% !TeX program = pdflatex
%=====================================================================
% XII-D.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XII-D.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XII-D.tex
%
% Selected source and inverse:
%   J_D = sigma J_3 + D_sigma,
%   u_D = -P E_0^R J_D.
%
% Debugging recorded explicitly:
%   1. The transposed unweighted root need not be supported at t >= 0.
%      A compact ambient collar root is constructed. Its effective
%      weighted coefficients are compact and supported at t >= 0.
%   2. The scalar slot includes the switched scalar-response support,
%      not only the observation support.
%   3. With u_D = G^R J_D, the finite two-jet response law has a
%      MINUS two-jet term. The action's perturbative metric coefficient
%      would solve L r = -J_D and is therefore -u_D.
%      No inverse normalization is changed.
%   4. An exact signed plateau identity is proved.
%      Unqualified invariance of the cubic insertion alone is not claimed.
%
% Proved here:
%   local cutoff-independent smooth response;
%   completed first-order scalar/coordinate root;
%   compact completed cubic kernel and its microcausal cone bound;
%   carrier admission and ordering stability of the enlarged named sector;
%   exact finite plateau variation and a conditional companion cancellation.
%
% Not constructed:
%   an interacting differential, a quantum retarded product, a state,
%   a loop, a complete second-order relational observable, or a sharp limit.
%
% The original XI-G exclusion is retained as predecessor input.
%
% No successful compilation, proof-assistant verification,
% independent referee review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{inputstatement}[theorem]{Input statement}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\PP}{\mathcal P}
\newcommand{\RR}{\mathcal R}
\newcommand{\Lie}{\mathcal L}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\Pmass}{P_m}
\newcommand{\ER}{E_0^{\mathrm R}}
\newcommand{\EA}{E_0^{\mathrm A}}
\newcommand{\ERm}{E_m^{\mathrm R}}
\newcommand{\EAm}{E_m^{\mathrm A}}
\newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Fsol}{\mathscr F^{\mathrm{sol}}_{\mathrm{mc,pol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\EalgP}{\mathscr E_{H'}}
\newcommand{\Afree}{\mathscr A_0}
\newcommand{\Sect}{\mathscr S_H}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\CD}{\mathcal C_{3,\mathrm D}}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\newcommand{\DS}{\mathcal D_\sig}
\newcommand{\RD}{\mathfrak R_{\sig}}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Hcal}{\mathcal H}
\newcommand{\Lag}{\mathfrak L}
\newcommand{\Vp}{\widetilde V_+}
\newcommand{\Vm}{\widetilde V_-}
\newcommand{\Edist}{\mathcal E'}
\newcommand{\alphad}{\alpha_d}
\newcommand{\balphad}{\bar\alpha_d}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

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\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 XII-D}\\[0.4em]
Retarded response and carrier admission\\
for the cubic-only density-weighted Einstein source}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
At fixed regulator width, we apply the XI-track inverse
$-\mathcal P E_0^{\mathrm R}$ to the actual cubic source frozen
in XII-C, including its ramp-derivative terms. Finite propagation
gives a cutoff-independent smooth response without assuming
source conservation. We then construct the first-order completion
from both the divergence-form coordinate leg and the switched
scalar-response leg.

The transposed root is smooth but need not be compact before
localization. A compact ambient collar root is constructed;
multiplication by the actual source coefficients produces compact
effective roots supported at nonnegative times. The completed
degree-$(2,1)$ kernel has a covector-sum-zero bound. This proves
membership in XII-A's polynomial microcausal domain and gives
carrier admission. Smooth changes of ordering add only regular
linear fields, so the formally completed sector generated by the
new insertion is ordering-stable.

The plateau calculation requires a sign correction. With the
prescribed forcing response $u_{\mathrm D}=G_{\mathrm{gr}}^{\mathrm R}
\mathcal J_{\mathrm D}$, the finite two-jet term enters with a
minus sign. We prove the corresponding exact variation, display
the gauge-fixing companion and Einstein-shell term separately,
and give a conditional companion cancellation. This is not an
unqualified invariance theorem for the cubic insertion alone.

The new admission neither changes XI-G's original exclusion nor
repairs XII-C's empty fixed-input jet. No quantum retarded product,
interacting differential, state, loop, or sharp-width limit 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) {Excluded cubic class};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Compact root};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Newly admitted kernel};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Passing this admission does not restore the excluded class.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The new kernel does not restore the old class. 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-XII-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Excluded cubic class};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Newly admitted kernel};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Passing this admission does not restore the excluded class.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The new kernel does not restore the old class. 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-XII-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Excluded cubic class};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Compact root};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Newly admitted kernel};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Passing this admission does not restore the excluded class.};
\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-XII-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Excluded cubic class};
\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) {Newly admitted kernel};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Compact root. Passing this admission does not restore the excluded class.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Excluded cubic class is not Newly admitted kernel. 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-XII-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Excluded cubic class};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Compact root};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Newly admitted kernel};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Passing this admission does not restore the excluded class.};
\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-XII-D}
\end{figure}

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


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



\scopebox{Release and proof boundary}{%
\textbf{Constructed:} the prescribed classical response, its
completed compact root, the compact microcausal cubic kernel,
carrier admission, ordering transport on the enlarged named
sector, and the exact signed plateau variation.

\smallskip
\textbf{Corrections:}
a compact root generally requires an ambient time collar;
its effective source-weighted coefficients, not the raw
transposed root, have nonnegative-time support.
The prescribed positive-source forcing response has a
negative two-jet variation. A positive two-jet law would belong
to the opposite forcing sign.

\smallskip
\textbf{Not claimed:}
unqualified plateau invariance of $\CD$ alone, construction of
the complete second-order relational observable, or repair of
the invariant coupled jet.

\smallskip
The original detected unramped classes remain excluded.
No carrier enlargement, quantum retarded product, state, loop,
interacting differential, or $\eps\to0$ limit is opened.}

\tableofcontents

%=====================================================================
\section{Release criteria and copied inputs}
\label{sec:inputs}
%=====================================================================

The endings are \textbf{Constructed}, \textbf{Proved obstructed},
and \textbf{Unresolved}. They apply to the object actually named.
Failure of a sufficient distribution-product hypothesis would
not prove nonexistence. An empty coupled solution fiber would
not obstruct a prescribed gauge-fixed response.

\[
 \boxed{\text{XII-D first applies }\Gret\text{ to the source }\JD
 \text{ frozen in XII-C}.}
\]

\begin{inputstatement}[XII-C: the chosen action and source]
The selected model weights the specified cubic density only:
\[
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}
   +\kappa\int_{\MT}(\sig-1)\ell_3\dd^4x,
 \qquad
 \ell_3=-2(r_3+b_1r_2+b_2r_1).
\]
The scalar action, free gauge-fixing coefficient, and higher
unweighted gravitational coefficients are unchanged.
The actual cubic Euler source is
\begin{equation}
 \JD^{\mu\nu}
 =\sig J_3^{\mu\nu}+\DS^{\mu\nu},
 \qquad
 \DS^{\mu\nu}
 =2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sig
                  -2M_2^{\mu\nu\rho}\partial_\rho\sig.
 \label{eq:source-input}
\end{equation}
In the pure-trace sector,
\[
 \eta_{\mu\nu}\DS^{\mu\nu}\big|_{\gamma=\psi\eta}
 =-6(\partial_\mu\sig)\psi\partial^\mu\psi.
\]
The layer coefficients are smooth and flat at their endpoints;
they are not hypersurface distributions \cite{XIIC}.
\end{inputstatement}

\begin{inputstatement}[XII-C: jet and successor boundary]
For the named inputs $\phi_m$, the invariant coupled jet
$\Jcal_{g,\sig}^{\mathrm D}(\phi_m)$ is empty. The proof uses a
late switch defect, not a failure of a Green operator.
The zero-field vacuum prevents universal jet emptiness.
The completed insertion and its generated root were not
constructed, and the inverse was recorded but not applied
\cite{XIIC}.
\end{inputstatement}

\begin{inputstatement}[XII-A-v2, XI-B, and XI-G]
XII-A supplies the polynomial domain, shell quotient, product,
ordering maps, and regular embedding:
\[
 \Ealg=\Fsol[[\hbar,\kappa]],\qquad
 \Fsol=\Fpol/\Nsol.
\]
Its detected original unramped classes remain excluded.
XI-B supplies
\[
 \Gret=-\PP\ER,\qquad
 A+2D^*D=-\Box\PP,\qquad \PP D^*=-\RR,\qquad K=2\RR.
\]
XI-G's source-ramping remark is not an admission theorem for
\eqref{eq:source-input} \cite{XIIA,XIB,XIG}.
\end{inputstatement}

XII-B's weighting audit is not reopened.
No X-track normalization or response identification is imported.
The predecessor statements are kept separate from the new
response, root, and cone calculations.

The analytic input consists of smooth wave Cauchy theory and
finite propagation \cite{BGP}, together with the distributional
product and proper-pushforward calculus \cite{Hormander}.
The functional and ordering framework is comparison background
only \cite{FR}; XII-A's established maps are used without
reconstructing its carrier.

%=====================================================================
\section{D0: the frozen sheet and the source convention}
\label{sec:sheet}
%=====================================================================

Use
\[
 M=\R^4,\quad
 \eta=\operatorname{diag}(1,-1,-1,-1),\quad
 \Box=\partial_t^2-\Delta_{\mathbf x},\quad
 \Pmass=\Box+m^2,\quad m\ge0,
\]
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad 0<\eps<T.
\]
Fields are ambient smooth configurations. No spatial decay is
required. All finite-slab constructions depend only on their
restrictions to a neighborhood of the slab.

Retain the fixed ramp
\[
 \sig(t)=\chi(t/\eps),\qquad
 \chi(s)=
 \begin{cases}
 0,&s\le0,\\[2pt]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
 1,&s\ge1,
 \end{cases}
\]
and
\[
 g\in C_c^\infty(\MTo;[0,1]),\qquad
 \overline U\subset\Int\{g=1\}\cap\{\eps<t<T\},\qquad
 \ga=C_c^\infty(U;TM).
\]
We assume $U\ne\varnothing$, as in the plateau handoff.
Neither external function is varied.

Trace reversal and divergence are
\[
 \PP r=r-\tfrac12\eta\,\tr_\eta r,\qquad
 D^\nu r=\partial_\mu(\PP r)^{\mu\nu}.
\]
Parentheses include $\tfrac12$.

For completeness, the coefficients in
\eqref{eq:source-input} have the finite prescription
\[
 u_0^{\mu\nu}=\eta^{\mu\nu},\quad
 u_1^{\mu\nu}=-\gamma^{\mu\nu},\quad
 u_2^{\mu\nu}=\gamma^\mu{}_\lambda\gamma^{\lambda\nu},
\]
\[
 b_0=1,\quad b_1=\tfrac12\tr_\eta\gamma,\quad
 b_2=\tfrac18(\tr_\eta\gamma)^2
                         -\tfrac14\gamma_{\mu\nu}\gamma^{\mu\nu},
\]
\begin{align}
 C_r^{\mu\nu\alpha\beta}
 &=
 \sum_{\substack{i+j+k=r\\i,j,k\ge0}}
 b_i\left(
 u_j^{\mu\nu}u_k^{\alpha\beta}
                   -u_j^{\mu(\alpha}u_k^{\beta)\nu}\right),
 \qquad r=0,1,2,\\
 \Gamma_n^\rho{}_{\alpha\beta}
 &=
 \tfrac12u_{n-1}^{\rho\lambda}
 \left(\partial_\alpha\gamma_{\beta\lambda}
      +\partial_\beta\gamma_{\alpha\lambda}
      -\partial_\lambda\gamma_{\alpha\beta}\right),
 \qquad n=1,2,\\
 M_2^{\mu\nu\rho}
 &=C_0^{\mu\nu\alpha\beta}\Gamma_2^\rho{}_{\alpha\beta}
       +C_1^{\mu\nu\alpha\beta}\Gamma_1^\rho{}_{\alpha\beta}.
 \label{eq:source-coefficients}
\end{align}
Only the displayed coefficients occur in the sums.
The original $J_3$ is the quadratic coefficient
$2[\kappa^2](\sqrt{-q}G^{\mu\nu})$, with
$q=\eta+\kappa\gamma$ and XII-C's curvature convention.

Thus $\JD$ is a quadratic differential polynomial of total
field-derivative order at most two, with smooth coefficients
built from $\sig,\sig',\sig''$.
Its polarization is
\[
 \JD(\gamma_1,\gamma_2)
 =
 \tfrac12\bigl(
 \JD[\gamma_1+\gamma_2]-\JD[\gamma_1]-\JD[\gamma_2]\bigr).
\]
It vanishes for $t\le0$ and is flat there.
For $t\ge\eps$, it equals $J_3$.

The prescribed response is
\begin{equation}
 \boxed{u_{\mathrm D}[\gamma]=-\PP\ER\bigl(\JD[\gamma]\bigr).}
 \label{eq:response}
\end{equation}
This is not the response to $\sig J_3$ alone.

\begin{remark}[Forcing sign versus action coefficient]
\label{rem:forcing-sign}
Put $L=-\Box\PP$.
Equation~\eqref{eq:response} solves
$Lu_{\mathrm D}=\JD$.
The gravitational contribution to a perturbative coefficient
of the action equation instead solves $Lr=-\JD$.
That coefficient is $r=-u_{\mathrm D}$.
This distinction changes a forcing sign, not the coefficient
$-1$ in the inverse of $L$.
It is essential in Section~\ref{sec:plateau}.
\end{remark}

%=====================================================================
\section{D1: the smooth classical response}
\label{sec:response}
%=====================================================================

\begin{theorem}[Local construction and cutoff independence]
\label{thm:response}
For every smooth $\gamma$, \eqref{eq:response} defines a unique
smooth zero-initial-data solution on the slab.
For every compact observation region $K\subset\MTo$, it can
be computed using a fixed spatial cutoff equal to one near
\[
 C_K=J^-(K)\cap\{t\ge0\}.
\]
The result on a neighborhood of $K$ is independent of that
cutoff. Moreover,
\begin{equation}
 \supp u_{\mathrm D}
 \subset J^+(\supp\JD[\gamma])\cap\MT.
 \label{eq:response-support}
\end{equation}
The support assertion is not a layer-support assertion.
\end{theorem}

\begin{proof}
The set $C_K$ is compact. Choose a spatial cutoff $\zeta$ equal
to one near its spatial projection and replace $\gamma$ by
$\zeta\gamma$ for this local calculation.
The modified source is spatially compact on bounded time
intervals and vanishes smoothly for $t\le0$.
Smooth wave Cauchy theory gives its zero-data response.

All derivatives of the two metric configurations agree near
$C_K$, so their local differential sources agree there.
Finite propagation implies equality of the resulting responses
near $K$. Two admissible cutoffs therefore give the same result.
Applying this argument on compact regions defines a consistent
smooth response throughout the slab.

The source is past compact in the causal sense: its intersection
with the causal past of any compact set is compact.
Consequently the ambient retarded prescription agrees with
the locally constructed zero-data solution.
Finite propagation gives \eqref{eq:response-support};
homogeneous zero-data uniqueness gives uniqueness.
\end{proof}

The source need not be spatially compact before localization,
and it need not vanish above the ramp. The cutoff is an
auxiliary local construction, not a change of the model.

\subsection{The response divergence}

We record the relevant XII-C identity before using it.
For a symmetric contravariant density $v$, define
\[
 N_{0,\nu}(v)=2\eta_{\rho\nu}\partial_\mu v^{\mu\rho},
 \qquad
 Q_\nu(v)
 =2\partial_\mu(v^{\mu\rho}\gamma_{\rho\nu})
                       -v^{\mu\rho}\partial_\nu\gamma_{\mu\rho}.
\]
The actual source obeys
\begin{equation}
 N_{0,\nu}(\JD)
       =-\sig Q_\nu(A\gamma)+\RD_\nu[\gamma],
 \label{eq:source-divergence}
\end{equation}
where
\begin{equation}
 \begin{split}
 \RD_\nu={}&
 2\eta_{\rho\nu}(\partial_\mu\sig)J_3^{\mu\rho}\\
 &+4\eta_{\rho\nu}
 \left[
  (\partial_\mu C_2^{\mu\rho\alpha\beta})
                          \partial_\alpha\partial_\beta\sig
 -(\partial_\mu M_2^{\mu\rho\lambda})\partial_\lambda\sig
 -M_2^{\mu\rho\lambda}\partial_\mu\partial_\lambda\sig
 \right].
 \end{split}
 \label{eq:ramp-defect}
\end{equation}
The apparent third ramp derivative cancels in XII-C's
coefficient identity. No conservation assumption is made.

\begin{proposition}[Existing, possibly nonzero divergence]
The response satisfies
\begin{equation}
 D^\nu u_{\mathrm D}
 =
 -\ER\bigl(\partial_\mu\JD^{\mu\nu}\bigr)
 =
 \tfrac12\ER\left(
 \sig Q^\nu(A\gamma)-\RD^\nu[\gamma]\right).
 \label{eq:response-divergence}
\end{equation}
\end{proposition}

\begin{proof}
Use $\PP^2=\id$ in \eqref{eq:response}, commute constant
derivatives with the locally defined wave inverse, and apply
\eqref{eq:source-divergence}.
\end{proof}

A nonzero value in \eqref{eq:response-divergence} is not a
Green-operator obstruction. It does not turn the response
into an invariant coupled Einstein solution.

%=====================================================================
\section{D2: both completion legs and the compact root}
\label{sec:root}
%=====================================================================

The completion is written explicitly so that no unspecified
predecessor formula is needed.
Set
\[
 \mathfrak a^{\mu\nu,ab}
 =\tfrac12\eta^{\mu\nu}\eta^{ab}
                         -\eta^{\mu(a}\eta^{b)\nu},
 \qquad
 a_1^{\mu\nu}[r]=\mathfrak a^{\mu\nu,ab}r_{ab},
 \qquad
 b_1[r]=\tfrac12\eta^{ab}r_{ab}.
\]

The divergence-form coordinate coefficient and the scalar
equation coefficient are
\begin{align}
 X_1^\mu[r]
 &=-\ER\partial_\lambda
                  \bigl(\sig a_1^{\lambda\mu}[r]\bigr),\\
 V_{1,g}[r]\phi
 &=\partial_\mu\left(g a_1^{\mu\nu}[r]\partial_\nu\phi\right)
                            +m^2g b_1[r]\phi,\\
 \phi_1[r;\phi]&=-\ERm\bigl(V_{1,g}[r]\phi\bigr).
 \label{eq:completion-legs}
\end{align}
They follow from the inherited coordinate prescription and
the minimally coupled switched scalar action.
In particular, derivatives of $g$ are retained inside the
displayed divergence.

For $f\in C_c^\infty(\MTo)$ define
\begin{equation}
 \Aone[f,g](r;\phi)
 =
 \int f(x)\left(
 \phi_1[r;\phi](x)-X_1^\mu[r](x)\partial_\mu\phi(x)
                    \right)\dd^4x.
 \label{eq:Aone}
\end{equation}
The scalar field is arbitrary and smooth at this stage.

\begin{definition}[Completed new insertion]
\label{def:completed}
The classical completed cubic insertion is
\begin{equation}
 \boxed{
 \CD(f,g)[\gamma,\phi]
       =\Aone[f,g]\bigl(u_{\mathrm D}[\gamma];\phi\bigr).
 }
 \label{eq:CD}
\end{equation}
It is quadratic in the metric and linear in the scalar.
Equation~\eqref{eq:CD}, not an independently chosen source
pairing, is its definition.
\end{definition}

\subsection{The first root and its transpose}

Put $v_f=\EAm f$.
Compact integration by parts in \eqref{eq:Aone} gives
\begin{equation}
 \Aone[f,g](r;\phi)=\int\alpha_{f,g}[\phi]^{ab}(z)r_{ab}(z)\dd^4z,
 \label{eq:alpha-pair}
\end{equation}
where
\begin{align}
 \alpha_{\mathrm{co}}^{ab}(z)
 &=-\sig(z)\mathfrak a^{\lambda\mu,ab}
                 \partial_\lambda\EA(f\partial_\mu\phi)(z),\\
 \alpha_{\mathrm{sc}}^{ab}(z)
 &=g(z)\mathfrak a^{\mu\nu,ab}
                   \partial_\mu v_f(z)\partial_\nu\phi(z)
       -\tfrac{m^2}{2}g(z)\eta^{ab}v_f(z)\phi(z),\\
 \alpha_{f,g}&=\alpha_{\mathrm{co}}+\alpha_{\mathrm{sc}}.
 \label{eq:alpha}
\end{align}
Both legs are present.

If $K=\supp f$, define the corrected scalar support set
\begin{equation}
 O_{f,g}
 =K\cup\bigl(\supp g\cap J^-(K)\bigr).
 \label{eq:scalar-support}
\end{equation}
The map $\phi\mapsto\alpha_{f,g}[\phi]$ depends only on the
scalar near $O_{f,g}$.
Its range is smooth, compact, and supported in $C_K$.

Transpose the metric inverse with its prescribed sign:
\begin{equation}
 \beta_{f,g}[\phi]
       =-\EA\bigl(\PP\alpha_{f,g}[\phi]\bigr).
 \label{eq:beta}
\end{equation}
The minus sign follows from $\Gret=-\PP\ER$ and
\eqref{eq:alpha-pair}; it is not an independent convention.

\subsection{Why the raw root needs a collar}

The smooth advanced function \eqref{eq:beta} may extend to
negative times. In particular, it need not vanish at $t=0$.
One cannot generally multiply it by a smooth cutoff supported
in $t\ge0$ which is identically one at every positive time.

Choose $\delta>0$ and
$\vartheta\in C_c^\infty((-\delta,T+\delta))$ with
$\vartheta=1$ on a neighborhood of $[0,T]$.
Set
\begin{equation}
 a_{f,g}[\phi](y)=\vartheta(y^0)\beta_{f,g}[\phi](y).
 \label{eq:compact-root}
\end{equation}

\begin{theorem}[Completed compact root]
\label{thm:root}
For fixed $f,g,\vartheta$, the map
\[
 a_{f,g}:C^\infty(M)\longrightarrow
 C_c^\infty(M;\operatorname{Sym}^2T^*M)
\]
is continuous and linear, with range in a fixed compact support
contained in
\[
 J^-(K)\cap\{t\ge-\delta\}.
\]
It reproduces the completed insertion:
\begin{equation}
 \CD(f,g)[\gamma,\phi]
 =
 \int a_{f,g}[\phi]_{\mu\nu}(y)
                         \JD^{\mu\nu}[\gamma](y)\dd^4y.
 \label{eq:root-identity}
\end{equation}
The effective roots
$\sig a_{f,g}$, $\sig'a_{f,g}$, and $\sig''a_{f,g}$
are compact and supported in $C_K$.
The insertion is independent of the collar cutoff.
\end{theorem}

\begin{proof}
The first root \eqref{eq:alpha} is supported in $C_K$.
Advanced support gives $\supp\beta\subset J^-(C_K)
\subset J^-(K)$. Intersecting this past with the bounded
time collar makes its support compact.

Multiplication by fixed smooth coefficients, differentiation,
and wave propagation on fixed causal compact sets are
continuous smooth-function operations. This proves continuity
of the root map into its fixed compact support stage.

Transpose the metric response in \eqref{eq:CD}.
The resulting integral is $\int\beta\,\JD$.
Only nonnegative times below the latest observation time
contribute: the source vanishes for $t\le0$, and the advanced
root vanishes above the observation.
Hence insertion of $\vartheta$ changes nothing and gives
\eqref{eq:root-identity}.
All three effective roots vanish smoothly at nonpositive
times and have support in $C_K$.
\end{proof}

Thus the independent source pairing of XII-C now becomes a
derived equality. It was not renamed as the insertion.
The raw-root support requirement has been corrected explicitly,
without adding a contact term or altering the source.

The set $O_{f,g}$ controls the scalar slot.
It does not control the entire functional support:
the metric slots can occur in its causal past.
This distinction prevents a false support claim for a
nonlocal response functional.

%=====================================================================
\section{D3: the completed-kernel cone calculation}
\label{sec:admission}
%=====================================================================

All covectors are compared using the fixed Minkowski coordinate
trivialization. For $n\ge2$, write
\[
 Z_n=
 \left\{(x_1,p_1;\ldots;x_n,p_n):
                  p_1+\cdots+p_n=0\right\}\setminus0.
\]
Since the future and past causal cones are pointed, $Z_n$
does not meet $\Vp^{\,n}\cup\Vm^{\,n}$.

\subsection{The root kernels}

Write
\[
 \alpha_{f,g}[\phi](z)=\int A_{f,g}(z,x)\phi(x)\dd^4x.
\]
Its two components are explicitly
\begin{align}
 A_{\mathrm{co}}^{ab}(z,x)
 &=
 \sig(z)\mathfrak a^{\lambda\mu,ab}
 \partial_\mu^x\left[
       f(x)\partial_\lambda^zE_0^{\mathrm A}(z,x)\right],\\
 A_{\mathrm{sc}}^{ab}(z,x)
 &=
 -g(z)\mathfrak a^{\mu\nu,ab}\partial_\mu v_f(z)
                      \partial_\nu^x\delta(x-z)
 \nonumber\\
 &\quad-\tfrac{m^2}{2}g(z)\eta^{ab}v_f(z)\delta(x-z).
 \label{eq:A-kernels}
\end{align}
In the first term of $A_{\mathrm{sc}}$, the coefficient depends
on $z$, so differentiation acts only on the displayed delta.

Translation invariance of the Minkowski Green kernel gives
\begin{equation}
 \WF(A_{f,g})\subset Z_2.
 \label{eq:A-WF}
\end{equation}
This includes arbitrary diagonal Green covectors; no
positive-frequency restriction is used.

Let $B_{f,g}(y,x)$ be the kernel of the compact root
\eqref{eq:compact-root}:
\begin{equation}
 B_{f,g}(y,x)
 =
 -\vartheta(y^0)\int
 E_0^{\mathrm A}(y,z)\PP A_{f,g}(z,x)\dd^4z.
 \label{eq:B-kernel}
\end{equation}

\begin{lemma}[Root composition]
\label{lem:root-kernel}
The composition \eqref{eq:B-kernel} exists, has compact
support, and satisfies
\[
 \WF(B_{f,g})\subset Z_2.
\]
Its scalar projection is contained in $O_{f,g}$.
\end{lemma}

\begin{proof}
The $z$ support of $A$ is compact.
A forbidden product cancellation would have zero covectors
in the external variables $y,x$.
A Green wavefront covector has the form $(p,-p)$, so the
zero $y$ covector forces that Green covector to vanish.
Then \eqref{eq:A-WF} and the zero $x$ covector force the
root covector to vanish as well. The product is therefore
defined.

The $z$ projection is proper on the relevant support.
For a pushforward wavefront point, cancellation of the
internal $z$ covectors leaves output covectors $(p,-p)$.
This proves the bound.
The time collar makes the support compact; the scalar
projection follows directly from \eqref{eq:A-kernels}.
\end{proof}

\subsection{The actual local source kernel}

Let $j_{\mathrm D}(y;y_1,y_2)$ be the symmetric bilinear
kernel defined by
\[
 \pair{j_{\mathrm D}(y;\cdot,\cdot)}
             {\gamma_1\otimes\gamma_2}
       =\JD(\gamma_1,\gamma_2)(y).
\]
It is a finite sum of differentiated diagonal deltas with
smooth coefficients built from $\sig,\sig',\sig''$.
Consequently
\begin{equation}
 \begin{split}
 \supp j_{\mathrm D}&\subset\{y=y_1=y_2,\ t_y\ge0\},\\
 \WF(j_{\mathrm D})&\subset
 \{(y,q;y_1,p_1;y_2,p_2):q+p_1+p_2=0\}\setminus0.
 \end{split}
 \label{eq:j-WF}
\end{equation}
Flatness at zero means that these are ambient smooth
coefficients. No step function is inserted.

The polarized completed kernel is
\begin{equation}
 t_{\mathrm D}(y_1,y_2,x)
 =
 \pi_{y*}\left[
 B_{f,g}(y,x)_{\mu\nu}
                 j_{\mathrm D}^{\mu\nu}(y;y_1,y_2)\right].
 \label{eq:t-kernel}
\end{equation}

\begin{theorem}[Compact microcausal completed kernel]
\label{thm:admission}
The product and pushforward in \eqref{eq:t-kernel} exist.
The kernel is compact and symmetric in its metric slots, with
\begin{equation}
 \supp t_{\mathrm D}
 \subset
 \{y_1=y_2\in C_K\}\times O_{f,g},
 \qquad
 \WF(t_{\mathrm D})\subset Z_3.
 \label{eq:t-bounds}
\end{equation}
After species symmetrization it defines
$\CD(f,g)\in\Fpol$.
\end{theorem}

\begin{proof}
A forbidden cancellation in the product would have zero
external covectors in $x,y_1,y_2$.
The root bound forces its $y$ covector to be zero.
The source bound then forces its $y$ covector to be zero.
There is no nonzero total cancellation, so the product exists.

The root is compact and the source is supported on the local
diagonal, making the $y$ projection proper.
For an output wavefront point, write the root covectors as
$(p,-p)$ and the source covectors as $(q,p_1,p_2)$.
Internal cancellation gives $q=-p$; the local source relation
gives $p_1+p_2=p$. The scalar output covector is $-p$.
Thus the three output covectors sum to zero.

The support assertion follows from the root support, the
diagonal source support, and its nonnegative-time coefficients.
Permutation of the three species slots preserves the sum-zero
bound. A nonzero tuple wholly in either causal cone cannot
sum to zero. This is precisely XII-A's microcausal exclusion.
\end{proof}

The theorem includes the $\sigma''C_2$ and $\sigma'M_2$
kernels, not only the $\sigma J_3$ kernel.
The proof is an explicit product--pushforward calculation;
it is not an inference from smoothness of the response.
There is no failed collinear product hypothesis to reinterpret
as a no-go.

The bound is special to this finite Minkowski composition:
constant fiber maps, translation-invariant Green kernels,
local differential vertices, and smooth external weights
supply covector balance. No corresponding theorem for an
arbitrary nonlocal or curved-background kernel is claimed.

This admission concerns the changed functional
\eqref{eq:CD}. It does not supply a new representative of
XI-G's original excluded shell class.

%=====================================================================
\section{D4: descent and ordering on the enlarged named sector}
\label{sec:descent}
%=====================================================================

\begin{corollary}[Carrier admission]
The new insertion has a well-defined class
\[
 C_{\mathrm D,H}(f,g)=Q_H\CD(f,g)\in\Ealg.
\]
It has even parity and ghost number zero.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:admission} supplies an off-shell microcausal
polynomial lift. XII-A's already proved free-shell descent
then applies. The insertion uses two even metric slots and
one even scalar slot.
\end{proof}

No conservation constraint on $\JD$ is needed for this
quotient operation. Admission to the gauge-fixed free-shell
carrier is not solvability of the invariant coupled equation.

Let $T_{\mathrm D}$ denote the fully species-symmetric
degree-three kernel, normalized so that
\[
 \CD(\Phi)=\pair{T_{\mathrm D}}{\Phi^{\otimes3}}
\]
with no additional factorial.

\begin{proposition}[Regular ordering correction]
\label{prop:ordering}
For an admissible smooth change $d=H'-H$,
\begin{equation}
 \alphad\CD=\CD+\hbar L_d,\qquad
 L_d(\Phi)=\pair{\ell_d}{\Phi},
 \quad
 \ell_d^C
 =
 3\,\pi_{3*}\left[
 d_{AB}(x_1,x_2)T_{\mathrm D}^{ABC}(x_1,x_2,x_3)
 \right].
 \label{eq:ordering}
\end{equation}
Every component of $\ell_d$ is compact and smooth.
Only admissible metric--metric and metric--scalar contractions
can contribute.
\end{proposition}

\begin{proof}
The second functional derivative of a cubic contributes
$3\cdot2$; the ordering generator contains $\hbar/2$.
No higher even derivative remains.

A nonzero wavefront point of the projected linear kernel
would require a tuple $(0,0,p)$ in the cubic wavefront set.
The sum-zero bound forces $p=0$, a contradiction.
Compactness follows from proper projection of the compact
cubic support. There is only one scalar slot, and no odd
species, so the stated pair types exhaust the possibilities.
\end{proof}

Define $\Sect_{\mathrm D,H}$ to be the coefficientwise
$(\hbar,\kappa)$-adic completion of the algebra generated by
XII-A's named sector and all $C_{\mathrm D,H}(f,g)$.
This is formal completion, not analytic closure.

\begin{theorem}[Ordering-stable enlarged named sector]
The inherited ordering isomorphism maps
$\Sect_{\mathrm D,H}$ onto the corresponding
$\star_{H'}$-sector generated by the unchanged new formulas:
\[
 \balphad(\Sect_{\mathrm D,H})
                   =\Sect_{\mathrm D,H'}.
\]
\end{theorem}

\begin{proof}
XII-A transports the old sector onto its counterpart.
By Proposition~\ref{prop:ordering}, each new generator is
transported to the unchanged new generator plus a regular
linear field. Regular linear fields already belong to the
old sector. The inverse ordering map gives the reverse
inclusion. Formal-order preservation gives equality of the
coefficientwise completions.
\end{proof}

Continuity of the inherited ordering and quotient maps is
used only at XII-A's released level. No completeness or global
product hypocontinuity claim is added. No interacting
differential is inferred from this sector construction.

%=====================================================================
\section{D5: the exact signed plateau variation}
\label{sec:plateau}
%=====================================================================

\subsection{Defects before imposing plateau support}

For reference, XII-C's changed action identity is
\[
 \mathcal N_\nu(e_{\mathrm D})
 =
 \kappa\left[
 E_\phi\partial_\nu\phi-\mathcal B_{g,\nu}
 -(\sig-1)Q_\nu(A\gamma)+\RD_\nu
                  +\kappa Q_\nu(Z_\sig)\right],
\]
where
\[
 Z_\sig=(\sig-1)J_3+\DS,
\]
\[
 \mathcal B_{g,\nu}
 =
 (\partial_\mu g)T_0^\mu{}_\nu
 -(1-g)(\Pmass\phi)\partial_\nu\phi
             +(\partial_\nu g)(\Lag_\eta-\Lag_q).
\]
The complete ramp defect is \eqref{eq:ramp-defect}.
The gauge-fixed identity has the additional term
$\mathcal N_\nu(e_{\mathrm{gf}})$.
These terms are displayed before setting $g=\sig=1$
on the parameter support.

For $\xi\in\ga$, locality gives
\begin{equation}
 \JD[\gamma+K\xi]-\JD[\gamma]
                  =J_3[\gamma+K\xi]-J_3[\gamma].
 \label{eq:layer-variation}
\end{equation}
Indeed every term in the difference contains a derivative
of $\xi$ or $\xi$ itself, hence is supported in $U$.
There $\sig=1$ and all ramp derivatives vanish.
The layer has not been discarded from the response:
its \emph{variation under these parameters} is zero.

\subsection{The finite cubic identity and its sign}

For a contravariant rank-two density define
\[
 (\Lie_\xi^{\mathrm{dens}}v)^{\mu\nu}
 =
 \xi^\rho\partial_\rho v^{\mu\nu}
 -v^{\rho\nu}\partial_\rho\xi^\mu
 -v^{\mu\rho}\partial_\rho\xi^\nu
                  +(\partial_\rho\xi^\rho)v^{\mu\nu}.
\]
Set
\[
 s_\xi[\gamma]
       =\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi.
\]
This tensor is compactly supported in $U$.

\begin{lemma}[Finite source variation]
\label{lem:finite-source}
For every smooth $\gamma$ and every $\xi\in\ga$,
\begin{equation}
 \JD[\gamma+K\xi]-\JD[\gamma]
 =
 \Lie_\xi^{\mathrm{dens}}(A\gamma)-A s_\xi[\gamma].
 \label{eq:finite-source}
\end{equation}
\end{lemma}

\begin{proof}
Equation~\eqref{eq:layer-variation} reduces the assertion to
the unweighted local coefficient identity.
For the flow generated by $\kappa\xi$,
\[
 \frac{F^*q-\eta}{\kappa}
 =\gamma+K\xi+\kappa s_\xi[\gamma]+O(\kappa^2).
\]
Naturality of the Einstein tensor density gives the same
pullback for its Euler derivative. Comparing the coefficient
of $\kappa$ in
\[
 e_{\mathrm{EH}}=A\gamma+\kappa J_3[\gamma]+O(\kappa^2)
\]
gives
\[
 A s_\xi+J_3[\gamma+K\xi]
       =J_3[\gamma]+\Lie_\xi^{\mathrm{dens}}(A\gamma).
\]
This proves \eqref{eq:finite-source}.
\end{proof}

Define the zero-data gauge-fixing companion parameter
\begin{equation}
 \rho_\xi[\gamma]=\ER D s_\xi[\gamma].
 \label{eq:rho}
\end{equation}
It is a propagated compact-source field, not a compact
residual plateau parameter.

\begin{theorem}[Actual response law]
\label{thm:response-law}
The prescribed response obeys
\begin{equation}
 \boxed{
 \begin{split}
 u_{\mathrm D}[\gamma+K\xi]-u_{\mathrm D}[\gamma]
 ={}&-s_\xi[\gamma]+K\rho_\xi[\gamma]\\
 &-\PP\ER\Lie_\xi^{\mathrm{dens}}(A\gamma).
 \end{split}}
 \label{eq:response-law}
\end{equation}
\end{theorem}

\begin{proof}
Apply $\Gret$ to \eqref{eq:finite-source}.
For compact $s$, the identities in the input sheet imply
\[
 \Gret A s
 =\Gret(L-2D^*D)s
 =s-K\ER Ds.
\]
The term with $\Lie_\xi^{\mathrm{dens}}(A\gamma)$ is
$-\PP\ER\Lie_\xi^{\mathrm{dens}}(A\gamma)$.
Substitution gives \eqref{eq:response-law}.
\end{proof}

The positive-two-jet formula sometimes associated with the
action's coefficient belongs to $-u_{\mathrm D}$, not to
\eqref{eq:response}. Multiplying the whole law by minus one
recovers that distinction; no factor $-4$ or change of
Green normalization is involved.

\begin{remark}[The sign difference can be detected locally]
The sign is not merely a choice of names.
Choose a compact $\xi\in\ga$ which near an interior point is
\[
 \xi=\lambda(t-t_0)(x^2-x_0^2)\partial_1,\qquad \lambda\ne0,
\]
and put $\gamma=0$.
In that neighborhood, writing
$F=\lambda(t-t_0)(x^2-x_0^2)$ gives
$s_\xi=-\partial F\otimes\partial F$.
At the central point,
$(A s_\xi)^{11}=(A s_\xi)^{33}=-2\lambda^2$.
Equation~\eqref{eq:finite-source} gives
$J_3[K\xi]=-A s_\xi$.
The proposed positive-two-jet response
$s_\xi-K\ER Ds_\xi$ has $L$-image $A s_\xi$,
whereas the prescribed response has $L$-image $-A s_\xi$.
They cannot agree.
This diagnoses an incompatible response law, not a failure
of kernel admission.
\end{remark}

\subsection{The ledger before addition}

Keep the scalar field fixed in this finite metric shift.
By linearity of $\Aone$ in its metric slot,
\[
 \Delta_\xi\CD
 =\CD[\gamma+K\xi,\phi]-\CD[\gamma,\phi].
\]
The three contributions are
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}X@{}}
\toprule
Contribution & Exact value\\
\midrule
Two-jet response &
$-\Aone[f,g](s_\xi;\phi)$\\[0.3em]
Gauge-fixing companion &
$+\Aone[f,g](K\rho_\xi;\phi)$\\[0.3em]
Einstein-shell term &
$-\Aone[f,g](\PP\ER\Lie_\xi^{\mathrm{dens}}(A\gamma);\phi)$\\
\bottomrule
\end{tabularx}
\end{center}
No row has been assigned an automatic numerical value.

\begin{corollary}[Exact completed plateau identity]
\label{cor:plateau}
For every $\xi\in\ga$,
\begin{equation}
 \boxed{
 \begin{split}
 \Delta_\xi\CD+\Aone[f,g](s_\xi;\phi)
 ={}&\Aone[f,g](K\rho_\xi;\phi)\\
 &-\Aone[f,g](\PP\ER
             \Lie_\xi^{\mathrm{dens}}(A\gamma);\phi).
 \end{split}}
 \label{eq:plateau-identity}
\end{equation}
\end{corollary}

This is the proved plateau variation.
The algebraic companion on the left is explicit; it has
not been identified with an unconstructed full second-order
observable. The Einstein-shell term stays named whenever
$A\gamma$ does not vanish near the parameter support.

\subsection{When the gauge-fixing companion vanishes}

A support hypothesis on observations, not a shrinkage of
$\ga$, makes a useful conditional cancellation precise.

\begin{proposition}[Causal plateau condition]
\label{prop:causal-plateau}
Assume $\Pmass\phi=0$ and that $g=1$ on a neighborhood of
\[
 J^-(K)\cap\supp(K\rho_\xi).
 \label{eq:causal-plateau-condition}
\]
Then
\[
 \Aone[f,g](K\rho_\xi;\phi)=0.
\]
If also $A\gamma=0$ on a neighborhood of $\supp\xi$, then
\[
 \Delta_\xi\CD+\Aone[f,g](s_\xi;\phi)=0.
\]
\end{proposition}

\begin{proof}
The source $Ds_\xi$ is supported above $\eps$, so
$\rho_\xi$ and its derivatives vanish before that time.
Consequently the coordinate ramp is one wherever
$K\rho_\xi$ can contribute. The coordinate equation gives
\[
 X_1[K\rho_\xi]=\ER\Box\rho_\xi=\rho_\xi
\]
with zero past data.

For any smooth vector $v$, direct differentiation of
\eqref{eq:completion-legs} with $g=1$ gives
\[
 V_{1,1}[Kv]\phi
 =
 -\Pmass(v^\mu\partial_\mu\phi)
 +v^\mu\partial_\mu(\Pmass\phi)
                    +(\partial_\mu v^\mu)\Pmass\phi.
\]
On the free scalar shell this is
$-\Pmass(v^\mu\partial_\mu\phi)$.
The support condition \eqref{eq:causal-plateau-condition}
makes the switched and unswitched scalar sources agree
throughout the causal past relevant to $K$.
Finite propagation and zero past data therefore give
\[
 \phi_1[K\rho_\xi;\phi]
            =\rho_\xi^\mu\partial_\mu\phi
\]
on $K$. The two legs in \eqref{eq:Aone} cancel.

If $A\gamma$ vanishes near $\supp\xi$, the Einstein-shell
source in \eqref{eq:plateau-identity} vanishes.
The final assertion follows.
\end{proof}

Without the stated causal condition the gauge-fixing row is
not deleted. Without the Einstein condition the last row is
not deleted. In particular, no universal zero variation of
$\CD$ alone is asserted.

These are classical identities for named polynomial lifts.
The compact shift $K\xi$ need not preserve the gauge-fixed
free shell, so no action of $\ga$ by automorphisms of
$\Ealg$ is inferred. There is no interacting differential
or quantum Ward identity hidden in the ledger.

%=====================================================================
\section{Decision and successor package}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}
                              p{.23\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
D0 & Frozen &
Actual density source and XI-track inverse retained.\\[0.4em]
D1 & Constructed &
Smooth cutoff-independent positive-source forcing response;
support and divergence proved.\\[0.4em]
D2 & Constructed, with support correction &
Both completion legs and a compact collar root; effective
weighted roots have the required nonnegative-time support.\\[0.4em]
D3 & Constructed &
Compact completed kernel with covector-sum-zero bound,
including all layer terms.\\[0.4em]
D4 & Constructed &
Admission to the inherited shell carrier and ordering
stability of the enlarged named sector.\\[0.4em]
D5 & Constructed as a signed identity &
Exact three-term variation and a conditional companion
cancellation. The positive-two-jet version is incompatible
with the prescribed forcing sign.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Response and admission release]
For the source and completion specified in this manuscript,
$u_{\mathrm D}$ and its completed root exist,
$\CD(f,g)\in\Fpol$, and $Q_H\CD(f,g)\in\Ealg$.
The enlarged named sector is stable under admissible ordering
transport. Its classical finite plateau variation is
\eqref{eq:plateau-identity}, with the additional cancellation
hypotheses stated in Proposition~\ref{prop:causal-plateau}.
\end{theorem}

\begin{proof}
The response is Theorem~\ref{thm:response}, the compact root
is Theorem~\ref{thm:root}, and the cone verdict is
Theorem~\ref{thm:admission}. Section~\ref{sec:descent} applies
the inherited carrier and ordering maps.
The finite variation is Theorem~\ref{thm:response-law} and
Corollary~\ref{cor:plateau}.
\end{proof}

Thus the corrected response, admission, and signed-identity
target is \textbf{Constructed}.
An alternative requirement of unqualified invariance of
$\CD$ alone is not part of this release.
The literal positive-two-jet response law is not silently
transferred to the opposite forcing sign.

The successor receives:
\begin{enumerate}
\item $u_{\mathrm D}=-\PP\ER(\sig J_3+\DS)$;
\item the completed two-leg functional \eqref{eq:CD};
\item the generated root \eqref{eq:compact-root}, its collar
qualification, and its effective support bounds;
\item the explicit compact-kernel admission theorem;
\item the inherited quotient class and regular ordering correction;
\item the signed plateau ledger and its exact conditions.
\end{enumerate}

It receives no interacting BRST class, full second-order
relational construction, loop value, quantum retarded product,
state, or sharp limit.

\subsection*{Retained boundaries}

\begin{itemize}
\item The original detected unramped shell classes remain excluded.
The admitted functional is a different model-dependent insertion.
\item XII-C's empty fixed-input invariant jet is not repaired.
The constructed object solves a prescribed gauge-fixed forcing equation.
\item The layer source was included before evolution. No extra
ramp, contact prescription, subtraction, or moved initial surface
was introduced.
\item The inverse coefficient is $-1$, not $-4$.
No identification with an X-track mixed map is made.
\item Propagated support is not layer support.
No $\eps\to0$ limit is taken.
\item No state, loop, antibracket, master identity, cone token,
or Enter record is imported.
\end{itemize}

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 u_{\mathrm D}=-\PP\ER(\sig J_3+\DS),\qquad
 \CD=\Aone[u_{\mathrm D};\phi]\in\Fpol,\\
 \WF(t_{\mathrm D})\subset Z_3,\qquad
 Q_H\CD\in\Ealg,\\
 \Delta_\xi\CD+\Aone[s_\xi;\phi]
 =
 \Aone[K\rho_\xi;\phi]
 -\Aone[\PP\ER\Lie_\xi^{\mathrm{dens}}(A\gamma);\phi].
 \end{gathered}}
\]

The source pairing is a derived identity, not a renamed
definition. Smooth response and carrier admission have
separate proofs. Neither is an Einstein-jet repair.

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

\paragraph{Classical response scope.}
The retarded map is a map on smooth configurations, defined
locally by finite propagation. It is not a quantum retarded
product or an interacting field map.

\paragraph{Admission scope.}
The cone calculation proves membership of the new compact
polynomial kernel in XII-A's domain. It neither enlarges
that domain nor removes the predecessor obstruction.

\paragraph{Root correction.}
The raw advanced root can extend below the initial surface.
A compact ambient collar representative is used only in
pairings with the flat, nonnegative-time source coefficients.
No sharp cutoff or boundary distribution is introduced.

\paragraph{Plateau correction.}
The forcing convention $Lu_{\mathrm D}=\JD$ is distinguished
from the action coefficient convention $Lr=-\JD$.
The signed variation is proved for the former.
A complete second-order relational observable has not been
constructed by naming its algebraic companion.

\paragraph{Dependency scope.}
XII-C supplies the action and actual source. XII-A supplies
the carrier and ordering maps. XI-B supplies the inverse
normalization. XI-G supplies the unreopened old exclusion.
The completed root and new cone bound are proved here.

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

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

\bibitem{XIIA}
N.~Cheung,
\emph{Landmark XII-A: One graded polynomial carrier for the
finite-ramp multiplet, with ordering-stable composites and
a robust observation exclusion}.
Supplied manuscript \texttt{XII-A-v2.tex}.
Carrier, shell descent, regular sector, and ordering inputs.

\bibitem{XIIB}
N.~Cheung,
\emph{Landmark XII-B: Admission and one retarded response of the
ramped cubic Einstein source, at fixed regulator width:
handoff audit, variational consistency, and the unresolved release}.
Supplied manuscript \texttt{XII-B.tex}.
Density/source distinction and pre-freeze stop rule.

\bibitem{XIIC}
N.~Cheung,
\emph{Landmark XII-C: One declared cubic weighting:
actual Euler source, changed Noether identity,
and the coupled-jet question}.
Supplied manuscript \texttt{XII-C.tex}.
Selected action, complete local source, changed identity,
and fixed-input jet verdict.

\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}.
Inverse normalization, free identities, and layer qualifications.

\bibitem{XIG}
N.~Cheung,
\emph{Landmark XI-G: The free-shell class of the cubic Einstein
kernel: the onset covector is inherited by every representative}.
Supplied manuscript \texttt{XI-G.tex}.
Original class exclusion; no source-ramping admission
statement 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.
Smooth Cauchy theory, causal Green operators, and finite propagation.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
second edition, Springer, 1990.
Wavefront bounds for differential operators, distribution
products, kernel composition, and proper pushforward.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428.
Functional and deformation-quantization background;
the present admission is proved for its explicit compact
polynomial kernel.

\end{thebibliography}

\end{document}