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% XIV-A.tex
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%
% Frozen sheet (A0):
%   action      S_D = S_EH + kappa int (sigma-1) ell_3 + S_g^sc   (XII-C);
%   inverse     G_gr^R = -P E_0^R, coefficient -1                 (XI-B);
%   backgrounds g, sigma_eps fixed and inert, no spurion;
%   algebra     g_W = C_c^infty(W;TM), closure(W) in P_g cap {eps<t<T};
%   observable  O = Phi_D^(2) + A_1(r_g[phi];phi)                 (XII-E).
%
% Proved here:
%   - antifields, antibracket, kinematic rule with XI-B normalization;
%   - exact antibracket square
%         (S_ext,S_ext) = kappa int c^nu K_nu,
%     K_nu = XII-C's changed Noether density, every switch and ramp
%     term displayed;
%   - a nonzero value of that functional on the unrestricted ghost
%     space (late-switch witness recomputed in these conventions):
%     the unrestricted master equation is Proved obstructed for the
%     frozen class;
%   - the relative master equation on W-supported ghosts holds exactly,
%     its differential is nilpotent, and XI-B's gauge fermion preserves
%     it;
%   - the relative complex meets XII-A's free ghost shell only at c=0;
%   - under (UCP_W): Delta_xi O|_{C_sol} = B_xi, zero on C_W, equal to
%     -8I on the copied witness; the relative free differential of O on
%     the bosonic shell IS the auxiliary pairing. The master equation
%     does not decide XII-F.
%
% Not constructed:
%   R_1^(1), a quantum gauge defect, a reduced algebra, an interacting
%   differential on E_H, a completion of the auxiliary pairing, a
%   spurion sheet, a density-switched variational sheet, a state, a
%   loop, 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{lmodern}
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\usepackage{enumitem}
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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{\Gret}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\Cbos}{\mathscr C_{\mathrm{sol}}^{\mathrm{bos}}}
\newcommand{\CW}{\mathscr C_W}
\newcommand{\CWbos}{\mathscr C_W^{\mathrm{bos}}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\NW}{\mathscr N_W}
\newcommand{\Ealg}{\mathscr E_H}
\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{\Ocal}{\mathcal O}
\newcommand{\Aux}{\mathfrak B}
\newcommand{\gW}{\mathfrak g_W}
\newcommand{\Kbrk}{\mathfrak K}
\newcommand{\Kden}{\mathcal K}
\newcommand{\Sext}{S_{\mathrm{ext}}}
\newcommand{\Sgf}{S_{\Psi}}
\newcommand{\SD}{S_{\mathrm D}}
\newcommand{\abr}[2]{\left(#1,#2\right)}
\newcommand{\Fsp}{\mathfrak F}
\newcommand{\FW}{\mathfrak F_W}
\newcommand{\Rampdef}{\mathfrak R}
\newcommand{\Zsource}{Z_\sig}
\newcommand{\Dsource}{\mathcal D_\sig}
\newcommand{\Bsw}{\mathcal B}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Rquant}{R_1^{(1)}}
\newcommand{\sfree}{s_0}
\newcommand{\sext}{s_{\mathrm{ext}}}
\newcommand{\Kclass}{\mathfrak K_{\mathrm{loc,R}}}
\newcommand{\Sclass}{\mathfrak S_{\mathrm{kin}}}
\newcommand{\Lag}{\mathfrak L}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\gh}{gh}

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

\title{\textbf{Landmark XIV-A}\\[0.4em]
Classical master equation and observable ledger\\
for one declared Einstein--scalar sheet}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We freeze one sheet before any antifield is written: XII-C's
cubic-only density weighting, XI-B's inverse with coefficient $-1$,
the switch and ramp as fixed inert backgrounds, the compact plateau
algebra $\gW$, and the assembled second-order relational coefficient
$\Ocal$. The convolution-regulated sheet and the source-switched sheet
are recorded with their own obstructions and are not used.

Antifields and the antibracket are introduced, and the extended action
generates the kinematic diffeomorphism rule with the inherited ghost
normalization $c=2\xi$. Its antibracket square is computed exactly:
$(\Sext,\Sext)=\kappa\int c^\nu\Kden_\nu$, where $\Kden_\nu$ is
XII-C's changed Noether density with the scalar-switch defect, the
cubic-only bulk mismatch, and every ramp-derivative term displayed. On
the unrestricted ghost space this functional is nonzero; a late-switch
witness is recomputed in the present conventions. The unrestricted
master equation is therefore Proved obstructed for the frozen class, at
$\hbar^0$, independently of higher antifield terms.

On the relative space whose ghosts lie in $\gW$, the square vanishes
identically, the differential is nilpotent, and gauge fixing by XI-B's
fermion preserves the identity. That relative complex meets XII-A's
free ghost shell only at $c=0$; it induces no interacting differential
on the inherited carrier.

The observable ledger is separate. Under the uniform causal plateau
hypothesis the finite variation of $\Ocal$ on the complete free shell
equals the auxiliary pairing, vanishes on the evaluation set $\CW$,
and equals $-8I<0$ on the copied witness. We prove that the relative
free differential of $\Ocal$ on the bosonic shell is exactly that
pairing. Thus the master equation does not decide XII-F. The ending is
Constructed on the relative domain; the unrestricted sheet is Proved
obstructed; no completion, quantum retarded coefficient, reduced
algebra, or sharp limit is opened.
\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) {Frozen cubic sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Antifield shift};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Quantum master equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The relative symmetry does not construct a quantum master equation.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Relative symmetry is not a master equation. 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-XIV-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) {Frozen cubic sheet};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Quantum master equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The relative symmetry does not construct a quantum master equation.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Relative symmetry is not a master equation. 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-XIV-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) {Frozen cubic sheet};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Antifield shift};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Quantum master equation};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The relative symmetry does not construct a quantum master equation.};
\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-XIV-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) {Frozen cubic sheet};
\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 master equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Antifield shift. The relative symmetry does not construct a quantum master equation.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Frozen cubic sheet is not Quantum master equation. 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-XIV-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: Frozen cubic sheet};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Antifield shift};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Quantum master equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The relative symmetry does not construct a quantum master equation.};
\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-XIV-A}
\end{figure}

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


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



\scopebox{Release and proof boundary}{%
\textbf{Frozen (A0):} cubic-only sheet, inverse $-1$, inert $g$ and
$\sig$, $\gW$, observable $\Ocal$. One route: relative symmetry.

\smallskip
\textbf{Constructed:} antifield sector and antibracket; the exact
breaking functional $(\Sext,\Sext)=\kappa\int c^\nu\Kden_\nu$; the
relative master equation on $\FW$ and its nilpotent differential;
gauge fixing by XI-B's fermion; the evaluation identity on $\CW$ with
the auxiliary pairing printed; the identification of the relative free
differential of $\Ocal$ with that pairing.

\smallskip
\textbf{Proved obstructed:} the unrestricted master equation for every
extension in the frozen kinematic class $\Sclass$ (nonzero $\hbar^0$
breaking); unqualified invariance of $\Ocal$ on $\Csol$ (copied
witness); completion inside $\Kclass$ (copied from XIII-D).

\smallskip
\textbf{Outside the claim:} a completion of the auxiliary pairing in
any class not frozen here; $\Rquant$; a quantum gauge defect; a
reduced algebra; an interacting differential on $\Ealg$; a spurion
or density-switched variational sheet; $\eps\to0$.

\smallskip
\textbf{Ending for the frozen relative sheet: Constructed.} The
unrestricted sheet is Proved obstructed and is recorded as the
boundary of the claim. The release is relative; it is not an
interacting quantum algebra.}

\tableofcontents

%=====================================================================
\section{Release criteria and the non-identification sheet}
\label{sec:inputs}
%=====================================================================

The endings are \textbf{Constructed}, \textbf{Proved obstructed}, and
\textbf{Unresolved}, applied to the object actually named on the
domain actually named. A missing map is not a no-go. An $\hbar^0$
defect is not removed by an $\hbar$ counterterm. XIII-C's unresolved
product is not an anomaly.

\[
 \boxed{\text{XIV-A freezes one sheet and tests }(\Sext,\Sext)
 \text{ and }\Delta_\xi\Ocal\text{ on it. It does not construct }\Rquant.}
\]

The following statements are used with their predecessor scope. They
are not one model. Each is copied so that the later sections can refer
to it without reconstruction.

\begin{inputstatement}[X-A, convolution regulator: release]
\label{input:XAconv}
For the frozen switched action, spacetime-convolution regulator,
Cauchy prescription, and plateau-supported gauge algebra, the
relational scalar fails gauge invariance at order $\kappa$. The
decisive identity is $E_0C\,T_{g,\eps}(K\xi)=S_\eps\xi\ne\xi$, with
witness smearing $c_\eps>0$. There is no cleared classical
gauge-invariant handoff. A changed regulator is a new problem
\cite{XAconv}.
\end{inputstatement}

\begin{inputstatement}[X-G: two separate releases]
\label{input:XG}
On the X-E/X-F normalization $P_{\mathrm{gr}}=-\tfrac14\PP\Box$,
$\Delta_{\mathrm R}^{\mathrm{gr}}=-4\PP\ER$, the free multiplet with
independent $b$, gauge fermion $\Psi_0=\int\bar c(C-b)$,
$s\bar c=b$, and $b=C[h]/2$ is constructed; off-shell nilpotence uses
the independent auxiliary field. Separately, an exact source-preserving
variational upgrade of X-F's mixed map
$k_{\mathrm{mix}}[w;u]=-\Delta^{\mathrm{gr}}_{\mathrm R}(\sig Q_wu)$
is Proved obstructed: a compact pure-gauge ramp background gives the
reciprocity pairing
\[
 \frac{5M_\chi}{8}\int_0^\eps\zeta(\sig'')^2\dd t>0
\]
on free metric solutions. Source switching is not density switching.
The null-collinear set $\mathcal N$ is a failed sufficient product
criterion, not nonexistence. The unconditional one-loop background
identity is Unresolved \cite{XG}.
\end{inputstatement}

\begin{inputstatement}[XII-C: the selected action]
\label{input:XIIC}
Density before variation, at cubic order only:
\begin{equation}
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}+\kappa\int_{\MT}(\sig-1)\ell_3\dd^4x,
 \qquad
 \ell_3=-2(r_3+b_1r_2+b_2r_1),
 \qquad
 S_{\mathrm{EH}}=-\frac2{\kappa^2}\int_{\MT}\sqrt{-q}\,R[q]\dd^4x.
 \label{eq:action}
\end{equation}
Full Einstein--Hilbert weighting is rejected: it produces a
flat-background Euler term and changes the free sheet. The actual
cubic source is $\JD=\sig J_3+\Dsource$. With
$e_{\mathrm D}=\delta\SD/\delta\gamma$, $E_\phi=\delta\SD/\delta\phi$,
$w=\sig-1$, and $\Zsource=wJ_3+\Dsource$, the changed Noether identity
holds off shell:
\begin{equation}
 \mathcal N_\nu(e_{\mathrm D})
 =\kappa\bigl[
 E_\phi\partial_\nu\phi-\Bsw_{g,\nu}
 -wQ_\nu(A\gamma)+\Rampdef_{\sig,\nu}
 +\kappa Q_\nu(\Zsource)\bigr].
 \label{eq:noether-input}
\end{equation}
For every declared switch, the fixed-input invariant jet
$\Jcal^{\mathrm D}_{g,\sig}(\phi_m)$ is empty: a solution-fiber
obstruction, not an observable no-go. The quadratic coefficients are
XI-A/XI-B's; the inverse is $\Gret=-\PP\ER$ with coefficient $-1$
\cite{XIIC,XIB}.
\end{inputstatement}

\begin{inputstatement}[XII-E and XII-F: the coefficient and its witness]
\label{input:XIIF}
The scalar-linear coefficient is
$\PhiD=\Asc[f,g](\gamma,0;\phi)-\CD$, with the slot
$\rD=-u_{\mathrm D}=\PP\ER\JD$; the full displayed coefficient is
$\Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi)$,
$\rg=-\tfrac12\PP\ER(gT_0[\phi])$. For $\xi$ compact in the plateau
and $\Pmass\phi=0$,
\begin{equation}
 \Delta_\xi\Ocal\big|_{\Csol}
 =-\Aone[f,g](K\rho_\xi;\phi)+\Aux_\xi[b,\phi],
 \qquad
 \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])\cdot D^*b\dd^4x,
 \label{eq:ledger-input}
\end{equation}
with $s_\xi=\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi$,
$\rho_\xi=\ER Ds_\xi$. The companion vanishes under the causal plateau
condition (CP). On the compact free-shell tuple
$m=0$, $\psi=t^2+(x^1)^2$, $\gamma=\psi\eta$, $\phi=\psi$,
$b=D\gamma=(-2t,2x^1,0,0)$, with $\xi_*=a(t)\zeta(\mathbf x)\partial_t$
and (CP) holding,
\begin{equation}
 \Delta_{\xi_*}\PhiD(u_*)=\Aux_{\xi_*}(u_*)
 =-8\int f(t,\mathbf x)\int_0^t(t-s)^2a(s)\dd s\,\dd^4x
 =:-8I<0.
 \label{eq:witness-input}
\end{equation}
A rule with $Y^\gamma_\xi=Y^\phi_\xi=0$ changes the lift by zero. No
cancelling map was supplied \cite{XIIE,XIIF}.
\end{inputstatement}

\begin{inputstatement}[XIII-A: evaluation, not quotient]
\label{input:XIIIA}
Fix nonempty $W$ with $\overline W\subset\Int\{g=1\}\cap\{\eps<t<T\}$,
$\gW=C_c^\infty(W;TM)$, $\CW=\{u\in\Csol:D^*b=0\text{ on }W\}$. Under
\begin{equation}
 \mathrm{(UCP}_W\mathrm)\qquad
 g=1\ \text{on a neighbourhood of}\
 J^-(\supp f)\cap J^+(\overline W)\cap\MT,
 \label{eq:UCP}
\end{equation}
one has $\Delta_\xi\Ocal|_{\CW}=0$ for every $\xi\in\gW$. The
coefficientwise kernel $\NW$ is not a two-sided ideal: compact
smearings give
$[L_q,G_h]_{\star_H}=i\hbar C(q,h)\mathbf 1$ with $C(q,h)>0$, and an
admissible rank-one ordering change gives
$\alpha_d(L_q^2)=L_q^2+\hbar L_q(u)^2$. No reduced algebra was handed
off; $\CW$ is an evaluation set \cite{XIIIA}.
\end{inputstatement}

\begin{inputstatement}[XIII-C: the unopened quantum gate]
\label{input:XIIIC}
The coefficient of $\lambda\hbar$ is Unresolved. A successor must
supply: a prescription across the noncoincident characteristic
configurations, or a proved cancellation in the selected pair; uniform
transverse estimates on the three- and four-vertex strata; compatible
partial-diagonal contacts; the output wavefront estimate after
pushforward. The free differential is not applied to $\Rquant$
\cite{XIIIC}.
\end{inputstatement}

\begin{inputstatement}[XIII-D: one bounded completion class excluded]
\label{input:XIIID}
On the detecting data, every shell-descending action in the
parameter-local/one-retarded class $\Kclass$ induces the zero
representation on $\Fpol/\Nsol$, and
\[
 \theta=[h\zeta\partial_t,(t-t_*)h\zeta\partial_t]=h^2\zeta^2\partial_t,
 \qquad
 \Aux_\theta(u_*)=-8I<0 .
\]
Ending: Proved obstructed for that class only. The cochain
$\mathcal C_\xi=-\Aux_\xi[D\gamma,\phi]$ is admitted and is not an
integrated completion. Nonlocal Cauchy-data actions remain undecided
\cite{XIIID}.
\end{inputstatement}

\begin{inputstatement}[XII-A and XI-B: carrier and free rule]
\label{input:XIIA}
The carrier is $\Ealg=(\Fpol[[\hbar,\kappa]],\star_H)/\Nsol[[\hbar,\kappa]]$
with complete shell
$\Csol=\{\Box\gamma=0,\ \Pmass\phi=0,\ \Box c=\Box\bar c=0,\ b=D\gamma\}$.
The frozen free differential is $\sfree\gamma=\RR c$,
$\sfree\bar c=ib$, $\sfree\phi=\sfree c=\sfree b=0$, with $\RR=K/2$;
it is a nilpotent derivation of $\star_H$ descending to $\Ealg$. The
ghost normalization $c=2\xi$ recovers $K\xi$. The identities
$A+2D^*D=-\Box\PP$, $\PP D^*=-\RR$, $DK=\Box$, $A\RR=0$ hold. Compact
plateau residual parameters are zero:
$\gW\cap\{\Box\xi=0\}=\{0\}$ \cite{XIIA,XIB}.
\end{inputstatement}

%=====================================================================
\section{A0: the frozen sheet}
\label{sec:sheet}
%=====================================================================

\subsection{Geometry, backgrounds, and the parameter algebra}

Use
\[
 M=\R^4,\quad
 \eta=\operatorname{diag}(1,-1,-1,-1),\quad
 \Box=\partial_t^2-\Delta_{\mathbf x},\quad
 \Pmass=\Box+m^2,\ m\ge0,
\]
\[
 \MT=[0,T]\times\R^3,\qquad \MTo=(0,T)\times\R^3,\qquad 0<\eps<T.
\]
Parentheses include $\tfrac12$. Indices are moved with $\eta$.
The ramp $\sig(t)=\chi(t/\eps)$ is the inherited smooth function,
zero for $t\le0$, one for $t\ge\eps$, flat at both endpoints. The
switch is $g\in C_c^\infty(\MTo;[0,1])$ with $P_g=\Int\{g=1\}\ne\varnothing$.
Both are fixed external functions; neither is varied, neither is a
spurion, and no transformation law is assigned to them.

Fix once a nonempty open, relatively compact $W$ with
\begin{equation}
 \overline W\subset P_g\cap\{\eps<t<T\},
 \qquad
 \gW=C_c^\infty(W;TM),
 \label{eq:W}
\end{equation}
with the ordinary vector-field bracket
$[\xi,\zeta]^\mu=\xi^\nu\partial_\nu\zeta^\mu-\zeta^\nu\partial_\nu\xi^\mu$.
The observation $f\in C_c^\infty(\MTo;\R)$ and the switch are chosen so
that \eqref{eq:UCP} holds. All of $(f,g,m,\eps,T,W)$ are fixed before
any parameter, ghost, or antifield is selected, and $\gW$ is not
reduced afterwards.

\subsection{Fields, operators, and the inverse}

The independent fields are $(\gamma_{\mu\nu},\phi,c^\mu,\bar c_\mu,b_\mu)$
with parities $(0,0,1,1,0)$ and ghost numbers $(0,0,1,-1,0)$. Odd
identities are coefficientwise in finite Grassmann algebras, with odd
coefficients placed on the left. Put
\[
 \PP r=r-\tfrac12\eta\tr_\eta r,\qquad
 D^\nu r=\partial_\mu(\PP r)^{\mu\nu},\qquad
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,\qquad
 \RR=\tfrac12K,
\]
\[
 (D^*v)_{\mu\nu}=-\partial_{(\mu}v_{\nu)}+\tfrac12\eta_{\mu\nu}\partial_\rho v^\rho,
 \qquad
 A=2G_{\mathrm{lin}},\qquad A+2D^*D=-\Box\PP .
\]
The inverse frozen on this sheet is
\begin{equation}
 \boxed{\ \Gret=-\PP\ER,\qquad\text{coefficient }-1.\ }
 \label{eq:inverse}
\end{equation}
X-G's $-4\PP\ER$ inverts the differently normalized Hessian
$-\tfrac14\PP\Box$; it is not converted into \eqref{eq:inverse} by
vocabulary and is not used.

The classical action is XII-C's
\begin{equation}
 \SD=S_{\mathrm{grav,D}}+S^{\mathrm{sc}}_g,
 \qquad
 S^{\mathrm{sc}}_g=\int_{\MT}\bigl((1-g)\Lag_\eta[\phi]+g\Lag_q[\phi]\bigr)\dd^4x,
 \qquad q=\eta+\kappa\gamma,
 \label{eq:SD}
\end{equation}
with $S_{\mathrm{grav,D}}$ from \eqref{eq:action} and
$\Lag_q=\tfrac12\sqrt{-q}(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-m^2\phi^2)$.
It is a formal bulk variational functional tested with compact
interior variations; its Euler derivatives $e_{\mathrm D}^{\mu\nu}$,
$E_\phi$ are the exact expressions of XII-C. The gauge-fixing term
$S_{\mathrm{gf}}=\int D_\nu\gamma D^\nu\gamma$ is generated below by
XI-B's fermion; it is not added by hand.

\subsection{The observable}

The observable is the full displayed gauge-fixed second-order
relational coefficient of XII-E,
\begin{equation}
 \Ocal(f,g)[\gamma,\phi]
 =\PhiD(f,g)[\gamma,\phi]+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD(f,g)[\gamma,\phi].
 \label{eq:O}
\end{equation}
It has metric and scalar arguments only, degree $(2,1)$ plus the
degree-$(0,3)$ scalar-sourced slot, and $Q_H\Ocal\in\Ealg$ by XII-E.
The sign $-1$ on $\CD$ is XII-E's and is not reopened. The evaluation
domains are
\[
 \Cbos=\{(\gamma,\phi,b):\Box\gamma=0,\ \Pmass\phi=0,\ b=D\gamma\},
 \qquad
 \CWbos=\{u\in\Cbos:D^*b=0\text{ on }W\}.
\]
Since $\Ocal$ has no ghost argument, its evaluations depend only on
the bosonic part of a configuration.

\subsection{Routes, and the one selected}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.24\linewidth}X p{.26\linewidth}@{}}
\toprule
Route & Written before the witness & Legal release\\
\midrule
Relative symmetry on the cubic-only sheet \textbf{(selected)} &
Symmetry supported where $g=\sig=1$; XII-C's plateau contraction
copied; observable domains $\Cbos$, $\CWbos$ named &
Constructed only if the master identity and the observable ledger
both close on that domain\\[0.3em]
Spurion &
A declared transformation of $g$ or $\sig$, entered in the extended
action &
A new bookkeeping; not a repair of the fixed-background defect\\[0.3em]
New variational sheet &
Density switched before variation, layer terms retained, source and
Noether identity recomputed; X-G's mixed map not reused &
A new model\\[0.3em]
Exit from $\Kclass$ &
Nonlocal Cauchy data, nonzero past data, or a different integration
interface, frozen before $\theta$ is reused &
XIII-D's triviality theorem does not apply until that change is
written\\[0.3em]
Nonvariational &
Source-switched response kept, X-G's obstruction cited &
Recorded; no master equation\\
\bottomrule
\end{tabularx}
\end{center}

Completion inside $\Kclass$ is closed by XIII-D and is not reopened.
No spurion, new sheet, or exit class is frozen here; those routes are
recorded as not taken.

\subsection{Why two other sheets are not used}

\begin{itemize}
\item The convolution regulator of Input statement~\ref{input:XAconv}
is not the ramp. Its defect $S_\eps\xi\ne\xi$ concerns a different
coordinate prescription; it is neither an input to the ramp master
equation nor repaired by it.
\item X-G's source-switched mixed map is not XII-C's density-weighted
source. The reciprocity theorem blocks an exact upgrade that keeps
that field map on every background; it says nothing about the
density-switched action \eqref{eq:action}, whose source and Noether
identity were recomputed in XII-C.
\item XII-C's empty jet is not emptiness of the gauge-fixed shell and
not a failure of $\Ocal$. XIII-A's evaluation identity is not a
quotient. XIII-D's cochain is not a completion. XIII-C's inventory is
not a product.
\end{itemize}

\subsection*{Stop-rule check}

Action: \eqref{eq:SD}. Inverse: \eqref{eq:inverse}. Switch law: $g$
and $\sig$ fixed and inert, no transformation. Observable:
\eqref{eq:O}. Parameter algebra: \eqref{eq:W}. The sheet is named;
Section~\ref{sec:A1} may write $(S,S)$.

%=====================================================================
\section{A1: the classical master equation on the frozen sheet}
\label{sec:A1}
%=====================================================================

This gate concerns the action, not $\Ocal$.

\subsection{Antifields and the antibracket}

For each field $\Phi^A$ introduce an antifield $\Phi^*_A$ of opposite
parity and ghost number $\gh(\Phi^*_A)=-1-\gh(\Phi^A)$:
\[
\begin{array}{c|ccccc}
 \Phi^*_A & \gamma^{*\mu\nu} & \phi^* & c^*_\mu & \bar c^{*\mu} & b^{*\mu}\\ \hline
 \text{parity} & 1 & 1 & 0 & 0 & 1\\
 \gh & -1 & -1 & -2 & 0 & -1
\end{array}
\]
Antifields are smooth sections compactly supported in $\MTo$; on the
relative space below, $c^*$ is a smooth section over $W$. For
functionals $F,G$ of fields and antifields, the antibracket is
\begin{equation}
 \abr FG
 =\int\left[
 \frac{\delta^{\mathrm R}F}{\delta\Phi^A}
 \frac{\delta^{\mathrm L}G}{\delta\Phi^*_A}
 -\frac{\delta^{\mathrm R}F}{\delta\Phi^*_A}
 \frac{\delta^{\mathrm L}G}{\delta\Phi^A}
 \right]\dd^4x,
 \label{eq:antibracket}
\end{equation}
with the left/right functional derivatives and the graded Jacobi
identity of \cite[Ch.~15]{HT}; on local functionals the derivatives
are Euler derivatives and the bracket is again local
\cite{GPS,FRBV}. Antifield degree is polynomial degree in the
$\Phi^*_A$.

\subsection{The kinematic rule}

\begin{definition}[Kinematic diffeomorphism rule, XI-B normalization]
\label{def:kin}
Let $c$ be the odd vector field $c^\mu\partial_\mu$ and write, with
odd coefficients on the left,
\[
 (\Lie_c\gamma)_{\mu\nu}
 =c^\rho\partial_\rho\gamma_{\mu\nu}
 +(\partial_\mu c^\rho)\gamma_{\rho\nu}
 +(\partial_\nu c^\rho)\gamma_{\mu\rho},
 \qquad \Lie_c\eta=Kc .
\]
Define the odd, ghost-number-one derivation $s$ of the field algebra by
\begin{equation}
 \begin{aligned}
 s\gamma&=\RR c+\tfrac{\kappa}{2}\Lie_c\gamma
         =\tfrac12\Lie_c(\eta+\kappa\gamma)\big/1,
 &s\phi&=\tfrac{\kappa}{2}\,c^\mu\partial_\mu\phi,\\
 sc^\mu&=\tfrac{\kappa}{2}\,c^\nu\partial_\nu c^\mu,
 &s\bar c_\mu&=ib_\mu,\qquad sb_\mu=0 .
 \end{aligned}
 \label{eq:kin}
\end{equation}
Equivalently $sq=\tfrac{\kappa}{2}\Lie_cq$ for $q=\eta+\kappa\gamma$:
the ghost generates the vector field $\tfrac12 c$, so that $c=2\xi$
recovers $\delta_\xi\gamma=K\xi+\kappa\Lie_\xi\gamma$ and
$\delta_\xi\phi=\kappa\xi^\mu\partial_\mu\phi$. The backgrounds $g$,
$\sig$ are $s$-inert. At $\kappa=0$ the rule is XII-A's frozen $\sfree$.
\end{definition}

The notation $\tfrac12\Lie_c(\eta+\kappa\gamma)/1$ in \eqref{eq:kin}
is a reminder that $\kappa s\gamma=sq=\tfrac\kappa2\Lie_cq$; the
displayed first form is the one used.

\begin{lemma}[Kinematic nilpotence]
\label{lem:nil}
$s^2=0$ on the independent fields and on their graded polynomial
differential algebra.
\end{lemma}

\begin{proof}
Put $w^\mu=c^\nu\partial_\nu c^\mu$. The graded identity
$\Lie_c\Lie_c=\Lie_w$ holds: for a scalar $u$,
$c^\nu\partial_\nu(c^\mu\partial_\mu u)=w^\mu\partial_\mu u+c^\nu c^\mu\partial_\nu\partial_\mu u$
and the last term vanishes by anticommutativity; for covectors one
extra term is treated identically, and the identity extends to tensors
by the graded Leibniz rule (X-G's kinematic lemma, normalization
independent). Then, since $s$ is an odd derivation and $\Lie_cq$ is
linear in $c$,
\[
 s(sq)=\tfrac\kappa2\bigl(\Lie_{sc}q-\Lie_c(sq)\bigr)
 =\tfrac\kappa2\bigl(\tfrac\kappa2\Lie_wq-\tfrac\kappa2\Lie_c\Lie_cq\bigr)=0,
\]
hence $s^2\gamma=0$. Likewise
$s^2\phi=\tfrac{\kappa^2}{4}(w^\mu\partial_\mu\phi-\Lie_c\Lie_c\phi)=0$.
For the ghost,
\[
 s(c^\nu\partial_\nu c^\mu)
 =\tfrac\kappa2\bigl(c^\rho\partial_\rho c^\nu\,\partial_\nu c^\mu
 -c^\nu\partial_\nu c^\rho\,\partial_\rho c^\mu
 -c^\nu c^\rho\partial_\nu\partial_\rho c^\mu\bigr)=0,
\]
the first two terms cancelling after renaming and the third vanishing
by antisymmetry. Finally $s^2\bar c=i\,sb=0$ and $s^2b=0$. The square
of an odd derivation is an even derivation, so vanishing on generators
suffices.
\end{proof}

\subsection{The extended action and its antibracket square}

\begin{definition}[Extended action in the frozen class]
\label{def:Sext}
\begin{equation}
 \Sext=\SD[\gamma,\phi]
 +\int\bigl[\gamma^{*\mu\nu}s\gamma_{\mu\nu}+\phi^*s\phi
 +c^*_\mu sc^\mu+\bar c^{*\mu}s\bar c_\mu\bigr]\dd^4x .
 \label{eq:Sext}
\end{equation}
There is no $b^*$ term because $sb=0$. The frozen class $\Sclass$
consists of all $S'=\Sext+S'_{\ge2}$ with $S'_{\ge2}$ a local even
ghost-number-zero functional of antifield degree at least two, with
the backgrounds inert and the rule \eqref{eq:kin} unchanged.
\end{definition}

\begin{lemma}[Antifield-degree decomposition]
\label{lem:decomp}
For $S'\in\Sclass$,
\begin{equation}
 \abr{S'}{S'}\big|_{\text{antifield degree }0}
 =\abr\Sext\Sext=2\,s\SD,
 \qquad
 s\SD:=\int\bigl(e_{\mathrm D}^{\mu\nu}s\gamma_{\mu\nu}+E_\phi\,s\phi\bigr)\dd^4x .
 \label{eq:decomp}
\end{equation}
\end{lemma}

\begin{proof}
$\SD$ depends only on the even fields $\gamma,\phi$, so its left and
right derivatives coincide with the ordinary Euler derivatives
$e_{\mathrm D},E_\phi$. In \eqref{eq:antibracket} with $F=G=\Sext$,
the antifield-degree-zero part is produced only by pairing a
derivative of $\SD$ with $\delta^{\mathrm L}\Sext/\delta\Phi^*_A=s\Phi^A$,
or with $\delta^{\mathrm R}\Sext/\delta\Phi^*_A$; moving the odd
antifield past the odd $s\Phi^A$ for even $\Phi^A$ gives the sign that
makes the two terms equal, yielding $2\int(e_{\mathrm D}\cdot s\gamma+E_\phi s\phi)$.
The antifield-degree-one part is a sum of terms
$\pm2\int\Phi^*_A\,s^2\Phi^A$, which vanish by Lemma~\ref{lem:nil}.
There is no degree-two part because $\Sext$ is linear in antifields.
Adding $S'_{\ge2}$ changes $(S',S')$ only in antifield degree
$\ge1$. This is the standard decomposition of
\cite[Ch.~17]{HT}, \cite{GPS}.
\end{proof}

\begin{theorem}[The breaking functional, with every term displayed]
\label{thm:breaking}
On the field space $\Fsp$ of smooth multiplets with ghosts
$c,\bar c$ compactly supported in $\MTo$,
\begin{equation}
 \boxed{\
 \abr\Sext\Sext
 =\Kbrk[\gamma,\phi,c]
 :=\kappa\int_{\MT}c^\nu\,\Kden_\nu[\gamma,\phi]\dd^4x,\ }
 \label{eq:breaking}
\end{equation}
exactly as a formal series in $\kappa$, where
\begin{align}
 \Kden_\nu
 &=\Bsw_{g,\nu}+wQ_\nu(A\gamma)-\Rampdef_{\sig,\nu}-\kappa Q_\nu(\Zsource),
 \label{eq:Kden}\\
 \Bsw_{g,\nu}
 &=(\partial_\mu g)T_0^{\mu}{}_\nu[\phi]
 -(1-g)(\Pmass\phi)\partial_\nu\phi
 +(\partial_\nu g)\bigl(\Lag_\eta[\phi]-\Lag_q[\phi]\bigr),
 \nonumber\\
 Q_\nu(v)&=2\partial_\mu(v^{\mu\rho}\gamma_{\rho\nu})-v^{\mu\rho}\partial_\nu\gamma_{\mu\rho},
 \qquad w=\sig-1,\qquad \Zsource=wJ_3+\Dsource,
 \nonumber\\
 \Dsource^{\mu\nu}
 &=2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sig-2M_2^{\mu\nu\rho}\partial_\rho\sig,
 \nonumber\\
 \Rampdef_{\sig,\nu}
 &=2\eta_{\rho\nu}(\partial_\mu\sig)J_3^{\mu\rho}
 +4\eta_{\rho\nu}\Bigl[(\partial_\mu C_2^{\mu\rho\alpha\beta})\partial_\alpha\partial_\beta\sig
 -(\partial_\mu M_2^{\mu\rho\lambda})\partial_\lambda\sig
 -M_2^{\mu\rho\lambda}\partial_\mu\partial_\lambda\sig\Bigr],
 \nonumber
\end{align}
with $T_0^{\mu}{}_\nu=T_0^{\mu\rho}\eta_{\rho\nu}$ and $C_2,M_2,J_3$
XII-C's finite coefficients. Equivalently
$\mathcal N_\nu(e_{\mathrm D})=\kappa[E_\phi\partial_\nu\phi-\Kden_\nu]$.
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:decomp}, $(\Sext,\Sext)=2s\SD$. Write $\xi=c/2$,
an odd compact vector field. Then $s\gamma=K\xi+\kappa\Lie_\xi\gamma$
and $s\phi=\kappa\xi^\mu\partial_\mu\phi$, and
$s\SD=\int e_{\mathrm D}\cdot(K\xi+\kappa\Lie_\xi\gamma)+\kappa\int E_\phi\,\xi\cdot\partial\phi$
is the first variation of $\SD$ along XII-C's scaled variation,
evaluated coefficientwise in the Grassmann algebra. XII-C's proof of
\eqref{eq:noether-input} computes this variation in two ways for an
arbitrary compact interior parameter. Compact integration by parts
gives
$s\SD=\int\xi^\nu[-\mathcal N_\nu(e_{\mathrm D})+\kappa E_\phi\partial_\nu\phi]$.
Direct variation gives: zero from the natural Einstein--Hilbert
density; $\kappa\int\xi^\nu\Bsw_{g,\nu}$ from the switched scalar
action; and
$-\kappa\int\xi^\nu[N_{0,\nu}(\Zsource)+\kappa Q_\nu(\Zsource)]$ from
the added cubic functional, where XII-C's Lemma on the cubic
divergence gives $N_{0,\nu}(\Zsource)=-wQ_\nu(A\gamma)+\Rampdef_{\sig,\nu}$
with the apparent third ramp derivative cancelled. Hence
$s\SD=\kappa\int\xi^\nu\Kden_\nu$ with $\Kden$ as in \eqref{eq:Kden},
and comparison of the two forms is \eqref{eq:noether-input}.
Substituting $\xi=c/2$ and doubling gives \eqref{eq:breaking}. All
steps are identities of local densities modulo total derivatives,
integrated against the compact ghost.
\end{proof}

Every term of $\Kden_\nu$ carries $1-g$, $\partial g$, $w=\sig-1$,
$\partial\sig$, or $\partial^2\sig$. The term $wQ_\nu(A\gamma)$ is the
bulk breaking from weighting only the cubic coefficient; it is not a
ramp-derivative term, and it is not removed by setting ramp derivatives
to zero. It is removed only where $\sig=1$.

\subsection{The unrestricted master equation is obstructed}

\begin{lemma}[A falling switch region above the ramp]
\label{lem:fall}
There are $c_0>0$ and a nonempty open $O\subset\{t>\eps\}$ on which
$\partial_tg\le-c_0$. Moreover $O\cap P_g=\varnothing$.
\end{lemma}

\begin{proof}
Choose $(t_*,\mathbf x_*)\in W$, so $t_*>\eps$ and $g(t_*,\mathbf x_*)=1$.
Compact support gives $t_1\in(t_*,T)$ with $g(t_1,\mathbf x_*)=0$; the
mean value theorem gives a point between them with $\partial_tg<0$,
and smoothness gives the open neighbourhood and $c_0$ (XII-C's lemma).
On $P_g$ all derivatives of $g$ vanish, so $O\cap P_g=\varnothing$.
\end{proof}

\begin{proposition}[Nonzero breaking on the unrestricted space]
\label{prop:witnessA1}
Let $\phi=\phi_m$, with $\phi_m=\sin(mt)/m$ for $m>0$ and $\phi_0=t$,
let $\gamma=0$, and let $c=\vartheta\,\zeta(t,\mathbf x)\,\partial_t$
with $\vartheta$ an odd generator and $0\ne\zeta\in C_c^\infty(O)$,
$\zeta\ge0$. Then
\begin{equation}
 \Kbrk[0,\phi_m,c]
 =\frac\kappa2\,\vartheta\int\zeta\,\partial_tg\dd^4x,
 \qquad
 \int\zeta\,\partial_tg\dd^4x\le-c_0\int\zeta\dd^4x<0 .
 \label{eq:witnessA1}
\end{equation}
In particular $(\Sext,\Sext)\ne0$ on $\Fsp$.
\end{proposition}

\begin{proof}
At $\gamma=0$: $A\gamma=0$, and $J_3$, $C_2$, $M_2$, $\Dsource$,
$\Zsource$, $\Rampdef_\sig$ are quadratic in $\gamma$ and vanish;
$\Lag_q=\Lag_\eta$; $\Pmass\phi_m=0$. Hence
$\Kden_\nu=(\partial_\mu g)T_0^{\mu}{}_\nu[\phi_m]$. For $\phi_m$,
$\partial_t\phi_m=\cos(mt)$, $(\partial\phi)^2-m^2\phi^2=\cos(2mt)$,
so $T_0^{00}=\cos^2(mt)-\tfrac12\cos(2mt)=\tfrac12$ and
$T_0^{0i}=0$. Thus $c^\nu\Kden_\nu=c^0(\partial_\mu g)T_0^{\mu0}
=\tfrac12\vartheta\zeta\,\partial_tg$, and \eqref{eq:witnessA1}
follows from Lemma~\ref{lem:fall}. The Grassmann coefficient is a
strictly negative real number.
\end{proof}

\begin{theorem}[Unrestricted master equation: Proved obstructed in $\Sclass$]
\label{thm:obstructA1}
For every $S'\in\Sclass$, the antifield-degree-zero component of
$(S',S')$ equals $\Kbrk$, which is a nonzero functional on $\Fsp$ at
order $\hbar^0$. Hence no $S'\in\Sclass$ satisfies $(S',S')=0$ on the
unrestricted field space, and no correction beginning at order $\hbar$
changes this conclusion.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:decomp}, Theorem~\ref{thm:breaking}, and
Proposition~\ref{prop:witnessA1}. Higher antifield terms and
$\hbar$-corrections do not affect the antifield-degree-zero
$\hbar^0$ coefficient.
\end{proof}

The witness ghost in Proposition~\ref{prop:witnessA1} is supported in
the falling region, outside $P_g$. Leaving $\Sclass$ means changing
the rule, transforming a background, or changing the action: the
spurion, new-sheet, or nonvariational routes of Section~\ref{sec:sheet},
none of which is taken here. A second, independent breaking lives in
the ramp layer through $wQ_\nu(A\gamma)$ and $\Rampdef_\sig$; one
witness suffices for the theorem.

\subsection{The relative master equation}

\begin{definition}[Relative field space]
\label{def:FW}
$\FW$ is the graded space of multiplets
$(\gamma,\phi,c,\bar c,b)$ with $c\in\gW\otimes\Lambda^{\mathrm{odd}}$
and the other fields unrestricted smooth sections, together with
antifields as above, $c^*$ being a smooth section over $W$. The
antibracket \eqref{eq:antibracket} is the canonical one for this
pairing and satisfies the graded Jacobi identity.
\end{definition}

The rule \eqref{eq:kin} preserves $\FW$: $sc$ is supported in $W$
because $\gW$ is closed under the bracket, and $s\gamma$, $s\phi$ are
supported in $W$ apart from the field-independent term absent at this
order. So $\Sext$ is a well-defined functional on $\FW$.

\begin{lemma}[Vanishing of the breaking density on the plateau]
\label{lem:Kzero}
$\Kden_\nu[\gamma,\phi](x)=0$ for every $x\in P_g\cap\{t>\eps\}$ and
every smooth $(\gamma,\phi)$. In particular $\Kden_\nu=0$ on a
neighbourhood of $\overline W$.
\end{lemma}

\begin{proof}
On the open set $P_g$, $g\equiv1$, so $1-g$ and all derivatives of
$g$ vanish: $\Bsw_{g,\nu}=0$. For $t>\eps$, $\sig\equiv1$, so
$w=0$ and all derivatives of $\sig$ vanish: $wQ_\nu(A\gamma)=0$,
$\Dsource=0$, $\Zsource=wJ_3+\Dsource=0$, and every term of
$\Rampdef_{\sig,\nu}$ carries $\partial\sig$ or $\partial^2\sig$.
By \eqref{eq:W}, $\overline W\subset P_g\cap\{\eps<t<T\}$, which is
open.
\end{proof}

\begin{theorem}[Relative master equation]
\label{thm:relative}
On $\FW$,
\begin{equation}
 \boxed{\ \abr\Sext\Sext=0\quad\text{identically},\ }
 \label{eq:relativeME}
\end{equation}
exactly in $\kappa$, at $\hbar^0$. Consequently
$\sext:=(\Sext,\,\cdot\,)$ is a nilpotent, odd, ghost-number-one
derivation of the local functionals on $\FW$, and the same holds for
every $S'\in\Sclass$ whose higher antifield terms satisfy the
corresponding degree-$\ge1$ equations.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:breaking}, $(\Sext,\Sext)=\kappa\int c^\nu\Kden_\nu$.
For $c\in\gW\otimes\Lambda$ the integrand vanishes pointwise:
$c^\nu(x)=0$ for $x\notin W$ and $\Kden_\nu(x)=0$ for $x\in W$ by
Lemma~\ref{lem:Kzero}. The functional derivatives of $\Kbrk$ along
$\FW$ vanish as well: the $\gamma$- and $\phi$-derivatives retain the
factor $c^\nu$, and the $c$-derivative is $\kappa\Kden_\nu$ restricted
to $W$, which is zero. Hence $\Kbrk\equiv0$ on $\FW$ together with its
derivatives, and the graded Jacobi identity gives
$\sext^2F=\tfrac12((\Sext,\Sext),F)=0$. Theorem~\ref{thm:obstructA1}
shows the antifield-degree-zero equation is satisfied by every
$S'\in\Sclass$ on $\FW$; the remaining degree-$\ge1$ equations are the
usual conditions on $S'_{\ge2}$ and are satisfied, for example, by
$S'_{\ge2}=0$.
\end{proof}

This is XII-C's plateau contraction written in antifield form: the
relative symmetry is the compact algebra $\gW$ supported where both
backgrounds are trivial, and the master equation holds because the
breaking density vanishes there. It is an action statement. It does
not cancel a propagated observable remainder; Section~\ref{sec:A2}
shows that the remainder survives.

\begin{proposition}[Gauge fixing by XI-B's fermion]
\label{prop:gf}
Let $\Psi=-i\int\bar c_\mu(2D^\mu\gamma-b^\mu)\dd^4x$ and
$\Sgf[\Phi,\Phi^*]:=\Sext[\Phi,\Phi^*+\delta\Psi/\delta\Phi]$. Then
$(\Sgf,\Sgf)=\Kbrk$ on $\Fsp$ and $(\Sgf,\Sgf)=0$ on $\FW$. At
antifield degree zero,
\begin{equation}
 \Sgf\big|_{\Phi^*=0}
 =\SD+\int\bigl[2b_\mu D^\mu\gamma-b_\mu b^\mu
 +i\bar c_\mu\bigl(\Box c^\mu+\kappa D^\mu(\Lie_c\gamma)\bigr)\bigr]\dd^4x,
 \label{eq:Sgf}
\end{equation}
whose quadratic part is XI-B's free gauge-fixed action with auxiliary
equation $b=D\gamma$ and ghost term $i\bar c\Box c$.
\end{proposition}

\begin{proof}
The shift $\Phi^*\mapsto\Phi^*+\delta\Psi/\delta\Phi$ is a canonical
transformation of the antibracket \cite[Ch.~18]{HT}, so
$(\Sgf,\Sgf)$ is $(\Sext,\Sext)$ evaluated at shifted antifields.
Since $\Kbrk$ does not depend on antifields, both statements follow
from Theorems~\ref{thm:breaking} and \ref{thm:relative}. For
\eqref{eq:Sgf}, $\Sext$ at $\Phi^*=\delta\Psi/\delta\Phi$ equals
$\SD+s\Psi$, and with $DK=\Box$,
$s(2D^\mu\gamma-b^\mu)=\Box c^\mu+\kappa D^\mu(\Lie_c\gamma)$, so
$s\Psi=\int[b\cdot(2D\gamma-b)+i\bar c\cdot(\Box c+\kappa D\Lie_c\gamma)]$.
The quadratic part is XI-B's $S_{\mathrm{FP}}+S_\phi+\sfree\Psi$,
since the cubic-only modification does not change the free Hessian
(XII-C).
\end{proof}

\subsection{The carrier boundary}

\begin{proposition}[The relative complex meets the free ghost shell only at zero]
\label{prop:carrier}
A ghost configuration $c\in\gW\otimes\Lambda$ with $\Box c=0$ is zero.
Consequently the restriction of the relative complex $(\FW,\sext)$ to
XII-A's complete shell $\Csol$ has trivial ghost sector. The relative
master equation induces no action on $\Ealg$ beyond the inherited
free differential $\sfree$, and $\sfree$ on $\Ealg$ acts on
unrestricted free ghosts, which are not elements of $\gW$.
\end{proposition}

\begin{proof}
A compactly supported solution of the homogeneous wave equation with
support in $W\subset\{t>\eps\}$ has zero Cauchy data on a surface
$\{t=t_0\}$, $\eps<t_0<\min_{\overline W}t$, and vanishes by
uniqueness, coefficientwise \cite{XIB,BGP}. The remaining assertions
restate XII-A's construction: its carrier ghosts are free fields with
arbitrary Cauchy data.
\end{proof}

Thus A1 ends: \textbf{Constructed} for the relative master equation
on $\FW$; \textbf{Proved obstructed} for the unrestricted master
equation in $\Sclass$. No interacting differential on $\Ealg$ is
inferred from the relative identity.

%=====================================================================
\section{A2: the observable ledger}
\label{sec:A2}
%=====================================================================

The master equation does not decide XII-F. This section computes the
observable variation on the domain named in A0 with the auxiliary
pairing printed, and relates the relative differential of
Section~\ref{sec:A1} to it.

\subsection{The copied ledger and the evaluation identity}

For $\xi\in\gW$ and a free scalar, \eqref{eq:ledger-input} is the
finite variation of \eqref{eq:O} under $\gamma\mapsto\gamma+K\xi$ at
fixed $\phi$; the scalar-sourced slot has no metric argument and
contributes zero to $\Delta_\xi$. The two rows are the propagated
companion and the auxiliary pairing; nothing else.

\begin{theorem}[Evaluation identity and the printed pairing]
\label{thm:eval}
Assume \textup{(UCP$_W$)}. For every $\xi\in\gW$ and every
$u=(\gamma,\phi,b)\in\Cbos$,
\begin{equation}
 \Delta_\xi\Ocal(u)=\Aux_\xi[b,\phi]
 =-2\int(D\Lie_\xi\beta_{f,g}[\phi])_\nu b^\nu\dd^4x .
 \label{eq:eval}
\end{equation}
In particular $\Delta_\xi\Ocal=0$ on $\CWbos$, and on the detecting
tuple of Input statement~\ref{input:XIIF}, realized with
$\xi_*\in\gW$ and a switch satisfying \textup{(UCP$_W$)},
$\Delta_{\xi_*}\Ocal(u_*)=-8I<0$.
\end{theorem}

\begin{proof}
Every term of $Ds_\xi[\gamma]$ is supported in $\supp\xi\subset W$,
so $\supp(K\rho_\xi)\subset J^+(\overline W)\cap\MT$, and
\textup{(UCP$_W$)} implies (CP) for every smooth $\gamma$; XII-D's
causal plateau proposition then gives $\Aone(K\rho_\xi;\phi)=0$ for
free $\phi$. This is XIII-A's argument and yields \eqref{eq:eval} from
\eqref{eq:ledger-input}. On $\CWbos$ the compact tensor
$\Lie_\xi\beta_{f,g}[\phi]$ is supported in $W$, where $D^*b=0$, and
the second form of \eqref{eq:ledger-input} vanishes. The witness value
is \eqref{eq:witness-input}; XIII-A records that $W$ can be chosen to
contain $\supp\xi_*$ and $g$ to satisfy \textup{(UCP$_W$)} within
XII-F's admissible class.
\end{proof}

The release statement is: evaluation on $\CWbos$, with the pairing
\eqref{eq:eval} printed on $\Cbos$. It is not closure on $\Csol$. The
witness $u_*$ lies outside XIII-A's defect set, and $\gW$ is not
shrunk after it.

\subsection{The relative differential of the observable is the pairing}

Since $\Ocal$ depends on $(\gamma,\phi)$ only and is
antifield-independent,
\begin{equation}
 \sext\Ocal=\abr\Sext\Ocal
 =\pair{\Ocal'_\gamma}{s\gamma}+\pair{\Ocal'_\phi}{s\phi}
 =\sfree\Ocal+\frac\kappa2\Bigl[\pair{\Ocal'_\gamma}{\Lie_c\gamma}
 +\pair{\Ocal'_\phi}{c^\mu\partial_\mu\phi}\Bigr],
 \qquad
 \sfree\Ocal:=\pair{\Ocal'_\gamma}{\RR c}.
 \label{eq:sextO}
\end{equation}

\begin{proposition}[Relative free differential equals the auxiliary pairing]
\label{prop:relO}
Assume \textup{(UCP$_W$)}. For every $c\in\gW\otimes\Lambda^{\mathrm{odd}}$
and every $u=(\gamma,\phi,b)\in\Cbos$,
\begin{equation}
 \boxed{\
 \sfree\Ocal(u)=\Aux_{c/2}[b,\phi]
 =-\int\bigl(D\Lie_c\beta_{f,g}[\phi]\bigr)_\nu b^\nu\dd^4x
 =\int\bigl(\Lie_{c/2}\beta_{f,g}[\phi]\bigr)\cdot A\gamma\dd^4x,\ }
 \label{eq:relO}
\end{equation}
coefficientwise, where $\Aux_{c/2}$ is defined by linearity from
$\Aux_\xi$. It vanishes on $\CWbos$. For $c=2\vartheta\xi_*$ on the
detecting tuple, $\sfree\Ocal(u_*)=-8I\,\vartheta\ne0$. Moreover
\begin{equation}
 \pair{\Ocal''_{\gamma\gamma}}{K\xi\otimes K\xi}(u)=0
 \qquad(\xi\in\gW,\ u\in\Cbos).
 \label{eq:quadzero}
\end{equation}
\end{proposition}

\begin{proof}
Fix $\xi\in\gW$ and $u\in\Cbos$. Since $\PhiD$ is a polynomial of
degree two in $\gamma$ and the scalar-sourced slot has no metric
argument, $\lambda\mapsto\Delta_{\lambda\xi}\Ocal(u)$ is a real
polynomial of degree at most two:
\[
 \Delta_{\lambda\xi}\Ocal(u)
 =\lambda\pair{\Ocal'_\gamma}{K\xi}(u)
 +\tfrac{\lambda^2}{2}\pair{\Ocal''_{\gamma\gamma}}{K\xi\otimes K\xi}(u).
\]
For every real $\lambda$, $\lambda\xi\in\gW$, so Theorem~\ref{thm:eval}
gives $\Delta_{\lambda\xi}\Ocal(u)=\Aux_{\lambda\xi}[b,\phi]=\lambda\Aux_\xi[b,\phi]$,
the pairing being linear in the parameter through $\Lie_\xi\beta$.
Comparing coefficients of $\lambda$ and $\lambda^2$ gives
$\pair{\Ocal'_\gamma}{K\xi}(u)=\Aux_\xi[b,\phi]$ and
\eqref{eq:quadzero}. For $c=\sum_i\vartheta_i\xi_i$ with odd
generators $\vartheta_i$ and $\xi_i\in\gW$, linearity of
$\RR c=K(c/2)$ and of $\Ocal'_\gamma$ gives
$\sfree\Ocal(u)=\sum_i\vartheta_i\Aux_{\xi_i/2}[b,\phi]=\Aux_{c/2}[b,\phi]$.
The three forms in \eqref{eq:relO} are XII-F's shell rewriting
$A\gamma=-2D^*b$ on $\Cbos$ and the flat adjoint. Vanishing on
$\CWbos$ and the witness value follow from Theorem~\ref{thm:eval}
with $\xi=\xi_*$ and $c=2\vartheta\xi_*$.
\end{proof}

\begin{corollary}[The master equation does not decide XII-F]
\label{cor:notdecide}
On $\FW$, $(\Sext,\Sext)=0$ and $\sext$ is nilpotent
(Theorem~\ref{thm:relative}), while $\sfree\Ocal$, the $\kappa^0$ part
of $\sext\Ocal$ in the kinematic rule, is the nonzero functional
\eqref{eq:relO} on $\Cbos$. The relative symmetry of the action is
compatible with a nonzero relative variation of the displayed
observable, and that variation is exactly the auxiliary pairing.
\end{corollary}

Proposition~\ref{prop:relO} is a statement about the off-shell ghost
$c\in\gW$ evaluated on a bosonic shell configuration. It is not an
evaluation in $\Ealg$: by Proposition~\ref{prop:carrier}, a nonzero
relative ghost is not a free-shell configuration, and XIII-A's display
of $\sfree\Ocal$ for unrestricted free ghosts retains the coordinate
ramp and scalar switching terms, which vanish identically for
$c\in\gW$. The two displays are different evaluations of the same
functional $\pair{\Ocal'_\gamma}{\RR c}$.

\subsection{Printed $\kappa$-corrections and the bare/completed caveat}

The bracket in \eqref{eq:sextO} is printed and not evaluated. With
XIII-D's expansion of the scalar derivative,
\begin{equation}
 \pair{\Ocal'_\phi}{Y}
 =\pair{(\PhiD)'_\phi}{Y}
 +\Aone[f,g]\bigl(\rg'_\phi[Y];\phi\bigr)
 +\Aone[f,g]\bigl(\rg[\phi];Y\bigr),
 \qquad
 \rg'_\phi[Y]=-\tfrac12\PP\ER\bigl(gT_0'[\phi][Y]\bigr),
 \label{eq:scalarvar}
\end{equation}
with $Y=\tfrac\kappa2c^\mu\partial_\mu\phi$. Since $\Ocal$ is the
coefficient of $\kappa^2$ in the relational expansion, these terms
enter at $\kappa^3$ there; a proposal acting on the lower coefficient
$\kappa\Aone$ must print that coupling power separately. Moreover
$\Ocal$ already contains the classical scalar response $p_1,p_2$ and
the slot $\rD+\rg$; its identification with a bare field observable of
the BV complex, on which $\sext$ acts through the single off-shell
scalar, is X-G's bare-versus-completed distinction and is not made
here. Proposition~\ref{prop:relO} does not depend on that
identification: it concerns $\pair{\Ocal'_\gamma}{\RR c}$ alone.

\subsection{Completion status}

No new completion class is frozen in this manuscript. Hence:
\begin{itemize}
\item inside $\Kclass$, completion is Proved obstructed
(Input statement~\ref{input:XIIID}); the cochain
$\mathcal C_\xi=-\Aux_\xi[D\gamma,\phi]$ is admitted, cancels
\eqref{eq:eval} algebraically on $\Cbos$, and is not the variation of
one fixed functional under the forced representation; citing it as the
completion would fail this gate;
\item a pure auxiliary or ghost rule changes $\Ocal$ by zero
(Input statement~\ref{input:XIIF});
\item constant compact metric translations commute and do not preserve
$\Nsol$; they do not supply the representation needed for a cocycle
test (XIII-A);
\item a coefficientwise quotient by vanishing on $\CW$ is incompatible
with the inherited brackets (Input statement~\ref{input:XIIIA}); the
evaluation identity of Theorem~\ref{thm:eval} is not promoted to a
quotient;
\item nonlocal Cauchy-data actions, nonzero past data, and other
integration interfaces remain undecided; none is frozen or tested here.
\end{itemize}

A2 ends: \textbf{Constructed} as an evaluation identity on $\CWbos$
with the pairing printed on $\Cbos$; \textbf{Proved obstructed} for
unqualified invariance on $\Csol$ and for completion inside $\Kclass$;
the completion in any unfrozen class is outside the claim.

%=====================================================================
\section{A3: recorded, not opened}
\label{sec:A3}
%=====================================================================

\begin{itemize}
\item XIII-C's four obligations (Input statement~\ref{input:XIIIC})
remain the gate in front of any quantum gauge defect. $\sfree$ and
$\sext$ are not applied to $\Rquant$; no $R_1$ coefficient is formed;
no Hadamard line is contracted against $\JD$.
\item X-G's reciprocity theorem is a non-identification: the sheet
selected in A0 is density switched, and X-G's mixed map is not its
field map. Had A0 selected the source-switched sheet, the master
equation would already be blocked by X-G's definition of an exact
upgrade; that ending would concern that upgrade only.
\item The convolution defect $S_\eps\xi\ne\xi$ is a different coordinate
prescription; it is not identified with the ramp edge and is not an
input to Theorem~\ref{thm:breaking}.
\item $\Jcal^{\mathrm D}_{g,\sig}(\phi_m)=\varnothing$ stands. The slot
$\rD+\rg$ solves a gauge-fixed forcing equation; it does not fill the
invariant coupled jet, and the empty jet is not a failure of $\Ocal$.
\item The XI-G classes with covectors $\pm(k_1,k_2,0)$,
$k_1+k_2=2\dd t$, remain excluded; $\Ocal$ is not a representative.
\item No reduced algebra, physical cohomology class, state, loop,
antibracket on $\Ealg$, interacting differential on $\Ealg$, or
$\eps\to0$ limit is constructed. The antibracket of
Section~\ref{sec:A1} is the classical off-shell antibracket on local
functionals of $\Fsp$ and $\FW$.
\end{itemize}

%=====================================================================
\section{Decision, release, and handoff}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.27\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
A0 & Frozen &
Cubic-only action \eqref{eq:SD}; inverse $-1$; inert $g,\sig$;
$\gW$; observable \eqref{eq:O}; relative route. Convolution and
source-switched sheets recorded, not used.\\[0.35em]
A1 & Constructed (relative); Proved obstructed (unrestricted) &
$(\Sext,\Sext)=\kappa\int c^\nu\Kden_\nu$ exactly, all terms displayed;
nonzero late-switch value on $\Fsp$ for every $S'\in\Sclass$;
zero identically on $\FW$ with nilpotent $\sext$; XI-B's fermion
preserves both statements; relative ghosts meet the free shell only
at $0$.\\[0.35em]
A2 & Constructed on $\CWbos$ (evaluation); Proved obstructed on $\Csol$ &
Under (UCP$_W$), $\Delta_\xi\Ocal|_{\Cbos}=\Aux_\xi$, zero on
$\CWbos$, $-8I$ on $u_*$; $\sfree\Ocal=\Aux_{c/2}$ for relative
ghosts; cochain not promoted; $\Kclass$ excluded by XIII-D.\\[0.35em]
A3 & Recorded &
XIII-C's obligations; X-G reciprocity and $-4$ as non-identifications;
convolution defect; empty jet; XI-G exclusion; no $\sfree$ on $\Rquant$.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[XIV-A release]
\label{thm:release}
For the sheet frozen in Section~\ref{sec:sheet}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item On the unrestricted field space $\Fsp$, every extension in the
kinematic class $\Sclass$ has antibracket square
$\kappa\int c^\nu\Kden_\nu$ at antifield degree zero, with $\Kden_\nu$
given by \eqref{eq:Kden}; this functional is nonzero
(Proposition~\ref{prop:witnessA1}). The unrestricted classical master
equation is Proved obstructed in $\Sclass$ at $\hbar^0$.
\item On the relative space $\FW$ of $W$-supported ghosts,
$(\Sext,\Sext)=0$ exactly, $\sext$ is nilpotent, and XI-B's gauge
fixing preserves the identity. The relative complex meets XII-A's free
ghost shell only at $c=0$.
\item Under \textup{(UCP$_W$)}, the finite variation of $\Ocal$ on
$\Cbos$ equals the auxiliary pairing $\Aux_\xi[b,\phi]$; it vanishes on
$\CWbos$ and equals $-8I<0$ on the copied witness. The relative free
differential of $\Ocal$ on $\Cbos$ is $\Aux_{c/2}[b,\phi]$.
\end{enumerate}
The ending for the frozen relative sheet, namely $(\FW,\sext)$
together with evaluation of $\Ocal$ on $\CWbos$, is
\textbf{Constructed}. The unrestricted versions of both interfaces are
Proved obstructed and are the boundary of the claim. The auxiliary
pairing is restricted to an evaluation set; it is neither cancelled
nor completed.
\end{theorem}

\begin{proof}
Theorems~\ref{thm:breaking}, \ref{thm:obstructA1}, \ref{thm:relative},
\ref{thm:eval}; Propositions~\ref{prop:gf}, \ref{prop:carrier},
\ref{prop:relO}.
\end{proof}

The constructed release is relative. It opens XIV-B's antifield
enlargement only as a continuation on $\FW$, with the ghost-sector
boundary of Proposition~\ref{prop:carrier} recorded, and not as an
interacting differential on $\Ealg$. XIII-C's four obligations remain
the gate in front of any Ward defect.

\subsection*{Handoff}

A successor receives:
\begin{enumerate}
\item the frozen sheet: action \eqref{eq:SD}, inverse $-1$, inert
backgrounds, $\gW$, observable \eqref{eq:O};
\item the antifield sector, antibracket, and kinematic rule
\eqref{eq:kin} with $c=2\xi$;
\item the exact breaking functional \eqref{eq:breaking}--\eqref{eq:Kden}
and its late-switch witness \eqref{eq:witnessA1};
\item the relative master equation \eqref{eq:relativeME}, the
gauge-fixed form \eqref{eq:Sgf}, and the carrier boundary;
\item the evaluation identity \eqref{eq:eval} with its domain, the
witness value, and the identification \eqref{eq:relO} of the relative
differential of $\Ocal$ with the auxiliary pairing;
\item the printed, unevaluated $\kappa$-corrections
\eqref{eq:sextO}--\eqref{eq:scalarvar} and the bare/completed caveat.
\end{enumerate}
It receives no reduced algebra, completed quantum binary datum,
physical cohomology class, interacting differential on $\Ealg$,
completion of the auxiliary pairing, spurion law, or sharp limit.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \abr\Sext\Sext=\kappa\int c^\nu\Kden_\nu,\qquad
 \Kden_\nu=\Bsw_{g,\nu}+wQ_\nu(A\gamma)-\Rampdef_{\sig,\nu}-\kappa Q_\nu(\Zsource),\\[2pt]
 \Kbrk[0,\phi_m,\vartheta\zeta\partial_t]=\tfrac\kappa2\vartheta\!\int\!\zeta\,\partial_tg\ne0
 \ \text{on }\Fsp;\qquad
 \Kbrk\equiv0\ \text{on }\FW,\quad \sext^2=0;\\[2pt]
 \gW\cap\{\Box c=0\}=\{0\};\qquad
 \text{(UCP}_W)\Rightarrow
 \Delta_\xi\Ocal|_{\Cbos}=\Aux_\xi,\quad
 \Delta_\xi\Ocal|_{\CWbos}=0,\quad
 \Delta_{\xi_*}\Ocal(u_*)=-8I<0;\\[2pt]
 \sfree\Ocal=\Aux_{c/2}[b,\phi]\ \text{for }c\in\gW\otimes\Lambda^{\mathrm{odd}}\text{ on }\Cbos .
 \end{gathered}}
\]
A reader can see which action was frozen, why the convolution sheet
and the source-switched sheet were not used, and that the auxiliary
pairing was restricted to an evaluation set rather than cancelled. A
constructed master equation is an identity of the extended action on
its named ghost space. An unresolved product remains unresolved.
Neither is an interacting quantum algebra.

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

\paragraph{Classical scope.}
All identities are classical, at $\hbar^0$, on smooth off-shell
configurations or on the named bosonic shell. The antibracket is the
classical antibracket of local functionals. No quantum antibracket,
quantum master equation, quantum Ward identity, or quantum retarded
coefficient is constructed.

\paragraph{Domain scope.}
The relative master equation is an identity on $\FW$, whose ghosts are
supported where both backgrounds are trivial. Its restriction to
XII-A's free shell has trivial ghost sector. The unrestricted master
equation in the frozen kinematic class fails by an explicit nonzero
$\hbar^0$ functional. The observable identity is an evaluation on
$\CWbos$, not a quotient and not closure on $\Csol$.

\paragraph{Non-identification scope.}
The convolution-regulated sheet, the source-switched sheet with
inverse $-4$, and the cubic-only sheet with inverse $-1$ are three
different models. Only the third is frozen. No obstruction proved on
another sheet is imported as a theorem about this one; the breaking
functional is recomputed from XII-C's identity in the present
conventions.

\paragraph{Dependency scope.}
XII-C supplies the action, its Euler derivatives, and the changed
Noether identity. XI-B supplies the inverse normalization, fermion,
ghost normalization, and the compact residual lemma. XII-A supplies the
carrier and the frozen free rule. XII-E and XII-F supply the observable,
its finite ledger, and the witness value. XIII-A supplies the uniform
plateau hypothesis and the evaluation identity. XIII-D supplies the
class exclusion. X-G supplies the normalization-independent kinematic
nilpotence lemma. The antifield sector, the exact antibracket square,
its late-switch witness, the relative master equation, the carrier
boundary, and the identification of the relative differential of
$\Ocal$ with the auxiliary pairing are proved here.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant check,
independently executed symbolic computation, independent referee
review, or journal acceptance is asserted. The witness values are
analytic; the value $-8I$ is copied from XII-F and not recomputed.

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

\bibitem{XIIC}
N.~Cheung,
\emph{Landmark XII-C: One declared cubic weighting: actual Euler
source, changed Noether identity, and the coupled-jet question}.
Supplied manuscript \texttt{XII-C.tex}.
Selected action, Euler derivatives, changed Noether identity, plateau
contraction, late-switch lemma, empty jet.

\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}.
Causal plateau proposition and roots.

\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}.
Assembled coefficient, sign, finite ledger, scalar-sourced slot.

\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}.
Auxiliary pairing, shell rewriting, compact witness and its value.

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

\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 $-1$, gauge fermion, ghost normalization, compact residual
intersection.

\bibitem{XIIIA}
N.~Cheung,
\emph{Landmark XIII-A: Classical cancellation on a fixed subclass, and
the obstruction to a coefficientwise restriction quotient}.
Supplied manuscript \texttt{XIII-A.tex}.
Uniform plateau hypothesis, evaluation identity, non-ideal theorem,
ordering counterexample.

\bibitem{XIIIC}
N.~Cheung,
\emph{Landmark XIII-C: The joint two-line extension of one selected
binary pair, on the unchanged carrier}.
Supplied manuscript \texttt{XIII-C.tex}.
Unresolved coefficient and successor obligations.

\bibitem{XIIID}
N.~Cheung,
\emph{Landmark XIII-D: One bounded classical completion of the
auxiliary pairing, or its obstruction on the unchanged sheet}.
Supplied manuscript \texttt{XIII-D.tex}.
Class exclusion, commutator parameters, admitted cochain, scalar
derivative expansion.

\bibitem{XAconv}
N.~Cheung,
\emph{Landmark X-A: Classical covariance of a fixed-regulator switched
relational scalar through order $\kappa^2$: an explicit regulator
obstruction}.
Supplied manuscript \texttt{X-A-conv.tex}.
Convolution-regulator defect; recorded as a non-identification.

\bibitem{XG}
N.~Cheung,
\emph{Landmark X-G: The normalized BRST multiplet and the one-loop
background gate}.
Supplied manuscript \texttt{X-G.tex}.
Kinematic nilpotence lemma; reciprocity obstruction and inverse $-4$
recorded as non-identifications.

\bibitem{HT}
M.~Henneaux and C.~Teitelboim,
\emph{Quantization of Gauge Systems},
Princeton University Press, 1992.
Chapters~15, 17, 18: antibracket, master equation and its antifield
expansion, canonical transformations and gauge fixing.

\bibitem{GPS}
J.~Gomis, J.~Par\'is, and S.~Samuel,
``Antibracket, antifields and gauge-theory quantization,''
\emph{Physics Reports} \textbf{259} (1995), 1--145,
arXiv:hep-th/9412228.
Antifield-number decomposition of the classical master equation.

\bibitem{FRBV}
K.~Fredenhagen and K.~Rejzner,
``Batalin--Vilkovisky formalism in the functional approach to
classical field theory,''
\emph{Communications in Mathematical Physics}
\textbf{314} (2012), 93--127, arXiv:1101.5112.
Antibracket of local functionals on smooth configuration spaces.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
ESI Lectures in Mathematics and Physics,
European Mathematical Society, 2007, arXiv:0806.1036.
Smooth Cauchy uniqueness and finite propagation.

\end{thebibliography}

\end{document}