% !TeX program = pdflatex
%=====================================================================
% XI-C.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XI-C.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XI-C.tex
%
% Essential corrections to the proposed sheet:
%
% 1. XI-B constructs the algebra generated by compactly smeared
%    linear fields and their finite products. It expressly does not
%    construct a microcausal Wick-polynomial extension.
%    A microcausal kernel bound is therefore not admission into
%    XI-B's algebra.
%
% 2. The fixed-width insertion and its one-graviton tree are
%    constructed below as classical distributional polynomial
%    symbols. Their quantization in the unchanged XI-B algebra
%    is not asserted.
%
% 3. The quantity called T below has the positive dressing sign
%    specified in the production plan. The retarded response of
%    the actual bare insertion B=-int f X_1 partial phi is -T.
%
% 4. The usual forward background variation gives
%       delta A_2 = -A_1[Lie_xi gamma]
%    for the explicitly defined restricted candidate.
%    The backward defect A_2(old)-A_2(new) has the requested
%    positive sign. A full quantum A_2 was not handed off.
%
% 5. Failure of Lie_xi to preserve the free equation does not,
%    by itself, prove that an observable is not an equation zero.
%    An explicit free-field witness is supplied.
%
% 6. A pure-gauge tensor is invisible to G_lin, not generally to
%    the gauge-fixed operator. The tree is independent of an
%    external metric tensor. No pure-gauge vanishing theorem
%    for this tree is asserted.
%
% 7. Recovering the nonzero scalar-only maps requires replacing
%    the metric by a prescribed classical background before
%    taking the scalar response. Setting that background to
%    zero gives the zero scalar response.
%
% 8. No cubic Einstein vertex is contracted. Its omission is a
%    scope restriction, not a leg-count vanishing theorem.
%
% Ending:
%   The fixed-width classical symbol tree is constructed.
%   The requested output in the frozen quantum algebra, and the
%   corresponding quantum second-order identity, remain unresolved.
%
% No successful compilation, mechanized verification, independent
% review, Hadamard selection, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\PP}{\mathcal P}
\newcommand{\RR}{\mathcal R}
\newcommand{\Afree}{\mathscr A_0}
\newcommand{\Aoff}{\mathscr P_{\mathrm{off}}}
\newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}}
\newcommand{\Pfin}{\mathscr P_{\mathrm{fin}}}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Coord}{\mathsf X}
\newcommand{\Xone}{\mathsf X_1}
\newcommand{\Xtwo}{\mathsf X_2}
\newcommand{\ER}{\mathsf E_0^{\mathrm R}}
\newcommand{\EA}{\mathsf E_0^{\mathrm A}}
\newcommand{\ERm}{\mathsf E_m^{\mathrm R}}
\newcommand{\EAm}{\mathsf E_m^{\mathrm A}}
\newcommand{\Ggr}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\Lone}{\mathsf L_{1,g}}
\newcommand{\Ltwo}{\mathsf L_{2,g}}
\newcommand{\Tcal}{\mathcal T}
\newcommand{\Bcal}{\mathcal B}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Atwo}{\mathcal A_2}
\newcommand{\Lie}{\mathcal L}
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\newcommand{\Pfree}{P_m}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
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\DeclareMathOperator{\tr}{tr}
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\DeclareMathOperator{\id}{id}

\newcommand{\scopebox}[2]{%
\par\medskip\noindent
\fbox{\begin{minipage}{%
\dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
\textbf{#1}\par\smallskip #2
\end{minipage}}
\par\medskip}

\title{\textbf{Landmark XI-C}\\[0.4em]
One fixed-regulator graviton tree for the ramp relational scalar:\\
the retarded symbol, its normalization,\\
and the boundary of the free-algebra handoff}
\author{Ngo Cheung}
\date{}

\usepackage{tikz}
\begin{document}
\maketitle

\begin{abstract}
We construct the one-graviton retarded tree determined by the
signed gauge-fixed inverse of Landmark XI-B and the switched
stress-tensor vertex of Landmark XI-A. At fixed regulator width,
the bilinear coordinate insertion and the cubic scalar output
have compact distributional kernels satisfying the polynomial
microcausal exclusion. The contraction is defined directly on
smooth off-shell configurations, without a Hadamard covariance.

For a smooth free scalar and a switch above the ramp, the
coordinate response is minus one half of a double retarded wave
inverse applied to the divergence of the switched stress.
It vanishes exactly when that source is conserved. The named
XI-A input gives a nonzero response and does not repair its empty
Einstein jet.

An essential domain qualification prevents an unconditional
quantum release. XI-B constructed finite products of compactly
smeared linear fields, not an extension admitting the singular
polynomial kernels used here. We prove the distinction and do
not substitute a wavefront estimate for an admission map.
A restricted classical second-order candidate has an explicit,
generically nonzero background defect; its sign and domain are
specified. No quantum second-order coefficient, interacting
BRST class, mixed loop, Hadamard selection, or sharp limit is
constructed. The classical symbol tree is complete; its output
in the unchanged quantum algebra remains unresolved.
\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) {Joint fixed-width bound};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Classical retarded symbol};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Quantum admission};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The admission gate is left open. The symbol does not close it.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A retarded symbol is not quantum admission. 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-XI-C}
\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) {Joint fixed-width bound};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Quantum admission};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The admission gate is left open. The symbol does not close it.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A retarded symbol is not quantum admission. 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-XI-C}
\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) {Joint fixed-width bound};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Classical retarded symbol};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Quantum admission};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The admission gate is left open. The symbol does not close it.};
\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-XI-C}
\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) {Joint fixed-width bound};
\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) {Quantum admission};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Classical retarded symbol. The admission gate is left open. The symbol does not close it.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Joint fixed-width bound is not Quantum admission. 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-XI-C}
\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: Joint fixed-width bound};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Classical retarded symbol};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Quantum admission};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The admission gate is left open. The symbol does not close it.};
\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-XI-C}
\end{figure}

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


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



\scopebox{Release: a constructed retarded symbol, with the quantum
admission gate left explicit}{%
The joint fixed-width kernel bound, the one-graviton classical
retarded contraction, its cubic output symbol, causal support,
and normalization are proved.

The literal quantum target in the unchanged XI-B algebra is
\textbf{Unresolved}: that algebra does not include a declared
admission of these distributional polynomial symbols.
A separate quantization or extension theorem is required.

The mixed loop is also unresolved. The empty fixed-input
Einstein jet is not repaired. Neither unresolved item is a
wave-equation obstruction to the retarded symbol.}

\tableofcontents

%=====================================================================
\section{Scope, the release theorem, and the domain correction}
\label{sec:scope}
%=====================================================================

The proposed calculation involves two logically separate steps:
constructing a retarded polynomial functional and admitting that
functional as an element of a quantum algebra. The first step
does not require a state. The second cannot be inferred solely
from the existence of a causal inverse.

XI-B's Definition of the free causal polynomial algebra uses
compact interior smearings of linear fields and their finite
products. Its construction expressly does not extend to all
local or microcausal polynomial functionals. The local vertex
and the nonlocal bilinear insertion in the present problem
therefore require more than the algebraic operations already
handed off.

We retain that limitation rather than silently replace
$\Afree$ by a Wick algebra. The additional space used below,
$\Pmc$, is a space of \emph{classical polynomial symbols}.
No quantum product on that space is selected. The usual
distinction between regular polynomial generators, local
polynomials, and microcausal functionals is relevant here
\cite{FR,HRV}.

\begin{theorem}[Fixed-width symbol release]
\label{thm:main}
For the conventions and support class below:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The bilinear insertion
$\Bcal_f=-\int f\,\Xone[\gamma]\cdot\partial\phi$
has a compact mixed kernel satisfying the polynomial
microcausal exclusion.
\item The one-graviton retarded contraction is a well-defined
continuous cubic classical scalar polynomial with a compact
microcausal kernel.
\item Its intermediate metric uses precisely
$\Ggr=-\PP\ER$.
For free scalar input and a switch above the ramp, its
coordinate response is the double retarded integral displayed
in Theorem~\ref{thm:normalization}.
\item The classical restricted second-order candidate defined
in Section~\ref{sec:boundary} has an explicitly nonzero
background defect for suitable free inputs.
\end{enumerate}
These statements do not establish an admission map into
$\Afree$, a quantum second-order Ward identity, or an
interacting differential.
\end{theorem}

The alternative endings remain distinct. There is no wavefront
obstruction to the classical tree constructed here. There is
also no proved theorem excluding every possible quantum
extension. The unmet step is admission into the frozen target,
not an unnamed propagator normalization.

%=====================================================================
\section{Copied handoff and frozen conventions}
\label{sec:handoff}
%=====================================================================

\subsection{Slab, ramp, and admitted classical parameters}

Use
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad
 \Sigma=\{t=0\},
\]
\[
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta,\qquad
 \Pfree=\Box+m^2,\quad m\geq0.
\]
All classical configurations are smooth on a neighbourhood of
the closed slab. Parentheses include the factor $1/2$.

Fix $0<\eps<T$, with the same smooth ramp
$\sig(t)=\chi(t/\eps)$ as XI-A and XI-B:
\[
 \chi(s)=
 \begin{cases}
 0,&s\leq0,\\[2pt]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
 1,&s\geq1.
 \end{cases}
\]
Fix $g\in C_c^\infty(\MTo;[0,1])$, with nonempty plateau
$P_g=\Int\{g=1\}$, and
\[
 \overline U\subset P_g\cap\{\eps<t<T\},\qquad
 \ga=C_c^\infty(U;TM).
\]
The observation test is
$f\in C_c^\infty(\MTo;\R)$, and $K=\supp f$.
No limit of $\eps$ is taken.

\subsection{Signed free multiplet}

For a symmetric tensor $r$, put
\[
 \PP r=r-\tfrac12\eta\,\tr_\eta r,\qquad
 D^\mu r=\partial_\nu(\PP r)^{\nu\mu},
\]
\[
 (K\xi)_{\mu\nu}
 =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,\qquad
 \RR=\tfrac12K.
\]
XI-B supplies
\begin{equation}
 L_\gamma=-\Box\PP,\qquad
 \Ggr=-\PP\ER,\qquad
 \PP^2=\id.
 \label{eq:inverse}
\end{equation}
Here $\ER$ is the zero-initial-data inverse of $\Box$.
Its advanced counterpart has zero final data.
Their causal support and smooth Cauchy properties are those
used in XI-B \cite{XIB,BGP}.

The coefficient of $\PP\ER$ is $-1$, not $-1/2$.
The scalar equation inverse is $\mathsf E_m^\pm$; its
action-Hessian inverse is $-\mathsf E_m^\pm$. Thus
\[
 [\phi(x),\phi(y)]=-i\hbar\,\delta_m(x,y),
 \qquad
 \delta_m=\mathsf E_m^{\mathrm R}-\mathsf E_m^{\mathrm A}.
\]

The auxiliary field is retained, with
\[
 \sfree\gamma=\RR c,\qquad
 \sfree\phi=0,\qquad
 \sfree c=0,\qquad
 \sfree\bar c=ib,\qquad
 \sfree b=0.
\]
The classical XI-A parameter $K\xi$ corresponds to $c=2\xi$.
Eliminating $b$ gives only on-shell nilpotence of the reduced
antighost rule. Ghosts and antifield source slots are spectators
in the contraction below.

XI-B also gives
\[
 \ga\cap\{\xi:\Box\xi=0\}=\{0\}.
\]
Consequently the plateau algebra is not a nontrivial residual
action on free de Donder solutions. Its classical background
action and the free graded differential are different objects.

\subsection{Vertex and coordinate operator}

The only contracted vertex is
\begin{equation}
 V_{1,g}[\gamma,\phi]
 =-\tfrac12\int_{\MT}
       g\,\gamma_{\mu\nu}T_0^{\mu\nu}[\phi]\dd^4x,
 \label{eq:vertex}
\end{equation}
where
\[
 T_0^{\mu\nu}[\phi]
 =\partial^\mu\phi\partial^\nu\phi
 -\tfrac12\eta^{\mu\nu}
   \left(\partial_\alpha\phi\partial^\alpha\phi-m^2\phi^2\right).
\]
It is a classical local polynomial before any quantum admission.

Write
\[
 a_1^{\alpha\mu}[r]
 =\tfrac12\eta^{\alpha\mu}\tr_\eta r-r^{\alpha\mu}.
\]
The coordinate map is the linear operator
\begin{align}
 L_\eps r:=\Xone[r]
 &=-\ER\partial_\alpha(\sig a_1^{\alpha\cdot}[r])
 \nonumber\\
 &=\ER\left(\sig Dr-\sig'a_1^{0\cdot}[r]\right).
 \label{eq:coordinate}
\end{align}
The layer formula alone applies only when $Dr=0$.
It cannot be imposed on a sourced response with nonzero
divergence.

The cubic Einstein vertex is not contracted here.
Its coupling degree permits contributions at the same order
as the present tree. Its omission is a scope restriction, not
a consequence of counting one metric leg.

%=====================================================================
\section{The joint kernel at fixed regulator width}
\label{sec:joint}
%=====================================================================

\subsection{A named classical symbol space}

Let $\Pmc$ be the space of finite polynomial functionals on
smooth configurations whose compact distributional derivative
kernels satisfy
\begin{equation}
 \WF(u_n)\cap
 \left(\widetilde V_+^{\,n}\cup
       \widetilde V_-^{\,n}\right)=\varnothing.
 \label{eq:microcausal}
\end{equation}
Here $\widetilde V_\pm$ are the closed causal covector cones
with zero covectors included; the total zero covector is
omitted from a wavefront set. Field indices are finite
dimensional and do not change the criterion.

This is a \emph{classical} symbol declaration. No Hadamard
product, locally convex quantum completion, or inclusion into
$\Afree$ is contained in the definition.

Let $G_0^{\mathrm R}(x-z)$ be the retarded wave kernel and define
\[
 \mathfrak a^{\alpha\mu,ab}
 =\tfrac12\eta^{\alpha\mu}\eta^{ab}
       -\eta^{\alpha(a}\eta^{b)\mu}.
\]
Integration by parts, with the flat ramp removing the initial
boundary term, gives
\begin{equation}
 (L_\eps r)^\mu(x)
 =\int\Lambda_\eps^{\mu,ab}(x,z)r_{ab}(z)\dd^4z,
 \qquad
 \Lambda_\eps^{\mu,ab}(x,z)
 =\sig(z)\mathfrak a^{\alpha\mu,ab}
                  \partial_\alpha^zG_0^{\mathrm R}(x-z).
 \label{eq:Lambda}
\end{equation}
All integrals are restricted to the slab where necessary.

The bare insertion is
\begin{equation}
 \Bcal_f[\gamma,\phi]
 =-\int f(x)(L_\eps\gamma)^\mu(x)
                       \partial_\mu\phi(x)\dd^4x.
 \label{eq:B}
\end{equation}
Ordering the variables as scalar slot $x$ and metric slot $z$,
its kernel is
\[
 b_f^{ab}(x,z)
 =\partial_\mu^x
       \left(f(x)\Lambda_\eps^{\mu,ab}(x,z)\right).
\]

\begin{proposition}[Joint fixed-width bound]
\label{prop:joint}
The mixed kernel is compactly supported and satisfies
\begin{equation}
 \WF(b_f)\subset
 \{(x,p;z,-p):(x,z;p,-p)\in\WF(G_0^{\mathrm R})\}.
 \label{eq:WFjoint}
\end{equation}
In particular, $\Bcal_f\in\Pmc$.
Its metric support is contained in
\[
 \mathcal D_K=J^-(K)\cap\MT,
\]
and its scalar support is contained in $K$.
\end{proposition}

\begin{proof}
The retarded wavefront has opposite endpoint covectors:
diagonal conormals and the off-diagonal null propagation
relation. Smooth multiplication by $\sig$ and $f$, followed
by differentiation, introduces no new wavefront directions
\cite{BDH}.

If $p$ and $-p$ both belong to the same closed causal cone,
then $p=0$, which is excluded. Thus the mixed kernel obeys
\eqref{eq:microcausal}. Lower derivative kernels are obtained
by smearing one slot with a smooth configuration. Their
wavefront bounds have zero covector in that integrated slot,
and hence zero in the remaining slot; they are smooth.

Retarded support gives $z\in J^-(x)$ and the ramp removes
negative-time dependence. The causal past of a compact
observation set within a bounded-time closed slab is compact.
Therefore the mixed kernel is compactly supported.
\end{proof}

The factor $\sig$ is smooth even at $t=0$.
No sharp step, boundary delta, or reflected sharp-Cauchy
branch has been inserted. This proof uses the actual joint
kernel, not smooth scalar tests evaluated at a fixed metric.

\subsection{Why this is not admission into XI-B's algebra}

Let $\Pfin$ denote the off-shell polynomial algebra generated
by compact smooth linear smearings. Every degree-$n$ kernel
in that algebra is a finite sum of smooth tensor products:
\begin{equation}
 u_n(x_1,\ldots,x_n)
 =\sum_{j=1}^N
       u_{j1}(x_1)\cdots u_{jn}(x_n),
 \qquad u_{ji}\in C_c^\infty(\MTo).
 \label{eq:finite}
\end{equation}
The classical limit of XI-B's construction uses precisely
this finite-generator presentation before its equation
quotient.

\begin{proposition}[The admission distinction is substantive]
\label{prop:domain}
For suitable nonzero $f$ supported above the ramp, the actual
off-shell mixed kernel $b_f$ is not a kernel of the form
\eqref{eq:finite}. The local vertex \eqref{eq:vertex} likewise
does not have a smooth finite-rank off-shell kernel.
Consequently the joint bound does not place either input in
XI-B's declared off-shell generator algebra.
\end{proposition}

\begin{proof}
Choose $f$ nonzero near a point above $\eps$, where $\sig=1$.
Microlocally at a noncharacteristic diagonal covector, the
wave inverse has principal symbol proportional to
$(\eta^{\alpha\beta}p_\alpha p_\beta)^{-1}$.
The two endpoint derivatives in $b_f$ give principal symbol
proportional to
\[
 f(x)\,
 \frac{p_\mu p_\alpha\mathfrak a^{\alpha\mu,ab}}
      {\eta^{\rho\sigma}p_\rho p_\sigma}.
\]
For $p=\dd t$ and $ab=00$, the numerator is nonzero:
$\mathfrak a^{00,00}=-1/2$.
The kernel therefore has a nonzero diagonal singularity and
is not smooth. Every kernel in \eqref{eq:finite} is smooth.

The vertex kernel contains differentiated diagonal delta
distributions linking its three field slots. Its nonzero
principal derivative coefficients cannot be represented by
a smooth kernel. This gives the same distinction for the
vertex.
\end{proof}

The proposition is about actual off-shell representatives.
It does not prove that every possible on-shell cubic symbol
has no alternative finite-generator representation.
Such an assertion would require a separate argument.

It does prove that the production plan cannot use an
off-shell kernel bound as a substitute for a supplied
admission map. XI-B has neither a topology for taking these
kernel limits nor a declared Wick-polynomial extension.
No algebraic limit operation may be inferred from its CCR.

%=====================================================================
\section{The retarded tree and its output symbol}
\label{sec:tree}
%=====================================================================

\subsection{Definition on unrestricted smooth configurations}

Put
\begin{equation}
 S[\phi]=\tfrac12gT_0[\phi],\qquad
 r_g[\phi]=\Ggr S[\phi]
       =-\tfrac12\PP\ER(gT_0[\phi]).
 \label{eq:metricresponse}
\end{equation}
The source is smooth and compact for every smooth $\phi$.
The intermediate tensor is an actual zero-data sourced
solution:
\[
 -\Box\PP r_g=S[\phi].
\]
It is not a homogeneous free graviton class.
Its auxiliary response is $b_g=Dr_g$, not an imposed zero.

\begin{definition}[One-graviton dressing symbol]
\label{def:tree}
Define
\begin{equation}
 \Tcal(f,g)[\phi]
 =\int_{\MT}
       f(x)(L_\eps r_g[\phi])^\mu(x)
                        \partial_\mu\phi(x)\dd^4x.
 \label{eq:T}
\end{equation}
This is the positive dressing quantity specified in the
production sheet.
\end{definition}

The retarded response of a functional $F$ to a vertex $V$
has the classical sign
$-F'G^{\mathrm R}V'$.
Since $\delta_\gamma V_{1,g}=-S$ and the insertion
\eqref{eq:B} has a minus sign, its one-graviton component is
\begin{equation}
 R_{\gamma,\mathrm{tree}}(V_{1,g};\Bcal_f)
 =-\Bcal_{f,\gamma}\Ggr V_{1,g,\gamma}
 =-\Tcal(f,g).
 \label{eq:physicalsign}
\end{equation}
Thus $\Tcal$ and the contribution to the actual bare
relational coefficient have opposite signs.
The physical coupling degree of \eqref{eq:physicalsign}
is $\kappa^2$.

\subsection{Existence without a singular diagonal product}

\begin{theorem}[Cubic symbol and one-line contraction]
\label{thm:tree}
The functional $\Tcal(f,g)$ is a continuous homogeneous cubic
polynomial on smooth scalar configurations. Its polarized
kernel is compact and obeys \eqref{eq:microcausal}.
Hence it belongs to the classical cubic symbol space
$\Pmc^{(3)}$.
The construction requires no Hadamard covariance.
\end{theorem}

\begin{proof}
For smooth $\phi$, the compact smooth stress source in
\eqref{eq:metricresponse} has a unique smooth retarded
response. Applying $L_\eps$ gives a smooth coordinate
response. Multiplication by the smooth scalar derivative
and integration against $f$ therefore defines a number.
Smooth Cauchy continuity on fixed causal compacts proves
continuity. Homogeneity is quadratic in the stress and
linear in the remaining scalar factor.

For the distributional kernel, compose the coordinate
kernel and the graviton inverse:
\begin{equation}
 Q_\eps^{\mu}{}_{ab}(x,y)
 =-\int
      \Lambda_\eps^{\mu,cd}(x,z)
      \PP_{cd,ab}G_0^{\mathrm R}(z-y)\dd^4z.
 \label{eq:Q}
\end{equation}
Before integrating $z$, each propagator has endpoint
covectors summing to zero. The two factors share only $z$.
A violation of the product criterion would force the
unshared $x$ and $y$ covectors to vanish, and then force the
shared covector to vanish as well. The product is therefore
defined. Proper integration over the bounded causal
intermediate region gives a distribution with opposite
$x,y$ covectors.

Attach the local stress kernel at $y$, using independent
scalar slots $u,v$, and the final scalar slot at $x$.
The stress diagonal kernel has total covector sum zero.
Each attachment shares only one slot with the preceding
kernel, so the same product argument applies.
Derivatives do not change the zero-sum property.

After the proper pushforwards, every external covector
tuple in the cubic kernel satisfies
$p_1+p_2+p_3=0$ in the fixed Cartesian trivialization.
A nonzero tuple with all entries future causal or all
entries past causal cannot satisfy this identity.
This proves \eqref{eq:microcausal}. Permuting and
symmetrizing scalar slots preserves the conclusion.

The support cutoffs $f$ and $g$, and retarded propagation
inside the finite slab, make all these pushforwards proper.
The distributional operation criteria are used with their
support requirements, not merely as formal kernel
multiplication \cite{BDH}.
\end{proof}

In particular, a source-diagonal stress tensor does not
produce a square of a propagator on identified endpoints
in this tree. Two successive retarded kernels are composed
through an independent intermediate variable.

The output is not a linear test class in
$\mathcal D/\Pfree\mathcal D$. It is a cubic classical
polynomial symbol. A quantum admission into $\Afree$ is a
separate step not supplied by Theorem~\ref{thm:tree}.

\subsection{Why representative information is retained}

The response is evaluated before any equation quotient.
For unrestricted $\phi$,
\[
 \partial_\nu(gT_0^{\nu\mu}[\phi])
 =g(\Pfree\phi)\partial^\mu\phi
          +(\partial_\nu g)T_0^{\nu\mu}[\phi].
\]
Dropping the first term is legitimate only after selecting
a smooth free input for a classical reduction.

Likewise, replacing $r_g$ by an unspecified homogeneous
metric class discards its sourced equation and generally
changes $Dr_g$. These two warnings are the scalar
representative warning of X-B-v3 and the sourced-metric
warning of XI-B in their respective slots.

%=====================================================================
\section{Normalization to the de Donder violation}
\label{sec:normalization}
%=====================================================================

Assume for this section that
\begin{equation}
 \supp g\subset\{t\geq\delta\},
 \qquad \eps<\delta<T.
 \label{eq:above}
\end{equation}
This condition is automatic for sufficiently small fixed
width relative to a fixed compact interior switch, but is
not imposed retroactively on switches overlapping the ramp.

\begin{theorem}[Double-retarded normalization]
\label{thm:normalization}
For every smooth free scalar $\phi_0$,
\begin{align}
 Dr_g[\phi_0]
 &=-\tfrac12\ER
             \bigl((\partial_\nu g)T_0^{\nu\cdot}[\phi_0]\bigr),
 \label{eq:Dresponse}\\
 \Xone[r_g[\phi_0]]
 &=-\tfrac12\ER\ER
             \bigl((\partial_\nu g)T_0^{\nu\cdot}[\phi_0]\bigr).
 \label{eq:double}
\end{align}
The complete vector in \eqref{eq:double} vanishes identically
on the slab if and only if the switched free source is
conserved:
\[
 (\partial_\nu g)T_0^{\nu\mu}[\phi_0]=0
 \quad\text{for every }\mu.
\]
\end{theorem}

\begin{proof}
Trace reversal is involutive, so
$\PP r_g=-\frac12\ER(gT_0)$.
Differentiation commutes with the retarded inverse on these
compact interior sources. Taking the divergence and using
$\partial_\nu T_0^{\nu\mu}[\phi_0]=0$ proves
\eqref{eq:Dresponse}.

Retardation makes $r_g$ and $Dr_g$ zero below $\delta$.
Thus the layer term in \eqref{eq:coordinate} is zero,
and $\sig Dr_g=Dr_g$ wherever this response can be nonzero.
Consequently $\Xone[r_g]=\ER Dr_g$, proving \eqref{eq:double}.

Applying $\Box^2$ to \eqref{eq:double} recovers minus one
half of the displayed source. Zero output therefore implies
zero source. Conversely, zero source gives zero response by
the zero-data inverse.
\end{proof}

This is an identity of whole response fields. A fixed
smearing can vanish even when the response does not; the
``if and only if'' statement is not a universal
nonvanishing assertion for every $f$.

When \eqref{eq:above} fails, the correct formula is still
\eqref{eq:coordinate}, including both the factor $\sig$
and the layer term. The unrestricted tree remains defined,
but the simplified double-integral normalization need not
hold.

\subsection{The named XI-A witness}

For
\[
 \phi_m(t)=
 \begin{cases}\sin(mt)/m,&m>0,\\ t,&m=0,\end{cases}
\]
XI-A gives $T_0^{00}=1/2$ and $T_0^{0i}=0$. Hence
\begin{equation}
 \Xone^0[r_g[\phi_m]]
 =-\tfrac14\ER\ER(\partial_tg)\not\equiv0.
 \label{eq:emptywitness}
\end{equation}
A nonzero compact interior switch cannot be time independent.
Injectivity in Theorem~\ref{thm:normalization} proves the
nonzero claim.

Both $r_g$ and its divergence exist.
The failure is the conservation condition needed for the
ungauge-fixed Einstein equation. Therefore
$\Jcal_g(\phi_m)=\varnothing$ remains the XI-A result
\cite{XIA}. The tree does not provide a missing Einstein jet.

\subsection{An evaluated temporal check}

Take $m=0$, $\phi_0=t$, and a nonnegative temporal bump
$g_0$ above the ramp. Use a compact spatial cutoff equal to
one on a sufficiently large causal cylinder containing the
observation dependence region. In that cylinder,
\begin{equation}
 \Xone^0[r_g[t]](t)
 =-\tfrac18\int_0^t(t-s)^2g_0(s)\dd s,
 \qquad \Xone^i[r_g[t]]=0.
 \label{eq:temporal}
\end{equation}
Indeed,
$\ER F(t)=\int_0^t(t-s)F(s)\dd s$ for a spatially constant
source, and integration by parts in
$\ER\ER(\partial_tg_0)$ gives \eqref{eq:temporal}.

A nonnegative observation bump after the source detects a
strictly negative value of $\Tcal$.
The bare response $-\Tcal$ has the opposite sign.
Replacing the inverse coefficient $-1$ by $-1/2$ would
halve this number and fail the copied normalization.

This is a classical evaluation on a smooth free solution,
not a quantum expectation value or an admitted cubic
element of $\Afree$.

%=====================================================================
\section{Causal support and the limits of the diagnostics}
\label{sec:support}
%=====================================================================

\begin{proposition}[Intermediate and external support]
\label{prop:support}
In the tree kernel, the intermediate coordinate-source
variable satisfies
\begin{equation}
 z\in
 \supp\sig\cap J^-(K)\cap
 J^+(\supp g)\cap\MT.
 \label{eq:zsupport}
\end{equation}
The vertex variable satisfies
$y\in\supp g\cap J^-(K)$.
The external scalar support of $\Tcal$ is contained in
\begin{equation}
 K\cup\bigl(\supp g\cap J^-(K)\bigr).
 \label{eq:outputsupport}
\end{equation}
All support bounds are compact in the closed slab.
\end{proposition}

\begin{proof}
The first retarded propagation requires $y\preceq z$.
The coordinate propagation requires $z\preceq x$, with
$x\in K$. The factor $\sig$ gives the remaining
restriction in \eqref{eq:zsupport}. Transitivity gives the
vertex bound. Two scalar slots come from the vertex and
one from the observation, proving \eqref{eq:outputsupport}.
Finite time and compact endpoint supports give compactness.
\end{proof}

The closed support of the smooth ramp can meet $t=0$.
It is not a claim that $\sig$ is nonzero there.
Nor is \eqref{eq:zsupport} a ramp-layer support statement:
the tree can propagate and remain nonzero long after
$\eps$.

A pure-gauge diagnostic must be stated correctly:
\[
 G_{\mathrm{lin}}[K\xi]=0,\qquad D(K\xi)=\Box\xi.
\]
For compact plateau $\xi$, the second expression is not
generally zero. In fact
\[
 (-\Box\PP)(K\xi)=-\PP K(\Box\xi).
\]
Thus invisibility to the invariant linearized Einstein
tensor does not imply invisibility to the gauge-fixed
operator.

Moreover, the tree \eqref{eq:T} has no external metric
argument to set equal to $K\xi$. Its source is the scalar
stress. A claimed vanishing of this tree merely because an
unrelated classical background is pure gauge is therefore
not a diagnostic of the constructed contraction.

%=====================================================================
\section{The background order boundary on a named candidate}
\label{sec:boundary}
%=====================================================================

\subsection{First order and scalar-only comparison}

To state a background transformation, introduce a prescribed
classical tensor $h$, a second coefficient $k$, and an
independent smooth tensor configuration $\gamma$.
Put $H=h+\gamma$.

For this subsection the copied scalar-only completion is a
classical polynomial functional:
\begin{equation}
 \Aone[r;\phi]
 =\pair{c_1[r]-\Lone[r]\EAm f}{\phi},
 \qquad
 c_1[r]=\partial_\mu(f\Xone^\mu[r]),
 \label{eq:Aone}
\end{equation}
where
\[
 \Lone[r]u
 =\partial_\mu(ga_1^{\mu\nu}[r]\partial_\nu u)
                  +gm^2b_1[r]u,
 \qquad b_1[r]=\tfrac12\tr_\eta r.
\]
For free $\phi$ and $\xi\in\ga$,
\[
 \Aone[K\xi;\phi]=0.
\]
More explicitly, the off-shell difference is the scalar
equation test
$\pair{\Pfree(X_\xi\EAm f)}{\phi}$.
This is the first-order background identity proved in
X-B-v3 \cite{XB}; it is also verified by
$\mathsf L_{1,1}[K\xi]=D_\xi\Pfree-\Pfree X_\xi$,
with $D_\xi v=\partial_\mu(\xi^\mu v)$.

Replacing the quantum metric by a fixed classical $r$
\emph{before} taking the scalar variation gives
\[
 R_{\mathrm{scalar}}(V_{1,g}[r];F_f)
 =-\pair{\Lone[r]\EAm f}{\phi},
 \qquad
 \rho_{1,g}[r;\phi]=-\ERm\Lone[r]\phi.
\]
These are exactly the scalar-only maps.
Setting $r=0$ gives their zero specialization.
It does not recover a nonzero background response without
a retained background.

Conversely, $\Tcal$ is independent of an external $\gamma$.
Setting $\gamma=0$ after taking the graviton response does
not remove $\Tcal$. Restriction of a configuration and
taking its response do not commute in that fashion.

\subsection{A precise restricted second-order coefficient}

Use X-C-v3's completed scalar-only coefficient
\begin{align}
 \Atwo^{\mathrm{sc}}[H,k;\phi]
 =\pair{
 c_2[H,k]-\Lone[H]\EAm c_1[H]
 -\Ltwo[H,k]\EAm f
 +\Lone[H]\EAm\Lone[H]\EAm f
 }{\phi}.
 \label{eq:A2sc}
\end{align}
Here $c_2$ is the copied bare second-coordinate test, and
$\Ltwo[H,k]$ uses $a_1[k]+a_2[H]$ and $b_1[k]+b_2[H]$.
All entries are evaluated on actual smooth configurations.
This is not a quantum admission assertion.

Its dependence on the second metric coefficient is
\begin{equation}
 \Atwo^{\mathrm{sc}}[H,k+r;\phi]
 -\Atwo^{\mathrm{sc}}[H,k;\phi]
 =\Aone[r;\phi].
 \label{eq:klinear}
\end{equation}
Indeed, the $k$ contribution to $\Xtwo$ is $\Xone[k]$,
and the $k$ contribution to $\Ltwo$ is $\Lone[k]$.

Define the restricted candidate
\begin{equation}
 \Atwo^{\mathrm{cand}}[h,k,\gamma;\phi]
 =\Atwo^{\mathrm{sc}}[h+\gamma,k;\phi]-\Tcal(f,g)[\phi].
 \label{eq:candidate}
\end{equation}
It names the scalar-only completion together with the
single graviton component constructed here.
It is not declared to be the full interacting-gravity
coefficient.

\begin{theorem}[Held-fixed metric defect and its sign]
\label{thm:defect}
For smooth free $\phi$, hold $\gamma$ fixed and transform
the classical coefficients by
\[
 h'=h+K\xi,\qquad
 k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi,
 \qquad \xi,\zeta\in\ga.
\]
Then the ordinary forward variation is
\begin{equation}
 \Atwo^{\mathrm{cand}}[h',k',\gamma;\phi]
 -\Atwo^{\mathrm{cand}}[h,k,\gamma;\phi]
 =-\Aone[\Lie_\xi\gamma;\phi].
 \label{eq:forwarddefect}
\end{equation}
Equivalently, the backward defect has the production
sheet's positive sign:
\[
 \Atwo^{\mathrm{cand}}[h,k,\gamma;\phi]
 -\Atwo^{\mathrm{cand}}[h',k',\gamma;\phi]
 =\Aone[\Lie_\xi\gamma;\phi].
\]
\end{theorem}

\begin{proof}
The full transformation of $H=h+\gamma$ would require
\[
 H'=H+K\xi,\qquad
 k'_{\mathrm{full}}
 =k'+\Lie_\xi\gamma.
\]
The supplied X-C-v3 background identity applies to these
smooth coefficients and the free scalar
\cite{XC}. It identifies
$\Atwo^{\mathrm{sc}}[H',k'_{\mathrm{full}};\phi]$
with $\Atwo^{\mathrm{sc}}[H,k;\phi]$.
Subtracting the omitted second coefficient and using
\eqref{eq:klinear} gives \eqref{eq:forwarddefect}.
The tree is independent of $h,k,\gamma$ and has zero
variation under this particular background operation.
\end{proof}

Thus the sign depends on the explicitly named orientation
of the difference. Writing a positive forward variation
without changing that convention would be incorrect.
A hatted quantum coefficient with this identity has not
been constructed.

\subsection{A nonzero defect on free configurations}

\begin{proposition}[The defect is not generally an equation zero]
\label{prop:defectwitness}
There are admissible plateau data and smooth free multiplet
configurations for which
$\Aone[\Lie_\xi\gamma;\phi]\neq0$.
Therefore this classical defect does not belong to the
ideal of functionals vanishing on all free configurations.
\end{proposition}

\begin{proof}
Take $m=0$ and the free tensor and scalar
\[
 \gamma=t\eta,\qquad
 \phi=t^2+\tfrac13|\mathbf x|^2.
\]
They satisfy $-\Box\PP\gamma=0$ and $\Box\phi=0$.
Set $b=D\gamma$ and the ghosts to zero; these choices
satisfy the auxiliary free equations.

Choose a nonnegative nonzero $a\in C_c^\infty((\eps,T))$.
On a sufficiently large causal cylinder let
$\xi=a(t)\partial_t$, with a compact spatial cutoff outside
the observation dependence region. Choose $U$ and $g=1$
near its support, and observe after the support of $a$.
All calculations in that region are unaffected by the
cutoff, by finite propagation.

Put $r=\Lie_\xi\gamma$. Its components there are
\[
 r_{00}=a+2ta',\qquad r_{ij}=-a\delta_{ij},\qquad r_{0i}=0.
\]
Consequently
\[
 a_1^{00}[r]=a-ta',\qquad
 a_1^{ij}[r]=-(a+ta')\delta^{ij},\qquad
 D^0r=ta''.
\]
There is no layer contribution, and
\[
 \Xone^0[r]=ta(t)-2\int_0^t a(s)\dd s,\qquad
 \Xone^i[r]=0.
\]
Direct differentiation gives
\[
 \Lone[r]\phi=-4ta'-2t^2a'',
\]
and its scalar retarded response is
\[
 \rho_{1,g}[r;\phi]
 =2t^2a(t)-4\int_0^t s\,a(s)\dd s.
\]
Thus the completed first coefficient has density
\[
 \rho_{1,g}[r;\phi]
      -\Xone[r]\cdot\partial\phi
 =4\int_0^t(t-s)a(s)\dd s.
\]
A nonnegative nonzero $f$ in the described observation
region gives a strictly positive pairing.
An equation-ideal functional would vanish on these free
configurations, proving the assertion.
\end{proof}

Failure of $\Lie_\xi$ to preserve the wave equation is
also visible in this example, but that failure alone would
not prove the proposition. The evaluated observable is the
necessary witness.

The shift
$\gamma\mapsto\gamma+\kappa\Lie_\xi\gamma$
does not generally preserve the free gauge-fixed solution
space. It therefore cannot simply be declared an
automorphism of $\Afree$ to cancel the defect.
The ghost rule $\sfree\gamma=\frac12Kc$ is not that shift.

%=====================================================================
\section{The mixed loop and the comparison checklist}
\label{sec:loop}
%=====================================================================

The insertion is bilinear, while the vertex is cubic.
After one graviton cross-contraction, a scalar
cross-contraction is still available. Two contractions
consume four of the five legs and leave a scalar-linear
coefficient. With the retarded prefactor, this is a
potential $\hbar^1\kappa^2$ contribution.

A schematic term involves
\[
 f(x)g(y)\Lambda_\eps(x,z)
 G_{\mathrm{gr}}^{\mathrm R}(z,y)
 \,\partial_x\partial_y H_m(x,y),
\]
with the appropriate stress derivatives and mass terms.
Here $H_m$ denotes a \emph{not selected} scalar Hadamard
bisolution. A full retarded-product construction would
also have to specify its covariance matrix and ordering
conventions. XI-B supplied none.

We do not evaluate this expression. Its off-diagonal
wavefront compatibility and its extension near
observation-vertex coincidence require separate analysis.
No divergence, subtraction, or unique diagonal extension
is inferred from the drawing. Missing Hadamard data are
a stop for this loop, not a no-go for
Theorem~\ref{thm:tree}.

In particular, tree-exactness cannot be imported from
X-C-v3, whose insertions were scalar-linear.
The present one-line coefficient is designated a tree
because it is the classical retarded coefficient.
It is not a claim that the full response of
$V_{1,g}$ and $\Bcal_f$ has no positive-$\hbar$ terms.

There is also a distinct one-scalar-line classical response
of this pair, with metric fields remaining.
It is not the single graviton component defined by
\eqref{eq:physicalsign}. Equation \eqref{eq:A2sc} names
that scalar sector for the background comparison;
one graviton component alone cannot establish a complete
second-order Ward theorem.

\subsection{VII-C requirements remain unmet}

The comparison concerns the supplied VII-C, not an
identification of kernels or signatures \cite{VIIC}.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}X p{.27\linewidth}@{}}
\toprule
 & VII-C requirement & XI-C status\\
\midrule
E1 &
Its polynomial Wick algebra embeds into a Hausdorff locally
convex extension with its fixed Wick product unchanged. &
Unmet. No such extension or Hadamard product is selected.\\[0.4em]
E2 &
The exact sharp insertions are admitted on all smooth
off-shell configurations. &
Unmet. Only fixed-width classical symbols are constructed.\\[0.4em]
E3 &
Admission agrees with regular inclusion and is sequentially
continuous in its stated normal-topology kernel stages. &
Unmet. No quantum admission map or such stage is supplied.\\[0.4em]
E4 &
Multiplication is hypocontinuous on bounded sets and the
field-independent $\hbar^3$ coefficient extraction is
continuous. &
Unmet. No corresponding quantum extension is constructed.\\
\bottomrule
\end{tabularx}
\end{center}

The word $B_f\star W_g\star B_{f'}$ is not formed.
No reflected light-ray residue is subtracted.
X-D's sharp-limit theorem is not used.
Neither the present kernel bounds nor the absence of a
chosen state establish VII-C's four assumptions.

%=====================================================================
\section{Decision and the qualified XI-D handoff}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.55\linewidth}X@{}}
\toprule
Gate & Result\\
\midrule
Signed graviton inverse & $-\PP\ER$, coefficient fixed\\
Joint fixed-width microcausal bound & Proved\\
Classical one-graviton contraction & Constructed\\
Cubic classical output domain & $\Pmc^{(3)}$, explicitly named\\
Intermediate metric and auxiliary response & Actual sourced solutions\\
Double-retarded normalization above ramp & Proved\\
Nonzero XI-A witness & Proved; jet not repaired\\
Causal support & Proved as inclusions\\
Actual insertion in XI-B's off-shell finite-generator domain &
Not admitted; singular-kernel mismatch proved\\
Output in unchanged quantum algebra & Unresolved\\
Classical first-order background identity & Retained\\
Restricted classical second-order defect & Proved with sign convention\\
Quantum first/second-order composite identity & Not constructed here\\
Mixed loop & Named, unresolved\\
Cubic Einstein contraction & Omitted by scope, not proved zero\\
Interacting differential or BRST class & Not constructed\\
Sharp limit & Not taken\\
\bottomrule
\end{tabularx}
\end{center}

The literal frozen-algebra target ends at
\textbf{Unresolved}, not \textbf{Constructed}.
The reason is concrete: its singular polynomial inputs and
output need an admission theorem beyond XI-B's finite
linear-generator algebra. A wavefront bound establishes
eligibility for an appropriate extension; it is not that
extension.

The classical retarded problem, by contrast, is completely
constructed. It has one normalized graviton line, a
nonzero de Donder dressing for the named source, and an
explicit background order boundary.

XI-D receives:
\begin{enumerate}
\item the cubic classical tree symbol and the sign
$R_{\gamma,\mathrm{tree}}=-\Tcal$ for the actual bare
insertion;
\item the compact kernel bounds and causal support;
\item the double-retarded source-divergence normalization;
\item the restricted candidate and its forward defect
$-\Aone[\Lie_\xi\gamma]$, or the equivalent positive
backward defect;
\item the demonstrated need for quantum composite admission;
\item the statement that $\sfree$ is a free differential,
not an interacting differential.
\end{enumerate}

It does not receive an admitted quantum cubic coefficient,
a Hadamard bisolution, a closed insertion, a loop
extension, or a sharp limit. Supplying any of those would
be an additional construction rather than a reinterpretation
of the causal inverse.

\subsection*{Conclusion}

The proved implication is
\[
 \boxed{
 \begin{gathered}
 \text{fixed smooth ramp}
 +\text{signed gauge-fixed inverse}
 +\text{classical stress vertex}\\
 \Longrightarrow
 \text{one compact microcausal retarded tree symbol}.
 \end{gathered}}
\]
The stronger implication to the unchanged quantum algebra
has not been established. Meanwhile the empty Einstein
jet and the unresolved mixed loop remain separate facts.
No construction is certified by silently changing its
output algebra.

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

\paragraph{Proof scope.}
The polynomial kernel bounds, causal contraction,
normalization, support, and classical defect are proved
on smooth configurations. Quantum composite admission is
explicitly not included.

\paragraph{Dependencies.}
XI-A supplies the signed interaction and the Einstein-jet
obstruction. XI-B supplies the multiplet, normalized causal
inverse, residual lemma, and limited algebraic domain.
X-B-v3 and X-C-v3 supply the scalar-only comparison and
completed classical background identity. X-D is not used.

\paragraph{Corrections.}
The insertion sign, forward versus backward defect sign,
finite-generator versus microcausal domain, pure-gauge
diagnostic, and scalar reduction are distinguished.
The cubic Einstein vertex is not eliminated by a graph
count. A second contraction is not declared absent.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, executed symbolic calculation, independent
referee review, or journal acceptance is asserted.

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

\bibitem{XIA}
N.~Cheung,
\emph{Landmark XI-A: A switched Einstein--scalar action for
the ramp relational scalar: sign conventions, Noether
identities, and an empty fixed-input Einstein jet}.
Supplied manuscript \texttt{XI-A.tex}.
Used for the signed vertex and the Einstein-jet obstruction.

\bibitem{XIB}
N.~Cheung,
\emph{Landmark XI-B: The free de Donder multiplet for the
ramp Einstein--scalar sheet: causal propagators, residual
gauge, and the layer dressing}.
Supplied manuscript \texttt{XI-B.tex}.
Used for the causal inverse, auxiliary multiplet,
free differential, and stated finite-generator algebra.

\bibitem{XB}
N.~Cheung,
\emph{Landmark X-B: Fixed-regulator admission and one
retarded correction for the ramp relational scalar, with
the metric kept classical}.
Supplied manuscript \texttt{X-B-v3.tex}.
Used for the scalar-only first response and background
identity, with its representative warning.

\bibitem{XC}
N.~Cheung,
\emph{Landmark X-C: The missing second-order retarded maps
for the ramp relational scalar, with the metric kept
classical}.
Supplied manuscript \texttt{X-C-v3.tex}.
Used for the explicitly named scalar-only completed
coefficient and its classical background identity.

\bibitem{VIIC}
\emph{Landmark VII-C: A mixed-Wick obstruction to a
normal-topology boundary extension for sharp Cauchy
harmonic insertions}.
Supplied manuscript \texttt{VII-C.tex}.
Comparison only; no fixed Wick matrix or sharp insertion
is imported.

\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, 2007.
arXiv:0806.1036.
Used for smooth Cauchy theory, finite propagation, and
causal Green operators.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
``Continuity of the fundamental operations on distributions
having a specified wave front set
(with a counter example by Semyon Alesker),''
\emph{Studia Mathematica}
\textbf{232} (2016), 201--226.
arXiv:1409.7662.
Used for wavefront multiplication and pushforward criteria.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428v2, 2013.
Used for the distinction between regular polynomial
generators and Hadamard-based polynomial extensions;
no such extension is constructed here.

\bibitem{HRV}
E.~Hawkins, K.~Rejzner, and B.~Visser,
\emph{A novel class of functionals for perturbative
algebraic quantum field theory},
arXiv:2312.15203v2, 2024.
Background for keeping polynomial kernel statements
separate from broader functional and continuity claims.

\end{thebibliography}

\end{document}