% !TeX program = pdflatex
%=====================================================================
% XII-F.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XII-F.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XII-F.tex
%
% Result:
%   The companion vanishes under the copied causal plateau condition.
%   The Einstein/auxiliary remainder need not vanish on the
%   gauge-fixed free shell, even when that condition holds.
%
% Explicit main witness, in a compactly realized dependence cylinder:
%   m = 0,
%   psi = t^2 + (x^1)^2,
%   gamma = psi eta,
%   phi = psi,
%   b = D gamma = (-2t,2x^1,0,0),
%   xi = a(t) partial_t, a >= 0 a nonzero compact bump.
%
% Its auxiliary remainder is
%   Delta_xi Phi_D^(2)
%     = -8 int f(t,x) int_0^t (t-s)^2 a(s) ds d^4x < 0.
%
% Debugging:
%   1. CP is sufficient, not logically necessary for every individual
%      companion cancellation. The off-CP diagnostic proves that a
%      universal cancellation cannot omit an appropriate support
%      hypothesis.
%   2. The support of the switched companion display is the support
%      of its sampling terms after transposition, not the full
%      functional support of its metric argument.
%   3. Pointwise configuration-dependent CP is distinguished from a
%      uniform CP hypothesis when an identity of carrier classes
%      is asserted.
%   4. Local A gamma = 0 on U is a sufficient condition for this
%      remainder. It does not decide XI-G's global Einstein-subshell
%      representative question.
%   5. Pure auxiliary/ghost variations leave the off-shell formula
%      Phi_D^(2)[gamma,phi] unchanged. A prospective cancelling rule
%      would need a specified induced action on gamma or phi, or a
%      separately constructed observable completion.
%
% Retained:
%   XII-E's minus sign on C_{3,D};
%   XII-D's inverse, root, admission, and signed response identity;
%   XII-C's source and empty fixed-input invariant jet;
%   XII-A's carrier and XI-G's original-class exclusion.
%
% Not constructed:
%   a quantum retarded product of J_D, an interacting differential,
%   a loop, a state, an antifield completion, or a sharp-width limit.
%
% No successful compilation, proof-assistant verification,
% independently executed symbolic computation, independent referee
% review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
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\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

\allowdisplaybreaks
\setlength{\emergencystretch}{3em}
\setlist{leftmargin=*,itemsep=3pt,topsep=5pt}
\numberwithin{equation}{section}
\setcounter{tocdepth}{1}

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

\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{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Afree}{\mathscr A_0}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}}
\newcommand{\CD}{\mathcal C_{3,\mathrm D}}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}}
\newcommand{\rD}{r_{\mathrm D}}
\newcommand{\rg}{r_g}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Comp}{\mathfrak C_\xi}
\newcommand{\Ein}{\mathfrak E_\xi}
\newcommand{\Aux}{\mathfrak B_\xi}
\newcommand{\Vp}{\widetilde V_+}
\newcommand{\Vm}{\widetilde V_-}
\newcommand{\Edist}{\mathcal E'}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Sym}{Sym}

\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-F}\\[0.4em]
The gauge-fixed remainder of $\Phi^{(2)}_{\mathrm D}$:\\
companion support, the auxiliary pairing,\\
and what is not a Ward identity}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
XII-E assembled the scalar-linear second-order relational coefficient
and left its gauge-fixed remainder explicit. We decide that remainder
without changing the source, inverse, insertion sign, or parameter
space.

The propagated companion has no ramp-layer contribution. Its switching
display is retained, its cancellation under the copied causal plateau
condition is stated with that hypothesis, and an explicit off-condition
diagnostic shows why universal cancellation cannot omit a support
assumption.

The Einstein remainder equals a compact auxiliary pairing on the
gauge-fixed free shell. It vanishes on configurations satisfying
$D^*b=0$ near the parameter support, but it does not vanish on the whole
shell. We exhibit a massless free scalar, a free pure-trace metric with
affine auxiliary field, a compact plateau parameter, and an observation
satisfying the causal plateau condition for which the assembled variation
is strictly negative. The calculation also fixes the value of the
auxiliary pairing. Its compact microcausal kernel defines a nonzero
class in the inherited carrier.

Unqualified classical invariance of the named coefficient is therefore
Proved obstructed. A cancelling interacting co-transformation is not
supplied by the free differential; its interface is recorded, not
constructed. The scalar-sourced slot is included once and has zero
metric variation. No quantum retarded product of the cubic source,
state, loop, interacting differential, sharp-width limit, old-class
repair, or Einstein-jet repair is introduced.
\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) {Companion support};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Off-plateau diagnostic};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Repair of the obstruction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Cancellation on the named configuration is not a repair of every extension.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{An off-plateau diagnostic is not a repair. 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-F}
\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) {Companion support};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Repair of the obstruction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Cancellation on the named configuration is not a repair of every extension.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{An off-plateau diagnostic is not a repair. 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-F}
\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) {Companion support};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Off-plateau diagnostic};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Repair of the obstruction};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Cancellation on the named configuration is not a repair of every extension.};
\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-F}
\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) {Companion support};
\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) {Repair of the obstruction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Off-plateau diagnostic. Cancellation on the named configuration is not a repair of every extension.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Companion support is not Repair of the obstruction. 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-F}
\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: Companion support};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Off-plateau diagnostic};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Repair of the obstruction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Cancellation on the named configuration is not a repair of every extension.};
\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-F}
\end{figure}

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


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



\scopebox{Release and proof boundary}{%
\textbf{Constructed:} the companion support statement, an explicit
off-causal-plateau diagnostic, the shell rewriting of the Einstein
row, and cancellation on the named local configuration subclass.

\smallskip
\textbf{Proved obstructed:} unqualified invariance of
$\Phi^{(2)}_{\mathrm D}$ on the complete gauge-fixed free shell.
A compact plateau witness satisfies the copied causal plateau
condition and has a strictly nonzero auxiliary pairing.

\smallskip
\textbf{Unresolved as a separate future problem:} an additional
co-transformation or observable completion cancelling that pairing.
Its absence is not promoted to a general no-go theorem.

\smallskip
No quantum retarded product, interacting differential, state, loop,
antibracket, sharp limit, or coupled-jet repair is claimed.
The inherited inverse coefficient and insertion sign are unchanged.}

\tableofcontents

%=====================================================================
\section{The question, the endings, and the copied handoff}
\label{sec:inputs}
%=====================================================================

The three endings are \textbf{Constructed},
\textbf{Proved obstructed}, and \textbf{Unresolved}.
They apply to the named problem, not to every possible completion.
A missing map is not a no-go. A nonzero variation of a specified
classical lift is not removed by subsequently reducing the parameter
space.

The closed chain is
\[
 \text{XII-C source}
 \longrightarrow
 \text{XII-D response and insertion}
 \longrightarrow
 \text{XII-E assembled coefficient}.
\]
The remaining question is whether the auxiliary row of that
coefficient vanishes on the gauge-fixed free shell under the copied
causal plateau condition.

\begin{inputstatement}[XII-E: assembly and sign]
\label{input:assembly}
The scalar-linear coefficient is
\begin{equation}
 \begin{split}
 \PhiD(f,g)[\gamma,\phi]
 &=\Asc[f,g](\gamma,\rD[\gamma];\phi)\\
 &=\Asc[f,g](\gamma,0;\phi)-\CD(f,g)[\gamma,\phi],
 \end{split}
 \label{eq:assembly-input}
\end{equation}
where
\[
 \rD=-u_{\mathrm D}=\PP\ER\JD,\qquad
 u_{\mathrm D}=-\PP\ER\JD,\qquad
 Lu_{\mathrm D}=\JD,\qquad L=-\Box\PP.
\]
It belongs to $\Fpol$ and has class $Q_H\PhiD\in\Ealg$.
The opposite-sign assembly has the doubled two-jet defect already
detected in XII-E. That sign question is not reopened \cite{XIIE}.
\end{inputstatement}

\begin{inputstatement}[XII-E: the finite remainder]
\label{input:remainder}
For $\xi\in\ga$ and $\Pmass\phi=0$, define
\[
 s_\xi[\gamma]
 =\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi,\qquad
 \rho_\xi[\gamma]=\ER Ds_\xi[\gamma],
\]
\[
 e_\xi[\gamma]
 =\PP\ER\Lie_\xi^{\mathrm{dens}}(A\gamma).
\]
With
$\Delta_\xi F=F[\gamma+K\xi,\phi]-F[\gamma,\phi]$,
\begin{equation}
 \Delta_\xi\PhiD
 =-\Aone[f,g](K\rho_\xi;\phi)
   +\int(\Lie_\xi\beta_{f,g}[\phi])\cdot A\gamma\dd^4y.
 \label{eq:remainder-input}
\end{equation}
The second term also equals $\Aone[f,g](e_\xi;\phi)$.
On $\Csol$, $A\gamma=-2D^*b$, so its auxiliary form is
\[
 -2\int(D\Lie_\xi\beta_{f,g}[\phi])_\nu b^\nu\dd^4y.
\]
These are classical identities of the specified lifts, not a quantum
Ward identity or an action on the quotient algebra \cite{XIIE}.
\end{inputstatement}

\begin{inputstatement}[XII-D through XII-E: causal plateau]
\label{input:CP}
Let $K_f=\supp f$. If $\Pmass\phi=0$ and
\begin{equation}
 \text{\rm(CP)}\qquad
 g=1\ \text{on a neighbourhood of }\
 J^-(K_f)\cap\supp(K\rho_\xi[\gamma]),
 \label{eq:CP}
\end{equation}
then $\Aone[f,g](K\rho_\xi;\phi)=0$.
The companion is retarded from the compact parameter support above
the ramp. Its switching display and the associated sampling supports
are those in Section~\ref{sec:companion}. No new ramp term is attached
to it \cite{XIID,XIIE}.
\end{inputstatement}

\begin{inputstatement}[XII-D: completed insertion]
The insertion is $\CD=\Aone[u_{\mathrm D};\phi]$, with both legs
retained. Its compact collar root and compact polarized kernel are
constructed, $\WF(t_{\mathrm D})\subset Z_3$, and
$Q_H\CD\in\Ealg$. The sign $Lu_{\mathrm D}=\JD$ differs from the
action-coefficient sign $L\rD=-\JD$ \cite{XIID}.
\end{inputstatement}

\begin{inputstatement}[Source, carrier, and free normalization]
XII-C fixes
\[
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}+\kappa\int(\sig-1)\ell_3\dd^4x,\qquad
 \JD=\sig J_3+\mathcal D_{\sig}.
\]
Its pure-trace layer contraction is
$-6(\partial_\mu\sig)\psi\partial^\mu\psi$.
The named invariant jet
$\Jcal^{\mathrm D}_{g,\sig}(\phi_m)$ remains empty by a late-switch
compatibility defect, not by failure of a Green operator
\cite{XIIC}.

XII-A supplies
\[
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                      \Nsol[[\hbar,\kappa]],
\]
with the complete free shell
\[
 \Csol=\{\Box\gamma=0,\ \Pmass\phi=0,\
 \Box c=\Box\bar c=0,\ b=D\gamma\}.
\]
The equation and action operators have this same kernel.
The original detected XI-G classes remain excluded
\cite{XIIA,XIG}.

XI-B fixes
\[
 \Gret=-\PP\ER,\qquad
 A+2D^*D=-\Box\PP,\qquad
 \PP D^*=-\RR,\qquad \RR=K/2.
\]
The plateau residual intersection is $\{0\}$, and propagated support
is not layer support. Its ghost rule $s_0\gamma=\RR c$ is not a
nonlinear metric co-transformation \cite{XIB}.
\end{inputstatement}

X-A's scalar-completion covariance has already been consumed by
XII-E. We do not repeat it or import any quantum or sharp-limit
claim. The new calculations use smooth wave uniqueness and finite
propagation to realize compact local witnesses \cite{BGP}.

\subsection{F0: entering status}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.39\linewidth}X@{}}
\toprule
Object & Status entering XII-F\\
\midrule
$\PhiD=\Asc(\gamma,0;\phi)-\CD$ &
Constructed and admitted; insertion included once with coefficient
$-1$\\[0.3em]
Companion $-\Aone(K\rho_\xi;\phi)$ &
Displayed; zero under (CP); propagated above the ramp\\[0.3em]
Einstein row $\int\Lie_\xi\beta\cdot A\gamma$ &
Displayed; zero under the local Einstein condition\\[0.3em]
Auxiliary row $-2\int D\Lie_\xi\beta\cdot b$ &
Printed; its general free-shell value not decided\\[0.3em]
Scalar-sourced slot $\Aone(\rg[\phi];\phi)$ &
Admitted; metric variation zero\\[0.3em]
$Q_H\PhiD$ &
Class of a classical insertion, not a retarded product\\[0.3em]
Old exclusion, empty jet, inverse coefficient &
Retained; not reopened\\
\bottomrule
\end{tabularx}
\end{center}

F0 is \textbf{Recorded}. In particular, the auxiliary row is not
assumed to vanish.

%=====================================================================
\section{F1: the sheet and the four-row ledger}
\label{sec:sheet}
%=====================================================================

Use
\[
 M=\R^4,\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta,\qquad \Pmass=\Box+m^2,
\]
\[
 \MT=[0,T]\times\R^3,\qquad \MTo=(0,T)\times\R^3.
\]
The inherited ramp $\sig$ is smooth, zero for $t\le0$, and one for
$t\ge\eps$, with $0<\eps<T$. The switch and parameter space are
\[
 g\in C_c^\infty(\MTo;[0,1]),\qquad
 \overline U\subset P_g\cap\{\eps<t<T\},\qquad
 P_g=\Int\{g=1\},\qquad
 \ga=C_c^\infty(U;TM).
\]
Fields are ambient smooth configurations; no decay is required.
All tensor pairings use the flat Lorentz identifications.
Parentheses include $\tfrac12$.

For a symmetric covariant tensor $r$,
\[
 \PP r=r-\tfrac12\eta\tr_\eta r,\qquad
 D^\nu r=\partial_\mu(\PP r)^{\mu\nu},
\]
\[
 (Kv)_{\mu\nu}=\partial_\mu v_\nu+\partial_\nu v_\mu,\qquad
 (D^*v)_{\mu\nu}
 =-\partial_{(\mu}v_{\nu)}
       +\tfrac12\eta_{\mu\nu}\partial_\rho v^\rho.
\]
Define
\[
 a_1^{\mu\nu}[r]
 =\tfrac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},\qquad
 b_1[r]=\tfrac12\tr_\eta r.
\]
The inherited first completion is
\begin{align}
 X_1^\mu[r]
 &=-\ER\partial_\lambda
                    \bigl(\sig a_1^{\lambda\mu}[r]\bigr),\\
 V_{1,g}[r]\phi
 &=\partial_\mu\bigl(g a_1^{\mu\nu}[r]\partial_\nu\phi\bigr)
                       +m^2g b_1[r]\phi,\\
 p_1[r;\phi]&=-\ERm V_{1,g}[r]\phi,\\
 \Aone[f,g](r;\phi)
 &=\int f\bigl(p_1[r;\phi]-X_1[r]\cdot\partial\phi\bigr)\dd^4x.
 \label{eq:first-completion}
\end{align}
In particular, $\partial g$ remains inside the scalar divergence.

The root notation is fixed by
\begin{equation}
 \Aone[f,g](r;\phi)=\int\alpha_{f,g}[\phi]\cdot r\dd^4z,
 \qquad
 \beta_{f,g}[\phi]=-\EA(\PP\alpha_{f,g}[\phi]).
 \label{eq:roots}
\end{equation}
These are XII-D's roots, not new choices. The raw advanced root can
extend below $t=0$; multiplication by the compact parameter in $U$
makes all pairings used here compact without a new boundary
prescription.

For a contravariant density $v$,
\[
 (\Lie_\xi^{\mathrm{dens}}v)^{\mu\nu}
 =\xi^\rho\partial_\rho v^{\mu\nu}
 -v^{\rho\nu}\partial_\rho\xi^\mu
 -v^{\mu\rho}\partial_\rho\xi^\nu
 +(\partial_\rho\xi^\rho)v^{\mu\nu}.
\]
The Lie derivative of $\beta$ is the ordinary covariant tensor Lie
derivative.

\begin{definition}[The two remainder functionals]
For fixed $\xi,f,g$ set
\begin{align}
 \Comp[\gamma,\phi]
 &:=\Aone[f,g](K\rho_\xi[\gamma];\phi),\\
 \Ein[\gamma,\phi]
 &:=\int(\Lie_\xi\beta_{f,g}[\phi])\cdot A\gamma\dd^4y,\\
 \Aux[b,\phi]
 &:=-2\int(D\Lie_\xi\beta_{f,g}[\phi])_\nu b^\nu\dd^4y.
 \label{eq:remainder-names}
\end{align}
\end{definition}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}X@{}}
\toprule
Ledger row & Exact meaning\\
\midrule
Companion & $-\Comp$, with the switching display below\\[0.3em]
Einstein row & $+\Ein$\\[0.3em]
Shell rewriting & $\Ein|_{\Csol}=\Aux|_{\Csol}$; an alternative
form of the same row, not an additional summand\\[0.3em]
Scalar-sourced slot & Metric variation $0$; recorded once in
Section~\ref{sec:scalar-slot}\\
\bottomrule
\end{tabularx}
\end{center}

\begin{lemma}[The shell rewriting]
\label{lem:shell}
On $\Csol$,
\[
 A\gamma=-2D^*b,\qquad
 \Ein=\Aux.
\]
\end{lemma}

\begin{proof}
$A\gamma=-\Box\PP\gamma-2D^*D\gamma=-2D^*b$.
For the compact tensor $q=\Lie_\xi\beta_{f,g}[\phi]$,
$\int q\cdot D^*b=\int(Dq)_\nu b^\nu$ by the flat adjoint identity.
\end{proof}

Thus the copied variation is
\begin{equation}
 \Delta_\xi\PhiD=-\Comp+\Ein,\qquad
 \Delta_\xi\PhiD\big|_{\Csol}=-\Comp+\Aux.
 \label{eq:four-row-result}
\end{equation}
Neither $K\rho_\xi$ nor $\Lie_\xi\gamma$ is replaced by a ghost
variation. No new Noether identity is derived.

%=====================================================================
\section{F2: companion support and the off-condition diagnostic}
\label{sec:companion}
%=====================================================================

Let
\[
 S_\xi=\supp\xi,\qquad
 \mathcal D_f=J^-(K_f)\cap\MT.
\]
Since every term of $s_\xi$ contains $\xi$ or a derivative of it,
\[
 \supp(Ds_\xi)\subset S_\xi\Subset U.
\]
Finite propagation gives
\begin{equation}
 \supp\rho_\xi\subset J^+(S_\xi)\cap\MT.
 \label{eq:rho-support}
\end{equation}
The minimum time of $S_\xi$ is strictly greater than $\eps$.
Consequently $\rho_\xi$ and all its derivatives vanish before that
time.

\begin{lemma}[No ramp contribution]
\label{lem:no-ramp}
For the handed companion,
\[
 X_1[K\rho_\xi]=\rho_\xi.
\]
\end{lemma}

\begin{proof}
The tensor $K\rho_\xi$ is supported above the ramp, so
$\sig K\rho_\xi=K\rho_\xi$.
Using $DK=\Box$ gives
\[
 X_1[K\rho_\xi]=\ER DK\rho_\xi=\ER\Box\rho_\xi=\rho_\xi,
\]
where the last equality follows from zero past data.
\end{proof}

This removes a ramp factor from this companion calculation. It does
not assert that the propagated field is supported in the parameter
region or in the ramp layer.

\subsection{The copied switching display}

For free $\phi$, put $q_\xi=\rho_\xi^\mu\partial_\mu\phi$ and
$v_f=\EAm f$. The display from XII-E is
\begin{equation}
 \begin{split}
 \Comp={}&
 \int f(g-1)q_\xi\dd^4x\\
 &-\int v_f\left(
 [\Pmass,g]q_\xi+
 (\partial_\mu g)a_1^{\mu\nu}[K\rho_\xi]\partial_\nu\phi
 \right)\dd^4x,
 \end{split}
 \label{eq:companion-display}
\end{equation}
where
\[
 [\Pmass,g]q=(\Box g)q+2(\partial^\mu g)\partial_\mu q.
\]
It follows from the free-scalar identity
$V_{1,1}[K\rho]\phi=-\Pmass(\rho\cdot\partial\phi)$,
the zero-data relation
$\ERm\Pmass(gq_\xi)=gq_\xi$, and
Lemma~\ref{lem:no-ramp}. We use it as the handed switching formula,
not as a new cubic response.

\begin{proposition}[Sampling support of the companion]
\label{prop:companion-support}
After transposition onto $v_f$, the terms in
\eqref{eq:companion-display} sample $\rho_\xi$ and its derivatives
only in
\begin{equation}
 \left(
 \supp\dd g\ \cup\ \supp((1-g)f)
 \right)
 \cap J^+(S_\xi)\cap\mathcal D_f.
 \label{eq:sampling-support}
\end{equation}
This set is above the ramp. Under (CP), $\Comp=0$.
\end{proposition}

\begin{proof}
The local term is supported in
$\supp((1-g)f)\cap\supp\rho_\xi$.
The advanced test $v_f$ is supported in $J^-(K_f)$.
Every coefficient of the second term contains a derivative of $g$,
whose support lies in $\supp\dd g$; derivatives of $\rho_\xi$ have
support contained in $\supp\rho_\xi$.
Apply \eqref{eq:rho-support}.
Cancellation under (CP) is Input statement~\ref{input:CP}.
\end{proof}

The set \eqref{eq:sampling-support} is not a claim that the full
functional dependence on $\gamma$ is localized there:
$\rho_\xi$ was itself propagated from $S_\xi$.
Nor is it a support bound for the scalar response before smearing.

The condition (CP) uses $\supp(K\rho_\xi)$, not
$\supp\rho_\xi$. Thus its cancellation need not arise by deleting
each term of \eqref{eq:companion-display} separately.
For example, a locally constant tail of $\rho_\xi$ can make the
local term and a propagated switching term cancel.

A useful configuration-independent sufficient condition is
\begin{equation}
 \text{\rm(UCP)}\qquad
 g=1\ \text{near }\ J^-(K_f)\cap J^+(S_\xi)\cap\MT.
 \label{eq:UCP}
\end{equation}
It implies (CP) for every $\gamma$ for this fixed parameter and
observation. We use (UCP) in the decisive witness.

\subsection{A nonzero off-(CP) check}

\begin{proposition}[Off-causal-plateau diagnostic]
\label{prop:offCP}
There are admissible compact data, a free scalar, and a free metric
for which
\[
 \supp((1-g)f)\cap J^+(S_\xi)\ne\varnothing,\qquad
 \Comp\ne0.
\]
The example violates (CP), not the cancellation theorem under (CP).
\end{proposition}

\begin{proof}
Work first in a dependence cylinder where the spatial cutoffs are
one. Let $m=0$, $\phi=t$, and $\gamma=t\eta$.
Choose $a\ge0$, $a\not\equiv0$, supported in a time interval
$(t_a,t_b)$ above the ramp.
Let
\[
 v=a(t)\cos(kx^1)\partial_t,\qquad \xi_\lambda=\lambda v,
 \qquad k>0.
\]
A compact spatial cutoff will be inserted below.

Write $r=\Lie_v(t\eta)$. Its nonzero local components are
\[
 r_{00}=(a+2ta')\cos(kx^1),\qquad
 r_{0\,1}=-tka\sin(kx^1),\qquad
 r_{ij}=-a\cos(kx^1)\delta_{ij}.
\]
Direct differentiation gives
\[
 (Dr)^0=t(a''+k^2a)\cos(kx^1),\qquad (Dr)^i=0.
\]
Hence the linear-in-$\lambda$ companion parameter is
\[
 \rho^{(1)}=R(t)\cos(kx^1)\partial_t,\qquad
 R''+k^2R=t(a''+k^2a),
\]
with zero initial data. Since
$(\partial_t^2+k^2)(ta)=t(a''+k^2a)+2a'$,
\begin{equation}
 R(t)=ta(t)-2\int_0^t\cos(k(t-s))a(s)\dd s.
 \label{eq:R-diagnostic}
\end{equation}

Choose a temporal switch $g_0$ equal to one on the support of $a$,
falling smoothly and nontrivially from one to zero in
$(t_c,t_d)$ with $t_b<t_c$, and zero near an observation time
$t_o>t_d$.
Its earlier rise occurs before $\supp a$.
Choose $k$ sufficiently small that
\[
 0<k(t-s)<\pi
\]
whenever $t\in[t_c,t_o]$ and $s\in\supp a$.
Above $\supp a$,
\[
 R'(t)=2k\int_0^t\sin(k(t-s))a(s)\dd s>0.
\]

For $\rho^{(1)}$, the free scalar $\phi=t$ gives
\[
 V_{1,g_0}[K\rho^{(1)}]\phi
 =-\left(g_0(R''+k^2R)+g_0'R'\right)\cos(kx^1).
\]
The coordinate leg is $\rho^{(1)}$ by
Lemma~\ref{lem:no-ramp}.
Because $g_0=1$ on the support of $R''+k^2R$,
the linear companion integrand is
\begin{equation}
 d(t)\cos(kx^1),\qquad
 d(t)=\int_0^t\frac{\sin(k(t-s))}{k}\,g_0'(s)R'(s)\dd s.
 \label{eq:d-diagnostic}
\end{equation}
For $t$ just after $t_d$, every nonzero source term in this integral
has negative sign: $g_0'\le0$, $R'>0$, and the sine kernel is
positive. Thus $d(t)<0$.

Choose $f\ge0$, nonzero, supported there and near $x^1=0$, where
$\cos(kx^1)>0$. Its linear companion value $J$ is strictly negative.
The exact finite parameter has
\[
 s_{\xi_\lambda}
 =\lambda\Lie_v\gamma+\tfrac{\lambda^2}{2}\Lie_v Kv.
\]
Linearity of the propagation and of $\Aone$ therefore gives
\[
 \Comp[\gamma,\phi]=\lambda J+\lambda^2J_2
\]
for a finite constant $J_2$ independent of $\lambda$.
It is nonzero for all sufficiently small positive $\lambda$.

Now replace $v$ by a compact spatial realization equal to the
printed expression throughout the observation dependence region.
Take $g=g_0(t)\zeta_g(\mathbf x)$, with $\zeta_g=1$ on the same
larger cylinder and compactly supported beyond it. Choose $U$ to
contain $\supp\xi_\lambda$, with closure in the early plateau
where $g_0=1$. The cutoffs are chosen before the calculation and
large enough that their transition regions cannot influence
$\supp f$. Finite propagation preserves
\eqref{eq:R-diagnostic}--\eqref{eq:d-diagnostic} on the observation
region \cite{BGP}.

At the observation $g=0$, and it lies in the causal future of the
parameter support. During the falling switch,
$(K\rho_{\xi_\lambda})_{00}$ has leading term
$2\lambda R'(t)\cos(kx^1)$, nonzero in the relevant causal past.
Thus (CP) fails.
The fields $\gamma=t\eta$ and $\phi=t$ are free; one may set
$b=D\gamma$ and the ghosts to zero.
\end{proof}

This proves that a universal companion cancellation cannot simply
discard its support hypothesis. It does not prove that (CP) is
necessary for every individual zero value: accidental cancellations
and special fields are possible. F2 ends \textbf{Constructed} as a
support statement and diagnostic, not Proved obstructed for the
on-(CP) identity.

%=====================================================================
\section{F3: cancellation on a subclass and a nonzero shell witness}
\label{sec:auxiliary}
%=====================================================================

\subsection{The local cancellation subclass}

\begin{proposition}[Cancellation on the named configuration subclass]
\label{prop:subclass}
Suppose $\Phi\in\Csol$, (CP) holds, and
\[
 D^*b=0\quad\text{on a neighbourhood of }S_\xi.
 \label{eq:local-closed}
\]
Then $\Delta_\xi\PhiD=0$.
For a free-shell configuration, \eqref{eq:local-closed} is equivalent
to $A\gamma=0$ on that neighbourhood.
In particular, constant $b$ gives zero auxiliary pairing.
\end{proposition}

\begin{proof}
Under (CP), the companion vanishes.
By compact integration by parts,
\[
 \Aux=-2\int(\Lie_\xi\beta_{f,g}[\phi])\cdot D^*b\dd^4y.
\]
The compact tensor $\Lie_\xi\beta$ is supported in $S_\xi$.
The result follows from \eqref{eq:local-closed} and
Lemma~\ref{lem:shell}. A constant vector field has $D^*b=0$.
\end{proof}

This is a restriction of configurations, not of $\ga$.
It is smaller than the complete gauge-fixed shell.
It is not a theorem about representatives of XI-G's old cubic class
on its global Einstein subshell.

\subsection{Compact realization of the decisive witness}

Fix times
\[
 \eps<t_a<t_b<t_c<t_d<T.
\]
Choose a nonzero nonnegative bump
$a\in C_c^\infty((t_a,t_b))$.
Choose a nonzero nonnegative observation
\[
 f\in C_c^\infty((t_c,t_d)\times B_r;[0,\infty)),
\]
where $B_r$ is a spatial ball centered at the origin.

Choose a spatial cutoff $\zeta$ equal to one on $B_R$, where
\[
 R>r+t_d-t_a,
\]
and supported in a larger ball. Set
\begin{equation}
 \xi=a(t)\zeta(\mathbf x)\partial_t.
 \label{eq:main-parameter}
\end{equation}
Choose an open relatively compact $U$ containing $\supp\xi$ with
$\overline U\subset\{\eps<t<T\}$.
Choose $g\in C_c^\infty(\MTo;[0,1])$ equal to one on a neighbourhood
of both $\overline U$ and
\[
 J^-(K_f)\cap J^+(S_\xi)\cap\MT.
 \label{eq:witness-diamond}
\]
These sets are compact and separated from the temporal endpoints,
so such a switch exists. This freezes an admissible tuple
$(f,g,U,\xi)$ before evaluation. It satisfies (UCP), hence (CP).

Put
\begin{equation}
 m=0,\qquad
 \psi=t^2+(x^1)^2,\qquad
 \gamma_{\mu\nu}=\psi\eta_{\mu\nu},\qquad
 \phi=\psi,\qquad
 b^\nu=D^\nu\gamma,\qquad c=\bar c=0.
 \label{eq:main-fields}
\end{equation}

\begin{lemma}[The witness is on the complete free shell]
\label{lem:main-shell}
The configuration \eqref{eq:main-fields} belongs to $\Csol$.
Its auxiliary field and its Einstein tensor coefficient are
\begin{equation}
 b=(-2t,2x^1,0,0),\qquad
 A\gamma=\operatorname{diag}(-4,-4,0,0).
 \label{eq:main-Ab}
\end{equation}
In particular $D^*b\ne0$.
\end{lemma}

\begin{proof}
$\Box\psi=2-2=0$, hence $\Box\gamma=0$ and $\Box\phi=0$.
Trace reversal gives $\PP(\psi\eta)=-\psi\eta$, so
$b^\nu=-\partial^\nu\psi$, which is \eqref{eq:main-Ab}'s first
formula. Its divergence is zero, and
\[
 (D^*b)_{00}=2,\qquad (D^*b)_{11}=2,
\]
with the other components zero.
Lemma~\ref{lem:shell} gives the second formula.
The ghost equations hold because the ghosts vanish.
\end{proof}

The affine $b$ is permitted: no decay or compact-support requirement
was imposed on free configurations. All test pairings remain compact.
The example is deliberately not a constant auxiliary field.

\subsection{Evaluation of the Einstein row}

Define the already handed compact-source response
\[
 w_\xi=\Lie_\xi^{\mathrm{dens}}(A\gamma),\qquad
 e_\xi=\PP\ER w_\xi.
\]
This evaluates XII-E's Einstein row. It is not a new application of
a propagator to $\JD$.

In the observation dependence region the spatial cutoff is one.
By \eqref{eq:main-Ab},
\begin{equation}
 w_\xi^{00}=4a'(t),\qquad
 w_\xi^{11}=-4a'(t),\qquad
 w_\xi^{\mu\nu}=0\quad\text{otherwise}.
 \label{eq:main-w}
\end{equation}
Indeed, for a constant diagonal contravariant density and a temporal
parameter, its $00$ component acquires $-a'v^{00}$ and its spatial
components acquire $+a'v^{ij}$.

Set
\[
 B(t)=\int_0^t a(s)\dd s,\qquad
 C(t)=\int_0^t(t-s)a(s)\dd s.
\]

\begin{lemma}[The completed Einstein response on the witness]
\label{lem:main-evaluation}
On the observation region,
\[
 (e_\xi)_{22}=(e_\xi)_{33}=4B(t),
 \qquad
 (e_\xi)_{\mu\nu}=0\quad\text{otherwise},
 \label{eq:main-e}
\]
and
\begin{equation}
 p_1[e_\xi;\phi]-X_1[e_\xi]\cdot\partial\phi
 =-8\int_0^t(t-s)^2a(s)\dd s.
 \label{eq:main-integrand}
\end{equation}
\end{lemma}

\begin{proof}
The trace of \eqref{eq:main-w} is $8a'$.
Consequently $\PP w_\xi$ has only its $22$ and $33$ components
nonzero, both equal to $4a'$.
The spatially homogeneous zero-data inverse satisfies
$\ER a'=B$, giving \eqref{eq:main-e}.

This local calculation equals the compact-source calculation:
all sources capable of influencing the observation have spatial
coordinates in $B_R$, where $\zeta=1$.
The transitions of $\zeta$ are outside that causal past.
Nested retarded operations do not enlarge the overall causal
dependence region. Finite propagation therefore justifies the
homogeneous formulas \cite{BGP}.

For \eqref{eq:main-e}, the relevant density coefficients are
\[
 a_1^{00}[e_\xi]=-4B,\qquad
 a_1^{11}[e_\xi]=4B,\qquad
 a_1^{22}[e_\xi]=a_1^{33}[e_\xi]=0.
\]
The entire response is supported after $t_a>\eps$.
Thus the coordinate ramp is one and
\[
 X_1^0[e_\xi]=4C(t),\qquad X_1^i[e_\xi]=0.
\]
On the causal region where $e_\xi$ contributes, $g=1$ by
\eqref{eq:witness-diamond}. Since
$\partial_t\phi=2t$ and $\partial_1\phi=2x^1$,
\[
 V_{1,1}[e_\xi]\phi
 =\partial_t(-8tB)+\partial_1(8x^1B)
 =-8ta(t).
\]
Hence
\[
 p_1[e_\xi;\phi]
 =8\int_0^t(t-s)s\,a(s)\dd s,
 \qquad
 X_1[e_\xi]\cdot\partial\phi=8tC(t).
\]
Subtracting proves \eqref{eq:main-integrand}.
\end{proof}

\begin{theorem}[Nonzero auxiliary pairing under causal plateau]
\label{thm:obstruction}
For the compact admissible tuple
\eqref{eq:main-parameter}--\eqref{eq:main-fields}, (CP) holds and
\begin{equation}
 \boxed{
 \Delta_\xi\PhiD
 =\Ein=\Aux
 =-8\int f(t,\mathbf x)
       \int_0^t(t-s)^2a(s)\dd s\,\dd^4x<0.
 }
 \label{eq:main-value}
\end{equation}
Equivalently,
\begin{equation}
 \boxed{
 \int(D\Lie_\xi\beta_{f,g}[\phi])_\nu b^\nu\dd^4y
 =4\int f(t,\mathbf x)
       \int_0^t(t-s)^2a(s)\dd s\,\dd^4x>0.
 }
 \label{eq:pairing-value}
\end{equation}
Thus unqualified invariance of $\PhiD$ on the complete gauge-fixed
free shell is false.
\end{theorem}

\begin{proof}
The switch satisfies (UCP), so the companion is zero.
The shell membership is Lemma~\ref{lem:main-shell}.
By the copied root identity,
$\Ein=\Aone[f,g](e_\xi;\phi)$.
Lemma~\ref{lem:main-evaluation} computes this value.
For every observation time, $t>t_b$ and $a$ is a nonzero
nonnegative bump, so the inner integral is strictly positive.
The nonzero nonnegative $f$ makes the displayed value strictly
negative.
Lemma~\ref{lem:shell} then gives
\eqref{eq:pairing-value}.
\end{proof}

The theorem is an existential counterexample in the frozen admissible
class, already at $m=0$. It does not assert a nonzero value for every
mass, observation, switch, or parameter. One counterexample suffices
to exclude the unrestricted invariance theorem.

The parameter space has not been shrunk. The field
$D^*b$ is nonzero exactly where a nonconstant auxiliary field can be
detected. The constant-$b$ zero check in
Proposition~\ref{prop:subclass} remains consistent with the
integration by parts.

%=====================================================================
\section{The nonzero carrier class and its exact scope}
\label{sec:carrier}
%=====================================================================

The witness proves a classical nonzero value. We now identify the
carrier statement it supports, without defining a gauge action on
the quotient.

\begin{proposition}[The auxiliary row is admitted and nonzero]
\label{prop:aux-class}
For fixed $\xi,f,g$, $\Aux[b,\phi]$ is an even,
ghost-number-zero compact microcausal polynomial of degree $(1,1)$
in $(b,\phi)$.
For the tuple of Theorem~\ref{thm:obstruction},
\[
 Q_H\Aux\ne0.
\]
Also the named lift $\Delta_\xi\PhiD$ belongs to $\Fpol$, and
\[
 Q_H(\Delta_\xi\PhiD)\ne0
\]
for that tuple. These conclusions persist at classical-symbol order
under every admissible smooth change of ordering.
\end{proposition}

\begin{proof}
XII-D's compact root kernel $B_{f,g}(y,x)$ satisfies
$\WF(B_{f,g})\subset Z_2$.
On $U$, its collar factor is one, so
$\Lie_\xi\beta_{f,g}$ is obtained from that kernel by differential
operators and multiplication by fixed smooth compact parameter
coefficients.
Applying $D$ does not enlarge its wavefront set.
Its $y$ support is in $S_\xi$ and its scalar projection is in
the compact set
\[
 O_{f,g}=K_f\cup(\supp g\cap J^-(K_f)).
\]
The resulting bilinear kernel is compact with wavefront set in
$Z_2$, which excludes the two all-causal cones. Thus it belongs to
XII-A's domain.

The initial configuration in Theorem~\ref{thm:obstruction} is on
$\Csol$ and gives a nonzero value. Therefore $\Aux\notin\Nsol$.

The functional $\PhiD$ is already admitted. Substitution of the
smooth configuration $K\xi$ into some of its metric slots, followed
by subtraction, preserves compactness and microcausality. Hence
its named finite-difference lift is admitted. Its nonzero shell
evaluation proves it also lies outside $\Nsol$.

For the present switch, (UCP) holds independently of $\gamma$.
Thus the companion vanishes for every free scalar and metric,
and the two lifts $\Delta_\xi\PhiD$ and $\Aux$ have the same
restriction to $\Csol$.

Finally, the inherited ordering maps are the identity modulo
$\hbar$. A nonzero classical-symbol class cannot be annihilated
by such transport \cite{XIIA}.
\end{proof}

This assertion does not require a new admission theorem for the
cubic source. The only new kernel observation concerns the already
displayed auxiliary remainder.

For a merely configuration-dependent instance of (CP), cancellation
holds on that configuration. It must not automatically be asserted as
an identity on the entire shell quotient. The stronger (UCP) used
above supplies the uniform hypothesis needed for that quotient
identity.

The compact shift $K\xi$ generally does not preserve $\Csol$.
Consequently $\Delta_\xi$ here is the difference of two prescribed
off-shell evaluations, tested on an initial shell configuration.
We do not claim it is an automorphism or a derivation of $\Ealg$.
The nonzero result excludes invariance of the named classical lift
on all such initial free configurations.

%=====================================================================
\section{F4: the interface of a prospective cancelling rule}
\label{sec:interface}
%=====================================================================

Theorem~\ref{thm:obstruction} supplies the nonzero premise for this
section. It obstructs $\PhiD$ alone. It does not obstruct every
possible further observable completion.

A proposed rule on the existing multiplet would have to specify
\[
 Y_\xi\Phi=
 (Y_\xi^\gamma,Y_\xi^\phi,Y_\xi^c,
                  Y_\xi^{\bar c},Y_\xi^b)
\]
and its dependence on the parameter, fields, coupling order, and
Cauchy data. If it is intended to act on the carrier, preservation
of the equation ideal and compatibility with the inherited product
must also be supplied. Merely naming its auxiliary component does
not meet these requirements.

At the level of the named lift, a cancelling interface under a
uniform causal plateau hypothesis is
\begin{equation}
 \left\langle(\PhiD)'_\gamma,Y_\xi^\gamma\right\rangle
 +\left\langle(\PhiD)'_\phi,Y_\xi^\phi\right\rangle
 =-\Aux[b,\phi].
 \label{eq:cancelling-interface}
\end{equation}
For a finite rule, the corresponding finite coefficient law must
replace this infinitesimal display. If the proposed correction acts
at a lower coefficient in the full relational expansion, that
coefficient and its coupling order must be printed separately.
It cannot be added to the already assembled second metric slot a
second time.

There is an elementary slot restriction: the off-shell formula
$\PhiD[\gamma,\phi]$ has no independent $b,c,\bar c$ argument.
Therefore
\[
 Y_\xi^\gamma=Y_\xi^\phi=0
 \quad\Longrightarrow\quad
 \delta^{Y}_\xi\PhiD=0.
\]
Changing only auxiliary or ghost variables cannot cancel
\eqref{eq:main-value} in this formula. A more general proposal would
need an induced action on $\gamma$ or $\phi$, or a separately defined
observable term involving the additional slots.

XII-A's free rule
\[
 s_0\gamma=\RR c,\qquad s_0\bar c=ib,\qquad
 s_0\phi=s_0c=s_0b=0
\]
does not supply \eqref{eq:cancelling-interface}.
It has ghost number one and is not the even finite parameter
transformation used in the calculation. Substituting a compact
plateau parameter for a free ghost also changes the equation
domain; the nonzero compact residual intersection is absent by
XI-B.

No cancelling map is selected here. In particular, $b$ is not
replaced by an antifield, no contact term is chosen after the
witness is known, and no symbol $\widehat s_{\mathrm{int}}$ is
introduced. The existence of a suitable additional completion is
\textbf{Unresolved}. This separate status does not weaken the
proved counterexample to invariance of $\PhiD$ alone.

%=====================================================================
\section{F5: the scalar-sourced slot, recorded once}
\label{sec:scalar-slot}
%=====================================================================

XII-E's admitted scalar-sourced coefficient is
\[
 \Aone[f,g](\rg[\phi];\phi),\qquad
 \rg[\phi]=-\tfrac12\PP\ER(gT_0[\phi]).
\]
The full displayed gauge-fixed second-order coefficient is
\begin{equation}
 \PhiD(f,g)[\gamma,\phi]+\Aone[f,g](\rg[\phi];\phi).
 \label{eq:full-coefficient}
\end{equation}

\begin{corollary}[Metric variation of the scalar-sourced slot]
The metric variation of \eqref{eq:full-coefficient}, at fixed
$\phi$, equals $\Delta_\xi\PhiD$.
In particular the witness \eqref{eq:main-value} also detects that
variation of the full displayed coefficient.
\end{corollary}

\begin{proof}
XII-E's scalar-sourced-slot corollary states its admission and its
absence of a metric argument \cite{XIIE}. Its difference under
$\gamma\mapsto\gamma+K\xi$ is therefore zero.
\end{proof}

The tree is not recomputed. The slot enters once and is not an
additional auxiliary cancellation row.

A future identity omitting $\rg$ is a metric-sector identity, not
the full displayed second-order coefficient identity. A rule that
varies $\rg$ under a metric-only shift has introduced a metric
argument that this slot does not possess. A transformation of the
scalar would be a different, explicitly specified operation.

%=====================================================================
\section{F6: decision, exclusions, and successor package}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.24\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
F0 & Recorded &
Inherited coefficient, sign, remainder, carrier, and boundaries
separated.\\[0.4em]
F1 & Constructed &
Four rows printed; Einstein and auxiliary expressions identified
as the same row on the shell.\\[0.4em]
F2 & Constructed &
No-ramp statement, switching sampling support, and a nonzero
off-(CP) diagnostic.\\[0.4em]
F3 & Proved obstructed for unqualified invariance &
Local cancellation subclass proved; explicit free-shell
auxiliary witness under (CP) is nonzero.\\[0.4em]
F4 & Interface recorded; map Unresolved &
No extra cancelling differential or completion supplied.\\[0.4em]
F5 & Recorded &
Scalar-sourced slot admitted by XII-E, included once, with zero
metric variation.\\[0.4em]
F6 & Recorded &
No product, jet repair, old-class repair, loop, state, or limit
inferred.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Remainder release]
\label{thm:release}
For the inherited coefficient and parameter space:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The companion has the switching display
\eqref{eq:companion-display}, sampling support
\eqref{eq:sampling-support}, and no ramp-layer contribution.
It vanishes under the copied condition (CP).
\item On the complete gauge-fixed free shell, the remaining
Einstein row is the compact auxiliary pairing $\Aux$.
It vanishes when $D^*b=0$ near the parameter support.
\item It does not vanish universally on that shell under (CP).
The admissible witness of Theorem~\ref{thm:obstruction} has the
strictly negative value \eqref{eq:main-value}.
Its carrier class is nonzero.
\item Adding the scalar-sourced slot once leaves this metric
variation unchanged.
\end{enumerate}
Thus unqualified invariance of the named coefficient is
Proved obstructed. Existence of a further cancelling completion
is not decided.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:companion-support},
Lemma~\ref{lem:shell},
Proposition~\ref{prop:subclass},
Theorem~\ref{thm:obstruction},
Proposition~\ref{prop:aux-class}, and
Section~\ref{sec:scalar-slot}.
\end{proof}

The worst ending among F1--F3 is \textbf{Proved obstructed}.
F2 does not turn an off-condition diagnostic into an on-condition
obstruction. F4 does not turn absence of a map into a general no-go.
The actual obstruction is the explicit nonzero value of the named
coefficient on a permitted initial free configuration.

\subsection*{Non-identifications retained}

\begin{itemize}
\item $Q_H\PhiD$ is not a quantum retarded product of $\JD$.
No Hadamard line is contracted against $\JD$.
\item The original detected XI-G classes with
$\pm(k_1,k_2,0)$ and $k_1+k_2=2\dd t$ remain excluded.
$\PhiD$ is not a representative of those classes.
\item $\Jcal^{\mathrm D}_{g,\sig}(\phi_m)=\varnothing$ stands.
The slot $\rD+\rg$ solves a gauge-fixed forcing equation, not
the invariant coupled Einstein equation.
\item The inverse coefficient is $-1$, not $-4$.
No X-track mixed map, reciprocity test, or triangle residue
is imported.
\item No state, loop, antibracket, master identity, interacting
differential, or $\eps\to0$ limit is constructed.
\item The local pairing hypothesis $A\gamma=0$ near $S_\xi$
does not decide XI-G's global Einstein-subshell
representative question.
\item No cone, token, conical readout, or Enter record is used.
\end{itemize}

\subsection*{Successor package}

A successor receives:
\begin{enumerate}
\item the two remainder functionals and their shell identification;
\item the companion's switching display, its sampling support,
and the distinction between (CP) and a uniform carrier hypothesis;
\item the local cancellation subclass $D^*b=0$ near $S_\xi$;
\item the explicit compact plateau witness, with auxiliary
pairing value \eqref{eq:pairing-value};
\item the nonzero carrier class and the qualification that the
finite parameter shift is not a quotient automorphism;
\item the prospective cancelling interface, without an asserted
solution, and the scalar-sourced slot recorded once.
\end{enumerate}
It receives no Ward identity, quantum retarded product, interacting
BRST class, or completed auxiliary co-transformation.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \Delta_\xi\PhiD\big|_{\Csol}
 =-\Aone(K\rho_\xi;\phi)
   -2\int(D\Lie_\xi\beta_{f,g}[\phi])\cdot b,\\[2pt]
 \text{\rm(CP)}\Longrightarrow
 \Delta_\xi\PhiD\big|_{\Csol}
 =-2\int(D\Lie_\xi\beta_{f,g}[\phi])\cdot b,\\[2pt]
 D^*b=0\text{ near }S_\xi
 \Longrightarrow \Delta_\xi\PhiD=0,\\[2pt]
 \text{explicit free-shell witness under (CP):}\qquad
 \Delta_\xi\PhiD
 =-8\int f(t,\mathbf x)\int_0^t(t-s)^2a(s)\dd s\,\dd^4x<0.
 \end{gathered}}
\]

The local vanishing conditions are sufficient, not asserted to be
necessary for every individual zero value. What is excluded is
invariance without an additional configuration condition or a
separately constructed cancelling completion. A classical admitted
class has not been renamed as a retarded product.

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

\paragraph{Proof scope.}
The new results concern the propagated companion's sampling support,
two compactly realized diagnostics, the local cancellation subclass,
and an explicit nonzero auxiliary pairing on the complete free shell.
The inherited admission and source calculations are not repeated.

\paragraph{Obstruction scope.}
The counterexample obstructs unqualified classical invariance of the
specified coefficient. It does not obstruct every possible additional
observable term, consistent co-transformation, or interacting
construction.

\paragraph{Carrier scope.}
The auxiliary remainder has a compact microcausal bilinear kernel.
Its nonzero value on a complete free configuration proves that its
class is not removed by the inherited null ideal.
No gauge action on the quotient is inferred.

\paragraph{Support correction.}
The switching display's support describes where the propagated
companion is sampled. It is not the complete support of the
functional's dependence on the original metric field.
The copied (CP) is a sufficient condition; the off-condition example
does not promote it to an if-and-only-if condition.

\paragraph{Dependency scope.}
XII-E supplies the assembly, sign, variation, and scalar-sourced slot.
XII-D supplies the completion roots and causal plateau proposition.
XII-A supplies the carrier and null ideal. XI-B supplies the free
operator identities. XII-C supplies only the retained source and jet
boundaries. The compact realizations use standard smooth Cauchy
uniqueness and finite propagation.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant check,
independently executed symbolic computation, independent referee
review, or journal acceptance is asserted. The witnesses and their
signs are established by the displayed analytic calculations.

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

\bibitem{XIIE}
N.~Cheung,
\emph{Landmark XII-E: Assembly of the second-order relational
coefficient after admission of the density-weighted cubic insertion}.
Supplied manuscript \texttt{XII-E.tex}.
Assembly sign, exact finite variation, companion display,
Einstein/auxiliary row, admission, and scalar-sourced slot.

\bibitem{XIID}
N.~Cheung,
\emph{Landmark XII-D: Retarded response and carrier admission for
the cubic-only density-weighted Einstein source}.
Supplied manuscript \texttt{XII-D.tex}.
Completed insertion, roots, compact kernel, signed response law,
and causal plateau proposition.

\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 source and retained fixed-input jet obstruction.

\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}.
Complete free shell, polynomial carrier, null ideal,
ordering transport, and retained exclusion package.

\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, adjoint identities, free differential,
and residual-gauge distinction.

\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 and the separate global
Einstein-subshell question; neither is reopened.

\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.
Chapter~3: smooth Cauchy theory, uniqueness, causal Green operators,
and finite propagation used to realize the local witnesses with
compact parameters.

\end{thebibliography}

\end{document}