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% XIII-A.tex
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%   pdflatex -interaction=nonstopmode -halt-on-error XIII-A.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIII-A.tex
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% Main results:
%   1. Classical cancellation on one fixed shell subclass C_W.
%   2. The b--gamma commutator gives a nonzero scalar:
%        [L_q,G_h] = i hbar int q^{mu nu}
%                                  partial_mu partial_nu(E_0 h).
%   3. For every nonempty W, compact q and h can make this scalar
%      nonzero. The coefficientwise restriction kernel is not a
%      two-sided ideal of the inherited product.
%   4. An admissible smooth rank-one change of ordering gives
%        alpha_d(L_q^2) = L_q^2 + hbar L_q(u)^2.
%
% Debugging incorporated:
%   - C_def is indexed by the frozen f,g,W.
%   - Strictness of C_def is observation-qualified, not universal.
%   - Nonemptiness respects the shell relation b=D gamma.
%   - Free-differential tangency is an odd tangent-vector statement.
%   - The commutator uses Delta_{b gamma}, not Delta_{gamma gamma}.
%   - Formal vanishing means coefficientwise vanishing.
%   - A quotient forcing hbar*1=0 is distinguished from a zero
%     quotient and from a torsion-free formal quantum algebra.
%   - A nonabelian cocycle test requires a specified Lie algebra
%     action; compact metric translations do not supply that action.
%
% No reduced quantum algebra, observable completion, R_1,
% time-ordered product, interacting differential, state, loop,
% antifield construction, master equation, or sharp-width limit
% is constructed.
%
% No successful compilation, proof-assistant verification,
% independently executed symbolic computation, independent referee
% review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\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{\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{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\CW}{\mathscr C_W}
\newcommand{\Cdef}{\mathscr C_{\mathrm{def}}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\NW}{\mathscr N_W}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\EalgP}{\mathscr E_{H'}}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}}
\newcommand{\CD}{\mathcal C_{3,\mathrm D}}
\newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}}
\newcommand{\rD}{r_{\mathrm D}}
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\newcommand{\Aux}{\mathfrak B}
\newcommand{\sfree}{s_0}
\newcommand{\alphad}{\alpha_d}
\newcommand{\balphad}{\bar\alpha_d}
\newcommand{\gW}{\mathfrak g_W}
\newcommand{\Ocal}{\mathcal O}
\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 XIII-A}\\[0.4em]
Classical cancellation on a fixed subclass,\\
and the obstruction to a coefficientwise restriction quotient}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We distinguish restriction of classical evaluations from restriction
compatible with the inherited quantum product. One nonempty open set
$W$ is fixed before parameters are selected. On the complete
gauge-fixed free shell, impose $D^*b=0$ on $W$. Under a uniform causal
plateau hypothesis, XII-F's auxiliary remainder vanishes on this
subclass, including for the full displayed second-order coefficient.
This is an evaluation identity, not an action on a quantum quotient.

We then test the coefficientwise restriction kernel. The constraint
smearing $L_q=\int q^{\mu\nu}(D^*b)_{\mu\nu}$ is an auxiliary-field
functional off shell. Its commutator with a metric trace smearing must
therefore use the inherited $b\gamma$ causal block. We compute that
commutator and construct compact smearings with a strictly nonzero
scalar coefficient for every nonempty $W$. Consequently the restriction
kernel is not a two-sided ideal. A quotient killing all these
constraints while retaining the unchanged brackets forces
$\hbar\mathbf1=0$.

An admissible smooth rank-one change of ordering independently fails
to preserve coefficientwise vanishing on the subclass. We also display
the free differential of the coefficient, retaining its coordinate
ramp and scalar switching terms, and distinguish tangency from closure.
The copied XII-F witness remains a classical boundary of the claim,
not a quantum anomaly. A reduced algebra and an integrated cancelling
completion remain unresolved. No new quantum domain or retarded
product is released.
\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) {Fixed open set};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Classical cancellation on a subclass};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Quantum-compatible restriction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A classical evaluation subclass is not the inherited quantum restriction.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Classical restriction is not the quantum product. 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-XIII-A}
\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) {Fixed open set};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Quantum-compatible restriction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {A classical evaluation subclass is not the inherited quantum restriction.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Classical restriction is not the quantum product. 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-XIII-A}
\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) {Fixed open set};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Classical cancellation on a subclass};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Quantum-compatible restriction};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {A classical evaluation subclass is not the inherited quantum restriction.};
\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-XIII-A}
\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) {Fixed open set};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {Quantum-compatible restriction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Classical cancellation on a subclass. A classical evaluation subclass is not the inherited quantum restriction.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Fixed open set is not Quantum-compatible restriction. 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-XIII-A}
\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: Fixed open set};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Classical cancellation on a subclass};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Quantum-compatible restriction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {A classical evaluation subclass is not the inherited quantum restriction.};
\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-XIII-A}
\end{figure}

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


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



\scopebox{Release and proof boundary}{%
\textbf{Constructed:} one fixed classical evaluation subclass and its
conditional cancellation identity; the free-differential display;
the $b\gamma$ commutator; compact detecting smearings; and the
rank-one ordering calculation.

\smallskip
\textbf{Proved obstructed:} the coefficientwise restriction quotient
with the inherited linear fields, unchanged causal brackets, and
nonzero formal unit. The new vanishing space is not a two-sided
$\star_H$-ideal and is not preserved by every admissible ordering
transport.

\smallskip
\textbf{Unresolved:} a separately constructed reduced algebra or an
integrated observable completion cancelling the full-shell remainder.
Neither absence is a universal no-go theorem.

\smallskip
The inherited carrier is unchanged. No $R_1$, time ordering, quantum
Ward identity, interacting differential, physical BRST class, loop,
state, antibracket, master equation, or sharp-width limit is supplied.}

\tableofcontents

%=====================================================================
\section{Three readings and the named restriction problem}
\label{sec:readings}
%=====================================================================

The endings \textbf{Constructed}, \textbf{Proved obstructed}, and
\textbf{Unresolved} refer to a specified object and its stated
requirements. They are not ordered assertions about every possible
successor theory. In particular, a classical cancellation can be
constructed while its naive quantum implementation is obstructed.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.22\linewidth}X
                                      p{.22\linewidth}@{}}
\toprule
Reading & Object & Ending here\\
\midrule
Evaluation-only &
Evaluate prescribed classical lifts on $\CW$; retain the existing
carrier without a new quotient. &
Constructed under the support hypothesis.\\[0.4em]
Reduced algebra &
Select compatible observables and prove a separate reduction
theorem. &
Unresolved.\\[0.4em]
Completion &
Add a declared term or co-transformation cancelling the auxiliary
remainder on the full shell. &
Unresolved.\\
\bottomrule
\end{tabularx}
\end{center}

The principal question is whether the first reading can be implemented
by simply quotienting the entire inherited carrier by coefficientwise
vanishing on $\CW$. We give a negative answer to that particular
question. We do not select the observable subalgebra, equivalence
relation, or product of a different reduction.

The obstruction uses regular linear fields. It does not require a
singular composite product, an extension at coincident points, or
a renormalization condition. Its mechanism is the existing central
commutator. The ordering calculation concerns functionals, not movement
of configurations by a change of product data.

%=====================================================================
\section{A0: prerequisite sheet and inherited conventions}
\label{sec:inputs}
%=====================================================================

Five manuscripts supply mathematical inputs. Their roles are
separated below; other predecessor results are retained only as
boundaries.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.17\linewidth}X@{}}
\toprule
Manuscript & Input actually used\\
\midrule
XII-F &
The shell remainder, conditional companion cancellation, the
strictly negative witness, nonzero carrier class of that lift,
and absence of independent auxiliary or ghost arguments.\\[0.35em]
XII-E &
Assembly with coefficient $-1$, the displayed scalar completion,
and the scalar-sourced slot with zero metric variation.\\[0.35em]
XII-D &
The inherited roots and the causal plateau cancellation proposition;
no new root or response is constructed.\\[0.35em]
XII-A-v2 &
The polynomial carrier, free-shell quotient, causal brackets,
smooth ordering maps, and the frozen free differential's product
compatibility.\\[0.35em]
XI-B &
Free normalization, auxiliary shell relation, ghost rule, and
the distinction between compact plateau shifts and residual gauge.\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Geometry, pairings, and equations}

Use the supplied Minkowski conventions
\[
 M=\R^4,\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \Pmass=\Box+m^2,
\]
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad 0<\eps<T,\qquad m\ge0.
\]
The ramp $\sig$ is smooth, zero for $t\le0$, and one for $t\ge\eps$.
The switch is
$g\in C_c^\infty(\MTo;[0,1])$, with $P_g=\Int\{g=1\}$.
All fields are ambient smooth configurations. No spatial decay is
imposed. Tensor indices and dual tests are identified using $\eta$.

For a symmetric covariant tensor $r$ and a vector $v$,
\begin{align}
 \PP r&=r-\tfrac12\eta\tr_\eta r,
 &
 D^\nu r&=\partial_\mu(\PP r)^{\mu\nu},\\
 (Kv)_{\mu\nu}&=\partial_\mu v_\nu+\partial_\nu v_\mu,
 &
 \RR&=\tfrac12K,\\
 (D^*v)_{\mu\nu}
 &=-\partial_{(\mu}v_{\nu)}
                +\tfrac12\eta_{\mu\nu}\partial_\rho v^\rho.
 \label{eq:operators}
\end{align}
Parentheses include $\tfrac12$. The frozen identities are
\[
 \PP^2=\id,\qquad DK=\Box,\qquad
 A+2D^*D=-\Box\PP,\qquad \PP D^*=-\RR,
 \qquad A\RR=0.
\]
The inverse remains $G_{\mathrm{gr}}^{\mathrm R}=-\PP\ER$,
with coefficient $-1$ \cite{XIB,XIIA}.

The independent multiplet is $(\gamma,\phi,c,\bar c,b)$.
Only $c,\bar c$ are odd; their ghost numbers are respectively
$1,-1$. The other fields have ghost number zero. The shell is
\begin{equation}
 \Csol=
 \{\Box\gamma=0,\ \Pmass\phi=0,\ \Box c=\Box\bar c=0,\
                         b=D\gamma\}.
 \label{eq:shell}
\end{equation}
Odd identities mean coefficientwise identities in every finite
Grassmann algebra, as in XII-A. In particular, $b$ is an independent
off-shell coordinate. Equation~\eqref{eq:shell} is not an off-shell
substitution rule.

\subsection{The copied classical release}

\begin{inputstatement}[XII-F's remainder release]
\label{input:remainder}
With the inherited roots
\[
 \Aone[f,g](r;\phi)=\int\alpha_{f,g}[\phi]\cdot r\,\dd^4x,
 \qquad
 \beta_{f,g}[\phi]=-\EA(\PP\alpha_{f,g}[\phi]),
\]
put
\[
 s_\xi[\gamma]=\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi,
 \qquad
 \rho_\xi[\gamma]=\ER Ds_\xi[\gamma].
\]
For compact plateau parameters and a free scalar,
\begin{equation}
 \Delta_\xi\PhiD\big|_{\Csol}
 =-\Aone[f,g](K\rho_\xi;\phi)+\Aux_\xi[b,\phi],
 \label{eq:copied-remainder}
\end{equation}
where
\begin{align}
 \Delta_\xi F
   &=F[\gamma+K\xi,\phi]-F[\gamma,\phi],\\
 \Aux_\xi[b,\phi]
   &=-2\int(D\Lie_\xi\beta_{f,g}[\phi])_\nu b^\nu\,\dd^4x\\
   &=-2\int(\Lie_\xi\beta_{f,g}[\phi])^{\mu\nu}
                                  (D^*b)_{\mu\nu}\,\dd^4x.
 \label{eq:auxiliary}
\end{align}
The last tensor is supported in $\supp\xi$.
The companion has no ramp-layer contribution and vanishes if
\begin{equation}
 \text{\rm(CP)}\qquad
 g=1\ \text{near }J^-(\supp f)\cap\supp(K\rho_\xi).
 \label{eq:CP}
\end{equation}
There are admissible data satisfying (CP) for which $\Aux_\xi$ is
strictly nonzero. The corresponding admitted lift has nonzero class
under $Q_H$. An additional cancelling completion was not constructed
\cite{XIIF,XIID}.
\end{inputstatement}

This is the substance of XII-F's release box: a conditional
cancellation, a counterexample to unqualified invariance, and a
separately unresolved completion. We do not repeat its dependence
cylinder calculation.

XII-E fixes, without a sign change,
\begin{equation}
 \PhiD
 =\Asc[f,g](\gamma,0;\phi)-\CD(f,g)[\gamma,\phi],
 \qquad \rD=-u_{\mathrm D},
 \label{eq:assembly}
\end{equation}
and the coefficient evaluated below is
\begin{equation}
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \rg[\phi]=-\tfrac12\PP\ER(gT_0[\phi]).
 \label{eq:full-object}
\end{equation}
The added term has no metric argument. Neither $\Ocal$ nor $\PhiD$
has an independent $b,c,\bar c$ argument \cite{XIIE,XIIF}.

\subsection{The inherited product and ordering}

XII-A supplies
\begin{equation}
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                       \Nsol[[\hbar,\kappa]],
 \qquad
 \Nsol=\{F:F|_{\Csol}=0\},
 \label{eq:carrier}
\end{equation}
where $\Nsol[[\hbar,\kappa]]$ is a two-sided ideal and
\[
 F\star_HG=m\exp(\hbar\Gamma_H)(F\otimes G).
\]
For linear fields the causal relation is
\[
 [\Phi_A(u),\Phi_B(v)]_{\mathrm{gr}}
 =i\hbar\pair{u}{\Delta_{AB}v}\mathbf1.
\]
The relevant even blocks are
\begin{equation}
 \Delta_{\gamma\gamma}=-\PP E_0,\qquad
 \Delta_{\gamma b}=\RR E_0,\qquad
 \Delta_{b\gamma}=-D\PP E_0,\qquad
 \Delta_{bb}=0,
 \label{eq:causal-blocks}
\end{equation}
with $E_0=\ER-\EA$ the ambient causal wave operator.

An admissible ordering difference is a smooth graded-symmetric
bisolution $d$, respecting parity and ghost number. Its map is
\begin{equation}
 \alphad=
 \exp\left(\frac{\hbar}{2}
                \pair{d}{\frac{\delta^2}{\delta\Phi^2}}\right),
 \qquad H'=H+d.
 \label{eq:ordering-map}
\end{equation}
It acts on functionals, preserves the full-shell ideal, and descends
to an isomorphism $\balphad:\Ealg\to\EalgP$ \cite{XIIA}.
No preservation of the new vanishing space is included in that input.

\subsection{Boundaries, not additional calculations}

XII-C retains
\[
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}+\kappa\int(\sig-1)\ell_3\,\dd^4x
\]
and its named fixed-input invariant jet remains empty. Thus
$S_{\mathrm{EH}}$ has not been removed \cite{XIIC}.
The original detected XI-G classes, including their stated covectors
$\pm(k_1,k_2,0)$ with $k_1+k_2=2\,\mathrm dt$, stay excluded under the
inherited qualification \cite{XIIA}. Our restriction is not a new
representative theorem for those classes.

XII-B, XI-C--F, X-A--H, VII--IX, and Landmark I are not additional
prerequisites. In particular, no X-G reciprocity or Bogoliubov claim
is used. Smooth wave existence and finite propagation are the standard
analytic input for one new compact-smearing construction
\cite[Chapter~3]{BGP}.

%=====================================================================
\section{One frozen region and the observation-qualified defect set}
\label{sec:domains}
%=====================================================================

\begin{definition}[Fixed restriction data]
\label{def:domains}
Fix $f,g,m,\eps,T$ and a nonempty open set $W$ satisfying
\begin{equation}
 \overline W\subset P_g\cap\{\eps<t<T\}.
 \label{eq:W}
\end{equation}
This choice precedes all parameter selections. Since $g$ is compactly
supported, $\overline W$ is compact. Define
\[
 \gW=C_c^\infty(W;TM),
 \qquad
 \CW=\{u\in\Csol:D^*b=0\text{ on }W\},
\]
and
\begin{equation}
 \Cdef(W;f,g)=
 \{u\in\Csol:\Aux_\xi(u)=0\text{ for every }\xi\in\gW\}.
 \label{eq:defect-set}
\end{equation}
The dependence on $W,f,g$ is suppressed only after those data are
frozen.
\end{definition}

The plateau handoff is used with this declared parameter region.
There is no later reduction of its parameter class. Requiring one
condition throughout $W$ is different from choosing a new
configuration-dependent neighbourhood for each parameter.

\begin{proposition}[Nonemptiness and inclusion]
\label{prop:inclusion}
One has
\[
 \varnothing\ne\CW\subseteq\Cdef(W;f,g)\subseteq\Csol.
\]
\end{proposition}

\begin{proof}
Take $\gamma=b=0$, with arbitrary free scalar and ghost solutions.
These configurations satisfy the shell relation and belong to $\CW$.
For $u\in\CW$ and $\xi\in\gW$, the compact tensor
$\Lie_\xi\beta_{f,g}[\phi]$ is supported inside $W$.
Equation~\eqref{eq:auxiliary} therefore gives $\Aux_\xi(u)=0$.
\end{proof}

Nonemptiness does not mean that arbitrary metric data can be combined
with $b=0$: the equation $b=D\gamma$ must still hold. Scalar and ghost
data are unrestricted within their free smooth solution classes in
the example just given. No classification of admissible metric data
is needed for this proposition.

\subsection{The copied witness and a quantifier correction}

For the detecting data of XII-F, with a compact spatial realization,
\[
 m=0,\qquad
 \psi=t^2+(x^1)^2,\qquad
 \gamma=\psi\eta,\qquad
 \phi=\psi,\qquad
 b=(-2t,2x^1,0,0),\qquad c=\bar c=0.
\]
The parameter is a compact realization of
$a(t)\partial_t$, with $a\ge0$ a nonzero bump. XII-F proves
\begin{equation}
 \boxed{
 \Aux_\xi(u_*)=\Delta_\xi\PhiD(u_*)
 =-8\int f(t,\mathbf x)
       \int_0^t(t-s)^2a(s)\,\dd s\,\dd^4x<0.
 }
 \label{eq:copied-witness}
\end{equation}
It also proves that this lift has nonzero full-shell carrier class
\cite[Theorem on the nonzero auxiliary pairing]{XIIF}.
The value and its factor $8$ are copied, not recomputed.

The witness can be included in fixed restriction data by choosing
$W$ to contain its compact parameter support and choosing the switch
to cover the relevant causal compact sets before evaluation.
The stronger uniform plateau requirement below is compatible with
that realization: the switch may be chosen one near
$J^-(\supp f)\cap J^+(\overline W)$ as well as near $\overline W$.
This is still among XII-F's allowed witness switches, since it covers
the smaller causal set generated by $\supp\xi$.

For such detecting fixed data,
\begin{equation}
 \CW\subseteq\Cdef(W;f,g)\subsetneq\Csol,
 \qquad u_*\notin\Cdef(W;f,g).
 \label{eq:strict-detection}
\end{equation}
The first inclusion is not asserted to be equality.

\begin{remark}[Strictness is not automatic for every observation]
\label{rem:quantifier}
Equation~\eqref{eq:strict-detection} is observation-qualified.
For example, $f=0$ makes every auxiliary pairing zero and gives
$\Cdef(W;0,g)=\Csol$. A copied existential witness cannot prove
strictness for every already frozen observation and switch.
By contrast, Section~\ref{sec:ideal} proves that $\CW$ is proper and
that the naive quotient fails for every nonempty $W$, independently
of whether the chosen observation detects $\Aux$.
\end{remark}

%=====================================================================
\section{A1: the classical evaluation identity}
\label{sec:classical}
%=====================================================================

Let $K_f=\supp f$. Impose the configuration-independent hypothesis
\begin{equation}
 \text{\rm(UCP$_W$)}\qquad
 g=1\ \text{on a neighbourhood of }\
       J^-(K_f)\cap J^+(W)\cap\MT.
 \label{eq:uniform-plateau}
\end{equation}
A convenient stronger sufficient version replaces $W$ by
$\overline W$. The stronger set is compact and lies strictly between
the slab's temporal endpoints. Its use makes explicit that such
support data can be selected without reference to a particular
configuration.

\begin{theorem}[Cancellation on the fixed subclass]
\label{thm:classical}
Under \textup{(UCP$_W$)},
\begin{equation}
 \Delta_\xi\Ocal(u)=0
 \quad\text{for every }u\in\CW
 \text{ and every }\xi\in\gW.
 \label{eq:classical-cancellation}
\end{equation}
This is an identity of evaluations of the prescribed off-shell lift.
\end{theorem}

\begin{proof}
Every term of $Ds_\xi[\gamma]$ is supported in $\supp\xi$.
The inherited support statement gives
\[
 \supp(K\rho_\xi)\subset J^+(\supp\xi)\cap\MT
                         \subset J^+(W)\cap\MT.
\]
Thus \textup{(UCP$_W$)} implies (CP) for every smooth $\gamma$.
The companion vanishes for the free scalar in $u$.

The auxiliary term vanishes by Proposition~\ref{prop:inclusion}.
Finally, the additional scalar-sourced term in
\eqref{eq:full-object} is unchanged by a metric-only shift.
Equation~\eqref{eq:copied-remainder} proves the result.
\end{proof}

No carrier quotient is needed for this argument. Its uniform
quantifiers concern evaluations on the stated class; they do not
turn evaluation into a homomorphism for $\star_H$.

The finite shift does not preserve the shell. If one tries to maintain
the auxiliary relation after $\gamma\mapsto\gamma+K\xi$, the required
auxiliary change is
\[
 b\mapsto b+D(K\xi)=b+\Box\xi.
\]
This does not generally preserve the homogeneous free equations.
The compact residual intersection is zero \cite{XIB}.
Therefore the theorem compares a prescribed shifted off-shell
evaluation with its value at an initial shell configuration.
It is not a nontrivial residual symmetry of $\CW$.

%=====================================================================
\section{A2: the free differential, tangency, and boundary terms}
\label{sec:differential}
%=====================================================================

The frozen odd differential is
\[
 \sfree\gamma=\RR c,\qquad
 \sfree\bar c=ib,\qquad
 \sfree\phi=\sfree c=\sfree b=0.
\]
Its ghost number is one. The even finite parameter difference
$\Delta_\xi$ has ghost number zero. Matching a differentiated vector
field in the two expressions does not identify these operations.

\begin{lemma}[Odd tangency]
\label{lem:tangency}
For $u\in\CW$, the free variation is an odd tangent vector to $\CW$.
Consequently $\sfree\NW\subseteq\NW$ for the classical vanishing
space defined in Section~\ref{sec:ideal}.
\end{lemma}

\begin{proof}
On the shell,
\[
 \Box(\RR c)=\RR\Box c=0,\qquad
 \sfree(b-D\gamma)=-D\RR c=-\tfrac12\Box c=0.
\]
Also $\Box b=D\Box\gamma=0$, so
$\sfree(\Box\bar c)=i\Box b=0$.
The other free equations are unchanged.
The additional restriction satisfies $\sfree(D^*b)=0$.
These statements are interpreted with an external odd infinitesimal
parameter. Differentiating a polynomial identity on this linear
restriction space proves preservation of its vanishing space.
\end{proof}

Tangency is not the assertion $\sfree\Ocal=0$. We now give an exact
display rather than replacing the compact parameter by a free ghost.

\subsection{The inherited completion in differentiable form}

For a symmetric tensor $r$, set
\[
 a_1^{\mu\nu}[r]=\tfrac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},
 \qquad b_1[r]=\tfrac12\tr_\eta r,
\]
\[
 b_2[\gamma]=\tfrac18(\tr_\eta\gamma)^2
                         -\tfrac14\gamma_{\alpha\beta}\gamma^{\alpha\beta},
\]
\[
 a_2^{\mu\nu}[\gamma]
 =\gamma^\mu{}_\alpha\gamma^{\alpha\nu}
   -\tfrac12(\tr_\eta\gamma)\gamma^{\mu\nu}
   +b_2[\gamma]\eta^{\mu\nu}.
\]
Write
\[
 X=X_1[\gamma],\qquad p=p_1[\gamma;\phi],
 \qquad k=\rD[\gamma]+\rg[\phi].
\]
The inherited formulas are
\begin{align}
 X_1^\mu[r]
   &=-\ER\partial_\alpha(\sig a_1^{\alpha\mu}[r]),\\
 p_1[r;\phi]
   &=-\ERm\left(
       \partial_\mu(g a_1^{\mu\nu}[r]\partial_\nu\phi)
                       +m^2g b_1[r]\phi\right),\\
 X_2^\mu
   &=-\ER\left(
       \partial_\alpha(\sig a_1^{\alpha\beta}[\gamma]
                                      \partial_\beta X^\mu)
       +\partial_\alpha(\sig(a_1[k]+a_2[\gamma])^{\alpha\mu})
             \right),\\
 p_2
   &=-\ERm\left(
       V_{1,g}[\gamma]p+
       \partial_\mu(g(a_1[k]+a_2[\gamma])^{\mu\nu}\partial_\nu\phi)
                       +m^2g(b_1[k]+b_2[\gamma])\phi\right).
 \label{eq:completion-display}
\end{align}
Here $V_{1,g}[r]v$ is the divergence-form scalar operator appearing
in the formula for $p_1$, with $\phi$ replaced by $v$.
By XII-E's affine slot identity,
\begin{equation}
 \Ocal=\int f\left(
 p_2-X^\mu\partial_\mu p+
 (X^\nu\partial_\nu X^\mu-X_2^\mu)\partial_\mu\phi+
 \tfrac12X^\mu X^\nu\partial_\mu\partial_\nu\phi
                       \right)\dd^4x.
 \label{eq:O-expanded}
\end{equation}
No lower coefficient or additional metric slot has been inserted.

Put
\[
 z=\RR c,\qquad Z=X_1[z],\qquad p_z=p_1[z;\phi],
 \qquad \dot k=\rD'_\gamma[z]=2\rD(\gamma,z).
\]
The last equality uses the polarization of the already fixed
quadratic response. It is not a newly selected response or
co-transformation; $\sfree\rg[\phi]=0$.

Define the explicit quadratic derivatives
\begin{align}
 \dot b_2
 &=\tfrac14(\tr_\eta\gamma)(\tr_\eta z)
                         -\tfrac12\gamma_{\alpha\beta}z^{\alpha\beta},\\
 \dot a_2^{\mu\nu}
 &=z^\mu{}_\alpha\gamma^{\alpha\nu}
   +\gamma^\mu{}_\alpha z^{\alpha\nu}
   -\tfrac12\left(
       (\tr_\eta z)\gamma^{\mu\nu}
       +(\tr_\eta\gamma)z^{\mu\nu}\right)
   +\dot b_2\eta^{\mu\nu},\\
 \widehat a^{\mu\nu}&=a_1^{\mu\nu}[\dot k]+\dot a_2^{\mu\nu},
 \qquad
 \widehat b=b_1[\dot k]+\dot b_2.
 \label{eq:quadratic-derivatives}
\end{align}

\subsection{Terms remaining for homogeneous ghosts}

Suppose now that the multiplet is on the free shell. Then
$Dz=0$ and $\partial_\mu a_1^{\mu\nu}[\gamma]=-b^\nu$.
The first coordinate variation is
\begin{equation}
 \boxed{Z^\mu=-\ER\bigl(\sig'a_1^{0\mu}[z]\bigr).}
 \label{eq:ghost-coordinate}
\end{equation}
Thus the free ghost equation removes the $\sig Dz$ term, not the
coordinate layer term.

For $v=c^\mu\partial_\mu\phi$, direct differentiation gives
$V_{1,1}[z]\phi=-\tfrac12\Pmass v$ on the free scalar shell.
Because $gv$ is compactly supported in the slab interior,
\begin{equation}
 \boxed{
 p_z=\tfrac12gv-\ERm\left(
       \tfrac12(\Box g)v+
       (\partial^\mu g)\partial_\mu v+
       (\partial_\mu g)a_1^{\mu\nu}[z]\partial_\nu\phi
                         \right).
 }
 \label{eq:ghost-scalar}
\end{equation}
This formula retains the local switched term and both parts of
$[\Pmass,g]v$. It requires no compact support assumption on $c$.

For completeness, write $Y=\sfree X_2$ and $p_{2,z}=\sfree p_2$.
Expanding all external weight derivatives in the completion gives
\begin{align}
 Y^\mu=-\ER\Bigl\{&
 \sig\bigl(
   a_1^{\alpha\beta}[z]\partial_\alpha\partial_\beta X^\mu+
   a_1^{\alpha\beta}[\gamma]\partial_\alpha\partial_\beta Z^\mu
   -b^\beta\partial_\beta Z^\mu+
   \partial_\alpha\widehat a^{\alpha\mu}\bigr)
 \nonumber\\
 &+\sig'\bigl(
   a_1^{0\beta}[z]\partial_\beta X^\mu+
   a_1^{0\beta}[\gamma]\partial_\beta Z^\mu+
   \widehat a^{0\mu}\bigr)\Bigr\},
 \label{eq:ghost-X2}\\
 p_{2,z}=-\ERm\Bigl\{&
 g\bigl(
   a_1^{\mu\nu}[z]\partial_\mu\partial_\nu p+
   a_1^{\mu\nu}[\gamma]\partial_\mu\partial_\nu p_z
   -b^\nu\partial_\nu p_z
 \nonumber\\
 &\qquad
   +m^2(b_1[z]p+b_1[\gamma]p_z)
   +\widehat a^{\mu\nu}\partial_\mu\partial_\nu\phi
   +(\partial_\mu\widehat a^{\mu\nu})\partial_\nu\phi
   +m^2\widehat b\phi\bigr)
 \nonumber\\
 &+(\partial_\mu g)\bigl(
    a_1^{\mu\nu}[z]\partial_\nu p+
    a_1^{\mu\nu}[\gamma]\partial_\nu p_z+
    \widehat a^{\mu\nu}\partial_\nu\phi\bigr)\Bigr\}.
 \label{eq:ghost-p2}
\end{align}
All source-owned ramp terms remain inside the fixed derivative
$\dot k$; none is set to zero. This notation does not require a new
expansion of the gravitational Euler source. Equivalently, if
$t_{\mathrm D}$ is XII-E's inherited polarized insertion kernel,
its entire contribution is fixed by
\begin{equation}
 \Aone[f,g](\dot k;\phi)
 =-\sfree\CD
 =-2\pair{t_{\mathrm D}}{\gamma\otimes z\otimes\phi}.
 \label{eq:source-derivative-kernel}
\end{equation}
This specifies that contribution without discarding any of its
smooth source coefficients.

\begin{proposition}[Exact free-differential display]
\label{prop:sO}
With the preceding notation,
\begin{align}
 \sfree\Ocal=\int f\Bigl\{&
 p_{2,z}-Z^\mu\partial_\mu p-X^\mu\partial_\mu p_z
 \nonumber\\
 &+(Z^\nu\partial_\nu X^\mu+
                 X^\nu\partial_\nu Z^\mu-Y^\mu)\partial_\mu\phi
 \nonumber\\
 &+\tfrac12(Z^\mu X^\nu+X^\mu Z^\nu)
                         \partial_\mu\partial_\nu\phi
                          \Bigr\}\dd^4x.
 \label{eq:sO}
\end{align}
It is an admitted odd, ghost-number-one polynomial.
No vanishing on $\CW$ is asserted.
\end{proposition}

\begin{proof}
Differentiate each of the five terms of
\eqref{eq:O-expanded}. All undifferentiated coefficient fields are
even, so the displayed first variations acquire no additional
Koszul signs. Equations~\eqref{eq:ghost-X2}--\eqref{eq:ghost-p2}
are the product rule applied to \eqref{eq:completion-display}.
The shell equations justify only the simplifications stated above.
Admission follows from the inherited preservation of $\Fpol$ by
$\sfree$ \cite{XIIA}.
\end{proof}

The coordinate $\sig'$ terms and the scalar $\partial g,\Box g$
terms are visible. Homogeneous ghosts need not vanish in their
sampling regions. The uniform plateau hypothesis on compact
parameters consequently does not delete these terms. Furthermore,
$\tfrac12\Lie_\xi K\xi$ belongs to a finite two-jet transformation;
it is absent from the infinitesimal rule $\sfree\gamma=\RR c$.
Theorem~\ref{thm:classical} is therefore not a proof of free BRST
closure.

%=====================================================================
\section{A3: the coefficientwise restriction kernel}
\label{sec:ideal}
%=====================================================================

\begin{definition}[Coefficientwise vanishing]
Set
\[
 \NW=\{F\in\Fpol:F|_{\CW}=0\}.
\]
For a formal series, vanishing means that each coefficient belongs
to $\NW$. Thus its formal vanishing space is
$\NW[[\hbar,\kappa]]$, not a condition allowing cancellation between
different powers of the formal parameters.
\end{definition}

One has $\Nsol\subseteq\NW$. The space $\NW$ is an ideal for ordinary
pointwise polynomial multiplication, and the vector-space restriction
quotient exists. These facts do not establish its compatibility with
$\star_H$. We test that compatibility using two regular fields.

\subsection{The independent auxiliary smearing}

Let
\[
 q\in C_c^\infty(W;\Sym^2TM),\qquad
 h\in C_c^\infty(\MTo;\R),
\]
and define
\begin{equation}
 L_q=\int q^{\mu\nu}(D^*b)_{\mu\nu}\,\dd^4x,
 \qquad
 G_h=\int h\,\eta^{\alpha\beta}\gamma_{\alpha\beta}\,\dd^4x.
 \label{eq:linear-tests}
\end{equation}
Both are even, ghost-number-zero regular linear functionals.
Every $L_q$ belongs to $\NW$.

Compact integration by parts gives
\begin{equation}
 L_q=\int u_{q,\lambda}b^\lambda\,\dd^4x,
 \qquad
 u_{q,\lambda}
 =\eta_{\lambda\nu}\partial_\mu q^{\mu\nu}
       -\tfrac12\partial_\lambda(\eta_{\mu\nu}q^{\mu\nu}).
 \label{eq:uq}
\end{equation}
The metric dual smearing is
\begin{equation}
 v_h^{\alpha\beta}=\eta^{\alpha\beta}h.
 \label{eq:vh}
\end{equation}
These are the actual off-shell derivative kernels. In particular,
$L_q$ has no metric derivative kernel off shell. Replacing its
argument by $D\gamma$ before computing the product would change the
lift and obscure which block performs the contraction.

\begin{proposition}[The $b\gamma$ scalar]
\label{prop:commutator}
Put $\chi=E_0h$. Then
\begin{equation}
 \boxed{
 [L_q,G_h]_{\star_H}
 =i\hbar\,C(q,h)\mathbf1,\qquad
 C(q,h)=\int q^{\mu\nu}\partial_\mu\partial_\nu\chi\,\dd^4x.
 }
 \label{eq:commutator}
\end{equation}
The formula holds for every admissible replacement $H'$ with the
same causal part.
\end{proposition}

\begin{proof}
There is at most one contraction between linear functionals.
Their commutator is therefore exactly
\[
 i\hbar\pair{u_q}{\Delta_{b\gamma}v_h}\mathbf1.
\]
In covariant flat identification, $v_h$ is $\eta h$. Applying only
the prescribed block gives
\begin{equation}
 \Delta_{b\gamma}v_h
 =-D\PP(\eta\chi)
 =-\partial^\nu\chi.
 \label{eq:block-evaluation}
\end{equation}
Indeed $D$ already contains trace reversal, so $\PP^2=\id$
leaves the divergence of $\eta\chi$ with the displayed minus sign.

Using \eqref{eq:uq},
\begin{align*}
 \pair{u_q}{-\partial\chi}
 &=\int q^{\mu\nu}\partial_\mu\partial_\nu\chi\,\dd^4x
       -\tfrac12\int(\eta_{\mu\nu}q^{\mu\nu})\Box\chi\,\dd^4x.
\end{align*}
The causal solution satisfies $\Box\chi=0$, proving
\eqref{eq:commutator}. The calculation keeps the coefficient $-1$
in $\Delta_{b\gamma}$ throughout. Finally, admissible changes leave
the graded antisymmetric part unchanged.
\end{proof}

No $\Delta_{\gamma\gamma}$ contraction was used. The positive sign
of the final Hessian pairing results from integration by parts and
the action of $D^*$ on $-\partial\chi$; it is not obtained by
changing the sign of the causal metric block.

\subsection{Compact smearings detecting every nonempty region}

\begin{lemma}[A compact causal source with a local nonzero Hessian]
\label{lem:compact-pair}
For every nonempty $W$ satisfying \eqref{eq:W}, there are real compact
$q,h$ as above such that $C(q,h)>0$.
\end{lemma}

\begin{proof}
Choose $(t_0,\mathbf x_0)\in W$ and a small open dependence diamond
$V$ whose closure is contained in $W$. Consider the polynomial
\[
 \chi_{\mathrm{loc}}
 =(t-t_0)^2+(x^1-x_0^1)^2.
\]
It solves $\Box\chi_{\mathrm{loc}}=0$ and has
$\partial_t^2\chi_{\mathrm{loc}}=2$.

At $t=t_0$, multiply its spatial Cauchy value by a smooth compact
cutoff equal to one on a ball large enough to contain the Cauchy
domain of dependence of $\overline V$; use zero time derivative.
The smooth wave Cauchy problem produces a real solution $\chi$ with
compact Cauchy data. By finite propagation and uniqueness, it agrees
with $\chi_{\mathrm{loc}}$ on $V$. On every bounded time interval
its spatial support is bounded.

Choose a smooth temporal function $\theta$ which is zero before
one time and one after a later time, with its transition compactly
inside $(0,T)$. Set
\[
 h=\Box(\theta\chi)
   =\theta''\chi+2\theta'\partial_t\chi.
\]
This is a smooth compact test in $\MTo$. The functions $\theta\chi$
and $(\theta-1)\chi$ solve the sourced equation with respectively
zero past and zero future data. Uniqueness gives
\[
 \ER h=\theta\chi,\qquad
 \EA h=(\theta-1)\chi,\qquad E_0h=\chi.
\]
This is the compact time-slice construction in the supplied wave
conventions \cite{XIIA,BGP}.

Choose a nonzero nonnegative $\omega\in C_c^\infty(V)$ and put
$q^{00}=\omega$, with all other components zero. Then
\[
 C(q,h)=2\int\omega\,\dd^4x>0.
\]
This construction is independent of the observation $f$ and of
the scalar mass.
\end{proof}

This is a new compact-smearing construction, not a recalculation
of XII-F's cylinder integral. Its local polynomial is used only to
detect the Hessian in the existing linear causal bracket.

\begin{corollary}[Properness of the classical subclass]
\label{cor:proper}
For every nonempty $W$, $\CW\subsetneq\Csol$.
Moreover the constraint smearings used above are not elements of
$\Nsol$.
\end{corollary}

\begin{proof}
Using the solution $\chi$ in Lemma~\ref{lem:compact-pair}, take
\begin{equation}
 u=(\eta\chi,0,0,0,-\partial\chi).
 \label{eq:ordering-solution}
\end{equation}
The metric is a wave solution and
$D(\eta\chi)=-\partial\chi$, so $u\in\Csol$.
Since $\Box\chi=0$,
\[
 (D^*u_b)_{\mu\nu}=\partial_\mu\partial_\nu\chi,
 \qquad L_q(u)=C(q,h)>0.
\]
Thus $u\notin\CW$ and $L_q\notin\Nsol$.
\end{proof}

\subsection{The obstruction theorem}

\begin{theorem}[Failure of the naive restriction ideal]
\label{thm:nonideal}
For every nonempty $W$, the space
$\NW[[\hbar,\kappa]]$ is not a two-sided ideal of the inherited
product. Its image
\[
 \overline{\NW}[[\hbar,\kappa]]
   =(\NW/\Nsol)[[\hbar,\kappa]]
\]
is not a two-sided ideal in $\Ealg$ either.
These assertions hold for every admissible ordering.
\end{theorem}

\begin{proof}
Choose $q,h$ from Lemma~\ref{lem:compact-pair}.
If the formal vanishing space were a two-sided ideal, both
$L_q\star_HG_h$ and $G_h\star_HL_q$ would belong to it.
Their difference would then belong to it.
But \eqref{eq:commutator} is the formal scalar
$i\hbar C(q,h)\mathbf1$.

The zero configuration belongs to $\CW$, so a nonzero constant
cannot vanish on $\CW$. Its $\hbar$ coefficient is therefore
outside $\NW$. This is a contradiction.
The same commutator descends through the full-shell ideal, and
constants survive that quotient, proving the on-shell assertion.
The causal scalar is ordering-independent.
\end{proof}

\begin{corollary}[No coefficientwise quantum restriction quotient]
\label{cor:noquotient}
There is no associative product on
\[
 (\Fpol/\NW)[[\hbar,\kappa]]
\]
for which coefficientwise restriction is a unital homomorphism
from the inherited $\star_H$-algebra.

More generally, a unital $\C[[\hbar,\kappa]]$-algebra homomorphism
which kills every $L_q$ while preserving the inherited brackets
must satisfy $\hbar\mathbf1=0$ in its target. It cannot have a
nonzero unit with faithful formal scalar inclusion, or be a
nonzero torsion-free formal quantum algebra.
\end{corollary}

\begin{proof}
The kernel of a homomorphism is a two-sided ideal, contradicting
Theorem~\ref{thm:nonideal}. Alternatively, apply a proposed
homomorphism to \eqref{eq:commutator}. Its left side is zero because
the image of $L_q$ is zero. Since $C(q,h)$ is a nonzero scalar,
the result forces $\hbar\mathbf1=0$.
In the coefficientwise restriction module the class of $1$ is
nonzero, and multiplication by $\hbar$ merely shifts coefficients,
so this relation is impossible.
\end{proof}

Over a formal power-series ring, forcing $\hbar\mathbf1=0$ does
not by itself force $\mathbf1=0$. It may produce an
$\hbar$-torsion quotient, including a classical specialization.
After inverting $\hbar$, however, it forces the zero algebra.
Neither option is the intended coefficientwise quantum restriction.

The inherited equation ideal remains valid. The obstruction is
caused by adding a new condition which the causal contractions
do not annihilate. Thus it is not a contradiction of XII-A's
full-shell descent theorem. A smaller observable algebra, different
constraint implementation, or separately proved reduction is not
excluded by this argument.

%=====================================================================
\section{A4: ordering acts on functionals}
\label{sec:ordering}
%=====================================================================

Use the same $q,h,\chi,u$ as in
\eqref{eq:ordering-solution}. Write $C=C(q,h)>0$.
The multiplet $u$ is an ordinary real even solution with only
ghost-number-zero components.

\begin{lemma}[An admissible rank-one difference]
\label{lem:rankone}
The matrix
\begin{equation}
 d_{AB}(x,y)=u_A(x)u_B(y)
 \label{eq:rankone}
\end{equation}
is an admissible smooth ordering difference in XII-A's class.
\end{lemma}

\begin{proof}
It is smooth and symmetric under simultaneous exchange of species
and spacetime slots. Every nonzero species is even and has ghost
number zero. Since $u$ satisfies the complete shell equations,
the matrix satisfies those equations in each slot, including the
auxiliary relation $u_b=Du_\gamma$.
Its graded antisymmetric part is zero, so $H+d$ has the same causal
part as $H$. These are precisely the required conditions.
No positivity or compact support condition on $d$ is imposed
\cite{XIIA}.
\end{proof}

The nonzero $bb$ component of this difference is permitted. The
vanishing frozen $H_{bb}$ block is not a requirement on every
admissible replacement. Nor is a separate condition making the
unchanged symbol rule $s_0$ a derivation for $H+d$ required: XII-A
transports that differential by conjugation when necessary.

\begin{theorem}[Failure of coefficientwise ordering preservation]
\label{thm:ordering}
For the admissible difference \eqref{eq:rankone},
\begin{equation}
 \boxed{
 \alphad(L_q^2)=L_q^2+\hbar L_q(u)^2
                         =L_q^2+\hbar C^2.
 }
 \label{eq:ordering-square}
\end{equation}
Consequently
\[
 \alphad\bigl(\NW[[\hbar,\kappa]]\bigr)
             \not\subseteq\NW[[\hbar,\kappa]].
\]
The corresponding descended vanishing subspaces are not preserved
by $\balphad$ either.
\end{theorem}

\begin{proof}
Let $\ell=L_q'$ be its field-independent linear derivative kernel.
Then
\[
 (L_q^2)''=2\ell\otimes\ell,
 \qquad
 \pair{d}{\ell\otimes\ell}=L_q(u)^2=C^2.
\]
The factor $\hbar/2$ in \eqref{eq:ordering-map} cancels the factor
$2$. A second self-contraction would require a fourth derivative,
which is zero. This proves \eqref{eq:ordering-square} exactly,
not merely to first order.

Although $L_q^2$ vanishes on $\CW$, the $\hbar$ coefficient of its
transport is the nonzero constant $C^2$. Coefficientwise vanishing
therefore fails. On-shell descent gives the same conclusion because
that constant is not in $\Nsol$.
\end{proof}

The hypothesis $L_q(u)\ne0$ is essential. Choosing a rank-one
bisolution built from $u\in\CW$ would give zero in this particular
test. The theorem proves failure for an explicitly allowed ordering
change; it does not assert failure for every smooth difference.

No point of $\CW$ has been transformed by $\alphad$. The failed
statement is preservation of a space of functional symbols. In
contrast, preservation of the complete free-shell ideal by the same
ordering maps remains an inherited theorem. The rank-one solution
lies outside the newly imposed subclass, exactly where its
constraint contraction can detect it.

%=====================================================================
\section{A6: the completion interface and its integrability conditions}
\label{sec:completion}
%=====================================================================

The quotient obstruction does not answer whether an additional
classical term can cancel $\Aux_\xi$ on the full shell.
Conversely, a cancellation of one evaluated pairing would not
construct a reduced quantum algebra. We keep those problems separate.

\subsection{A conditional cocycle test}

Fix the ordinary vector-field bracket convention
\[
 [\xi,\zeta]^\mu
   =\xi^\nu\partial_\nu\zeta^\mu
                       -\zeta^\nu\partial_\nu\xi^\mu.
\]
For a proposed action on functionals, choose the representation
convention
\begin{equation}
 [\delta_\xi,\delta_\zeta]=\delta_{[\xi,\zeta]}.
 \label{eq:representation}
\end{equation}
If another pullback convention is used, its corresponding sign
must be changed consistently.

A necessary test for a correction $F$ satisfying
$\delta_\xi F=-\Aux_\xi$ is
\begin{equation}
 \delta_\xi\Aux_\zeta-\delta_\zeta\Aux_\xi
       \stackrel{?}{=}\Aux_{[\xi,\zeta]}
                         \pmod{\Nsol}.
 \label{eq:cocycle}
\end{equation}
Indeed, under \eqref{eq:representation}, the difference of the
first two terms is
$-[\delta_\xi,\delta_\zeta]F
=-\delta_{[\xi,\zeta]}F$.

This test is well posed on the shell quotient only after the
candidate $\delta$ preserves the equation ideal. That action has
not been supplied here. In particular, the constant off-shell
metric-shift derivatives
\[
 T_\xi F=\pair{F'_\gamma}{K\xi}
\]
commute with each other. They do not furnish the nonabelian
representation \eqref{eq:representation}, and they generally do not
preserve $\Nsol$. They therefore cannot be inserted into
\eqref{eq:cocycle} as if they were a declared action of the
vector-field Lie algebra on the shell.

For a specified action and allowed correction class, failure of
\eqref{eq:cocycle} excludes a correction of the stated affine type.
Success supplies only a necessary compatibility condition, not
solvability. Without the action, interpreting a formal failure as
a universal completion obstruction would be invalid.

\subsection{The affine metric-slot target}

A possible interface is
\begin{equation}
 \Aone[f,g](r_\xi;\phi)
             \stackrel{?}{=}-\Aux_\xi[b,\phi].
 \label{eq:slot-target}
\end{equation}
For a degree-matched candidate, $r_\xi$ should be even, have ghost
number zero, be linear in the even parameter and linear in the
chosen even field slots, and have no scalar dependence.
A retarded candidate may be required to have zero past data and
support in $J^+(\supp\xi)\cap\MT$ for each fixed smooth field input.

Those conditions on smooth outputs are not sufficient for admission.
The composed functional must also have a compact polarized
degree-two kernel, with its wavefront set disjoint from the two
all-causal cones. For example, a covector-sum-zero bound would be
a sufficient kernel condition, but no such new kernel is constructed
here. The existence of a source pairing on smooth fields is not
substituted for this obligation.

Equation~\eqref{eq:slot-target} records a range and compatibility
problem; it does not assume an inverse of $\Aone$ on arbitrary
auxiliary pairings. Even a pointwise assignment satisfying this
slot equation would have to be realized as the variation of one
declared correction, with consistent support, field degree, and
parameter composition. It is that observable-completion problem
which remains unresolved.

An infinitesimal solution must also be integrated into the finite
coefficient law. The finite term
$\tfrac12\Lie_\xi K\xi$ cannot be recovered by merely naming a
linear ghost substitution. If a correction operates on a lower
coefficient of the relational expansion, its coupling order must
be printed; it is not added to the already filled second metric
slot a second time.

\subsection{Co-transformations and the absence of auxiliary slots}

For a proposed multiplet rule
\[
 Y_\xi=(Y_\xi^\gamma,Y_\xi^\phi,Y_\xi^c,
                         Y_\xi^{\bar c},Y_\xi^b),
\]
the full-coefficient infinitesimal interface is
\begin{equation}
 \pair{\Ocal'_\gamma}{Y_\xi^\gamma}
       +\pair{\Ocal'_\phi}{Y_\xi^\phi}
                   \stackrel{?}{=}-\Aux_\xi.
 \label{eq:Y-interface}
\end{equation}
The lift in \eqref{eq:full-object} has no independent auxiliary
or ghost argument. Hence a rule with
$Y_\xi^\gamma=Y_\xi^\phi=0$ changes it by zero.
A pure auxiliary or ghost variation cannot cancel the copied
nonzero value.

A scalar variation is different: it also differentiates the
scalar-sourced slot. Explicitly,
\[
 \delta_\phi\Aone(\rg[\phi];\phi)
 =\Aone(\rg'_\phi[Y_\xi^\phi];\phi)
                     +\Aone(\rg[\phi];Y_\xi^\phi).
\]
Thus the zero metric variation of that slot does not license
dropping it from a scalar co-transformation calculation.

A correction containing $b$ explicitly requires a declared action
on $b$, matching parity, and its own admitted kernel. The displayed
shell relation alone supplies none of those transformation laws.
No correction is absorbed into $Q_H$, the inherited equation ideal,
or an unlabelled modification of $\rD$.

The witness excludes the original coefficient, and it excludes
proposals whose added variation is demonstrably zero at that
witness. A finite-dimensional class of corrections would require
its generators and variations to be frozen and tested.
An undefined space of all completions cannot be excluded by
one scalar pairing.

%=====================================================================
\section{A5--A7: decision, loss ledger, 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
A0 & Recorded &
Five mathematical inputs; remaining predecessors are boundaries.\\[0.35em]
A1 & Constructed &
Evaluation cancellation on $\CW$ under the uniform plateau
hypothesis.\\[0.35em]
A2 & Constructed as a display &
Odd tangency and the exact free-differential formula; no closure
claim.\\[0.35em]
A3 & Proved obstructed &
The coefficientwise vanishing space is not a two-sided ideal
for any admissible ordering.\\[0.35em]
A4 & Proved obstructed for universal preservation &
An explicit admissible rank-one ordering change fails to preserve
that vanishing space.\\[0.35em]
A5 & Evaluation-only reading selected &
No new quantum quotient or reduced algebra.\\[0.35em]
A6 & Interface recorded; completion Unresolved &
A Lie action, integrability, admission, and finite integration
would require separate proofs.\\[0.35em]
A7 & Recorded &
No $R_1$, retarded Ward defect, master equation, state, or loop
is opened.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Restriction release]
\label{thm:release}
For every nonempty fixed $W$ satisfying \eqref{eq:W}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\CW$ is a nonempty proper classical shell subclass.
Under \textup{(UCP$_W$)}, the full displayed coefficient satisfies
$\Delta_\xi\Ocal|_{\CW}=0$ for every $\xi\in\gW$.
\item There are compact regular smearings $q,h$ for which
$[L_q,G_h]_{\star_H}=i\hbar C(q,h)\mathbf1$ with $C(q,h)>0$.
Therefore the naive coefficientwise restriction quotient is
incompatible with the unchanged inherited brackets and faithful
formal scalar inclusion.
\item An admissible smooth ordering difference satisfies
$\alphad(L_q^2)=L_q^2+\hbar C(q,h)^2$.
Coefficientwise vanishing on $\CW$ is not universally
ordering-preserved.
\item For the copied detecting observation data, the XII-F witness
lies outside $\Cdef(W;f,g)$ and its full-shell remainder class is
nonzero. No quantum anomaly or general completion no-go is inferred.
\end{enumerate}
The release is evaluation-only. A quantum domain is not supplied.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:inclusion},
Corollary~\ref{cor:proper},
Theorem~\ref{thm:classical},
Proposition~\ref{prop:commutator},
Lemma~\ref{lem:compact-pair},
Theorem~\ref{thm:nonideal},
Corollary~\ref{cor:noquotient},
Theorem~\ref{thm:ordering}, and the copied witness
\eqref{eq:copied-witness}.
\end{proof}

\subsection*{The loss ledger}

The restriction retains one conditional classical identity.
It does not retain unqualified full-shell invariance of the
specified coefficient. Parameters outside the declared compact
region, or observations and switches violating the uniform support
hypothesis, are not covered by that identity. The free differential
is retained on the inherited carrier, but a physical or interacting
BRST class of $\Ocal$ is not constructed.

The ordinary restriction quotient remains available as a
vector-space or pointwise classical construction. Calling it a
restricted quantum quotient would conflict with the explicit
commutator theorem. The inherited product itself has not failed:
its incompatibility with the additional proposed quotient has
been proved.

\subsection*{Exact handoff}

A successor receives:
\begin{enumerate}
\item one fixed evaluation domain $\CW$, its nonemptiness and
properness, and the uniform support hypothesis;
\item the cancellation identity for the full displayed coefficient;
\item the observation-qualified defect set and the copied witness
outside it;
\item the correctly typed $b\gamma$ commutator and compact positive
scalar test;
\item the rank-one ordering counterexample;
\item the free-differential display, separated from finite plateau
variation;
\item the unsolved slot and co-transformation interfaces
\eqref{eq:slot-target} and \eqref{eq:Y-interface}.
\end{enumerate}

It receives no reduced algebra, integrated completion, quantum
restriction domain, retarded product, or interacting differential.
Before XIII-B opens a product calculation, its opening sentence must
declare whether it uses evaluation-only claims in the unchanged
carrier, a separately constructed reduction, or a completed
observable. The cancellation theorem alone supplies none of the
missing product-domain or gauge-consistency obligations.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \text{\rm(UCP$_W$)}
 \quad\Longrightarrow\quad
 \Delta_\xi\Ocal|_{\CW}=0
       \quad(\xi\in C_c^\infty(W;TM)),\\[2pt]
 [L_q,G_h]_{\star_H}
   =i\hbar\int q^{\mu\nu}
                \partial_\mu\partial_\nu(E_0h)\,\dd^4x\,\mathbf1,
 \qquad C(q,h)>0\ \text{for suitable compact tests},\\[2pt]
 \NW[[\hbar,\kappa]]
       \text{ is not a two-sided quantum ideal},\\[2pt]
 \alphad(L_q^2)=L_q^2+\hbar L_q(u)^2,\qquad L_q(u)\ne0.
 \end{gathered}}
\]

The pairing vanishes on a proper classical subclass. That fact does
not implement the subclass as the proposed coefficientwise quotient
of $\Ealg$; the naive implementation is obstructed. The successor
retarded-product gate remains closed.

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

\paragraph{Dependency scope.}
XII-F's witness and remainder are copied with their hypotheses.
XII-E supplies the assembled coefficient and the differentiable
completion formulas. XII-D supplies the inherited roots and causal
plateau proposition. XII-A-v2 supplies the product, full-shell ideal,
brackets, and ordering class. XI-B supplies the free normalization
and differential. The commutator, compact detecting smearings,
new-ideal obstruction, and rank-one ordering calculation are proved
here.

\paragraph{Quantifier correction.}
The defect set is indexed by the frozen observation, switch, and
parameter region. Its strictness is asserted only for detecting
data. Properness of $\CW$ and failure of the naive quantum quotient
hold for every nonempty $W$.

\paragraph{Completion scope.}
The conditional cocycle equation requires a specified representation
preserving the equation ideal. Constant compact metric translations
do not supply that representation. No conclusion about every possible
reduction or completion is claimed.

\paragraph{Verification.}
The manuscript is a standalone analytic proof relative to its stated
inputs. No successful \texttt{pdflatex} execution, proof-assistant
verification, independently executed symbolic computation, independent
referee review, or journal acceptance is asserted.

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

\bibitem{XIIF}
N.~Cheung,
\emph{Landmark XII-F: The gauge-fixed remainder of
$\Phi^{(2)}_{\mathrm D}$: companion support, the auxiliary pairing,
and what is not a Ward identity}.
Supplied manuscript \texttt{XII-F.tex}.
Primary remainder handoff, conditional cancellation, evaluated
witness, nonzero full-shell class, and completion boundary.

\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 with coefficient $-1$, completion formulas, affine slot
identity, 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}.
Inherited roots and causal plateau cancellation proposition;
no response or root is reconstructed here.

\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}.
Off-shell auxiliary coordinate, free-shell ideal, product, causal
blocks, ordering maps, and frozen free differential.

\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 coefficient $-1$, ghost normalization, auxiliary shell
relation, and trivial compact residual intersection.

\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}.
Boundary only: the Einstein--Hilbert action is retained and the
named invariant fixed-input jet is not repaired.

\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 existence, uniqueness, causal Green
operators, and finite propagation.

\end{thebibliography}

\end{document}