are
\[
 k_1=(1,\mathbf n),\qquad k_2=(1,-\mathbf n),\qquad
 \gamma_a^\lambda=\eta_+e^{i\lambda k_a\cdot y},
 \qquad
 k\cdot y=\eta_{+,\mu\nu}k^\mu y^\nu.
\]
Their product phase is \(e^{2i\lambda t}\).

We take the following \emph{explicit smooth source} as frozen
data:
\begin{align}
\label{eq:S}
 S_\lambda^{\mu\nu}(t)
 &=\left[\lambda^2\sigma(t)Q^{\mu\nu}
             +\lambda d_1^{\mu\nu}(t)\right]e^{2i\lambda t},\\
\label{eq:Q}
 Q^{00}&=-2,\qquad Q^{0i}=0,\qquad
 Q^{ij}=n^in^j-3\delta^{ij},\\
\label{eq:d1}
 d_1^{\mu\nu}
 &=-i\sigma'(t)
       \left(\eta_+^{\mu\nu}
                      +2\delta^\mu_0\delta^\nu_0\right).
\end{align}
Thus
\begin{equation}
\label{eq:S00}
 S_\lambda^{00}
 =\left[-2\lambda^2\sigma-3i\lambda\sigma'\right]
                                      e^{2i\lambda t},
 \qquad S_\lambda^{0i}=0.
\end{equation}
These formulas are the supplied XVI-B/XVI-D witness source
\cite[\sid{prop:pair}]{CheungXVIB}
\cite[\sid{eq:Slambda}]{CheungXVID}. Neither \(Q\) nor \(d_1\)
contains an additional frequency factor.
% Frozen source record; the calculations below use its printed definition.
% ([enter.grok.me](https://enter.grok.me/passage/XVI-B-proposition-prop-pair))

The source definition is sufficient for the present proof.
We do not derive the full gravitational source polynomial
\(J_D\), its Euler origin, or its complete tensor-input domain.

\subsection{Response, response sign, and coordinate candidate}

Define
\begin{equation}
\label{eq:V}
 V_\lambda^{\mu\nu}(t)
 =\int_0^t(t-s)S_\lambda^{\mu\nu}(s)\,\dd s,
 \qquad
 R_\lambda^{\mu\nu}=-(\mathcal P_+V_\lambda)^{\mu\nu}.
\end{equation}
Then
\[
 V_\lambda''=S_\lambda,\qquad
 V_\lambda(0)=V_\lambda'(0)=0,\qquad
 V_\lambda^{0i}=0.
\]
Because the source is ambient smooth and flat at \(t=0\),
extension of \(V_\lambda\) by zero for \(t\le0\) is smooth.

The sign \(R_\lambda=-\mathcal P_+V_\lambda\) is retained.
It is the polarized response \(U_D\) on the witness, also
explicitly recorded in XVI-B's account of XVI-D
\cite[\sid{eq:later-response}]{CheungXVIB}
\cite{CheungXVID}. The separate action-equation coefficient
\(r_D=-u_D\) is not silently substituted for the response argument.
% The witness response sign and its relationship to U_D.
% ([enter.grok.me](https://enter.grok.me/passage/XVI-B-section-eq-later-response))

For a contravariant response argument \(R\), its native
first density coefficient is
\[
 a_{1,+}^{\mu\nu}[R]
 =\frac12\eta_+^{\mu\nu}
                   \eta_{+,\alpha\beta}R^{\alpha\beta}
       -R^{\mu\nu}
 =-(\mathcal P_+R)^{\mu\nu}.
\]
Consequently,
\begin{equation}
\label{eq:a1-witness}
 a_{1,+}[R_\lambda]=V_\lambda.
\end{equation}
The prepared coordinate forcing and candidate are
\begin{align}
\label{eq:Fco}
 F_{\mathrm{co}}^\mu
 &=\partial_\alpha
           \left(\sigma a_{1,+}^{\alpha\mu}[R_\lambda]\right)
   =\partial_t(\sigma V_\lambda^{0\mu}),\\
\label{eq:Xcand}
 X_{\mathrm{cand}}^\mu(t)
 &=-\int_0^t\sigma(s)V_\lambda^{0\mu}(s)\,\dd s.
\end{align}
These are the supplied completion interface and candidate
\cite[\sid{eq:coordinate-leg}]{CheungXVIB}
\cite[\sid{lem:Xcand}]{CheungXVID}.
% The completion density and its prepared divergence.
% ([enter.grok.me](https://enter.grok.me/passage/XVI-B-section-eq-coefficients))

Direct differentiation, with spatial homogeneity, proves
\begin{equation}
\label{eq:candidate-PDE}
 \Boxp X_{\mathrm{cand}}^\mu=-F_{\mathrm{co}}^\mu,\qquad
 X_{\mathrm{cand}}^\mu(0)
   =\partial_tX_{\mathrm{cand}}^\mu(0)=0,\qquad
 X_{\mathrm{cand}}^i=0.
\end{equation}
This calculation does not identify an inherited Green symbol
outside its declared domain.

\subsection{The declared metric-sign transport}

Lower the response with \(\eta_+\):
\[
 (R_\lambda^\flat)_{\mu\nu}
 =\eta_{+,\mu\alpha}\eta_{+,\nu\beta}
                      R_\lambda^{\alpha\beta}.
\]
The comparison metric coefficient on VI's background is
\begin{equation}
\label{eq:htransport}
 h_{\lambda,\mu\nu}=-(R_\lambda^\flat)_{\mu\nu}.
\end{equation}
This is the transport induced by
\[
 q_{+,\kappa}=\eta_++\kappa R_\lambda^\flat,\qquad
 q_{-,\kappa}=-q_{+,\kappa}
             =\eta_-+\kappa h_\lambda.
\]
For an actual real metric, \(\Box_{-q}=-\Box_q\).
The same identity holds for the formal jets because the
determinants agree in four dimensions and the inverses differ
by a minus sign. The harmonic equation and ordinary
coordinate-time data are therefore transported without changing
the coordinate scalar functions.

Complex coefficients are used only through complex linear
extension. Real witnesses will be obtained by taking
\(\operatorname{Re}(\zeta\,\cdot)\), with \(\abs{\zeta}=1\).

\begin{lemma}[Trace reversal under the declared transport]
\label{lem:transport}
For \eqref{eq:htransport},
\begin{equation}
\label{eq:transport-trace}
 \overline h_{\lambda,-}^{\mu\nu}
 =-(\mathcal P_+R_\lambda)^{\mu\nu}
 =V_\lambda^{\mu\nu}.
\end{equation}
Hence
\[
 D_-(h_\lambda)^\mu
 =\partial_tV_\lambda^{0\mu}.
\]
\end{lemma}

\begin{proof}
Raising both indices of \(-R_\lambda^\flat\) with
\(\eta_-=-\eta_+\) gives \(-R_\lambda^{\mu\nu}\).
Its minus-signature trace is
\(\eta_{+,\alpha\beta}R_\lambda^{\alpha\beta}\).
Substitution in \eqref{eq:trace-reverse-cov} yields
\(-\mathcal P_+R_\lambda\).
Equation \eqref{eq:V} and
\(\mathcal P_+^2=\id\) give \(V_\lambda\).
Spatial homogeneity then gives the divergence formula.
\end{proof}

%=====================================================================
\section{Lemma B: the detecting forcing calculation}
\label{sec:lemmaB}
%=====================================================================

\begin{lemma}[Lemma B: witness forcing mismatch]
\label{lem:B}
On the same slab \(M_T\), with the transport
\eqref{eq:htransport} and equal ordinary zero Cauchy data,
VI's first coordinate equation becomes
\begin{equation}
\label{eq:VI-normalized}
 \Boxp w_\lambda^\mu
       =-\partial_tV_\lambda^{0\mu}.
\end{equation}
The prepared candidate instead satisfies
\eqref{eq:candidate-PDE}. Their normalized forcing difference is
\begin{equation}
\label{eq:forcing-residual}
 \mathfrak r_\lambda^\mu
 :=F_{\mathrm{co}}^\mu-\partial_tV_\lambda^{0\mu}
 =\partial_t\left((\sigma-1)V_\lambda^{0\mu}\right).
\end{equation}
For every \(\lambda>0\), its time component is nonzero as a
smooth function. It is supported in the temporal preparation
layer \([0,\varepsilon]\), but not in a compact spatial set.

There are a phase \(\zeta\in\C\), \(\abs{\zeta}=1\), and a
real nonnegative \(f\in C_c^\infty(M_T^\circ)\) such that
\begin{equation}
\label{eq:forcing-test}
 \int_{M_T}f\,
       \operatorname{Re}(\zeta\mathfrak r_\lambda^0)
                            \,\dd^4x>0.
\end{equation}
Thus a real smooth witness detects the forcing mismatch.
\end{lemma}

\begin{proof}
Equation \eqref{eq:VI-first} and
Lemma~\ref{lem:transport} give
\[
 \Boxm w_\lambda^\mu=\partial_tV_\lambda^{0\mu}.
\]
Multiplying by \(-1\) gives
\eqref{eq:VI-normalized}. Subtracting its forcing from the
candidate forcing gives
\[
 (-\partial_tV_\lambda^{0\mu})
       -(-F_{\mathrm{co}}^\mu)
 =\mathfrak r_\lambda^\mu,
\]
which proves \eqref{eq:forcing-residual}.
The sign normalization is essential: the native candidate has
\(\Boxm X_{\mathrm{cand}}=+F_{\mathrm{co}}\), not
\(-F_{\mathrm{co}}\).

Suppose \(\mathfrak r_\lambda^0\) were identically zero.
Then \((\sigma-1)V_\lambda^{00}\) would be constant.
Its value at \(0\) is zero, so
\[
 (1-\sigma)V_\lambda^{00}=0
 \quad\hbox{on }[0,T].
\]
Choose a nonempty open interval \(J\subset(0,\varepsilon)\)
with \(0<\sigma<1\).
On \(J\), the last equality implies
\(V_\lambda^{00}=0\), hence
\(S_\lambda^{00}=(V_\lambda^{00})''=0\).
But the coefficient
\[
 -2\lambda^2\sigma-3i\lambda\sigma'
\]
in \eqref{eq:S00} has strictly nonzero real part on \(J\).
The exponential never vanishes. This is a contradiction.

The residual vanishes for \(t\le0\), because \(V_\lambda=0\),
and for \(t\ge\varepsilon\), because \(\sigma=1\).
At a time where it is nonzero it is nonzero at every spatial
point; its support is therefore not spatially compact.

Choose \(t_c\in(0,\varepsilon)\) with
\(\mathfrak r_\lambda^0(t_c)\ne0\), and choose
\[
 \zeta=
 \frac{\overline{\mathfrak r_\lambda^0(t_c)}}
      {\abs{\mathfrak r_\lambda^0(t_c)}}.
\]
By continuity,
\(\operatorname{Re}(\zeta\mathfrak r_\lambda^0)>0\)
on a sufficiently small interval about \(t_c\).
Take a nonnegative nonzero time bump \(b\) supported there
and a nonnegative nonzero spatial bump \(c\).
Then \(f(t,\mathbf x)=b(t)c(\mathbf x)\) proves
\eqref{eq:forcing-test}.
All operations are real linear after taking
\(\operatorname{Re}(\zeta\,\cdot)\).
\end{proof}

\paragraph{What the calculation excludes.}
The equality required for this direct identification is, in
VI's native sign,
\[
 D_-(h_\lambda)=F_{\mathrm{co}}.
\]
Equivalently, in the common plus-sign normalization it is
\[
 -\partial_\alpha a_{1,+}^{\alpha\mu}[R_\lambda]
 =
 -\partial_\alpha
       \left(\sigma a_{1,+}^{\alpha\mu}[R_\lambda]\right).
\]
Lemma~\ref{lem:B} disproves that equality on the witness.
It does not use the compact interaction switch \(g\).

\subsection{The solution-jet membership test}

\begin{proposition}[The transported response is not a first VI solution-jet coefficient]
\label{prop:jet-domain}
For \(t\in(\varepsilon,T)\),
\begin{equation}
\label{eq:Einstein-witness}
 \mathcal G_{-,ij}(h_\lambda)
 =\frac{\lambda^2}{2}
             (n_in_j-\delta_{ij})e^{2i\lambda t}.
\end{equation}
Its other components vanish on that interval.
In particular, \(h_\lambda\) fails
\eqref{eq:VI-vacuum}. For every nonzero phase \(\zeta\), the real
coefficient
\(\operatorname{Re}(\zeta h_\lambda)\) also fails that equation
on the slab.

Consequently, placing this response as VI's first metric
coefficient does not produce an element of \(\mathcal J_T\),
whatever higher metric and scalar coefficients are chosen.
\end{proposition}

\begin{proof}
For a spatially homogeneous tensor \(h\) with \(h_{0i}=0\),
write \(H_s=\sum_kh_{kk}\).
Substitution in \eqref{eq:Einstein} gives
\[
 \mathcal G_{-,00}(h)=\mathcal G_{-,0i}(h)=0,\qquad
 \mathcal G_{-,ij}(h)
 =\frac12\left(h_{ij}''-\delta_{ij}H_s''\right).
\]
For example,
\(\overline h_-^{00}=(h_{00}+H_s)/2\);
the \(h_{00}''\) terms cancel from the spatial equation.
This verifies the formula without imposing any gauge condition.

From \eqref{eq:htransport} and \eqref{eq:V},
\(h_\lambda=(\mathcal P_+V_\lambda)^\flat\).
On the plateau,
\[
 S_\lambda^{00}=-2\lambda^2e^{2i\lambda t},\qquad
 S_\lambda^{ij}
 =\lambda^2(n^in^j-3\delta^{ij})e^{2i\lambda t},
\]
so
\[
 \eta_{+,\alpha\beta}S_\lambda^{\alpha\beta}
 =6\lambda^2e^{2i\lambda t},\qquad
 (\mathcal P_+S_\lambda)^{ij}
 =\lambda^2n^in^j e^{2i\lambda t}.
\]
It follows that
\[
 h_{\lambda,ij}''
 =\lambda^2n_in_j e^{2i\lambda t},\qquad
 H_s''=\lambda^2e^{2i\lambda t}.
\]
This proves \eqref{eq:Einstein-witness}.

The matrix \(n_in_j-\delta_{ij}\) is nonzero.
For every \(\zeta\ne0\),
\(\operatorname{Re}(\zeta e^{2i\lambda t})\) is not identically
zero on an open time interval. Thus the corresponding real
Einstein coefficient is nonzero.
The first metric equation in \(\mathcal J_T\) is independent
of higher coefficients, which proves the final assertion.
\end{proof}

\begin{remark}[Two different failures]
Proposition~\ref{prop:jet-domain} is an equation-space failure for
the descended on-shell observable. Lemma~\ref{lem:B} is a forcing
mismatch between well-defined off-shell coordinate prescriptions.
Neither is a failure of smooth coordinate existence.
\end{remark}

\subsection{Support is recorded separately}

\begin{proposition}[The compact-interior source theorem is insufficient]
\label{prop:noncompact}
Both \(\operatorname{Re}F_{\mathrm{co}}^0\) and
\(\operatorname{Im}F_{\mathrm{co}}^0\) are spatially noncompact
on the slab. In particular,
\(F_{\mathrm{co}}\notin C_c^\infty(M_T^\circ;\C^4)\).
\end{proposition}

\begin{proof}
On the plateau,
\[
 F_{\mathrm{co}}^0=(V_\lambda^{00})',\qquad
 (F_{\mathrm{co}}^0)'
 =-2\lambda^2e^{2i\lambda t}.
\]
Neither the real nor the imaginary part of the last expression
vanishes identically on \((\varepsilon,T)\).
Hence neither corresponding part of
\(F_{\mathrm{co}}^0\) can vanish identically there.
Each is spatially constant, and is nonzero at every spatial
point at some plateau time. It cannot have compact spatial support.
\end{proof}

This agrees with the supplied XVI-D support boundary
\cite[\sid{lem:coord}]{CheungXVID}. It is not the cause of the
forcing mismatch: Section~\ref{sec:lemmaC} constructs the smooth
no-decay solution operator. Extending a source domain cannot
replace one forcing by another.

%=====================================================================
\section{Lemma C: uniqueness and the scope of an extension}
\label{sec:lemmaC}
%=====================================================================

\begin{lemma}[Lemma C: smooth uniqueness closes only a matched problem]
\label{lem:C}
Let \(F_1,F_2\) be smooth scalar, vector, or tensor forcings on
\(M_T\). Suppose \(u_1,u_2\) are smooth solutions of
\[
 \Boxp u_a=F_a
\]
with equal ordinary coordinate-time Cauchy data.
Then
\[
 u_1=u_2\quad\Longleftrightarrow\quad F_1=F_2.
\]
The statement holds componentwise over \(\C\), without spatial
decay or finite total energy.

For different operators, uniqueness of each separate problem
does not by itself identify their solutions. A proposed common
solution must satisfy both equations.
\end{lemma}

\begin{proof}
Equality of solutions implies equality of forcings by application
of \(\Boxp\).

Conversely, if the forcings agree, the difference \(z=u_1-u_2\)
satisfies \(\Boxp z=0\) and zero data. It suffices to treat a real
scalar component. Fix \(0<t_*<T\), \(\mathbf x_*\in\R^3\), and
\(\rho>0\). Set
\[
 B_s=\{\mathbf x:\abs{\mathbf x-\mathbf x_*}
                       <t_*+\rho-s\},
 \qquad
 E(s)=\frac12\int_{B_s}
       \left(z_t^2+\abs{\nabla z}^2\right)\dd^3x.
\]
These are compact integration regions, so \(E(s)\) is finite.
The wave equation gives
\[
 \partial_t\frac{z_t^2+\abs{\nabla z}^2}{2}
 =\nabla\cdot(z_t\nabla z).
\]
The boundary moves inward with speed one. Therefore
\[
 E'(s)=\int_{\partial B_s}
    \left(z_t\partial_nz
         -\frac12z_t^2-\frac12\abs{\nabla z}^2\right)\dd S
 \le0.
\]
Zero initial position implies zero initial spatial gradient,
and hence \(E(0)=0\). Nonnegativity gives \(E(s)=0\).
Thus \(z_t(s,\mathbf x_*)=0\) for \(0\le s\le t_*\), and
\[
 z(t_*,\mathbf x_*)
 =z(0,\mathbf x_*)+\int_0^{t_*}z_t(s,\mathbf x_*)\,\dd s=0.
\]
The point was arbitrary. Smoothness includes the endpoints.
Real and imaginary parts, and then all components, give the
stated extensions.

For different operators, the difference of two solutions need
not satisfy one homogeneous equation. The preceding uniqueness
argument cannot be applied until the equations have been matched
or the candidate has been tested in both.
\end{proof}

This is the no-decay discipline of the supplied VI and XVI-D
coordinate statements
\cite{CheungVI,CheungXVID}. We have supplied the proof rather than
using the phrase ``no spatial decay'' as an identification theorem.

\subsection{A local solution operator for all smooth sources}

\begin{proposition}[The finite-time smooth inverse]
\label{prop:local-inverse}
For every smooth source \(F\), define
\begin{equation}
\label{eq:Kirchhoff}
 (\GG_+F)(t,\mathbf x)
 =\frac1{4\pi}\int_0^t(t-s)
      \int_{\mathbb S^2}
       F(s,\mathbf x+(t-s)\omega)\,\dd\omega\,\dd s.
\end{equation}
Then \(\GG_+F\) is the unique smooth zero-data solution of
\(\Boxp u=F\).
It requires no spatial decay and agrees with the local
compact-cutoff construction. Moreover,
\[
 \GG_-F=-\GG_+F
\]
is the corresponding zero-data inverse of \(\Boxm\).
For spatially homogeneous \(F(t)\),
\begin{equation}
\label{eq:homogeneous-inverse}
 (\GG_+F)(t)=\int_0^t(t-s)F(s)\,\dd s.
\end{equation}
\end{proposition}

\begin{proof}
Let \(M_rF(s,\mathbf x)\) denote the normalized spherical mean.
For smooth \(F\), the spherical-mean identity is
\[
 \partial_r^2(rM_rF)=r\Delta_{\mathbf x}M_rF.
\]
It follows by the divergence theorem, equivalently from
\(\partial_r^2M_rF+(2/r)\partial_rM_rF
 =\Delta_{\mathbf x}M_rF\), with its smooth value at \(r=0\).

Differentiate
\[
 u(t,\mathbf x)=\int_0^t
        (t-s)M_{t-s}F(s,\mathbf x)\,\dd s.
\]
The first endpoint term vanishes. The second derivative has
endpoint term \(F(t,\mathbf x)\), because
\(\partial_r(rM_rF)|_{r=0}=F\).
The spherical-mean identity gives
\[
 u_{tt}=F+\Delta u.
\]
The position and first time derivative vanish at \(0\).
Smoothness follows by differentiation on the compact spheres
and bounded time intervals. Lemma~\ref{lem:C} proves uniqueness.

For a compact observation set, the integration points in
\eqref{eq:Kirchhoff} lie in a compact truncated causal past.
Multiplication of \(F\) by a smooth spatial cutoff equal to one
on that region leaves the answer unchanged there. This proves
the local-cutoff statement. The sign relation follows from
\(\Boxm=-\Boxp\). A homogeneous source has spherical mean \(F\),
which proves \eqref{eq:homogeneous-inverse}.
\end{proof}

This is the finite-time interpretation stated in VI
\cite[\sid{eq:Gzero}]{CheungVI}; general wave Cauchy theory is
treated in \cite{BGP}.
% The registered no-decay inverse is a local-cutoff construction.
% ([enter.grok.me](https://enter.grok.me/passage/VI-section-eq-Gzero))

\subsection{Restricted extensions: what agreement does and does not mean}

Set
\[
 \Dint=C_c^\infty(M_T^\circ;\R),
\]
viewed by smooth ambient zero extension, and let \(\Hsrc\) consist
of ambient smooth homogeneous sources \(q(t)\) that vanish for
\(t\le0\). Their intersection is zero: a nonzero spatially
constant source cannot be compactly supported in space.

If \(E_{0,\mathrm{int}}^+\) is the compact-interior zero-data
inverse, define
\begin{equation}
\label{eq:restricted}
 E_{\mathrm{res}}^+(F+q)
 =E_{0,\mathrm{int}}^+F+\int_0^t(t-s)q(s)\,\dd s,
 \qquad F\in\Dint,\quad q\in\Hsrc.
\end{equation}
The direct-sum decomposition makes this well defined.
Proposition~\ref{prop:local-inverse} and
Lemma~\ref{lem:C} show that it is the restriction of
\(\GG_+\) to \(\Dint\oplus\Hsrc\).
This reproves the mathematical content of the supplied restricted
extension \cite[\sid{prop:restricted-extension}]{CheungXVID}.

\begin{remark}[Agreement on compact sources is not the missing hypothesis]
Agreement of two \emph{linear extensions} on \(\Dint\) alone
does not determine their values on \(\Hsrc\).
For example, for \(b\in C_c^\infty((0,T))\) and an interior time
\(t_*\), the map
\[
 \widetilde E(F+q)=E_{\mathrm{res}}^+(F+q)+b(t)q(t_*)
\]
agrees on \(\Dint\) and retains zero initial data.
In general it is not a right inverse of \(\Boxp\).

By contrast, two extensions that both return smooth solutions
of the same wave equation with the same data \emph{must}
agree on their common source domain, by Lemma~\ref{lem:C}.
The equation, smoothness, and initial data are the decisive
additional requirements.
\end{remark}

This is exactly why the conditional coordinate-compatibility
clause of XVI-D cannot be replaced by an appeal to agreement on
compact sources \cite[\sid{prop:coordinate-compatibility}]{CheungXVID}.
Here the smooth inverse is constructed, but its construction
does not rename every inherited occurrence of a Green symbol.

%=====================================================================
\section{The first-order mismatch theorem}
\label{sec:main}
%=====================================================================

\begin{theorem}[The specified VI--XVI-D coordinate comparison mismatches]
\label{thm:mismatch}
Retain the slab, source, response sign, metric-sign transport,
and ordinary coordinate-time data in
Sections~\ref{sec:native}--\ref{sec:witness}.
For every \(\lambda>0\), the direct comparison of VI's first
coordinate coefficient on \(h_\lambda\) with
\(X_{\mathrm{cand}}[R_\lambda]\) is a mismatch.

More precisely:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The two normalized flat forcings differ by the nonzero
      residual \eqref{eq:forcing-residual}.
\item Their unique smooth zero-data coordinate coefficients are
      \begin{align}
      \label{eq:w-explicit}
      w_\lambda^\mu(t)
       &=-\int_0^tV_\lambda^{0\mu}(s)\,\dd s,\\
      \label{eq:coordinate-difference}
      d_\lambda^\mu(t)
       :=w_\lambda^\mu-X_{\mathrm{cand}}^\mu
       &=-\int_0^t(1-\sigma(s))
                         V_\lambda^{0\mu}(s)\,\dd s.
      \end{align}
      The time component \(d_\lambda^0\) is nonzero somewhere
      in \((0,\varepsilon)\).
\item A real linear combination of the witness coefficients
      and a compact real test detect that coordinate difference.
\item The response used as a first metric coefficient is not in
      VI's solution-jet domain, as shown by
      \eqref{eq:Einstein-witness}.
\end{enumerate}

This theorem excludes the stated direct first-order identification.
It does not exclude a differently declared metric map,
preparation, coupling-order comparison, or observable construction.
It neither repairs nor contradicts XVI-D's conditional
inherited-coordinate clause.
\end{theorem}

\begin{proof}
Item (i) is Lemma~\ref{lem:B}, with both operators normalized to
\(\Boxp\).

The field in \eqref{eq:w-explicit} has zero position and velocity
at \(0\) and satisfies
\[
 \Boxp w_\lambda^\mu=-\partial_tV_\lambda^{0\mu}.
\]
Lemma~\ref{lem:C} identifies it with VI's smooth first coefficient.
Equation \eqref{eq:Xcand} gives the other field; subtraction
proves \eqref{eq:coordinate-difference}.

If \(d_\lambda^0\) vanished identically on \((0,\varepsilon)\),
then its derivative would give
\((1-\sigma)V_\lambda^{00}=0\) there.
On an interval with \(0<\sigma<1\), this would force
\(S_\lambda^{00}=0\), contradicting \eqref{eq:S00}, exactly as
in Lemma~\ref{lem:B}. Thus item (ii) holds.

Choose \(t_c\in(0,\varepsilon)\) with \(d_\lambda^0(t_c)\ne0\)
and a unit phase \(\zeta\) such that
\(\operatorname{Re}(\zeta d_\lambda^0(t_c))>0\).
A nonnegative compact test supported in a small neighborhood
where that real part remains positive has a strictly positive
pairing. Since the constructions are complex linear on the
printed source coefficients, this pairing is the coordinate
difference for the real metric coefficient
\(\operatorname{Re}(\zeta h_\lambda)\).
This proves item (iii).

Item (iv) is Proposition~\ref{prop:jet-domain}.
None of these calculations uses an interaction switch \(g\),
a scalar-switch identity, a quantum coefficient, or a theorem
from an unregistered sheet.
\end{proof}

\begin{corollary}[Second-order terms do not repair the fixed mismatch]
\label{cor:order}
A correction starting at second coordinate order cannot make
\[
 x+\kappa w_\lambda+\kappa^2v
 \quad\hbox{and}\quad
 x+\kappa X_{\mathrm{cand}}+\kappa^2z
\]
equal modulo \(\kappa^3\), while their displayed first coefficients
remain fixed.
\end{corollary}

\begin{proof}
Equality modulo \(\kappa^3\) requires equality of the coefficient
of \(\kappa\). Theorem~\ref{thm:mismatch} disproves that equality.
\end{proof}

\begin{remark}[Plateau equality of forcings is not full Cauchy equality]
For \(t\ge\varepsilon\), the forcing residual vanishes, but
\[
 d_\lambda^0(t)
 =-\int_0^\varepsilon(1-\sigma(s))V_\lambda^{00}(s)\,\dd s.
\]
Thus the field difference has a constant preparation-memory
value there. This formula does not assert that this integral is
nonzero at every frequency. The theorem detects the difference
on the common slab and does not require a universal late-time
nonvanishing claim.
\end{remark}

\begin{remark}[A changed metric argument is a different comparison]
Replacing \(R_\lambda\) by a prepared metric argument would alter
the linear density coefficient and its divergence. Such a
replacement is not allowed silently in the theorem.
Likewise, replacing \(R_\lambda\) by the opposite action-equation
coefficient changes the response argument.
The result concerns the printed response sign and transport,
not every possible map between the manuscripts.
\end{remark}

%=====================================================================
\section{The scalar-switch defect: a separate classical identity}
\label{sec:switch}
%=====================================================================

This section is not a hypothesis of
Theorem~\ref{thm:mismatch}.
It records why VI's switch defect is a different question, and
proves the identity directly.

Use \(\eta_-\), fix \(m>0\), and put
\[
 P=\Boxm-m^2.
\]
Let \(\phi_0\) satisfy \(P\phi_0=0\).
For a symmetric tensor \(a\), define the density variation
\begin{equation}
\label{eq:Qscalar}
 Q_a u
 =-\partial_\mu
       \left(\overline a_-^{\mu\nu}\partial_\nu u\right)
       -\frac12m^2(\tr_-a)u.
\end{equation}
If
\[
 L_a=-a_-^{\mu\nu}\partial_\mu\partial_\nu
           -D_-(a)^\nu\partial_\nu,
\]
then direct expansion gives
\begin{equation}
\label{eq:Qrelation}
 Q_a=L_a+\frac12(\tr_-a)P.
\end{equation}

For a compact smooth vector field \(\xi\) supported in
\(M_T^\circ\), set
\[
 (K\xi)_{\mu\nu}
 =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,\qquad
 X_\xi=\xi^\mu\partial_\mu.
\]
Calculation gives
\begin{equation}
\label{eq:puregauge}
 D_-(K\xi)^\mu=\Boxm\xi^\mu,\qquad
 L_{K\xi}=-[P,X_\xi].
\end{equation}
Consequently,
\[
 w[K\xi]=\xi
\]
for the unswitched coordinate problem, by zero-data uniqueness.

Let \(g\in C_c^\infty(M_T^\circ;\R)\) be a fixed scalar
interaction switch. Let \(G_{m,0}\) be the smooth zero-data
inverse of \(P\), and define
\[
 \phi_{1,g}[h]=-G_{m,0}Q_{gh}\phi_0,\qquad
 A_{f,g}^{(1)}[h;\phi_0]
 =\int f\left(\phi_{1,g}[h]-w[h]^\mu
                                  \partial_\mu\phi_0\right)\dd^4x.
\]
The massive Cauchy inverse is the standard smooth wave Cauchy
construction; its local uniqueness uses the preceding energy
proof with the additional nonnegative term \(m^2z^2\)
\cite{BGP}.

\begin{proposition}[Classical retarded scalar-switch identity]
\label{prop:switch}
For the displayed definitions,
\begin{align}
\label{eq:switch-defect}
 &A_{f,g}^{(1)}[h+K\xi;\phi_0]
                       -A_{f,g}^{(1)}[h;\phi_0]\nonumber\\
 &\quad=\int f\left[
    (g-1)X_\xi\phi_0+
    G_{m,0}\left(
       (\partial_\mu g)\overline{K\xi}_-^{\mu\nu}
                                     \partial_\nu\phi_0
       -[P,g]X_\xi\phi_0
            \right)\right]\dd^4x,
\end{align}
where
\[
 [P,g]u=(\Boxm g)u
                  +2(\partial^\mu g)\partial_\mu u.
\]
This is a classical scalar-response identity. It is neither a
loop anomaly nor an obstruction to the harmonic-coordinate
Cauchy problem.
\end{proposition}

\begin{proof}
Set \(u=X_\xi\phi_0\). Equations
\eqref{eq:Qrelation}--\eqref{eq:puregauge} and \(P\phi_0=0\)
give \(Q_{K\xi}\phi_0=-Pu\).
The density product rule gives
\[
 Q_{gK\xi}\phi_0
 =gQ_{K\xi}\phi_0
       -(\partial_\mu g)\overline{K\xi}_-^{\mu\nu}
                                            \partial_\nu\phi_0.
\]
Therefore the scalar-response change is
\[
 G_{m,0}\left(
    gPu+(\partial_\mu g)\overline{K\xi}_-^{\mu\nu}
                                           \partial_\nu\phi_0
       \right).
\]
Since \(gPu=P(gu)-[P,g]u\), and \(gu\) has zero data,
\(G_{m,0}P(gu)=gu\).
The coordinate change is \(\xi\), so the dressing subtracts
\(u\). Smearing proves \eqref{eq:switch-defect}.

The coordinate equation contained no \(g\). Its transformation
was established before the scalar response was introduced.
\end{proof}

Equation \eqref{eq:switch-defect} has the form of the supplied
VI switch identity \cite[\sid{thm:switch-defect},
\sid{thm:plateau-obstruction}]{CheungVI}, also copied in
X-A's prerequisite sheet \cite{CheungXA}.
If \(\xi\) is supported strictly where \(g=1\) and
\(\dd g=0\), the displayed defect vanishes.
For larger supports the expression need not vanish.
A correction beginning at second order cannot change its fixed
first coefficient.

The first cancellation, larger-support defect, and plateau
second-order intertwiner reported for VII-A belong to its own
equations and transformations \cite{CheungVIIA,CheungXA}.
We record the analogy; we do not transfer its intertwiner or its
nonvanishing witnesses to the comparison theorem.

%=====================================================================
\section{Observable comparison and the kernel boundary}
\label{sec:observable}
%=====================================================================

\subsection{Equal shape is not equal functional}

VI's formal smearing is
\begin{equation}
\label{eq:O}
 \mathcal O_f[\gamma,\phi]
 =\int f(X)\phi(X_\gamma^{-1}(X))\,\dd^4X
                  \pmod{\kappa^3}.
\end{equation}
Its first coefficient is
\[
 \mathcal O_f^{[1]}
 =\int f\left(\phi_1-w^\mu[h]\partial_\mu\phi_0\right)\dd^4x.
\]
The source records descend this expression on the named
solution-jet quotient \cite[\sid{thm:descent}]{CheungVI}.

The prepared witness completion instead has the shape
\[
 \mathcal A_1[f,g](R;\phi)
 =\int f\left(\phi_{1,D}[R;\phi]
                 -X_1^\mu[R]\partial_\mu\phi\right)\dd^4x
\]
on its own completion domain
\cite{CheungXVID,CheungXVIB}. A common integrand shape does not
identify the metric argument, coordinate field, scalar response,
smearing convention, or equation space.

\begin{proposition}[Detection at a fixed scalar leg]
\label{prop:observable-detection}
For the real witness selected in
Theorem~\ref{thm:mismatch}, there are a homogeneous real free
Klein--Gordon scalar \(\phi\) and a real compact test \(f\) for
which replacing \(w\) by \(X_{\mathrm{cand}}\), while keeping the
scalar first coefficient fixed, changes the first-order smearing.
\end{proposition}

\begin{proof}
Write \(d^0=\operatorname{Re}(\zeta d_\lambda^0)\), and choose
a detecting time \(t_c\) at which \(d^0(t_c)>0\).
Take
\[
 \phi(t)=\sin(m(t-t_c)).
\]
Then \(P\phi=0\) and \(\dot\phi(t_c)=m>0\).
For any common smooth first scalar coefficient \(\Phi_1\),
the difference of the two displayed expressions is
\[
 -\int f\,d^0\dot\phi\,\dd^4x.
\]
Choose a nonnegative nonzero compact test where both factors are
positive. This integral is strictly negative.
\end{proof}

The proposition isolates the coordinate contribution.
It does not show that independently defined scalar legs cannot
compensate a coordinate difference in a particular complete
observable. Nor does it apply VI's solution-jet descent theorem
to the inadmissible first metric coefficient of
Proposition~\ref{prop:jet-domain}.

Even a successful future coordinate match would leave a separate
observable comparison: one must match all inputs and both scalar
and coordinate legs.

\subsection{A response and a partial contraction are not a full kernel}

The fields \(R_\lambda\), \(V_\lambda\), and
\(X_{\mathrm{cand}}\) are smooth fields on the slab.
They are not compact distributions on a three-slot configuration
product. In particular,
\[
 U_D=-\mathcal P_+V_\lambda
\]
is the response tensor, not the coordinate field.

For completeness, the following elementary construction explains
the witness boundary used in XVIII-A
\cite[\sid{thm:witness},\sid{cor:smoothwitness}]{CheungXVIIIA}.
% Source record for the pure-trace non-determination boundary.
% ([enter.grok.me](https://enter.grok.me/passage/XVIII-A-theorem-thm-witness))

\begin{lemma}[Pure-trace evaluations do not determine a three-slot kernel]
\label{lem:kernel}
The family of evaluations on all pure-trace tensor pairs does not
determine a compact smooth three-slot kernel with symmetric tensor
slots.
\end{lemma}

\begin{proof}
Choose a nonzero constant symmetric contravariant tensor \(A\)
such that \(A^{\mu\nu}\eta_{+,\mu\nu}=0\), for example
\(A^{12}=A^{21}=1\) and all other components zero.
Choose nonzero real compact smooth functions \(b,c\) and set
\[
 K^{\mu\nu,\rho\sigma}(x_1,x_2,x_3)
 =A^{\mu\nu}A^{\rho\sigma}
                     b(x_1)b(x_2)c(x_3).
\]
This is a nonzero compact smooth kernel symmetric under exchange
of its tensor slots. Each contraction with
\(\eta_+\psi_1\otimes\eta_+\psi_2\otimes\phi\) vanishes because
\(A:\eta_+=0\). Hence \(K\) and zero have identical data on the
entire pure-trace family.
\end{proof}

This proves only non-determination from those evaluations.
It does not prove nonuniqueness of XVI-H's complete root-based
kernel construction. XVI-H's partial witness recovery is not
full-kernel identification, and its general-test construction
retains its own hypotheses \cite{CheungXVIH}.
No full-kernel wavefront or admission test is performed here.

\subsection{Formal quantum order does not cancel the classical residual}

\begin{lemma}[Formal-order separation]
\label{lem:hbar}
Let \(E_0\ne0\) belong to a complex vector space \(E\).
In \(E[[\hbar]]\), no correction in
\(\hbar E[[\hbar]]\) makes \(E_0\) vanish.
\end{lemma}

\begin{proof}
The coefficient of \(\hbar^0\) remains \(E_0\).
Equality of formal series requires that coefficient to vanish.
\end{proof}

Applied to the real tested residual in
Lemma~\ref{lem:B}, this rules out repair by a positive formal
quantum order while the classical prescription is frozen.
The statement does not concern a singular rescaling or a
different normalization that divides by \(\hbar\).
It is the coefficientwise distinction recorded in
XVIII-A \cite[\sid{prop:hbar}]{CheungXVIIIA},
not a quantum Ward calculation.

%=====================================================================
\section{Equation-space discipline and unchanged obligations}
\label{sec:boundaries}
%=====================================================================

\subsection{The III comparison}

The supplied III records distinguish an unreduced Cauchy
isomorphism, a reduced quotient, a past-trivial retarded section
after a forced scalar correction, and a free-product obstruction
\cite{CheungIII,CheungXA}.
These answer different equation-space questions.

In the present comparison:
\begin{itemize}[leftmargin=*]
\item the smooth coordinate Cauchy map is defined before a
      solution-jet quotient is imposed;
\item Proposition~\ref{prop:jet-domain} tests membership in VI's
      solution-jet domain;
\item the scalar-switch identity tests a different switched
      assignment;
\item no free-product or quantum-product obstruction is used
      to decide the coordinate forcing.
\end{itemize}
III's retarded section is not substituted for either
\(w[h_\lambda]\) or \(X_{\mathrm{cand}}\).
Likewise, V's fixed-time scalar response is a negative control,
not a coordinate jet \cite{CheungV}.

\subsection{What the comparison does not construct}

XVI-A's three retarded-slot wavefront obligations remain outside
this calculation \cite{CheungXVIA}.
The paper constructs neither their kernels nor their products.
A changed smooth source or coordinate field does not by itself
determine a changed compact free-shell class.

No theorem of XVI-E, XVI-F, or XVI-G is used.
In particular, the cited sentence
\(I_{\mathrm{cand}}=Q_\lambda\) is not supplied here as a theorem
of XVI-D. The inherited clause in item (6) of XVI-D's
\sid{thm:response-package} remains conditional, and
\sid{prob:legs} is not discharged
\cite{CheungXVID}.

The exclusions established in XIX-A remain exclusions from this
comparison \cite{XIXA}. Em02-E's lapse, X-A's regulator ramp,
and the VIII incidence parameter are not substitute coordinate
data. This does not call the inflaton clock a non-clock:
XIX-A's named non-clock was the prescribed ramp.

The supplied Family 271 record remains documentary; its Lean
scope names 271-04 only. VI is not 271-04.
Family 282's free Ward action and Family 069's finite Ward defect
are not classical coordinate Cauchy data
\cite{CatalogueRecord}.

The reported scoped two-point result of XVI-Q is not a coordinate
identification \cite{CheungXVIQ}. Its scope is not enlarged here.
In particular, the separately recorded
\sid{O.1 = NOT_CLOSED} status outside that scope is unchanged.

No inherited quantum-admission exit is opened.
XVI-N and Route B are not discharged
\cite{CheungXVIIIA,CheungXVIN,CheungXVB}.
No quantum coefficient is admitted.

%=====================================================================
\section{Exact release}
\label{sec:release}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.34\linewidth}p{.17\linewidth}Y@{}}
\toprule
Question & Ending here & Exact extent\\
\midrule
XIX-A density versus VI scalar wave
& Proved equivalent
& Same metric jet and coordinate-time data; factorial conversion
  included
\\[4pt]
Witness forcing comparison
& Mismatch
& Nonzero residual
  \(\partial_t((\sigma-1)V_\lambda^{0\mu})\)
\\[4pt]
Smooth no-decay coordinate fields
& Constructed
& Explicit fields and finite-time smooth inverse
\\[4pt]
Direct first-coordinate identification
& Excluded
& For the printed response sign and metric-sign transport
\\[4pt]
Witness as a first VI solution-jet coefficient
& Excluded
& Explicit nonzero linearized Einstein coefficient
\\[4pt]
VI scalar-switch identity
& Separate identity
& Not a coordinate obstruction or theorem hypothesis
\\[4pt]
Native complete-observable identification
& Not established
& Scalar legs, domains, and complete maps still require comparison
\\[4pt]
XVI-D inherited coordinate compatibility
& Still conditional
& Its declared hypothesis is not replaced by this different route
\\[4pt]
Full-kernel, quantum, and characteristic admission
& Outside release
& No coefficient, product, or admission theorem constructed
\\
\bottomrule
\end{tabularx}
\end{center}

The relationship of this paper to VI and XVI-D is a separation
for the declared direct comparison:
\[
 \boxed{\text{declared path: null}.}
\]
Neither manuscript becomes a premise of the other.
The mathematical reason is the computed first forcing residual,
not the scalar-switch defect.

A successor proposing another match must specify the new metric
argument and coupling placement, normalize the operators, compare
all forcings and initial data, and establish the source-domain
realization. Only after those steps can smooth uniqueness identify
the coordinate coefficients. Observable equality requires its
own subsequent proof.

%=====================================================================
\section*{Declarations and source qualification}
\addcontentsline{toc}{section}{Declarations and source qualification}
%=====================================================================

\paragraph{Self-contained mathematical inputs.}
The proof uses the displayed scalar-wave definition, trace
reversal, witness source, time integrals, and fixed Cauchy data.
The gravitational derivation of the frozen source is not
reconstructed. The comparison is a theorem about that explicit
source and the two explicit coordinate prescriptions.

\paragraph{Source verification limits.}
VI's coordinate and solution-jet records, XVI-B's explicit source,
completion, and response records, and the supplied XVI-D passages
are distinguished. The response formula
\(R_\lambda=-\mathcal P_+V_\lambda\), omitted from the initial
comparison table, is explicitly recorded in XVI-B's
\sid{eq:later-response}. This sign is needed for Lemma B.
The available XVI-D passages suffice to state its candidate;
unsupplied proofs of its cited successor sheets are not imported.

\paragraph{Authorship and submission metadata.}
The document is anonymous. Citation of N.~Cheung does not attribute
authorship of this manuscript to that author. A signed submission
must supply authorized authorship metadata and stable copies or
archival locations for unpublished source records.

\paragraph{Verification.}
No successful pdfLaTeX run, proof-assistant verification,
independent numerical execution, peer-review decision, or
acceptance of this manuscript is asserted. The proofs supplied
are the printed arguments.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}
%=====================================================================

\bibitem{CheungVI}
N.~Cheung,
\emph{Relational Einstein--scalar observables at second order:
classical descent and a compact-switch obstruction},
Landmark VI, version 1, \sid{VI.tex}, unpublished manuscript,
n.d.
Enter Calculus Companion.
Relevant locators:
\sid{eq:DK}, \sid{def:jet-space}, \sid{eq:Gzero},
\sid{prop:coordinates}, \sid{lem:coordinate-transform},
\sid{thm:descent}, \sid{thm:switch-defect},
and \sid{thm:plateau-obstruction}.

\bibitem{CheungXVIB}
N.~Cheung,
\emph{Landmark XVI-B: Prepared-source diagnostics and the
complete-insertion boundary},
version 1, \sid{XVI-B.tex}, unpublished manuscript, n.d.
Enter Calculus Companion.
Relevant locators:
\sid{prop:trace}, \sid{prop:pair}, \sid{cor:tf},
\sid{eq:coordinate-leg}, \sid{eq:Aone},
and \sid{eq:later-response}.
The last record explicitly writes
\(R_\lambda=-\mathcal P V_\lambda\).

\bibitem{CheungXVID}
N.~Cheung,
\emph{Landmark XVI-D: Prepared witness response and completion domains},
version 1, \sid{XVI-D.tex}, unpublished manuscript, n.d.
Enter Calculus Companion; supplied passages.
Relevant locators:
\sid{eq:Slambda}, \sid{lem:coord}, \sid{lem:Xcand},
\sid{lem:uniqueness}, \sid{prop:restricted-extension},
\sid{prop:coordinate-compatibility}, \sid{prob:legs},
and \sid{thm:response-package}.
Item (6) of the response package retains its conditional hypothesis.

\bibitem{XIXA}
\emph{XIX-A: Clock Separation Before Coordinate Identification:
Four Labelled Objects and Five Non-Identifications},
anonymous draft supplied in the present manuscript exchange,
\sid{XIX-A.tex}, n.d.
Relevant locators:
\sid{def:harmonic}, \sid{lem:harmonic-existence},
and the release table.
No registered authorship or independent verification is inferred.

\bibitem{CheungVIIA}
N.~Cheung,
\emph{Plateau-compatible relational scalars:
a second-order classical intertwiner},
Landmark VII-A, version 1, \sid{VII-A.tex},
unpublished manuscript, n.d.
Enter Calculus Companion.
Method and documentary comparison only; the supplied
X-A prerequisite record is cited separately.

\bibitem{CheungXA}
N.~Cheung,
\emph{X-A},
version 1, \sid{X-A.tex}, unpublished manuscript, n.d.
Enter Calculus Companion.
``Prerequisite sheet'', including its separately copied
VI, VII-A, and III interfaces.
Copied statements are not transferred into the present
coordinate comparison.

\bibitem{CheungIII}
N.~Cheung,
\emph{Descent of first-order relational scalars and the free-product
obstruction: Classical equation spaces, retarded sections, and
compact Ward observables},
Landmark III, version 2, \sid{III.tex},
unpublished manuscript, n.d.
Enter Calculus Companion.
Documentary comparison of the unreduced Cauchy problem,
reduced quotient, forced retarded section, and product question.

\bibitem{CheungV}
N.~Cheung,
\emph{Admissible currents and first retarded corrections for a
fixed-time scalar response},
Landmark V, version 1, \sid{V.tex},
unpublished manuscript, n.d.
Enter Calculus Companion.
Negative control: a fixed-time scalar response is not a
coordinate jet.

\bibitem{CheungXVIIIA}
N.~Cheung,
\emph{Landmark XVIII-A: Prepared Relational Observables:
Non-Determination, Normalization Compatibility, and Conditional
Ultraviolet Matching},
version 2, \sid{XVIII-A-v2.tex},
unpublished manuscript, n.d.
Enter Calculus Companion.
Relevant locators:
\sid{thm:witness}, \sid{cor:smoothwitness},
\sid{prop:lateresponse}, \sid{prop:hbar},
and \sid{thm:release}.

\bibitem{CheungXVIH}
N.~Cheung,
\emph{Landmark XVI-H: Compact polarized kernel and witness recovery},
version 1, \sid{XVI-H.tex},
unpublished manuscript, n.d.
Enter Calculus Companion; supplied passages.
The general-test compact kernel and its partial witness
contraction are distinct from full-kernel admission.

\bibitem{CheungXVIA}
N.~Cheung,
\emph{Landmark XVI-A: Fixed-width retarded coefficient obligations},
version 1, \sid{XVI-A.tex},
unpublished manuscript, n.d.
Enter Calculus Companion.
The three retarded-slot obligations remain separate.

\bibitem{CheungXVIN}
N.~Cheung,
\emph{XVI-N},
version r12, \sid{XVI-N.tex},
unpublished manuscript, n.d.
Enter Calculus Companion; supplied passages.
Its conditional massless obstruction and unresolved Route B
are not coordinate Cauchy conclusions.

\bibitem{CheungXVIQ}
N.~Cheung,
\emph{XVI-Q},
release Q11, \sid{XVI-Q.tex},
unpublished manuscript, n.d.
Enter Calculus Companion; supplied passages.
Relevant documentary locator: \sid{rem:o1}.
Its scoped two-point closure is not enlarged or independently
verified here.

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B},
version 2, \sid{XV-B.tex},
unpublished manuscript, n.d.
Enter Calculus Companion; supplied passages.
The relative-lifting results do not construct the inherited
full characteristic map or open its downstream admission exit.

\bibitem{CatalogueRecord}
\emph{OpenAI catalogue and companion scope records},
supplied documentary records, including Family 271,
record dated 7 October 2026, and the supplied Family 282 and
Family 069 comparisons.
The reported Lean scope for Family 271 names 271-04 only.
Catalogue attribution, proof validity, and Lean execution are
not independently verified in this paper.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026),
article 1954943.
Publisher accepted-article record, accepted 28 September 2026.
DOI: \sid{10.3389/fphy.2026.1954943}.
Operational context only; no result is imported as coordinate data.

\bibitem{BFHPR}
R.~Brunetti, K.~Fredenhagen, T.-P.~Hack, N.~Pinamonti,
and K.~Rejzner,
``Cosmological perturbation theory and quantum gravity,''
\emph{Journal of High Energy Physics} \textbf{08} (2016), 032.
DOI: \sid{10.1007/JHEP08(2016)032}.
arXiv:1605.02573.
Relational-observable context, not an identification of the
two Cauchy prescriptions compared here.

\bibitem{Frob}
M.~B.~Fr\"ob,
``Gauge-invariant quantum gravitational corrections to
correlation functions,''
\emph{Classical and Quantum Gravity} \textbf{35} (2018), 055006.
DOI: \sid{10.1088/1361-6382/aaa74c}.
arXiv:1710.00839.
External relational-coordinate context; its quantum calculation
is not transferred to the present classical witness.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
ESI Lectures in Mathematics and Physics,
European Mathematical Society, Z\"urich, 2007.
DOI: \sid{10.4171/037}.
arXiv:0806.1036.

\end{thebibliography}

\end{document}