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% XI-B.tex
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%   pdflatex -interaction=nonstopmode -halt-on-error XI-B.tex
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%
% Scope:
% One free gauge-fixed graded multiplet, its causal Green operators,
% a free nilpotent differential, a residual-gauge lemma, and the
% classical fixed-width layer formula.
%
% Corrections to the proposed production sheet:
% 1. Parentheses include a factor 1/2. Consequently s_0 gamma=Kc/2.
%    The ghost normalization is fixed explicitly: c=2 xi reproduces
%    XI-A's metric variation K xi.
% 2. The scalar equation operator is Box+m^2, whereas its action
%    Hessian is -(Box+m^2). Their inverses are distinguished.
% 3. The independent auxiliary field is retained for off-shell
%    nilpotence. Eliminating it gives only on-shell nilpotence.
% 4. A propagated layer source is not supported only in the layer.
% 5. Nonzero tensor layer data need not give a nonzero dressing:
%    the relevant projection is sigma' a_1^{0 mu}.
% 6. The divergence of the switched retarded metric response exists.
%    What fails is its required vanishing in the Einstein equation.
% 7. Product compatibility is checked on the graded commutation
%    relations; it is not inferred from a formal Leibniz rule alone.
%
% No Hadamard two-point function, state, time-ordered product,
% interacting differential, tree, mixed loop, sharp-ramp limit,
% boundary-extension topology, or relational BRST class is constructed.
%
% No successful compilation, mechanized verification, independent
% referee review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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% Correct the lemma heading without changing the shared counter.
\makeatletter
\expandafter\def\csname lemma\endcsname{%
  \@thm{\theorem@headerfont}{theorem}{Lemma}}
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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{\Gp}{\mathcal G_U}
\newcommand{\Lie}{\mathcal L}
\newcommand{\PP}{\mathcal P}
\newcommand{\RR}{\mathcal R}
\newcommand{\AA}{\mathcal A}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Coord}{\mathsf X}
\newcommand{\Lag}{\mathfrak L}
\newcommand{\Adens}{\mathfrak A}
\newcommand{\Ezero}{\mathsf E_0}
\newcommand{\Em}{\mathsf E_m}
\newcommand{\Egen}{\mathsf E_\mu}
\newcommand{\Ggr}{G^{\mathrm{gr}}}
\newcommand{\Gaux}{G^{\mathrm{aux}}}
\newcommand{\Ggh}{G^{\mathrm{gh}}}
\newcommand{\sfree}{s_0}
\newcommand{\FP}{\mathrm{FP}}
\newcommand{\gf}{\mathrm{gf}}
\newcommand{\gh}{\mathrm{gh}}
\newcommand{\aux}{\mathrm{aux}}
\newcommand{\inv}{\mathrm{inv}}
\newcommand{\Aoff}{\mathscr P_{\mathrm{off}}}
\newcommand{\Afree}{\mathscr A_0}
\newcommand{\Pfree}{P_m}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\ghnum}{gh}

\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-B}\\[0.4em]
The free de Donder multiplet for the ramp Einstein--scalar sheet:\\
causal propagators, residual gauge, and the layer dressing}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
Starting from the signed quadratic operator supplied by Landmark
XI-A, we construct a free Einstein--scalar gauge-fixed multiplet on
a finite Minkowski slab. One gauge fermion reproduces exactly the
frozen de Donder term, fixes the ghost kinetic normalization, and
introduces an independent Nakanishi--Lautrup field. The auxiliary
field is retained for off-shell nilpotence.

We derive retarded and advanced Green operators for the graviton,
scalar, ghost, and auxiliary blocks. The graviton inverse is minus
trace reversal times the zero-data wave inverse; its coefficient is
fixed by the handed-off action, with the Newtonian sign providing
an independent check. A state-independent graded polynomial field
algebra is constructed from the resulting causal kernels. The free
differential is shown to preserve its equations and commutation
relations.

The plateau algebra has trivial intersection with residual de Donder
gauge. For classical de Donder fields, the ramp coordinate coefficient
is the retarded response to a layer source, not a field supported
only inside the layer. We give an explicit nonzero example and retain
the fixed regulator width. The empty fixed-input Einstein jet of
XI-A is not repaired: its gauge-fixed retarded response exists,
but has nonzero divergence. No Hadamard state, interacting
differential, tree, mixed loop, or relational BRST class 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) {Free graded multiplet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Nilpotent free differential};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Einstein jet};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Constructing the free gate does not repair the Einstein jet.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The free gate does not repair the Einstein jet. 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-B}
\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) {Free graded multiplet};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Einstein jet};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Constructing the free gate does not repair the Einstein jet.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The free gate does not repair the Einstein jet. 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-B}
\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) {Free graded multiplet};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Nilpotent free differential};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Einstein jet};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Constructing the free gate does not repair the Einstein jet.};
\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-B}
\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) {Free graded multiplet};
\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) {Einstein jet};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Nilpotent free differential. Constructing the free gate does not repair the Einstein jet.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Free graded multiplet is not Einstein jet. 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-B}
\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: Free graded multiplet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Nilpotent free differential};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Einstein jet};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Constructing the free gate does not repair the Einstein jet.};
\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-B}
\end{figure}

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


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



\scopebox{Release: the free gate is constructed; the Einstein jet is
not repaired}{%
The free graded multiplet, one gauge fermion, normalized causal
Green operators, a nilpotent free differential, the trivial
plateau residual algebra, and the classical layer formula are
constructed.

The operator inverted here is the \emph{gauge-fixed} operator.
The empty fixed-input Einstein jet remains empty.

No interaction is contracted with the dressing. No claim is made
that the plateau algebra acts by nontrivial residual transformations
on the free gauge-fixed graviton, or that the ramp scalar defines
an interacting BRST class.}

\tableofcontents

%=====================================================================
\section{Scope and the named construction}
\label{sec:scope}
%=====================================================================

This manuscript begins where XI-A stopped: with a signed quadratic
operator rather than a graviton two-point function. The construction
is performed at the free background $(\eta,0)$. A prescribed nonzero
free scalar used in a diagnostic is an input to a coupling expansion,
not a replacement background for the free Hessian.

There are three different objects:
\[
 \text{classical background pullbacks},\qquad
 \text{residual gauge transformations},\qquad
 \text{the free graded differential}.
\]
They must not be identified merely because they involve derivatives
of vector fields.

The polynomial algebra constructed below uses compactly smeared
linear fields and their finite products. It has causal commutation
relations, but no chosen Hadamard covariance or state. In particular,
it is not the Wick algebra used in VII-C. This restricted algebraic
construction suffices for the free multiplet and does not presume
admission of a nonlocal relational composite.

The standard distinction between free gauge fixing and interacting
BV quantization is respected \cite{BFR}. We construct the former,
including its free differential, without supplying an interacting
master equation.

\begin{theorem}[Free-multiplet release]
\label{thm:main}
For the XI-A quadratic normalization and the conventions of
Section~\ref{sec:handoff}, the following are established.
\begin{enumerate}[label=\textup{(\roman*)}]
\item One gauge fermion yields the frozen term
$S_{\gf}=\int D_\nu D^\nu$ after auxiliary-field elimination.
\item The full free operator has unique retarded and advanced
Green operators on compact interior sources, with the stated
causal support.
\item The free differential is nilpotent off shell with the
auxiliary field retained, and descends to the causal polynomial
field algebra.
\item The plateau algebra has zero intersection with residual
de Donder gauge.
\item The regulated first coordinate coefficient of a classical
de Donder field is a propagated layer functional.
\end{enumerate}
These statements do not construct an interacting or relational
BRST cohomology class.
\end{theorem}

The proofs are given in the successive sections. No tree calculation
is part of the theorem.

%=====================================================================
\section{The XI-A handoff and convention sheet}
\label{sec:handoff}
%=====================================================================

\subsection{Geometry and the plateau}

Retain
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad
 \Sigma_0=\{t=0\},
\]
\[
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta,\qquad
 \Pfree=\Box+m^2,\qquad m\geq0.
\]
Fields are smooth on a neighbourhood of the closed slab.
Indices are raised and lowered with $\eta$; parentheses contain
the factor $1/2$.

The switch satisfies
\[
 g\in C_c^\infty(\MTo;[0,1]),\qquad
 P_g=\Int\{g=1\}\neq\varnothing.
\]
The fixed ramp is
\[
 \sig(t)=\chi(t/\eps),\qquad 0<\eps<T,
\]
where
\[
 \chi(s)=
 \begin{cases}
 0,&s\leq0,\\[2pt]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
 1,&s\geq1.
 \end{cases}
\]
Choose $U$ open and relatively compact with
\[
 \overline U\subset P_g\cap\{\eps<t<T\},\qquad
 \ga=C_c^\infty(U;TM).
\]
Neither $g$ nor $\sig$ occurs in the free kinetic operator.

\subsection{Operators copied from XI-A}

For a symmetric tensor $\gamma$ define
\[
 \bar\gamma_{\mu\nu}
 =\gamma_{\mu\nu}
       -\tfrac12\eta_{\mu\nu}\tr_\eta\gamma,\qquad
 D^\nu\gamma=\partial_\mu\bar\gamma^{\mu\nu},
\]
\[
 (K\xi)_{\mu\nu}
       =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
\]
Let $\PP$ denote trace reversal, so $\bar\gamma=\PP\gamma$.
In four dimensions,
\begin{equation}
 \PP^2=\id,\qquad
 D(K\xi)=\Box\xi,\qquad
 G_{\mathrm{lin}}(K\xi)=0.
 \label{eq:basic}
\end{equation}

XI-A gives
\begin{equation}
 2G_{\mathrm{lin}}(\gamma)+E_{\gf}(\gamma)
                       =-\Box\PP\gamma,
 \qquad
 S_{\gf}=\int_{\MT}D_\nu\gamma D^\nu\gamma\dd^4x.
 \label{eq:handoff}
\end{equation}
This is not the normalization $-\frac12\Box\PP$.

The inherited interaction 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}
with
\[
 T_0^{\mu\nu}[\phi]
 =\partial^\mu\phi\partial^\nu\phi
  -\tfrac12\eta^{\mu\nu}
      \left(\partial_\rho\phi\partial^\rho\phi-m^2\phi^2\right).
\]
It is not inserted into the free action or into the free differential.

The invariant XI-A action has a fixed-background Noether identity;
its gauge-fixed action has an additional breaking term. The
switching contribution includes $(\partial_\mu g)T_0^{\mu\nu}$.
None of these facts is replaced by free gauge-fixed solvability
\cite{XIA}.

\subsection{Inputs not used}

VII-C's Hadamard graviton matrix is not an input. X-D is not used.
The text of X-B-v3 is not supplied here; no externally numbered
result from it is presumed. The scalar sector is constructed
directly below, with its off-shell and free-shell presentations
kept separate.

The independent fields and their gradings are:
\[
\begin{array}{c|ccccc}
 \text{field}&\gamma_{\mu\nu}&\phi&c^\mu&\bar c_\mu&b_\mu\\
 \hline
 \text{parity}&0&0&1&1&0\\
 \text{ghost number}&0&0&1&-1&0
\end{array}
\]
The algebra is complexified to accommodate the factor $i$ in
the antighost transformation. No positivity or ghost reality
condition is needed for this release.

%=====================================================================
\section{One gauge fermion and the auxiliary multiplet}
\label{sec:fermion}
%=====================================================================

\subsection{The normalization of the ghost}

Define
\begin{equation}
 \RR c=\tfrac12Kc,\qquad
 (\RR c)_{\mu\nu}=\partial_{(\mu}c_{\nu)}.
 \label{eq:R}
\end{equation}
Thus the requested parenthesis convention is retained.
XI-A's variation $K\xi$ is recovered by setting $c=2\xi$.
This fixes a ghost normalization; it introduces neither a new
metric gauge condition nor a gauge-parameter family.

The free odd derivation is specified on independent fields by
\begin{equation}
 \sfree\gamma=\RR c,\qquad
 \sfree\phi=0,\qquad
 \sfree c=0,\qquad
 \sfree\bar c_\mu=ib_\mu,\qquad
 \sfree b_\mu=0.
 \label{eq:s}
\end{equation}
It commutes with spacetime derivatives and raises ghost number
by one. Its action on products is the graded Leibniz rule.

\begin{lemma}[Off-shell nilpotence]
\label{lem:nilpotence}
The derivation \eqref{eq:s} satisfies $\sfree^2=0$ on the
independent fields and on their polynomial differential algebra.
\end{lemma}

\begin{proof}
The second variation of $\gamma$ is $\RR(\sfree c)=0$.
The second variation of $\bar c$ is $i\sfree b=0$.
The remaining fields already have zero first variation.
The square of an odd derivation is an even derivation; hence
vanishing on the generators proves vanishing on their polynomial
algebra.
\end{proof}

\subsection{Gauge fermion and free action}

The quadratic Fierz--Pauli action is copied in its signed form:
\begin{align}
 S_{\FP}[\gamma]=\frac12\int_{\MT}\Bigl(
 &\partial_\lambda\gamma_{\mu\nu}
       \partial^\lambda\gamma^{\mu\nu}
 -2(\partial_\mu\gamma^{\mu\nu})
       (\partial^\rho\gamma_{\rho\nu})
 \nonumber\\
 &+2(\partial_\mu\gamma^{\mu\nu})
       \partial_\nu\tr_\eta\gamma
 -\partial_\lambda\tr_\eta\gamma\,
       \partial^\lambda\tr_\eta\gamma
 \Bigr)\dd^4x.
 \label{eq:FP}
\end{align}
The free scalar action is
\[
 S_\phi=\tfrac12\int_{\MT}
       \left(\partial_\mu\phi\partial^\mu\phi-m^2\phi^2\right)\dd^4x.
\]

Choose exactly one fermion:
\begin{equation}
 \Psi=-i\int_{\MT}
           \bar c_\mu\left(2D^\mu\gamma-b^\mu\right)\dd^4x.
 \label{eq:Psi}
\end{equation}

\begin{proposition}[Recovery of the frozen gauge term]
\label{prop:fermion}
The free auxiliary action
\begin{equation}
 S_0=S_{\FP}+S_\phi+\sfree\Psi
 \label{eq:S0}
\end{equation}
has
\begin{equation}
 \sfree\Psi
 =\int_{\MT}
       \left(2b_\mu D^\mu\gamma-b_\mu b^\mu
                       +i\bar c_\mu\Box c^\mu\right)\dd^4x.
 \label{eq:sPsi}
\end{equation}
Its auxiliary equation is $b_\mu=D_\mu\gamma$.
Eliminating $b$ gives precisely $S_{\gf}$ and the ghost term
$i\int\bar c_\mu\Box c^\mu$.
\end{proposition}

\begin{proof}
Equation \eqref{eq:basic} gives
$\sfree D^\mu\gamma=\frac12\Box c^\mu$.
The minus sign from applying an odd derivation to the second
factor of $\bar c(2D-b)$ yields \eqref{eq:sPsi}.
The auxiliary variation is $2(D^\mu\gamma-b^\mu)$.
Finally,
\[
 2b_\mu D^\mu\gamma-b_\mu b^\mu
 =D_\mu\gamma D^\mu\gamma
       -(b-D\gamma)_\mu(b-D\gamma)^\mu.
\]
\end{proof}

All action identities are bulk variational identities, modulo
divergences and tested with compact interior variations. No
BRST boundary charge or boundary variational principle is asserted.

The Fierz--Pauli part is invariant under its linear gauge map,
and the scalar is unchanged by $\sfree$. Consequently
$\sfree S_0=0$ as a free bulk variational functional, since
$\sfree^2\Psi=0$. This does not contradict the XI-A breaking:
the bosonic gauge-fixing term alone is not gauge invariant.
The independent auxiliary and ghost terms produce the free
BRST-invariant completion.

\subsection{Why the auxiliary field cannot be discarded off shell}

After elimination,
$\sfree\bar c=iD\gamma$, and therefore
\[
 \sfree^2\bar c=\frac i2\Box c.
\]
This vanishes on the free ghost shell, not on unrestricted ghost
configurations. We retain $b$ in the off-shell multiplet.

For bookkeeping, name the antifields
\[
 \gamma^\ddagger,\quad \phi^\ddagger,\quad c^\ddagger,\quad
 \bar c^\ddagger,\quad b^\ddagger.
\]
An antifield has opposite parity and ghost number
$-1-\ghnum(\text{field})$. These names denote external source
slots only. They are not additional propagating generators here;
no antifield differential, BV antibracket, or interacting master
action is claimed.

%=====================================================================
\section{Causal Green operators and their normalization}
\label{sec:greens}
%=====================================================================

\subsection{Wave inverses on the slab}

For $\mu\geq0$, let
$\Omega_\mu=\sqrt{-\Delta+\mu^2}$. For a smooth source $F$ define
\begin{align}
 (\Egen^{\mathrm R}F)(t)
 &=\int_0^t
       \frac{\sin((t-s)\Omega_\mu)}{\Omega_\mu}F(s)\dd s,
 \label{eq:ER}\\
 (\Egen^{\mathrm A}F)(t)
 &=\int_t^T
       \frac{\sin((s-t)\Omega_\mu)}{\Omega_\mu}F(s)\dd s.
 \label{eq:EA}
\end{align}
The quotient has its continuous zero-frequency value. For
nondecaying smooth sources these formulas are interpreted locally
by finite propagation. On compact interior sources they are the
restrictions of the full Minkowski retarded and advanced operators.

\begin{lemma}[Slab Green identities]
\label{lem:Green}
The operators \eqref{eq:ER}--\eqref{eq:EA} invert
$\Box+\mu^2$ with, respectively, zero initial and zero final
Cauchy data. For $F\in C_c^\infty(\MTo)$,
\[
 (\Box+\mu^2)\Egen^\pm F=F,\qquad
 \Egen^\pm(\Box+\mu^2)F=F,
\]
and
\[
 \supp\Egen^{\mathrm R}F\subset J^+(\supp F)\cap\MT,\qquad
 \supp\Egen^{\mathrm A}F\subset J^-(\supp F)\cap\MT.
\]
These support bounds are compact in the \emph{closed} slab.
Moreover $(\Egen^{\mathrm R})^*=\Egen^{\mathrm A}$ on interior
test functions.
\end{lemma}

\begin{proof}
Differentiating the sine formulas twice verifies the equation
and the prescribed endpoint data. Uniqueness follows from
the homogeneous Cauchy problem. Applying this uniqueness to
$\Egen^\pm(\Box+\mu^2)F-F$ gives the left inverse on compact
interior tests. Finite propagation gives the support bounds.
The intersection of a causal future or past of a compact set
with a bounded-time closed slab is compact.
Green's identity proves transposition: the retarded solution
has zero data at $0$, the advanced solution has zero data at
$T$, and there is no spatial boundary contribution.
These are the standard smooth Cauchy properties
\cite[Chapter~3]{BGP}.
\end{proof}

The endpoint prescriptions select the inverses. They do not
require all free solutions or ghosts to have zero Cauchy data.

\subsection{Graviton and scalar blocks}

Under the flat Lorentz pairings, the graviton operator is
\[
 L_\gamma=-\Box\PP.
\]

\begin{theorem}[Normalized graviton inverse]
\label{thm:graviton}
Its retarded and advanced inverses are
\begin{equation}
 \boxed{\qquad
 \Ggr^{\mathrm R}=-\PP\Ezero^{\mathrm R},\qquad
 \Ggr^{\mathrm A}=-\PP\Ezero^{\mathrm A}.
 \qquad}
 \label{eq:Ggr}
\end{equation}
Thus the coefficient in
$c\,\PP\Ezero^{\mathrm R}$ is $c=-1$.
\end{theorem}

\begin{proof}
Trace reversal commutes with the wave operator and satisfies
$\PP^2=\id$. Hence
\[
 (-\Box\PP)(-\PP\Ezero^\pm)=\id
\]
and the left inverse follows on compact tests. Any coefficient
$c$ instead gives $-c\,\id$, forcing $c=-1$.
The support statement follows from Lemma~\ref{lem:Green}.
\end{proof}

This fixes the normalization from XI-A's Hessian. The same sign
reproduces its Newtonian comparison: for
$H=q-\eta$ and a static positive source,
\[
 H_{00}=2\Phi_N,\qquad
 \Phi_N(x)=-\frac{\kappa^2}{32\pi}
                  \int\frac{\rho(y)}{|x-y|}\dd^3y<0.
\]
A coefficient $-2$ would invert a different Hessian. The Newtonian
check is not permission to change the quadratic normalization.

The scalar \emph{equation} operator is $\Pfree$, with inverse
$\Em^\pm$. Its \emph{action Hessian} is $-\Pfree$, with inverse
$-\Em^\pm$. This distinction determines the scalar causal bracket
below.

\subsection{The independent auxiliary block}

Let $D^*$ be the formal adjoint of $D$. With flat index
identifications,
\begin{equation}
 (D^*v)_{\mu\nu}
 =-\partial_{(\mu}v_{\nu)}
        +\tfrac12\eta_{\mu\nu}\partial_\rho v^\rho.
 \label{eq:Dstar}
\end{equation}
Write $A=2G_{\mathrm{lin}}$. The even metric--auxiliary operator is
\begin{equation}
 \mathcal L_{\gamma b}
 =
 \begin{pmatrix}
 A&2D^*\\
 2D&-2\id
 \end{pmatrix}.
 \label{eq:Laux}
\end{equation}
Its source equation is
\[
 A\gamma+2D^*b=J,\qquad
 2D\gamma-2b=j.
\]

\begin{proposition}[Full auxiliary inverse]
\label{prop:aux}
The unique retarded and advanced inverses are
\begin{equation}
 \Gaux^\pm=
 \begin{pmatrix}
 \Ggr^\pm&\Ggr^\pm D^*\\
 D\Ggr^\pm&D\Ggr^\pm D^*-\frac12\id
 \end{pmatrix}
 =
 \begin{pmatrix}
 -\PP\Ezero^\pm&\RR\Ezero^\pm\\
 D(-\PP\Ezero^\pm)&0
 \end{pmatrix}.
 \label{eq:Gaux}
\end{equation}
The identities in the second matrix hold on compact interior
sources.
\end{proposition}

\begin{proof}
The auxiliary equation gives $b=D\gamma-j/2$. Substitution
reduces the metric equation to
\[
 (A+2D^*D)\gamma=J+D^*j.
\]
The handed-off identity is $A+2D^*D=-\Box\PP$.
Equation \eqref{eq:Ggr} therefore gives the first matrix and
uniqueness.

For the simplification, direct calculation gives
\[
 \PP D^*=-\RR,\qquad
 D\RR=\tfrac12\Box.
\]
Consequently
$\Ggr^\pm D^*=\RR\Ezero^\pm$ and
$D\Ggr^\pm D^*=\frac12\id$.
The last block cancels the algebraic contact term.
The source elimination proves both Green identities; derivatives
do not enlarge causal support.
\end{proof}

The vanishing $bb$ block does not make $b$ a discarded field:
the $\gamma b$ and $b\gamma$ blocks are nonzero. Nor is an
independent wave inverse for $b$ postulated.

\subsection{Ghost normalization and inverse}

Use left Grassmann variations. The ghost kinetic term yields
\[
 \frac{\delta^L S_0}{\delta c}=-i\Box\bar c,\qquad
 \frac{\delta^L S_0}{\delta\bar c}=i\Box c.
\]
In the order $(c,\bar c)$, with dual vector indices understood,
\begin{equation}
 \mathcal L_{\gh}
 =
 \begin{pmatrix}0&-i\Box\\ i\Box&0\end{pmatrix},
 \qquad
 \Ggh^\pm=
 \begin{pmatrix}0&-i\Ezero^\pm\\ i\Ezero^\pm&0\end{pmatrix}.
 \label{eq:Gghost}
\end{equation}
Matrix multiplication proves the two Green identities.
The scalar, auxiliary, and ghost blocks together give the full
free multiplet Green operator. No Feynman inverse is selected.

%=====================================================================
\section{The polynomial field algebra and the free differential}
\label{sec:algebra}
%=====================================================================

\subsection{Off-shell and causal presentations}

Let $\Aoff$ be the complex graded commutative polynomial algebra
generated by compact interior smearings of the five independent
fields. Products are finite. They have smooth finite-rank
multivariable kernels, so no coincident-field composite is
needed to define them.

This is an off-shell algebra: no field equation is used in
Lemma~\ref{lem:nilpotence}. A second, free-shell algebra is obtained
by imposing the linear equations and causal commutation relations.
Making that second presentation does not retroactively impose
free equations on classical relational functionals.

Put
\[
 \delta_\mu=\mathsf E_\mu^{\mathrm R}
                       -\mathsf E_\mu^{\mathrm A}.
\]
The full causal matrix is the retarded-minus-advanced inverse
of the action Hessian. Its nonzero blocks are
\begin{align}
 \Delta_{\gamma\gamma}&=-\PP\delta_0,
 &
 \Delta_{\phi\phi}&=-\delta_m,
 \label{eq:causal-even}\\
 \Delta_{\gamma b}&=\RR\delta_0,
 &
 \Delta_{b\gamma}&=D(-\PP\delta_0),
 \qquad \Delta_{bb}=0,
 \label{eq:causal-aux}\\
 \Delta_{c\bar c}&=-i\delta_0,
 &
 \Delta_{\bar c c}&=i\delta_0.
 \label{eq:causal-ghost}
\end{align}
The appropriate vector Kronecker symbols accompany the ghost
blocks. These are causal commutator kernels, not Hadamard
two-point functions.

\begin{definition}[Free causal polynomial algebra]
\label{def:algebra}
The algebra $\Afree$ is generated over $\C[[\hbar]]$ by the
compact interior smeared fields, subject to:
\begin{enumerate}[label=\textup{(\alph*)}]
\item the auxiliary free equations
\[
 A\gamma+2D^*b=0,\qquad b=D\gamma,\qquad
 \Pfree\phi=0,\qquad \Box c=\Box\bar c=0;
\]
\item the graded commutation relations
\begin{equation}
 [\Phi_A(f),\Phi_B(k)]_{\mathrm{gr}}
       =i\hbar\,\pair{f}{\Delta_{AB}k}\,\mathbf1.
 \label{eq:CCR}
\end{equation}
\end{enumerate}
Only finite polynomial degree is used at each formal
$\hbar$ coefficient.
\end{definition}

\begin{proposition}[Existence of the algebra]
\label{prop:algebra}
Definition~\ref{def:algebra} gives a nonzero associative graded
polynomial algebra. All pairings in \eqref{eq:CCR} exist.
\end{proposition}

\begin{proof}
Smooth compact tests can be paired with the displayed causal
distributions. Transposition of the Green operators implies
graded antisymmetry:
the even blocks are antisymmetric, while the odd--odd block is
symmetric under simultaneous exchange of its slots.

The equations annihilate the causal matrix on both sides, so
the pairing descends to the test space modulo equation tests.
It is a constant even graded antisymmetric form on that quotient.
The associated central graded Lie algebra satisfies the Jacobi
identity because every bracket is central. Its enveloping
algebra, with the central generator set to the unit and scaled
by $i\hbar$, supplies the required algebra.
Equivalently, symmetrization gives the constant graded Moyal
product on polynomials. This establishes associativity and the
polynomial vector-space presentation without choosing a state
\cite{FR}.
\end{proof}

No extension to all local or microcausal polynomial functionals
is asserted. In particular, \eqref{eq:vertex} is not being
multiplied in $\Afree$.

\subsection{Preservation of the free equations}

\begin{lemma}[Stability of the equation ideal]
\label{lem:equations}
The derivation $\sfree$ preserves the free equation ideal.
\end{lemma}

\begin{proof}
Since $A\RR=0$ and $\sfree b=0$,
\[
 \sfree(A\gamma+2D^*b)=0.
\]
Also,
\[
 \sfree(2D\gamma-2b)=\Box c.
\]
The ghost equation for $c$ is unchanged, while
\[
 \sfree(\Box\bar c)=i\Box b.
\]
Taking the divergence of the metric--auxiliary equation gives
$\Box b=0$: the divergence of $A\gamma$ is zero and the
divergence of $2D^*b$ is $-\Box b$.
The scalar equation is unchanged. Thus every transformed
equation belongs to the same linear equation ideal.
\end{proof}

The eliminated relations then give
$\sfree^2\bar c=\frac i2\Box c=0$ in the free-shell algebra,
while independent $b$ already provided off-shell nilpotence.

\subsection{Preservation of the causal product}

\begin{theorem}[Free differential on the causal algebra]
\label{thm:differential}
The field rule \eqref{eq:s} induces a ghost-number-one
nilpotent graded derivation of $\Afree$.
\end{theorem}

\begin{proof}
Equation stability is Lemma~\ref{lem:equations}.
It remains to check the commutation relations, not merely to
postulate a Leibniz rule.

The only nontrivial mixed check is the initially zero relation
between $\gamma$ and $\bar c$. The ghost block gives
\[
 \{c^\mu(x),\bar c_\nu(y)\}
      =\hbar\,\delta^\mu_\nu\delta_0(x,y).
\]
The auxiliary block gives
\[
 [\gamma(x),b(y)]
      =i\hbar\,\RR_x\delta_0(x,y),
\]
with its tensor indices understood. Therefore
\[
 \sfree[\gamma(x),\bar c(y)]
 =\{\RR c(x),\bar c(y)\}
           +[\gamma(x),ib(y)]
 =\hbar\RR_x\delta_0-\hbar\RR_x\delta_0=0.
\]
These are distributional identities tested with compact
smearings; no pointwise field multiplication is involved.

All other transformed relations contain only zero cross-blocks,
or $cc$ or $bb$ brackets, which vanish. The nonzero
$\gamma\gamma$, $\phi\phi$, and $c\bar c$ brackets have constant
right-hand sides and zero transformed left-hand sides.
Hence $\sfree$ preserves the relation ideal.
Nilpotence descends from Lemma~\ref{lem:nilpotence}.
\end{proof}

Thus the free algebra is a differential graded algebra. No
cohomology classification or interacting quantum differential
$\widehat s_{\mathrm{int}}$ follows from this statement.
The construction contains no stress-tensor vertex.

%=====================================================================
\section{Residual gauge and the switched-source diagnostic}
\label{sec:residual}
%=====================================================================

\subsection{The residual plateau algebra is trivial}

A classical de Donder field has $D\gamma=0$.
Under $\gamma\mapsto\gamma+K\xi$,
\[
 D(\gamma+K\xi)=D\gamma+\Box\xi.
\]
Its residual de Donder parameters therefore satisfy $\Box\xi=0$.

\begin{theorem}[Trivial plateau residual algebra]
\label{thm:residual}
\begin{equation}
 \boxed{\qquad
 \ga\cap\{\xi:\Box\xi=0\}=\{0\}.
 \qquad}
 \label{eq:residual}
\end{equation}
\end{theorem}

\begin{proof}
A vector field in $\ga$ vanishes on a neighbourhood of
$\Sigma_0$. Every component has zero initial Cauchy data.
If it also solves the homogeneous wave equation, Cauchy
uniqueness forces it to vanish everywhere on the slab.
\end{proof}

This does not say that all residual parameters on Minkowski
space vanish. Smooth noncompact wave solutions exist.
It says that the frozen plateau support class contains none
except zero.

Consequently $\ga$ is not a nontrivial residual action on
classical de Donder free solutions. More generally, a compact
shift $K\xi$ preserving the gauge-fixed free equation would
require $K\Box\xi=0$; the compactly supported vector
$\Box\xi$ would then be a Killing vector and hence zero,
again giving $\xi=0$.

The admitted plateau algebra remains an algebra acting on the
classical background prescription. It is not thereby promoted
to an interacting action on the free graviton algebra.
The graded transformation involving the ghost is a different
construction. In particular, ghosts are not required to be
compactly supported plateau wave solutions.

\subsection{Retarded response and its nonzero divergence}

For XI-A's source-response coefficient put
\[
 J^{\mu\nu}=\tfrac12gT_0^{\mu\nu}[\phi_0].
\]
The gauge-fixed retarded solution is
\begin{equation}
 \gamma_1^{\mathrm R}
     =-\tfrac12\PP\Ezero^{\mathrm R}
                          (gT_0[\phi_0]).
 \label{eq:response}
\end{equation}
Taking its divergence on the free scalar shell gives
\begin{equation}
 D^\nu\gamma_1^{\mathrm R}
 =-\tfrac12\Ezero^{\mathrm R}
               \bigl((\partial_\mu g)T_0^{\mu\nu}[\phi_0]\bigr),
 \qquad
 \Box D^\nu\gamma_1^{\mathrm R}
 =-\tfrac12(\partial_\mu g)T_0^{\mu\nu}[\phi_0].
 \label{eq:divergence}
\end{equation}

For the named input
\[
 \phi_m(t)=
 \begin{cases}\sin(mt)/m,&m>0,\\ t,&m=0,\end{cases}
\]
one has $T_0^{00}=1/2$ and $T_0^{0i}=0$. Thus
\begin{equation}
 D^0\gamma_1^{\mathrm R}
     =-\tfrac14\Ezero^{\mathrm R}(\partial_tg),
 \qquad
 \Box D^0\gamma_1^{\mathrm R}=-\tfrac14\partial_tg.
 \label{eq:witness}
\end{equation}
A nonzero compact interior switch has
$\partial_tg\not\equiv0$, so this divergence is nonzero.

Both \eqref{eq:response} and its divergence exist as smooth
retarded responses. The failed requirement is
$D\gamma_1=0$, not existence of the divergence.
The ungauge-fixed Einstein equation would require conservation
of $gT_0$ by its Bianchi identity. XI-A's empty
$\Jcal_g(\phi_m)$ is therefore unchanged. This diagnostic is
not a failure of the Green construction and uses no contraction
with the interaction.

%=====================================================================
\section{The ramp layer dressing}
\label{sec:layer}
%=====================================================================

\subsection{General and conditional formulas}

The density coefficient inherited from X-A is
\[
 a_1^{\mu\nu}[\gamma]
       =\tfrac12\eta^{\mu\nu}\tr_\eta\gamma-\gamma^{\mu\nu}.
\]
Its divergence gives exactly the de Donder combination:
\[
 C^\mu[\gamma]
       =-\partial_\alpha a_1^{\alpha\mu}[\gamma]
       =D^\mu\gamma.
\]
The regulated first coordinate coefficient, with zero initial
data, is therefore
\begin{equation}
 \Coord_1^\mu[\gamma]
 =\Ezero^{\mathrm R}
       \left(\sig D^\mu\gamma-\sig'a_1^{0\mu}[\gamma]\right).
 \label{eq:Xgeneral}
\end{equation}
This is the divergence-form ramp prescription, not a
multiplied-equation replacement.

\begin{theorem}[Classical layer formula]
\label{thm:layer}
For every smooth classical tensor satisfying $D\gamma=0$,
\begin{equation}
 \boxed{\qquad
 \Coord_1^\mu[\gamma]
       =-\Ezero^{\mathrm R}
                   \left(\sig'a_1^{0\mu}[\gamma]\right).
 \qquad}
 \label{eq:layer}
\end{equation}
Its source vanishes outside $0<t<\eps$.
If that source has spatially compact support, then
\[
 \supp\Coord_1
 \subset J^+\!\left(
       \supp(\sig'a_1^{0\cdot}[\gamma])\right)\cap\MT.
\]
The propagated coefficient need not vanish for $t>\eps$.
It is nonzero precisely when the source vector in
\eqref{eq:layer} is nonzero.
\end{theorem}

\begin{proof}
Set $D\gamma=0$ in \eqref{eq:Xgeneral}.
The derivative $\sig'$ is smooth, vanishes off the ramp,
and is flat at both endpoints. Finite propagation proves the
support inclusion. Since
$\Box\Coord_1=-\sig'a_1^{0\cdot}$, zero coefficient implies
zero source. The converse follows from zero-data uniqueness.
\end{proof}

Strictly, the closed support of $\sig'$ can meet both $0$ and
$\eps$, even though the function vanishes at those endpoints.
The support statement is about a smooth layer source, not a
hypersurface distribution.

A nonzero tensor in the layer can have
$a_1^{0\mu}=0$ and hence zero dressing. For example, a constant
purely spatial trace-free tensor has this property.
Thus ``nonzero layer data'' must refer to the projected source
in \eqref{eq:layer}, not to arbitrary nonzero metric components.

\subsection{An explicit field with persistent layer response}

\begin{proposition}[Nonzero de Donder diagnostic]
\label{prop:layerexample}
Let $\lambda\neq0$ and $\gamma_{\mu\nu}=\lambda\eta_{\mu\nu}$.
It solves the homogeneous gauge-fixed free equation and
$D\gamma=0$. Its coordinate coefficient is
\[
 \Coord_1^i=0,\qquad
 \Coord_1^0(t)
       =-\lambda\int_0^t\sig(s)\dd s.
\]
In particular, for $t\geq\eps$,
\begin{equation}
 \Coord_1^0(t)=-\lambda(t-\eps/2)\neq0.
 \label{eq:layerexample}
\end{equation}
\end{proposition}

\begin{proof}
The constant tensor has zero derivatives, and
$a_1^{\mu\nu}=\lambda\eta^{\mu\nu}$.
For its spatially constant source, the zero-frequency wave
formula gives
\[
 \Coord_1^0(t)
 =-\lambda\int_0^t(t-s)\sig'(s)\dd s
 =-\lambda\int_0^t\sig(s)\dd s.
\]
The ramp satisfies $\chi(1-s)=1-\chi(s)$, so
$\int_0^\eps\sig(s)\dd s=\eps/2$.
This yields \eqref{eq:layerexample}.
\end{proof}

No spatial decay was imposed on classical configurations.
This example is legitimate under the local smooth Cauchy
prescription and shows explicitly why the output cannot be
labelled layer-supported.

\subsection{Smearing, admission, and the limit not taken}

For a compact interior covector test $a_\mu$,
transposition gives the de Donder-layer identity
\begin{equation}
 \int a_\mu\Coord_1^\mu[\gamma]\dd^4x
 =-\int_{\MT}\sig'(z)
       (\Ezero^{\mathrm A}a)_\mu(z)
          a_1^{0\mu}[\gamma](z)\dd^4z.
 \label{eq:layerkernel}
\end{equation}
The resulting linear kernel is smooth at fixed $\eps$.
Its support is contained in the closed layer intersected with
the causal past of $\supp a$.

Equation \eqref{eq:layerkernel} is a classical restriction to
de Donder tensors. The full auxiliary multiplet is not restricted
by $D\gamma=0$: its equation is $b=D\gamma$. Imposing $D=0$
as an additional algebraic quotient would require a separate
analysis and is not done here.

A boundary-supported trace is suggested by formally shrinking
the layer. No convergence theorem, sharp limit, or identification
with a sharp Cauchy kernel is asserted. In particular, a smooth
closed-slab kernel need not be a compactly supported interior
test. This classical kernel fact is not an admission theorem
for a quantum relational composite.

Neither $\Coord_1$ nor its scalar dressing is declared
$\sfree$-closed. The plateau covariance of X-A does not prove
such a claim for unrestricted ghost Cauchy data.

%=====================================================================
\section{The VII-C non-import checklist}
\label{sec:checklist}
%=====================================================================

VII-C concerns a sharp zero-Cauchy insertion, a fixed Hadamard
Wick matrix in the opposite signature, and a specified
normal-topology extension. Those are not the data of the
causal polynomial algebra constructed here \cite{VIIC}.

For precision, its four extension requirements are recorded
below. ``Unmet'' means not established or not instantiated by
this manuscript, not disproved for every possible successor
construction.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}X p{.27\linewidth}@{}}
\toprule
 & VII-C requirement & Status in XI-B\\
\midrule
E1 &
The old microcausal Wick algebra embeds into a Hausdorff locally
convex $\C[[\hbar]]$-algebra, with its fixed Hadamard Wick product
unchanged. &
Unmet. We construct a causal polynomial algebra, not that
Hadamard Wick embedding.\\[0.5em]

E2 &
Sharp insertions $Q_\Sigma(B_f)$ are admitted, with
$[Q_\Sigma(B_f)]_{\hbar^0}=B_f$ on all smooth configurations. &
Unmet. No sharp insertion or quantum relational composite is
admitted.\\[0.5em]

E3 &
Admission on the stated degree-$(1,1)$ kernel stages is
sequentially continuous in the normal topology and agrees with
the old regular inclusion; the regularized insertions converge
to the sharp element. &
Unmet. No such admission map or topology is constructed.\\[0.5em]

E4 &
Multiplication is hypocontinuous on bounded sets, and the
field-independent $\hbar^3$ coefficient map is continuous and
agrees with old coefficient extraction. &
Unmet as a topological package. Our product is algebraic;
no specified locally convex continuity result is proved.\\
\bottomrule
\end{tabularx}
\end{center}

Formal coefficient extraction is available algebraically.
That does not establish E4. Likewise the tensor $\PP$ derived
from the XI-A operator is not an imported Hadamard block.
A causal inverse does not select a positive-frequency
bisolution.

We do not form the word
$B_f\star W_g\star B_{f'}$, do not evaluate a reflected
light-ray residue, and do not subtract one. There is no
conclusion here that the ramp resolves VII-C's obstruction,
nor that VII-C obstructs the present restricted free algebra.

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.55\linewidth}X@{}}
\toprule
Gate & Result\\
\midrule
One gauge fermion and frozen normalization & Constructed\\
Independent Nakanishi--Lautrup field & Retained\\
Ghost kinetic coefficient & $i\bar c_\mu\Box c^\mu$\\
Graviton causal inverse & $-\PP\Ezero^{\mathrm R/A}$\\
Scalar equation versus Hessian inverse & Distinguished\\
Full auxiliary and ghost Green blocks & Constructed\\
Free causal polynomial algebra & Constructed, without a state\\
Off-shell free nilpotence & Proved with independent $b$\\
Free product compatibility & Proved on the graded relations\\
Plateau residual algebra & $\{0\}$\\
De Donder ramp coefficient & Propagated layer functional\\
Fixed-input Einstein jet & Not repaired\\
Stress-tensor vertex versus dressing & Not contracted\\
Hadamard Wick extension or VII-C hypotheses & Not established\\
Tree, mixed loop, interacting differential & Not constructed\\
Sharp-ramp limit & Not taken\\
\bottomrule
\end{tabularx}
\end{center}

The ending is \textbf{Constructed} for the free gate.
The existence of its causal Green operators is not evidence
against XI-A's on-shell obstruction. Different equations are
being solved.

XI-C receives:
\begin{enumerate}
\item the signed operator $-\Box\PP$ and its causal inverses;
\item the single gauge fermion, auxiliary equations, ghost
normalization, and full free causal matrix;
\item the free graded polynomial algebra and its differential;
\item the residual-gauge result \eqref{eq:residual};
\item the classical layer formula, with its support qualifications;
\item the inherited vertex \eqref{eq:vertex}, still uncontracted;
\item the distinction between the classical plateau action and
a residual action on the gauge-fixed free graviton.
\end{enumerate}

It does not receive a Hadamard state, Feynman prescription,
tree, mixed loop, renormalized relational product, or
interacting BRST class. The first future contraction must name
its domain and its covariance; neither is supplied by the
word ``propagator.''

\subsection*{Conclusion}

The free construction closes:
\[
 \boxed{
 \text{XI-A signed operator}
 \ \longrightarrow\
 \text{one auxiliary graded multiplet}
 \ \longrightarrow\
 \text{causal inverses and a free differential}.
 }
\]
At the same time,
\[
 \ga\cap\{\Box\xi=0\}=\{0\},
 \qquad
 \Jcal_g(\phi_m)=\varnothing
 \quad\text{for the XI-A Einstein reading}.
\]
The former is a residual-gauge statement, the latter a coupled
source obstruction. Neither is a failure of the constructed
gauge-fixed Green operator.

%=====================================================================
\appendix
\section{A compact sign and normalization audit}
\label{app:signs}
%=====================================================================

\subsection{Three different minus signs}

The reduced graviton Hessian is $-\Box\PP$, so its inverse is
$-\PP\Ezero^\pm$.

The scalar action Hessian is $-(\Box+m^2)$, so its inverse is
$-\Em^\pm$, even though the equation inverse is $+\Em^\pm$.

The retarded-minus-advanced scalar Hessian kernel is therefore
$-\delta_m$. With the convention \eqref{eq:CCR},
\[
 [\phi(x),\phi(y)]=-i\hbar\delta_m(x,y).
\]
Since $\partial_{x^0}\delta_m$ at equal times is the spatial
delta, this gives
$[\phi,\partial_t\phi]=i\hbar\delta_{\R^3}$
in the canonical ordering. Thus the action and commutator signs
agree.

\subsection{Ghost and auxiliary cancellation}

The half-normalized metric differential is $\sfree\gamma=\RR c$.
The gauge fermion produces $i\bar c\Box c$, not
$2i\bar c\Box c$.

Its causal anticommutator is
\[
 \{c,\bar c\}=\hbar\delta_0,
\]
while
$[\gamma,b]=i\hbar\RR\delta_0$.
These exact coefficients produce the cancellation in
Theorem~\ref{thm:differential}. Replacing one coefficient
without rescaling the ghost would destroy that compatibility.

One could instead use the full ghost normalization
$\sfree\gamma=K\widetilde c$, with $c=2\widetilde c$.
Then the corresponding ghost kinetic coefficient changes.
That is an equivalent reparametrization, not a second gauge
chosen in this paper.

\subsection{The layer formula is conditional}

For an unrestricted tensor,
\[
 \Box\Coord_1^\mu
       =\sig D^\mu\gamma-\sig'a_1^{0\mu}.
\]
Only the classical condition $D\gamma=0$ removes the first
term. The gauge-fixed equation $-\Box\bar\gamma=0$
does not itself impose $D\gamma=0$.
The auxiliary field records that distinction.

%=====================================================================
\section{Dependencies and verification scope}
\label{app:dependencies}
%=====================================================================

The proofs use the explicit quadratic action, compact interior
variation, ordinary smooth wave Cauchy theory, algebraic trace
reversal, and graded central commutation relations. They do
not use a graviton Hadamard matrix from another manuscript.

The supplied XI-A is used for its signed action and the
Einstein-jet obstruction. The ramp coefficient is derived
from the displayed divergence-form prescription. VII-C is
used only to identify the separate hypotheses that are not
constructed here.

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

\paragraph{Construction level.}
The free algebra is a formal algebra of compactly smeared
linear fields and finite products. No Hilbert-space
representation or state is selected.

\paragraph{Boundary level.}
Initial and final data specify causal inverses. No independent
boundary algebra, BRST boundary charge, or admission topology
is constructed.

\paragraph{Interaction level.}
The switched stress-tensor vertex remains a named classical
interaction. It is not used in any contraction with the ramp
dressing and is not included in $\sfree$.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, executed symbolic computation, 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 companion manuscript.
Used for the signed quadratic operator, the switching
conventions, and the fixed-input Einstein-jet obstruction.

\bibitem{VIIC}
\emph{Landmark VII-C: A mixed-Wick obstruction to a
normal-topology boundary extension for sharp Cauchy harmonic
insertions}.
Supplied comparison manuscript.
Used only for the extension checklist; its free matrix and
products are not inputs to the present construction.

\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.
\href{https://arxiv.org/abs/0806.1036}{arXiv:0806.1036}.
Used for smooth Cauchy theory, causal Green operators, and
finite propagation.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428v2, 2013.
\href{https://arxiv.org/abs/1208.1428}{arXiv:1208.1428}.
Background for the state-independent polynomial deformation;
all block normalizations and compatibility checks are derived
in this manuscript.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
``Quantum gravity from the point of view of locally covariant
quantum field theory,''
\emph{Communications in Mathematical Physics}
\textbf{345} (2016), 741--779.
\href{https://arxiv.org/abs/1306.1058}{arXiv:1306.1058}.
Comparison framework for the distinction between free gauge
fixing and an interacting BV construction.

\end{thebibliography}

\end{document}