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% The prepared witness-response results are proved from explicit inputs.
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\hypersetup{
  pdftitle={Landmark XVI-D: Prepared witness response and completion domains},
  pdfsubject={Response proofs, no-decay uniqueness, and inherited-completion compatibility}
}

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\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

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\newtheorem{definition}[theorem]{Definition}
\newtheorem{problem}[theorem]{Open obligation}

\numberwithin{equation}{section}

\title{Landmark XVI-D\\[0.3em]
  \large Prepared witness response and completion domains\\
  \normalsize Revised proof manuscript
  --- inherited-completion programme unresolved}
\author{}
\date{}

\usepackage{tikz}
\begin{document}

\maketitle

\begin{abstract}
We establish the prepared response on the counter-propagating
pure-trace witness family. The source is smooth and past compact
relative to compact observation sets, but is not a compact-interior
source. Real polarization and complex bilinear extension produce
the response
$U_D=-\mathcal P V_\lambda$, where $V_\lambda$ is an explicit
zero-data time integral. A local energy proof supplies uniqueness
among all smooth zero-data solutions on the slab, without a spatial
decay assumption.

The spatial traceless response contraction is computed exactly.
For the completion, the switched scalar forcing is a compact-interior
source, whereas the coordinate forcing is spatially noncompact.
An explicit coordinate candidate is nevertheless a unique smooth
solution of the required Cauchy problem. We also construct a
restricted auxiliary extension of the compact-source wave operator,
while keeping its identification with the inherited coordinate
prescription conditional.

The single-metric root identity is not directly applied to a
polarized source. Its legitimate polarization, and the later
root--auxiliary compatibility result of XVI-F-v3, are recorded
separately. No compact class kernel, quantum retarded coefficient,
or new admission theorem is constructed.
\end{abstract}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Prepared witness response};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Completion domains};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Ultraviolet subtraction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {No ultraviolet subtraction is claimed.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The witness response claims no ultraviolet subtraction. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-XVI-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Prepared witness response};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Ultraviolet subtraction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {No ultraviolet subtraction is claimed.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The witness response claims no ultraviolet subtraction. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-XVI-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Prepared witness response};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Completion domains};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Ultraviolet subtraction};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {No ultraviolet subtraction is claimed.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-XVI-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Prepared witness response};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {Ultraviolet subtraction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Completion domains. No ultraviolet subtraction is claimed.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Prepared witness response is not Ultraviolet subtraction. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-XVI-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Prepared witness response};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Completion domains};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Ultraviolet subtraction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {No ultraviolet subtraction is claimed.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-XVI-D}
\end{figure}

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


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



\scopebox{Scope and release status.}{%
The width $\varepsilon>0$ is fixed.

\textbf{Proved here:}
smooth zero-data uniqueness without spatial decay;
support and domain distinctions;
real polarization and complexification;
the explicit witness response;
the spatial traceless response contraction;
the compact scalar-leg check;
the explicit coordinate candidate;
and conditional inherited-completion compatibility.

\textbf{Not inferred:}
a failed compact-source hypothesis is not nonexistence of a
solution or of an operator extension.

The inherited coordinate prescription must be matched to a
specified larger-domain realization before a direct inherited
insertion identification is claimed.
The class-kernel and quantum-admission programme remains
\Unresolved.
}

\section{Provenance and dependency structure}
\label{sec:provenance}

The available material comprises the supplied XVI-A, revised
XVI-B, original XVI-D, XVI-E, and XVI-F-v3 texts, together with
predecessor statements reproduced in those texts and the supplied
coverage material.

Complete XI-B, XII-D, and XVI-C sources are not supplied here.
References to their internal labels identify reported passages;
they do not certify a full-file or full-series audit.

Four kinds of statement are distinguished:
\begin{enumerate}
\item \textbf{Imported formula or definition:}
      retained with its stated convention and domain.
\item \textbf{Imported theorem:}
      used only within the explicitly recorded quantifier.
\item \textbf{Proof in this manuscript:}
      a calculation or argument given below.
\item \textbf{Compatibility obligation:}
      an identification not supplied by the preceding proof.
\end{enumerate}

The main dependency structure is noncircular:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.27\linewidth}Y Y@{}}
\toprule
Input or result & Provenance & Use here\\
\midrule
Prepared witness source
 & XVI-B, source-level formula
 & Input to the response calculation\\
Diagonal response existence
 & Reported XII-D response theorem
 & Defines real polarization\\
No-decay uniqueness
 & Proved in Section~\ref{sec:uniqueness}
 & Bilinearity and witness identification\\
Compact scalar Green theorem
 & Reported XI-B, \texttt{lem:Green}
 & Scalar completion on compact forcing\\
Coordinate candidate
 & Formula also present in XVI-E/F-v3
 & Reproved directly here\\
Root and later compatibility
 & XVI-E and XVI-F-v3
 & Separate later-results record\\
\bottomrule
\end{tabularx}
\end{center}

Only the source calculation of XVI-B is used as a premise.
Its later-results summary of XVI-D is not used to prove XVI-D.

General wave-equation background is available in
\cite{BGP}; the no-decay uniqueness argument needed here is
given explicitly below rather than left as an external premise.

\section{Setting, source input, and signs}
\label{sec:notation}

\subsection{Geometry and smoothness}

Let
\[
M=\mathbb R^4,\qquad
M_T=[0,T]\times\mathbb R^3,\qquad
M_T^\circ=(0,T)\times\mathbb R^3,
\qquad
0<\varepsilon<T,
\]
with
\[
\eta=\operatorname{diag}(1,-1,-1,-1),
\qquad
\Box=\partial_t^2-\Delta.
\]
Smoothness on the closed slab means smoothness up to its time
boundaries; all Cauchy traces used below are therefore defined.
No spatial boundedness, decay, or finite total-energy condition
is imposed on smooth solutions.

Spacetime indices are raised and lowered with $\eta$.
Spatial Euclidean contractions use $\delta_{ij}$.
The frequency parameter is $\lambda$.
Repeated spacetime derivative indices are denoted by
$\alpha,\mu,\nu$, not by $\lambda$.

Fix a real smooth switch
\begin{equation}
\sigma(t)=\sigma_\varepsilon(t)=\chi(t/\varepsilon),
\qquad
\chi\in C^\infty(\mathbb R;\mathbb R),
\qquad
\chi(s)=0\ (s\le0),\quad
\chi(s)=1\ (s\ge1).
\label{eq:switch}
\end{equation}
Thus $\sigma$ is flat at $0$, its positive-order derivatives
vanish at $\varepsilon$, and $\supp\sigma'\subset[0,\varepsilon]$.

\subsection{Trace reversal}

For a symmetric tensor $r$, define
\begin{equation}
(\mathcal P r)_{\mu\nu}
=r_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta r,
\qquad
\tr_\eta r=\eta^{\mu\nu}r_{\mu\nu}.
\label{eq:P}
\end{equation}
The analogous raised-index formula is used for contravariant
tensors. In four dimensions,
\begin{equation}
\tr_\eta(\mathcal P r)=-\tr_\eta r,
\qquad
\mathcal P^2=\mathrm{id}.
\label{eq:P2}
\end{equation}
These follow directly from $\eta^{\mu\nu}\eta_{\mu\nu}=4$.
Trace reversal has constant coefficients and commutes with $\Box$.

The selected free operator and compact-test inverse are
\begin{equation}
L=-\Box\mathcal P,
\qquad
G_{\mathrm{gr}}^{\mathrm R}
=-\mathcal P E_0^{\mathrm R}.
\label{eq:inverse}
\end{equation}
The coefficient is $-1$, not $-4$ \cite{XIB}.
The placement of $\mathcal P$ in this notation does not enlarge
the domain of the scalar operator.

\subsection{Prepared source and witness}

Write $J_D=\mathcal J_{\mathrm D}$ for the designated prepared
source. The imported source relation is
\begin{equation}
J_D[\gamma]=\sigma J_3[\gamma]+\mathcal D_\sigma[\gamma].
\label{eq:preparedsource}
\end{equation}
It is a real homogeneous quadratic differential polynomial.
Its real polarization is
\begin{equation}
J_D(a,c)
=\frac12\bigl(J_D[a+c]-J_D[a]-J_D[c]\bigr).
\label{eq:sourcepolarization}
\end{equation}
The complex extension is bilinear, not sesquilinear.

Let $|\mathbf n|=1$ and, for $\lambda>0$, set
\begin{equation}
k_1=(1,\mathbf n),\qquad
k_2=(1,-\mathbf n),\qquad
\gamma_a^\lambda=\eta e^{\ii\lambda k_a\cdot y},
\qquad
k\cdot y=\eta_{\mu\nu}k^\mu y^\nu.
\label{eq:witness}
\end{equation}
Both $k_a$ are null, so $\Box\gamma_a^\lambda=0$.

The source-level result imported from XVI-B is
\begin{equation}
\begin{aligned}
S_\lambda^{\mu\nu}
&:=J_D^{\mathbb C}(\gamma_1^\lambda,\gamma_2^\lambda)\\
&=\bigl[
 \lambda^2\sigma(t)\mathsf q^{\mu\nu}
 +\lambda\mathfrak d_1^{\mu\nu}(t)
 \bigr]e^{2\ii\lambda t},
\end{aligned}
\label{eq:Slambda}
\end{equation}
where
\begin{align}
\mathsf q^{00}&=-2,
&
\mathsf q^{0i}&=0,
&
\mathsf q^{ij}&=n^in^j-3\delta^{ij},
\label{eq:q}\\
\mathfrak d_1^{\mu\nu}
&=-\ii\sigma'
  \bigl(\eta^{\mu\nu}+2\delta^\mu_0\delta^\nu_0\bigr).
\label{eq:d1}
\end{align}
In particular,
\[
\mathfrak d_1^{00}=-3\ii\sigma',
\qquad
\mathfrak d_1^{0i}=0,
\qquad
\mathfrak d_1^{ij}=\ii\sigma'\delta^{ij}.
\]
Neither $\mathsf q$ nor $\mathfrak d_1$ contains an additional
factor of $\lambda$ \cite{XVIB}.

Equation~\eqref{eq:Slambda} defines an ambient smooth tensor.
It is not merely a slab formula extended discontinuously at
an endpoint.

\subsection{Object and sign register}

The diagonal response is denoted $u_{\mathrm D}[\gamma]$;
the polarized response is $U_D[\gamma_1,\gamma_2]$.
A prospective compact observable kernel is denoted $w_D$.
These are not interchangeable.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.23\linewidth}Y Y@{}}
\toprule
Object & Equation or definition & Status\\
\midrule
$u_{\mathrm D}[\gamma]$
 & $L u_{\mathrm D}[\gamma]=J_D[\gamma]$
 & Imported diagonal response\\
$r_{\mathrm D}[\gamma]$
 & $r_{\mathrm D}=-u_{\mathrm D}$;
   $Lr_{\mathrm D}=-J_D$
 & Action-equation convention\\
$V_\lambda$
 & $\Box V_\lambda=S_\lambda$, zero data
 & Explicit particular field\\
$R_\lambda$
 & $R_\lambda=-\mathcal P V_\lambda$;
   $LR_\lambda=S_\lambda$
 & Witness response proved below\\
$X_{\mathrm{cand}}$
 & $\Box X_{\mathrm{cand}}=-F_{\mathrm{co}}$
 & Explicit coordinate candidate\\
$w_D$
 & Compact kernel on three field slots
 & Not constructed here\\
\bottomrule
\end{tabularx}
\end{center}

The sign convention for a diagonal action-equation coefficient
is not an unlabelled identification of that coefficient with a
response on a different domain. If it is polarized, its
polarization is correspondingly the negative of the polarized
response.

\section{The two inherited interfaces and no-decay uniqueness}
\label{sec:uniqueness}

\subsection{Compact-interior Green theorem}

Define the real compact-interior source class
\[
\Dint=C_c^\infty(M_T^\circ;\mathbb R).
\]
The complex version consists of real and imaginary parts in
$\Dint$.

The reported XI-B compact-source theorem states, for
$F\in\Dint$,
\begin{equation}
(\Box+\mu^2)E_\mu^{\mathrm R}F=F,
\qquad
E_\mu^{\mathrm R}(\Box+\mu^2)F=F,
\label{eq:compactGreen}
\end{equation}
with zero initial data, and the corresponding advanced statement
with zero final data. It also gives the causal support bounds
\begin{equation}
\begin{aligned}
\supp E_\mu^{\mathrm R}F
&\subset J^+(\supp F)\cap M_T,\\
\supp E_\mu^{\mathrm A}F
&\subset J^-(\supp F)\cap M_T.
\end{aligned}
\label{eq:compact-support}
\end{equation}
These supports are compact in the closed slab \cite{XIB}.

The support can meet a time boundary.
Compactness in the closed slab is not compact support in the
open slab.

\subsection{Diagonal response theorem}

The supplied XII-D response statement is:
\begin{quote}
For every smooth $\gamma$, the response formula defines a unique
smooth zero-initial-data solution on the slab.
\end{quote}
The reported Section D0 also states:
\begin{quote}
No spatial decay is required.
\end{quote}
Its response formula is
\begin{equation}
u_{\mathrm D}[\gamma]
=-\mathcal P E_0^{\mathrm R}\bigl(J_D[\gamma]\bigr),
\qquad
Lu_{\mathrm D}[\gamma]=J_D[\gamma].
\label{eq:response}
\end{equation}
The reported construction is local near compact observation sets
and independent of its auxiliary spatial cutoff \cite{XIID}.

We use its existence assertion on real smooth metric
perturbations. We do not infer the scope of uniqueness solely
from the phrase concerning spatial decay: the needed uniqueness
class is established directly next.

\begin{lemma}[Smooth zero-data uniqueness without spatial decay]
\label{lem:uniqueness}
Let $\mu\ge0$ and let $z$ be a real smooth function on $M_T$.
If
\[
(\Box+\mu^2)z=0,
\qquad
z(0,\mathbf x)=\partial_tz(0,\mathbf x)=0
\quad\text{for every }\mathbf x,
\]
then $z=0$.

The same conclusion holds componentwise for complex functions,
vectors, and tensors. Consequently, a smooth solution of
$LZ=F$ with prescribed initial data is unique whenever it exists.
No spatial decay or finite total energy is required.
\end{lemma}

\begin{proof}
Fix $0<t_0<T$, $\mathbf x_0\in\mathbb R^3$, and $\rho>0$.
For $0\le s\le t_0$, let
\[
B_s=\{\mathbf x:
 |\mathbf x-\mathbf x_0|<t_0+\rho-s\},
\]
and define the local energy
\[
\mathcal E(s)
=\frac12\int_{B_s}
 \bigl(z_t^2+|\nabla z|^2+\mu^2z^2\bigr)\dd^3x.
\]
Every integration region lies in a fixed compact set.
Thus $\mathcal E(s)$ is finite even if the total energy on
$\mathbb R^3$ is infinite.

Writing
$e=(z_t^2+|\nabla z|^2+\mu^2z^2)/2$,
the equation gives
\[
\partial_te=\nabla\cdot(z_t\nabla z).
\]
Since the boundary of $B_s$ moves inward at speed one,
the divergence theorem and differentiation of the moving region
give
\[
\mathcal E'(s)
=\int_{\partial B_s}
 \bigl(z_t\partial_nz-e\bigr)\dd S.
\]
At each boundary point,
\[
z_t\partial_nz
\le\frac12\bigl(z_t^2+(\partial_nz)^2\bigr)
\le e.
\]
Hence $\mathcal E'(s)\le0$.

The zero initial data imply $\mathcal E(0)=0$:
the initial spatial gradient also vanishes because
$z(0,\cdot)$ is identically zero.
Nonnegativity gives $\mathcal E(s)=0$ for every $s\le t_0$.
Smoothness then implies $z_t=0$ throughout each $B_s$.
In particular $z_t(s,\mathbf x_0)=0$ for $0\le s\le t_0$,
so
\[
z(t_0,\mathbf x_0)
=z(0,\mathbf x_0)+\int_0^{t_0}z_t(s,\mathbf x_0)\dd s=0.
\]
The point was arbitrary. Continuity includes $t=T$.

Real and imaginary parts satisfy the same real equation.
The argument therefore applies componentwise.

If $LZ=0$, then $\Box(\mathcal P Z)=0$.
The tensor $\mathcal P Z$ has zero initial data whenever $Z$ does.
The componentwise result gives $\mathcal P Z=0$, and
$\mathcal P^2=\mathrm{id}$ gives $Z=0$.
Apply this to the difference of two solutions.
\end{proof}

This proves the unrestricted smooth uniqueness needed by the
polarization and identification arguments. Even if an inherited
construction theorem were formulated using a smaller construction
class, two smooth solutions of the present Cauchy problem could
not differ outside that class.

\section{Past compactness and the source-domain distinction}
\label{sec:domain}

\begin{definition}[Past compactness relative to observation sets]
\label{def:past}
An ambient field $F$ is called \emph{past compact relative to
compact observation sets} if
\[
\supp F\cap J^-(K)
\]
is compact for every compact $K\subset M$.
This is the meaning of ``past compact'' used in this manuscript.
\end{definition}

This property is distinct from $F\in\Dint$.
It does not, by itself, prove an operator identity or existence
of a retarded solution.

\begin{lemma}[Support bound for fields with an initial lower bound]
\label{lem:past}
Let $F$ be an ambient field with $\supp F\subset\{t\ge0\}$.
Suppose a compact observation set $K$ is contained in
$\{t\le b,\ |\mathbf x|\le R\}$, where $b,R\ge0$.
Then
\begin{equation}
\supp F\cap J^-(K)
\subset
\{0\le t\le b,\ |\mathbf x|\le R+b\},
\label{eq:pastbound}
\end{equation}
and the intersection is compact.
\end{lemma}

\begin{proof}
For $y\in\supp F\cap J^-(K)$, there is $x\in K$ such that
\[
0\le y^0\le x^0\le b,
\qquad
|\mathbf y-\mathbf x|\le x^0-y^0.
\]
Thus $|\mathbf y|\le R+b$.

The right side of \eqref{eq:pastbound} is compact.
The support is closed. Also $J^-(K)$ is closed:
if $y_j\to y$ and $y_j\in J^-(x_j)$ with $x_j\in K$,
a subsequence of $x_j$ converges to $x\in K$, and the closed
causal inequality passes to the limit.
The intersection is therefore a closed subset of a compact set.
\end{proof}

\begin{proposition}[Witness source support and noncompactness]
\label{prop:source-domain}
For $\lambda>0$, $S_\lambda$ is smooth, vanishes for $t\le0$,
is flat at $0$, and is past compact in the sense of
Definition~\ref{def:past}.
Nevertheless,
\[
S_\lambda|_{M_T}\notin C_c^\infty(M_T^\circ;\mathbb C).
\]
At $\lambda=0$, the displayed source is zero.
\end{proposition}

\begin{proof}
Smoothness, past vanishing, and flatness follow from
\eqref{eq:switch}, \eqref{eq:Slambda}, and \eqref{eq:d1}.
Lemma~\ref{lem:past} proves past compactness.

For $\varepsilon<t<T$,
\[
S_\lambda^{00}(t,\mathbf x)
=-2\lambda^2e^{2\ii\lambda t}.
\]
This is nonzero at every spatial point.
Thus the spatial support is unbounded and the source is not
compactly supported in the open slab.

Every term of \eqref{eq:Slambda} has a factor $\lambda$ or
$\lambda^2$, proving the assertion at zero frequency.
\end{proof}

The compact-source identity \eqref{eq:compactGreen} cannot be
applied to $S_\lambda$ merely by citing its compact-interior
quantifier. This is a limitation of that theorem, not a
nonexistence conclusion.

\section{Real polarization and complexification}
\label{sec:polarization}

\begin{lemma}[Real polarized response]
\label{lem:realB}
For real smooth symmetric tensor fields $a,c$, define
\begin{equation}
B(a,c)
=\frac12\bigl(
 u_{\mathrm D}[a+c]-u_{\mathrm D}[a]-u_{\mathrm D}[c]
\bigr).
\label{eq:B}
\end{equation}
Then $B$ is symmetric and real bilinear,
\[
LB(a,c)=J_D(a,c),
\qquad
B(a,c)|_{t=0}=\partial_tB(a,c)|_{t=0}=0.
\]
\end{lemma}

\begin{proof}
The imported response-existence assertion applies to
$a+c,a,c$. Linearity of $L$ and
\eqref{eq:sourcepolarization} give the displayed equation.
The initial data pass through the same linear combination.

The polarization of a homogeneous quadratic differential
polynomial is symmetric and bilinear. Therefore each additivity
or real-homogeneity defect of $B$ satisfies the homogeneous
equation $LZ=0$ with zero initial data.
Lemma~\ref{lem:uniqueness} makes every such defect zero.
Symmetry also follows directly from \eqref{eq:B}.
\end{proof}

The auxiliary localization in the inherited response construction
is not a cutoff imposed on the witness.

\begin{lemma}[Complex bilinear response extension]
\label{lem:BC}
For real smooth $a,b,c,d$, define
\begin{equation}
B_{\mathbb C}(a+\ii b,c+\ii d)
=B(a,c)-B(b,d)+\ii B(a,d)+\ii B(b,c).
\label{eq:BC}
\end{equation}
This is the unique complex bilinear extension of $B$.
It satisfies
\[
LB_{\mathbb C}=J_D^{\mathbb C}
\]
and preserves zero initial data.
\end{lemma}

\begin{proof}
Complex bilinearity forces the four-term expansion
\eqref{eq:BC}. Conversely, real bilinearity of $B$ verifies
additivity and multiplication by $\ii$ in each slot, so the
formula defines a complex bilinear map.

The operator $L$ has real coefficients. Apply $L$ to the
four terms and use Lemma~\ref{lem:realB}. The result is the
same four-term extension of the polarized source.
Initial data are preserved term by term.
\end{proof}

On the witness, write
\[
a_a=\eta\cos(\lambda k_a\cdot y),
\qquad
b_a=\eta\sin(\lambda k_a\cdot y),
\qquad
\gamma_a^\lambda=a_a+\ii b_a.
\]
The real inputs and their sums are smooth. Thus
$B_{\mathbb C}(\gamma_1^\lambda,\gamma_2^\lambda)$ is defined
using real response evaluations. It does not require invoking
a predecessor theorem directly on complex metric perturbations.

Set
\[
U_D=B
\quad\text{on real pairs},\qquad
U_D=B_{\mathbb C}
\quad\text{on complexified pairs}.
\]

\section{Explicit witness response and traceless channel}
\label{sec:response}

Define the ambient particular field
\begin{equation}
V_\lambda^{\mu\nu}(t)
=\int_0^t(t-s)S_\lambda^{\mu\nu}(s)\dd s,
\label{eq:Vlambda}
\end{equation}
and set
\begin{equation}
R_\lambda=-\mathcal P V_\lambda.
\label{eq:Rlambda}
\end{equation}
For negative $t$ the integral is zero, because its source is zero
there. This is a smooth past-zero extension, not a sharp
multiplication of a nonvanishing solution.

\begin{lemma}[Explicit zero-data solution]
\label{lem:R}
The fields $V_\lambda$ and $R_\lambda$ are smooth and vanish
flatly for $t\le0$. They satisfy
\[
\Box V_\lambda=S_\lambda,
\qquad
LR_\lambda=S_\lambda,
\]
and have zero initial position and velocity.
\end{lemma}

\begin{proof}
For $t\ge0$, differentiation gives
\[
\partial_tV_\lambda(t)=\int_0^tS_\lambda(s)\dd s,
\qquad
\partial_t^2V_\lambda(t)=S_\lambda(t).
\]
Spatial derivatives vanish. The initial traces are zero.
Higher time derivatives at $0$ are zero because $S_\lambda$
is flat there. This also proves smoothness of the past-zero
extension.

Using \eqref{eq:P2},
\[
LR_\lambda
=(-\Box\mathcal P)(-\mathcal P V_\lambda)
=\Box V_\lambda=S_\lambda.
\]
The constant pointwise map $\mathcal P$ preserves smoothness,
flatness, and the initial data.
\end{proof}

\begin{proposition}[Witness response identification]
\label{lem:identify}
On $M_T$,
\begin{equation}
U_D[\gamma_1^\lambda,\gamma_2^\lambda]
=R_\lambda=-\mathcal P V_\lambda.
\label{eq:identify}
\end{equation}
\end{proposition}

\begin{proof}
Lemma~\ref{lem:BC} and the source input
\eqref{eq:Slambda} show that the left side solves
$LU_D=S_\lambda$ with zero initial data.
Lemma~\ref{lem:R} gives the same equation and data for $R_\lambda$.
Their difference is a smooth complex homogeneous zero-data
solution. Lemma~\ref{lem:uniqueness} makes it zero.
\end{proof}

This identifies the particular field with the polarized response.
It does not, solely from the compact-interior theorem, identify
$V_\lambda$ with the named expression $E_0^{\mathrm R}S_\lambda$.

\subsection{Spatial traceless contraction}

Let
\[
h_{ij}=n_in_j-\frac13\delta_{ij},
\qquad
n_i=\delta_{ij}n^j,
\]
and define
\begin{equation}
F_\lambda(t)
=\int_0^t(t-s)\sigma(s)e^{2\ii\lambda s}\dd s.
\label{eq:Flambda}
\end{equation}

\begin{proposition}[Exact spatial traceless response contraction]
\label{lem:contraction}
The response satisfies
\begin{equation}
h_{ij}V_\lambda^{ij}
=\frac23\lambda^2F_\lambda,
\qquad
h_{ij}U_D^{ij}
=-\frac23\lambda^2F_\lambda.
\label{eq:contraction}
\end{equation}
The correction $\lambda\mathfrak d_1$ contributes zero to this
channel, including after trace reversal.

These are response contractions, not completed insertion values.
\end{proposition}

\begin{proof}
The Euclidean identities are
\[
h_{ij}\delta^{ij}=0,
\qquad
h_{ij}n^in^j=\frac23.
\]
Thus \eqref{eq:q} and \eqref{eq:d1} give
\[
h_{ij}S_\lambda^{ij}
=\frac23\lambda^2\sigma e^{2\ii\lambda t}.
\]
Integrating proves the first identity.

On raised spatial components,
\[
(\mathcal P V_\lambda)^{ij}
=V_\lambda^{ij}
 +\frac12\delta^{ij}\tr_\eta V_\lambda.
\]
The added term is annihilated by $h$. Apply
\eqref{eq:identify} to obtain the second identity.
Any correction entering the trace also multiplies
$\delta^{ij}$ and is annihilated by the same contraction.
\end{proof}

No assertion that $F_\lambda(t)$ is nonzero at every time or
frequency is required.

\section{Completion formulas and the detailed domain check}
\label{sec:legs}

Fix
\[
m\ge0,\qquad
f\in C_c^\infty(M_T^\circ;\mathbb R),\qquad
g\in C_c^\infty(M_T^\circ;[0,1]),
\]
and let $\phi$ be a real smooth scalar.
No scalar free equation is required for the existence checks in
this section.

The supplied completion coefficients are
\begin{equation}
\begin{aligned}
\mathfrak a^{\mu\nu,ab}
&=\frac12\eta^{\mu\nu}\eta^{ab}
  -\eta^{\mu(a}\eta^{b)\nu},\\
a_1^{\mu\nu}[r]
&=\mathfrak a^{\mu\nu,ab}r_{ab}
 =-(\mathcal P r)^{\mu\nu},\\
b_1[r]&=\frac12\eta^{ab}r_{ab}.
\end{aligned}
\label{eq:coefficients}
\end{equation}
The equality with trace reversal is a pointwise calculation.

The inherited legs are written
\begin{equation}
\begin{aligned}
X_1^\mu[r]
&=-E_0^{\mathrm R}
  \partial_\alpha\bigl(\sigma a_1^{\alpha\mu}[r]\bigr),\\
V_{1,g}[r]\phi
&=\partial_\mu
  \bigl(g a_1^{\mu\nu}[r]\partial_\nu\phi\bigr)
  +m^2g b_1[r]\phi,\\
\phi_1[r;\phi]
&=-E_m^{\mathrm R}\bigl(V_{1,g}[r]\phi\bigr).
\end{aligned}
\label{eq:completion-legs}
\end{equation}
These are inherited definitions on a domain where their operator
applications are justified \cite{XIID}.

The corresponding functional is
\begin{equation}
\mathcal A_1[f,g](r;\phi)
=\int_{M_T}f(x)
 \bigl(\phi_1[r;\phi](x)
       -X_1^\mu[r](x)\partial_\mu\phi(x)\bigr)\dd^4x.
\label{eq:Aone}
\end{equation}
The diagonal completed insertion is
\begin{equation}
\mathcal C_{3,\mathrm D}(f,g)[\gamma,\phi]
=\mathcal A_1[f,g](u_{\mathrm D}[\gamma];\phi)
\label{eq:CD}
\end{equation}
on its inherited domain.

A direct polarized-response reading would be
\[
C_D(f,g;\gamma_1,\gamma_2,\phi)
=\mathcal A_1[f,g](U_D[\gamma_1,\gamma_2];\phi),
\]
but only after both legs have been justified on that response.
Its witness value is denoted
$\mathcal A_D(\lambda;f,g,\phi)$.
This is not a definition of a compact class-kernel pairing.

\subsection{Coordinate forcing}

\begin{proposition}[Coordinate forcing and compact-domain miss]
\label{lem:coord}
On $R_\lambda$,
\begin{equation}
a_1[R_\lambda]=V_\lambda,
\qquad
F_{\mathrm{co}}^\mu
:=\partial_\alpha
  \bigl(\sigma a_1^{\alpha\mu}[R_\lambda]\bigr)
=\partial_t(\sigma V_\lambda^{0\mu}).
\label{eq:Fco}
\end{equation}
Moreover,
\[
V_\lambda^{0i}=F_{\mathrm{co}}^i=0.
\]
The forcing is smooth, flatly zero for $t\le0$, and past compact
relative to compact observation sets.

For $\lambda>0$ it is not in
$C_c^\infty(M_T^\circ;\mathbb C)$.
Both its real and imaginary time components are nonzero
spatially noncompact functions.
\end{proposition}

\begin{proof}
By \eqref{eq:coefficients} and \eqref{eq:P2},
\[
a_1[R_\lambda]
=-\mathcal P(-\mathcal P V_\lambda)=V_\lambda.
\]
All coefficients depend only on time, proving
\eqref{eq:Fco}.
Equations \eqref{eq:q} and \eqref{eq:d1} give
$S_\lambda^{0i}=0$, hence $V_\lambda^{0i}=0$.

Smoothness and flat past vanishing follow from
Lemma~\ref{lem:R}. Lemma~\ref{lem:past} proves past compactness.

On the nonempty interval $(\varepsilon,T)$,
\[
F_{\mathrm{co}}^0=\partial_tV_\lambda^{00},
\qquad
\partial_tF_{\mathrm{co}}^0
=-2\lambda^2e^{2\ii\lambda t}.
\]
The derivative is not identically zero, so
$F_{\mathrm{co}}^0$ is not identically zero.
Likewise its real and imaginary derivatives are
$-2\lambda^2\cos(2\lambda t)$ and
$-2\lambda^2\sin(2\lambda t)$.
Neither vanishes identically on that interval.
Thus both real and imaginary components are nonzero functions
independent of $\mathbf x$, and neither has compact spatial
support.
\end{proof}

Consequently the compact-interior theorem
\eqref{eq:compactGreen} alone does not justify the named coordinate
application $E_0^{\mathrm R}F_{\mathrm{co}}$.
This conclusion concerns the hypothesis of that theorem.
It does not say that the coordinate Cauchy problem has no
solution, or that no operator extension exists.

The factor $g$ does not occur in $F_{\mathrm{co}}$.
Compact support of the final test $f$ also does not put an
unmodified internal forcing into $\Dint$.

\subsection{Scalar forcing}

\begin{proposition}[Scalar completion is verified on the witness]
\label{lem:scalar}
The forcing
\[
V_{1,g}[R_\lambda]\phi
\]
has real and imaginary parts in $\Dint$ and support contained in
$\supp g$. Hence
\begin{equation}
\phi_{\mathrm{cand}}
:=-E_m^{\mathrm R}\bigl(V_{1,g}[R_\lambda]\phi\bigr)
\label{eq:phicand}
\end{equation}
is defined by complex linear extension of the compact-source
operator. It is smooth, has zero initial data, and satisfies
\begin{equation}
(\Box+m^2)\phi_{\mathrm{cand}}
=-V_{1,g}[R_\lambda]\phi.
\label{eq:phi-equation}
\end{equation}
\end{proposition}

\begin{proof}
Expanding the divergence gives
\[
\begin{aligned}
V_{1,g}[R_\lambda]\phi
={}&(\partial_\mu g)a_1^{\mu\nu}[R_\lambda]
       \partial_\nu\phi\\
 &+g(\partial_\mu a_1^{\mu\nu}[R_\lambda])
       \partial_\nu\phi\\
 &+g a_1^{\mu\nu}[R_\lambda]
       \partial_\mu\partial_\nu\phi\\
 &+m^2g b_1[R_\lambda]\phi.
\end{aligned}
\]
Every coefficient apart from $g$ and its derivatives is smooth.
The support of every derivative of $g$ is contained in
$\supp g$, a compact subset of $M_T^\circ$.
Thus every term is smooth and compactly supported there.

For real $F_1,F_2\in\Dint$, define
\[
E_m^{\mathrm R}(F_1+\ii F_2)
=E_m^{\mathrm R}F_1+\ii E_m^{\mathrm R}F_2.
\]
This is the unique complex linear extension.
The real differential operator $\Box+m^2$, the initial data,
and the support bounds pass to this extension.
Equation \eqref{eq:phi-equation} includes the minus sign in
\eqref{eq:phicand}.
\end{proof}

Derivatives of $g$ and the mass term have not been deleted.
The scalar-leg verification does not verify the coordinate leg.

\section{Coordinate candidate and a restricted auxiliary extension}
\label{sec:candidate}

\begin{proposition}[Explicit coordinate solution]
\label{lem:Xcand}
Define
\begin{equation}
X_{\mathrm{cand}}^\mu(t)
=-\int_0^t\sigma(s)V_\lambda^{0\mu}(s)\dd s.
\label{eq:Xcand}
\end{equation}
Then $X_{\mathrm{cand}}$ is smooth, flatly zero for $t\le0$,
has $X_{\mathrm{cand}}^i=0$, and satisfies
\begin{equation}
\Box X_{\mathrm{cand}}^\mu=-F_{\mathrm{co}}^\mu,
\qquad
X_{\mathrm{cand}}(0)=\partial_tX_{\mathrm{cand}}(0)=0.
\label{eq:boxX}
\end{equation}
It is the unique smooth solution of this Cauchy problem on $M_T$.
\end{proposition}

\begin{proof}
The integrand is smooth and flatly zero in the past.
Thus the integral has the asserted smoothness and past extension.
It is spatially independent, and
\[
\partial_tX_{\mathrm{cand}}^\mu=-\sigma V_\lambda^{0\mu},
\qquad
\partial_t^2X_{\mathrm{cand}}^\mu
=-\partial_t(\sigma V_\lambda^{0\mu})
=-F_{\mathrm{co}}^\mu.
\]
The initial traces vanish. The spatial components vanish by
Proposition~\ref{lem:coord}.
Uniqueness follows componentwise from
Lemma~\ref{lem:uniqueness}.
\end{proof}

This is the candidate formula also present in XVI-E and
XVI-F-v3. It is reproved here; no named Green-operator application
is used in its definition.

\subsection{An extension can be constructed without renaming it}

Let $\Hsrc$ be the real vector space of ambient smooth sources
$q(t,\mathbf x)=q(t)$, independent of space and zero for $t\le0$,
restricted to the slab.
Regard $\Dint$ as an ambient source by its smooth zero extension.
Then
\[
\Dint\cap\Hsrc=\{0\}.
\]
Indeed, a nonzero spatially constant function cannot be compactly
supported in space.

\begin{proposition}[Restricted extension on homogeneous and compact sources]
\label{prop:restricted-extension}
On $\Dint\oplus\Hsrc$, define
\begin{equation}
\mathscr E_{0,\mathrm{res}}^{\mathrm R}(F+q)
=E_0^{\mathrm R}F+\int_0^t(t-s)q(s)\dd s.
\label{eq:restricted-extension}
\end{equation}
This is a well-defined real linear extension of the
compact-interior operator. It gives the unique smooth zero-data
solution of
\[
\Box u=F+q.
\]
It extends uniquely by complex linearity and componentwise.

On the witness,
\begin{equation}
\mathscr E_{0,\mathrm{res}}^{\mathrm R}S_\lambda=V_\lambda,
\qquad
-\mathscr E_{0,\mathrm{res}}^{\mathrm R}F_{\mathrm{co}}
=X_{\mathrm{cand}}.
\label{eq:restricted-witness}
\end{equation}
\end{proposition}

\begin{proof}
The direct-sum decomposition is unique, so the definition is
unambiguous.
The compact part has the equation and initial data from
\eqref{eq:compactGreen}. The homogeneous part has them by direct
differentiation. The sum therefore solves the stated Cauchy
problem.
Lemma~\ref{lem:uniqueness} proves uniqueness.

Both witness sources belong to the complexification of $\Hsrc$.
The first formula in \eqref{eq:restricted-witness} is
\eqref{eq:Vlambda}.
For the second, put $H^\mu=\sigma V_\lambda^{0\mu}$.
Then $F_{\mathrm{co}}^\mu=(H^\mu)'$ and $H^\mu(0)=0$, so
\[
\int_0^t(t-s)F_{\mathrm{co}}^\mu(s)\dd s
=\int_0^tH^\mu(s)\dd s
\]
by integration by parts.
Taking the negative gives \eqref{eq:Xcand}.
\end{proof}

The new symbol
$\mathscr E_{0,\mathrm{res}}^{\mathrm R}$
is essential. Its existence is now proved, so existence of a
restricted solution-operator extension is not left open.
However, the construction is not a declaration that every
inherited occurrence of $E_0^{\mathrm R}$ already denotes this
extension.

The restricted domain contains the polarized witness forcing.
It need not contain every forcing arising before polarization
from real cosine and sine inputs and their sums.
That larger family requires its own domain statement if
family-level identification is claimed.

\subsection{Updated evidence about the inherited definition}

The original XVI-D reasoning must not be read as saying that
XI-B defines its Green symbols \emph{only} on compact-interior
sources.

The supplied XVI-F-v3 explicitly reports a preceding XI-B
sine-integral definition for smooth sources, with nondecaying
sources interpreted locally by finite propagation.
It uses that wider defining quantifier to identify the
\emph{advanced} operator on its boundary-flat source class.
This is separate from the narrower inversion theorem
\texttt{lem:Green} \cite{XVIFvThree}.

Accordingly:
\begin{itemize}
\item failure of the compact-interior hypothesis is proved;
\item absence of every wider inherited definition is not proved
      and is not asserted;
\item the remaining retarded-coordinate check is whether the
      inherited completion uses the reported smooth-source
      prescription on the required family, with the stated
      smoothness and zero initial data.
\end{itemize}

Under that verified wider reading, the following comparison
settles the field identity. No further uniqueness premise is
needed.

\begin{proposition}[Conditional inherited-coordinate compatibility]
\label{prop:coordinate-compatibility}
Suppose the inherited application
$E_0^{\mathrm R}F_{\mathrm{co}}$ is defined and is a smooth
solution of
\[
\Box E_0^{\mathrm R}F_{\mathrm{co}}=F_{\mathrm{co}}
\]
with zero initial data.
Then
\begin{equation}
X_1[R_\lambda]
=-E_0^{\mathrm R}F_{\mathrm{co}}
=X_{\mathrm{cand}}.
\label{eq:coordinate-compatible}
\end{equation}
Consequently the direct inherited completion on $R_\lambda$
exists and equals
\begin{equation}
\mathcal A_D(\lambda;f,g,\phi)
=\int_{M_T}f
 \bigl(\phi_{\mathrm{cand}}
       -X_{\mathrm{cand}}^\mu\partial_\mu\phi\bigr)\dd^4x.
\label{eq:direct-conditional}
\end{equation}
\end{proposition}

\begin{proof}
The first two expressions in \eqref{eq:coordinate-compatible}
refer to the inherited definition \eqref{eq:completion-legs}.
By the assumed operator property they solve
\eqref{eq:boxX}.
Proposition~\ref{lem:Xcand} and
Lemma~\ref{lem:uniqueness} give their equality.

The scalar field is already verified by
Proposition~\ref{lem:scalar}.
Thus both fields in the integrand of
\eqref{eq:direct-conditional} are smooth.
Multiplication by $f$ gives compact support and a finite integral.
Substitution in \eqref{eq:Aone} proves the formula.
\end{proof}

Agreement of two extensions on $\Dint$ alone would not prove
agreement at $F_{\mathrm{co}}$.
Agreement as smooth zero-data Cauchy solutions does prove it.

For comparison, the integral on the right of
\eqref{eq:direct-conditional} exists independently of the
hypothesis of Proposition~\ref{prop:coordinate-compatibility}.
When independently defined it is denoted $I_{\mathrm{cand}}$.
Its existence does not silently identify it with the inherited
completion.

\section{The root interface and the later compatibility result}
\label{sec:root}

\subsection{Direct application versus polarization}

The reported XII-D compact-root theorem supplies, for fixed
$f,g$ and collar $\vartheta$, a compact smooth root
$a_{f,g}[\phi]$ and the diagonal identity
\begin{equation}
\mathcal C_{3,\mathrm D}(f,g)[\gamma,\phi]
=\int_M a_{f,g}[\phi]_{\mu\nu}(y)
       J_D^{\mu\nu}[\gamma](y)\dd^4y
\label{eq:root-identity}
\end{equation}
on its stated domain.
The root has support in a fixed compact subset of
$J^-(\supp f)\cap\{t\ge-\delta\}$ \cite{XIID}.

A polarized source $S_\lambda$ is not itself a single metric
argument $\gamma$ of this theorem.
Thus substituting $S_\lambda$ for $\gamma$ is not a direct
application of \eqref{eq:root-identity}.
This does not prohibit polarization of valid diagonal values.

\begin{lemma}[Conditional polarized root representation]
\label{lem:root-polarization}
Suppose \eqref{eq:root-identity} holds on a real vector space
containing the cosine and sine witness inputs and their required
sums.
With $f,g,\phi$ and the root fixed, the real polarization
\[
T_D(a,c;\phi)
=\frac12\bigl(
 \mathcal C_{3,\mathrm D}[a+c,\phi]
 -\mathcal C_{3,\mathrm D}[a,\phi]
 -\mathcal C_{3,\mathrm D}[c,\phi]
\bigr)
\]
satisfies
\[
T_D(a,c;\phi)
=\int_M a_{f,g}[\phi]\cdot J_D(a,c)\dd^4y.
\]
Its complex bilinear extension has witness value
\begin{equation}
T^{\mathbb C}(\gamma_1^\lambda,\gamma_2^\lambda;\phi)
=Q_\lambda,
\qquad
Q_\lambda
:=\int_M a_{f,g}[\phi]_{\mu\nu}
             S_\lambda^{\mu\nu}\dd^4y.
\label{eq:Q}
\end{equation}
The integral defining $Q_\lambda$ exists whenever the stated
compact smooth root is available.
\end{lemma}

\begin{proof}
Apply the diagonal root identity to $a+c,a,c$ and polarize.
The common root is independent of the metric input, so linearity
of the integral gives the polarized source.

Complex extension uses the same four-term expansion as
\eqref{eq:BC}. The complexified source evaluates to
$S_\lambda$ by \eqref{eq:Slambda}, yielding \eqref{eq:Q}.
Finally, the product of the compact smooth root with the ambient
smooth source is compactly supported and integrable.
\end{proof}

This scalar argument does not independently exhibit an inherited
coordinate field. To pass from polarized diagonal values to the
direct response reading
$\mathcal A_1[f,g](R_\lambda;\phi)$, the domain and linearity of
the inherited completion must also hold on the required response
family.

\subsection{Latest supplied XVI-E/F results}

The supplied XVI-E retains
$T^{\mathbb C}=Q_\lambda$ through the real root-theorem values and
proves fixed-data rapid decay of that scalar root pairing
\cite{XVIE}.

The supplied \texttt{XVI-F-v3.tex} goes further than the earlier
XVI-F versions:
\begin{itemize}
\item it constructs $I_{\mathrm{cand}}$ independently;
\item it proves $I_{\mathrm{cand}}=Q_\lambda$ in
      \texttt{prop:compat};
\item it identifies the advanced root on the slab using the
      reported smooth-source sine-integral definition;
\item it proves direct fixed-data rapid decay of
      $I_{\mathrm{cand}}$;
\item it does not identify the auxiliary coordinate field with
      the inherited retarded coordinate prescription.
\end{itemize}

The earlier unresolved residual is therefore not repeated as the
current result of XVI-F-v3.
That version proves vanishing of the residual pairing; it does
not require the discrepancy tensor to vanish at every negative
time \cite{XVIFvThree}.

These are later results on named scalar evaluations.
They are not premises of the response proof in
Section~\ref{sec:response}.
They do not by themselves prove the retarded operator identity
in Proposition~\ref{prop:coordinate-compatibility}.

Their rapid-decay assertions hold at fixed regulator and fixed
real smooth $\phi$ independent of $\lambda$.
They are not pointwise response decay, a frequency-dependent
scalar estimate, or a class-admission theorem.

\section{Response theorem and class boundary}
\label{sec:release-theorem}

\begin{theorem}[Prepared witness-response package]
\label{thm:response-package}
Assume the explicit source input
\eqref{eq:Slambda}--\eqref{eq:d1}, the reported existence of the
real diagonal responses \eqref{eq:response}, and the
compact-interior scalar Green theorem
\eqref{eq:compactGreen}.

For every $\lambda>0$ and $0<\varepsilon<T$:
\begin{enumerate}
\item $S_\lambda$ is past compact relative to compact observation
      sets but is not a compact-interior source.
\item Real polarization has a unique complex bilinear extension,
      and its witness value is
      $U_D[\gamma_1^\lambda,\gamma_2^\lambda]
       =-\mathcal P V_\lambda$.
\item The exact spatial traceless response contraction is
      \eqref{eq:contraction}.
\item The scalar completion field is verified on the witness.
\item The coordinate Cauchy problem has the unique smooth
      solution \eqref{eq:Xcand}, and the restricted auxiliary
      extension \eqref{eq:restricted-extension} exists.
\item If the inherited retarded coordinate prescription satisfies
      the hypothesis of
      Proposition~\ref{prop:coordinate-compatibility}, its direct
      witness completion equals $I_{\mathrm{cand}}$.
\end{enumerate}
The conclusions require no spatial decay of the witness or of
the compared smooth solutions.
\end{theorem}

\begin{proof}
Assertion (1) is Proposition~\ref{prop:source-domain}.
Assertion (2) follows from Lemmas~\ref{lem:realB},
\ref{lem:BC}, and Proposition~\ref{lem:identify}, using the
independent uniqueness proof
Lemma~\ref{lem:uniqueness}.
Assertion (3) is Proposition~\ref{lem:contraction}.
Assertions (4) and (5) are
Propositions~\ref{lem:scalar}, \ref{lem:Xcand}, and
\ref{prop:restricted-extension}.
Assertion (6) is
Proposition~\ref{prop:coordinate-compatibility}.
\end{proof}

The theorem is about smooth responses and explicitly specified
completion routes. It is not a construction of a compact
observable kernel.

In particular, a response tensor $U_D$ is not a kernel $w_D$
on two metric slots and a scalar slot.
If such a compact kernel is later constructed, comparison with
the frozen XI-G kernel $w_0$ must use the relevant free-shell
null space
\[
\Isol
=\left\{z:
 \langle z,\gamma_1\otimes\gamma_2\otimes\phi\rangle=0
 \ \text{for all smooth }
 \Box\gamma_a=0,\ (\Box+m^2)\phi=0
\right\}
\]
within the specified compact kernel space.

The condition $w_D-w_0\in\Isol$ transfers the frozen obstruction.
Changing a source does not establish or exclude that condition.
A different-class proof would not itself prove admission
\cite{XVIA,XVIB,XIG}.

The separate characteristic quantum-product programme described
in the supplied coverage material is not discharged by these
smooth witness-response calculations.

\section{Status and minimal next steps}
\label{sec:status}

\begin{center}
\small
\renewcommand{\arraystretch}{1.1}
\begin{tabularx}{\linewidth}{@{}p{0.27\linewidth}Y@{}}
\toprule
Item & Exact status\\
\midrule
All-smooth zero-data uniqueness
 & Proved locally without spatial decay:
   Lemma~\ref{lem:uniqueness}.\\
\addlinespace
Source support and compact-domain distinction
 & Proved for $\lambda>0$:
   Proposition~\ref{prop:source-domain}.
   At $\lambda=0$ the source is zero.\\
\addlinespace
Polarization and complexification
 & Proved from diagonal response existence and local uniqueness:
   Lemmas~\ref{lem:realB} and~\ref{lem:BC}.\\
\addlinespace
Witness response and traceless contraction
 & Proved:
   Propositions~\ref{lem:identify} and~\ref{lem:contraction}.\\
\addlinespace
Scalar completion
 & Verified on compact-interior forcing:
   Proposition~\ref{lem:scalar}.\\
\addlinespace
Coordinate Cauchy problem
 & Candidate exists and is unique:
   Proposition~\ref{lem:Xcand}.
   No nonexistence is asserted.\\
\addlinespace
Restricted auxiliary extension
 & Constructed:
   Proposition~\ref{prop:restricted-extension}.
   Not silently identified with inherited notation.\\
\addlinespace
Inherited coordinate identity
 & Conditional:
   Proposition~\ref{prop:coordinate-compatibility}.
   The compact-source theorem alone is insufficient.\\
\addlinespace
Root and auxiliary witness values
 & XVI-E supplies the root reading;
   XVI-F-v3 proves $I_{\mathrm{cand}}=Q_\lambda$.
   The obsolete unresolved residual is not retained.\\
\addlinespace
High-frequency estimates
 & No new full inherited-insertion estimate here.
   Later fixed-data scalar estimates are recorded separately.\\
\addlinespace
Class kernel and quantum admission
 & Not constructed by this manuscript; programme unresolved.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{problem}[Inherited-coordinate interface]
\label{prob:legs}
Verify, for the inherited retarded prescription used in
\eqref{eq:completion-legs}, one of the following:
\begin{enumerate}
\item an applicable smooth-source definition or extension that
      gives a smooth zero-data solution at $F_{\mathrm{co}}$;
\item agreement there with the explicitly constructed restricted
      extension; or
\item a larger-domain theorem covering the real response family
      needed before polarization.
\end{enumerate}

The new restricted extension already proves that such an
extension is mathematically possible on the witness source
space. The remaining question is its match to the inherited
prescription and, when required, the prepolarization family.

The hypothesis of
Proposition~\ref{prop:coordinate-compatibility} is sufficient
for the direct witness reading.
Family-level polarization additionally requires the relevant
diagonal coordinate fields and linearity on their domain.

Status of that inherited identification in this manuscript:
\Unresolved.
\end{problem}

The minimal order of further work is:
\begin{enumerate}
\item verify the inherited retarded-coordinate realization on the
      required source domain;
\item compare it with $X_{\mathrm{cand}}$ using the already proved
      uniqueness theorem;
\item identify the direct inherited completion and its relation
      to the reported root and auxiliary scalar values;
\item formulate any further high-frequency, compact-kernel, and
      free-shell comparison tasks for that actual identified
      observable.
\end{enumerate}

There is no need to re-prove uniqueness or to leave
$I_{\mathrm{cand}}=Q_\lambda$ open after importing XVI-F-v3.
Conversely, neither result replaces the inherited retarded
domain check.

\section{Debugging and non-claims}
\label{sec:nonclaims}

The revision corrects the following points:
\begin{itemize}
\item Smooth uniqueness is proved in the relevant unrestricted
      class, rather than inferred from an ambiguous reading of a
      localization statement.
\item Past compactness is defined explicitly and is not confused
      with compact support in the open slab.
\item The repeated derivative index is $\alpha$, avoiding
      confusion with the frequency $\lambda$.
\item Coordinate forcing is proved spatially noncompact on the
      nonempty interval $(\varepsilon,T)$; both real and imaginary
      components are checked.
\item ``The compact-source theorem does not apply'' replaces any
      wording suggesting that every inherited construction fails.
\item The existence of a coordinate solution and of a restricted
      extension is affirmative, not left as a supposed
      nonexistence issue.
\item The broader smooth-source definition reported by XVI-F-v3
      is acknowledged; its retarded use must be matched to the
      inherited completion.
\item The root limitation is a limitation on direct substitution,
      not a prohibition on valid polarization.
\item The current XVI-F-v3 compatibility result replaces the
      earlier unresolved-residual status.
\end{itemize}

No claim is made of:
\begin{itemize}
\item repair of X-G or XI-G;
\item new admission into a microcausal polynomial class;
\item construction of a compact class kernel or a quantum
      retarded coefficient;
\item a general distributional operator-domain extension;
\item a cutoff-modified witness;
\item a sharp-width limit or uniform estimates as
      $\varepsilon\downarrow0$;
\item an interacting BRST/BV construction, loop, state, or
      ultraviolet subtraction.
\end{itemize}

The auxiliary extension is a specified smooth-source
construction, not a distributional extension theorem.
The inverse coefficient $-1$ is not $-4$.
A response contraction is not an insertion, and a smooth response
is not a class kernel.
No successful compiler execution or full-corpus verification is
represented.

\scopebox{Final release statement.}{%
The prepared witness-response theorem is proved, including
no-decay uniqueness, polarization, explicit identification, the
traceless contraction, and the scalar completion check.

The coordinate Cauchy problem has a unique explicit solution,
and a restricted auxiliary operator extension exists.
Their identification with the inherited coordinate prescription
is stated precisely and remains conditional.

The supplied XVI-F-v3 equality
$I_{\mathrm{cand}}=Q_\lambda$ is recorded as proved there.
It is not misreported as an unresolved residual, nor used to
silently rename the coordinate field.

The inherited-completion and class-admission programme remains
\Unresolved.
This manuscript is complete as a response proof and a
domain-sensitive compatibility interface, not as a proof of the
unresolved quantum coefficient.
}

\begin{thebibliography}{99}

\bibitem{XVIA}
\emph{Landmark XVI-A}.
Supplied revised \texttt{XVI-A.tex}.
Used for the separate retarded-product obligations and the
conditional free-shell class-transfer statement.

\bibitem{XVIB}
\emph{Landmark XVI-B}.
Supplied revised \texttt{XVI-B.tex}.
Source-level locators:
\texttt{prop:trace}, \texttt{prop:pair},
\texttt{eq:JDeval}, and \texttt{cor:tf}.
Only the source-level results are premises of the response proof;
its summary of later XVI-D results is not used circularly.

\bibitem{XVIE}
\emph{Landmark XVI-E}.
Supplied \texttt{XVI-E.tex}.
Locators:
\texttt{lem:Q}, \texttt{lem:TD}, \texttt{lem:TC},
\texttt{lem:Xcand}, \texttt{lem:compat}, and
\texttt{cor:decay}.
Used for the later root reading and its stated compatibility
boundary, not as a premise of witness-response identification.

\bibitem{XVIFvThree}
\emph{Landmark XVI-F}.
Supplied \texttt{XVI-F-v3.tex}.
Locators:
\texttt{prop:Icand}, \texttt{prop:kernel},
\texttt{prop:formula}, \texttt{prop:compat}, and
\texttt{prop:decay}.
This version proves witness root compatibility and reports the
wider smooth-source sine-integral definition.
Its result supersedes the earlier unresolved root--auxiliary
residual.

\bibitem{XIB}
\emph{Manuscript XI-B}, version 1.
Supplied identifier \texttt{862627cd}.
Reported locators:
\texttt{lem:Green}, \texttt{thm:main(ii)},
\texttt{thm:graviton}, and \texttt{eq:Ggr}.
The compact-interior Green statements are reproduced in the
supplied XVI texts.
The wider smooth-source defining paragraph is reported in
XVI-F-v3; the complete XI-B source is not independently certified.

\bibitem{XIID}
\emph{Manuscript XII-D}, version 1.
Supplied identifier \texttt{e3e3f67d}.
Reported locators:
Section D0, \texttt{eq:response}, \texttt{thm:response},
\texttt{rem:forcing-sign}, \texttt{eq:completion-legs},
\texttt{eq:Aone}, \texttt{def:completed},
\texttt{eq:CD}, \texttt{thm:root}, and
\texttt{eq:root-identity}.
Used only through the supplied statements and formulas.

\bibitem{XIG}
\emph{Manuscript XI-G}, version 1.
Supplied identifier \texttt{3cae2be9}.
Reported locators:
\texttt{lem:qvalues}, \texttt{eq:qcounter},
\texttt{def:class}, and \texttt{thm:obstruction}.
The frozen obstruction is not transferred to a source tensor.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
arXiv:0806.1036 [math.DG] (2008).
General background on wave equations, Green operators, and
Cauchy problems.
The local-energy uniqueness argument used in this manuscript is
proved explicitly in Lemma~\ref{lem:uniqueness}.

\end{thebibliography}

\end{document}