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% XIII-D.tex
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% Exact release:
%   - one parameter-local/one-retarded correction class is frozen;
%   - shell descent forces every action in that class to act trivially
%     on the complete free-shell quotient;
%   - the copied auxiliary pairing is nonzero on an explicit parameter
%     commutator, using a specialization of XII-F's witness family;
%   - the necessary cocycle condition therefore fails for every
%     shell-descending representation in the frozen class;
%   - an admitted parameter-dependent cancellation cochain is displayed,
%     but is not misidentified as an integrated completion;
%   - the integrated completion problem in the frozen class is
%     Proved obstructed.
%
% Scope qualification:
% Parameter locality of the unpropagated components is an explicit
% ansatz, frozen before the witness test. It makes one precise bounded
% class inside the requested envelope. No obstruction is asserted for
% arbitrary nonlocal actions on Cauchy data outside this class.
%
% XIII-C's joint quantum extension remains Unresolved.
% No R_1^(1), higher retarded product, quantum gauge defect,
% restriction quotient, interacting differential, master equation,
% state, antifield, physical cohomology class, or sharp-width limit
% is constructed.
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% No successful compilation, proof-assistant verification,
% independently executed symbolic computation, independent referee
% review, or journal acceptance is asserted.
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\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{\ER}{E_0^{\mathrm R}}
\newcommand{\EA}{E_0^{\mathrm A}}
\newcommand{\ERm}{E_m^{\mathrm R}}
\newcommand{\Pmass}{P_m}
\newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\GR}{G_{\mathrm{act}}^{\mathrm R}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Fsol}{\mathscr F^{\mathrm{sol}}_{\mathrm{mc,pol}}}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\NW}{\mathscr N_W}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\CW}{\mathscr C_W}
\newcommand{\gW}{\mathfrak g_W}
\newcommand{\Ocal}{\mathcal O}
\newcommand{\Vcal}{\mathcal V_g}
\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{\Bcal}{\mathfrak B}
\newcommand{\Kclass}{\mathfrak K_{\mathrm{loc,R}}}
\newcommand{\Ccal}{\mathcal C}
\newcommand{\Rtree}{R_1^{(0)}}
\newcommand{\Rquant}{R_1^{(1)}}
\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}}
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}

\title{\textbf{Landmark XIII-D}\\[0.4em]
One bounded classical completion of the auxiliary pairing,\\
or its obstruction on the unchanged sheet\\[0.4em]
\large A parameter-local/one-retarded class,\\
shell descent, and a classical cocycle obstruction}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We freeze the classical completion route for the displayed
second-order coefficient, without changing the action, the external
switches, or the inherited carrier. The tested correction class has
parameter-local unpropagated components and at most one new retarded
metric application. A successful correction must also supply an even
representation of the compact vector-field algebra preserving the
complete free-shell equation ideal.

These requirements have a decisive consequence. A causal,
zero-past field variation tangent to the homogeneous free shell
vanishes there by Cauchy uniqueness. Every shell-descending action
in the frozen class therefore induces the trivial representation
on the existing classical shell quotient.

We specialize the copied XII-F witness family so that its detecting
parameter is an explicit commutator of two allowed compact parameters.
The auxiliary pairing is nonzero on that commutator. Its necessary
cocycle condition consequently fails for every representation in
the frozen class, excluding both the slot equation and an integrated
compensating functional in that class.

An admitted parameter-dependent bilinear cochain cancels the copied
value algebraically. Its kernel is displayed, but it is not the
variation of a compensating functional under the declared action.
This distinction prevents algebraic cancellation from being released
as an integrated completion.

The ending is Proved obstructed for the explicitly frozen class,
not for arbitrary nonlocal actions outside it. The quantum
two-line extension remains Unresolved. No quantum gauge-consistency
defect or interacting differential 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) {Route A};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Route D, inside one class};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Every extension};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Route C is not opened. One route does not decide every extension.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{One route is not every extension. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Route A};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Every extension};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Route C is not opened. One route does not decide every extension.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{One route is not every extension. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Route A};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Route D, inside one class};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Every extension};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Route C is not opened. One route does not decide every extension.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-XIII-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Route A};
\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) {Every extension};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Route D, inside one class. Route C is not opened. One route does not decide every extension.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Route A is not Every extension. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Route A};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Route D, inside one class};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Every extension};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Route C is not opened. One route does not decide every extension.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-XIII-D}
\end{figure}

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


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



\scopebox{Release and scope}{%
\textbf{Route selected:} Route A, with Route D as the obstruction
alternative inside one class. Route B is the evaluation-only result
already released in XIII-A. Route C is not opened.

\smallskip
\textbf{Class frozen:} $\Kclass(f,g,W)$, the parameter-local/
one-retarded, shell-descending class of
Definition~\ref{def:class}. Parameter locality of its unpropagated
components is an explicit ansatz, not an inference about every
possible correction.

\smallskip
\textbf{Constructed:} the class definition, its forced trivial
shell representation, the necessary cocycle calculation, and one
admitted cancellation cochain.

\smallskip
\textbf{Proved obstructed:} an integrated cancellation in this
class for the detecting fixed data of
Section~\ref{sec:witness}. Every member satisfying the representation
and descent interfaces fails the cancellation target.

\smallskip
\textbf{Outside the class:} arbitrary nonlocal actions on Cauchy
data and other completions remain undecided. The result is not a
universal completion no-go.

\smallskip
$\CW$ remains an evaluation set. No quotient by $\NW$ is taken.
XIII-C's quantum extension and the quantum gauge-consistency gate
remain closed.}

\tableofcontents

%=====================================================================
\section{Copied prerequisite sheet}
\label{sec:inputs}
%=====================================================================

The three endings have their inherited meanings:
\begin{enumerate}
\item \textbf{Constructed}: the named correction exists, is admitted,
and satisfies every declared interface.
\item \textbf{Proved obstructed}: an explicit incompatibility excludes
every successful map in a previously frozen class.
\item \textbf{Unresolved}: a required map, representation, cocycle,
admission, or support theorem has not been established.
\end{enumerate}
A missing map is not a no-go. A sufficient wavefront criterion that
fails is not, by itself, a nonexistence theorem.

\subsection{XIII-A: evaluation and the non-ideal boundary}

\begin{inputstatement}[Restriction handoff]
Under $(\mathrm{UCP}_W)$,
\[
 \Delta_\xi\Ocal\big|_{\CW}=0,
 \qquad \xi\in C_c^\infty(W;TM),
\]
with
\[
 \overline W\subset\Int\{g=1\}\cap\{\eps<t<T\},
 \qquad
 \CW=\{u\in\Csol:D^*b=0\text{ on }W\}.
\]
The coefficientwise kernel $\NW$ is not a two-sided ideal:
compact tests give
\[
 [L_q,G_h]_{\star_H}
 =i\hbar C(q,h)\mathbf1,\qquad C(q,h)>0,
\]
and an admissible rank-one ordering change satisfies
\[
 \alpha_d(L_q^2)=L_q^2+\hbar L_q(u)^2.
\]
Forcing $\hbar\mathbf1=0$ does not force $\mathbf1=0$ over
$\C[[\hbar]]$. It does rule out a torsion-free formal quantum
quotient with the inherited brackets. No reduced algebra was
handed off \cite{XIIIA}.
\end{inputstatement}

Neither the commutator nor the ordering counterexample is
recomputed here.

\subsection{XIII-B: the tree datum}

\begin{inputstatement}[Classical binary handoff]
The vertex
\[
 \Vcal=-\frac12\int g\,\gamma_{\mu\nu}
                         T_0^{\mu\nu}[\phi]\,\dd^4x
\]
is admitted. The tree datum
\[
 \Rtree
 =-\pair{\Ocal'}{\GR\Vcal'}
 =\pair{\Ocal'_\gamma}{\rg[\phi]}
                   +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]}
\]
is constructed, proper, and microcausal. Ordered values of $R_1$
are $0$ or
$\frac{i}{\hbar}[\Ocal,\Vcal]_{\star_H}$.
The literal normalization
$R_1=\Vcal\star_H\Ocal$ is proved obstructed.
The joint two-line extension was left unresolved \cite{XIIIB}.
\end{inputstatement}

This tree datum is not the correction studied below.

\subsection{XIII-C: the quantum stop}

\begin{inputstatement}[Joint-extension handoff]
The eleven-branch inventory, the widened class
$\mathfrak C_{\mathrm{tot}}\subset\mathfrak C_{\mathrm{part}}$,
and the elementary parallel-line extensions are constructed.
The parallel kinetic primitive has scaling degree $8$ and
codimension $4$.

Collision exclusion and the $Z_3$ bounds do not establish the joint
off-collision product: distinct null-related vertices can carry a
nonzero wavefront cancellation. The coefficient linear in $\hbar$
remains Unresolved. That is a failed sufficient product criterion,
not a nonexistence theorem. The free differential was not applied
to $\Rquant$. No gauge-consistency defect was opened \cite{XIIIC}.
\end{inputstatement}

No quantum extension premise is used in the classical proof.

\subsection{XII-F: the retained pairing and witness}

\begin{inputstatement}[Auxiliary remainder]
On $\Csol$, under the copied causal-plateau condition,
\begin{equation}
 \Delta_\xi\PhiD
 =\Bcal_\xi[b,\phi]
 =-2\int
   \bigl(D\Lie_\xi\beta_{f,g}[\phi]\bigr)\cdot b\,\dd^4x.
 \label{eq:B-input}
\end{equation}
The companion vanishes under (CP). Local vanishing holds if
$D^*b=0$ near $\supp\xi$.

The compact free-shell witness satisfying (CP) has
\begin{equation}
 \Delta_\xi\PhiD
 =-8\int f(t,\mathbf x)
       \int_0^t(t-s)^2a(s)\,\dd s\,\dd^4x<0.
 \label{eq:witness-input}
\end{equation}
Its carrier class is nonzero. A cancelling co-transformation was
not supplied. The scalar-sourced slot has zero metric variation
and is included once \cite{XIIF}.
\end{inputstatement}

For the kernel calculation only, we also use XII-F's admission of
the auxiliary row: for fixed $\xi,f,g$ its bilinear kernel in
$(b,\phi)$ is compact and has wavefront set contained in $Z_2$.
This is the kernel used in its nonzero-class proof, not the cubic
kernel of $\CD$.

\subsection{The assembled coefficient and unchanged action}

The displayed lift is
\begin{equation}
 \Ocal
 =\PhiD+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD,
 \label{eq:O}
\end{equation}
with
\begin{equation}
 \rD=\PP\ER\JD=-u_{\mathrm D},
 \qquad
 \rg[\phi]=-\tfrac12\PP\ER(gT_0[\phi]).
 \label{eq:slots}
\end{equation}
The opposite insertion sign is already detected and is not
reopened \cite{XIIE}.

The action is XII-C's cubic-only density weighting,
\[
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}+\kappa\int(\sig-1)\ell_3\,\dd^4x.
\]
Its fixed-input jet
$\mathcal J_{g,\sigma}^{\mathrm D}(\phi_m)=\varnothing$
is retained as a boundary, not an interaction entry
\cite{XIIC}. The inverse coefficient remains $-1$.
The free rule $s_0\gamma=\RR c$ is odd and has ghost number one;
it is not the even rule $Y_\xi$ studied here \cite{XIIA,XIB}.

%=====================================================================
\section{D0: the route and the unchanged sheet}
\label{sec:freeze}
%=====================================================================

\[
 \boxed{\text{Route A is frozen; Route D is its obstruction alternative.}}
\]

Use
\[
 M=\R^4,\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \Pmass=\Box+m^2,
\]
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad 0<\eps<T.
\]
The backgrounds $g$ and $\sig$ are fixed; neither is a spurion.
The independent fields are
$(\gamma,\phi,c,\bar c,b)$, with no antifields. The complete 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}
Fields are ambient smooth configurations. No spatial decay is
required. Grassmann-valued identities are interpreted coefficientwise.

Fix $f,g,m,\eps,T,W$ before selecting correction maps, with
\[
 W\ne\varnothing,\qquad
 \overline W\subset P_g\cap\{\eps<t<T\},
 \qquad P_g=\Int\{g=1\},
\]
and
\[
 \gW=C_c^\infty(W;TM).
\]
Its bracket is the ordinary vector-field bracket
\begin{equation}
 [\xi,\zeta]^\mu
 =\xi^\nu\partial_\nu\zeta^\mu
                         -\zeta^\nu\partial_\nu\xi^\mu.
 \label{eq:bracket}
\end{equation}
This is not the free ghost solution space.

Impose the uniform hypothesis
\begin{equation}
 \mathrm{(UCP}_W\mathrm{)}\qquad
 g=1\text{ near }
 J^-(\supp f)\cap J^+(W)\cap\MT.
 \label{eq:UCP}
\end{equation}
Then the copied remainder, including the scalar-sourced slot, is
\begin{equation}
 \Delta_\xi\Ocal\big|_{\Csol}
       =\Bcal_\xi\big|_{\Csol}.
 \label{eq:full-remainder}
\end{equation}
The domain of the cancellation target is the complete shell.
$\CW$ is only a subsequent evaluation set.

The order is classical, $\hbar^0$, at the displayed $\kappa^2$
coefficient. One additional even correction is allowed.
No insertion counted by $\lambda$ is added, and no coefficient
of a quantum retarded product is constructed.

\subsection{Routes not reopened}

Route B is the conditional evaluation identity of XIII-A.
Its naive quantum quotient is already obstructed.
This paper does not construct a reduction.

Route C would change the problem to an interacting or quantum
construction. It is not selected. In particular, the unresolved
two-line coefficient is neither used nor differentiated.

\subsection{What makes the present class bounded}

The production envelope leaves the unpropagated parts of a rule
unspecified. We make one precise choice: those parts are
parameter-local, and the only new propagation is at most one
application of $\Gret=-\PP\ER$.

This is an explicit restriction defining the tested class.
It is made before the witness is used in a calculation.
The theorem below does not exclude arbitrary nonlocal
transformations of Cauchy data outside this realization.

%=====================================================================
\section{D1: the frozen correction class}
\label{sec:class}
%=====================================================================

For $\xi\in\gW$, write $S_\xi=\supp\xi$.
A parameter-local expression has the form
\[
 \ell_\xi^A[\Phi](x)
 =\sum_{\alpha,j}
   (\partial^\alpha\xi^j)(x)
                         L_{\alpha j}^A[\Phi](x),
\]
where the sum is finite and the coefficients are smooth polynomial
expressions in field jets and the fixed backgrounds.
Its support is contained in $S_\xi$.

\begin{definition}[The class $\Kclass(f,g,W)$]
\label{def:class}
A structural proposal in $\Kclass(f,g,W)$ consists of an even,
ghost-number-zero, parameter-linear field rule
\[
 Y_\xi=(Y_\xi^\gamma,Y_\xi^\phi,0,0,Y_\xi^b),
\]
possibly accompanied by one fixed compensating functional $F$
of the metric and scalar slots, with the following restrictions.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The rule has parameter-local parts and, at most, one new
retarded output
\begin{equation}
 r_\xi[\Phi]=\Gret\,j_\xi[\Phi],
 \qquad
 \supp j_\xi[\Phi]\subseteq S_\xi.
 \label{eq:one-retarded}
\end{equation}
The source is smooth, compact, polynomial in the fields and linear
in $\xi$.
\item Every nonlocal component of $Y_\xi$ is a finite differential
expression in this same $r_\xi$, multiplied by smooth polynomial
field coefficients. Thus
\begin{equation}
 Y_\xi^A=\ell_\xi^A+\mathsf M^A[\Phi]r_\xi,
 \qquad A\in\{\gamma,\phi,b\},
 \label{eq:Y-form}
\end{equation}
with zero retarded output permitted.
\item The output in \eqref{eq:one-retarded} has zero past data and,
on the slab, support in $J^+(S_\xi)\cap\MT$.
There is no second new retarded application.
\item A proposed cancellation effect must be an even,
ghost-number-zero homogeneous quadratic polynomial with a compact
polarized kernel bounded by $Z_2$.
For a direct slot correction, that effect is
\begin{equation}
 K_\xi^Y
 =\pair{\Ocal'_\gamma}{Y_\xi^\gamma}
                      +\pair{\Ocal'_\phi}{Y_\xi^\phi}.
 \label{eq:effect}
\end{equation}
For an integrated compensating-functional proposal, the effect is
$\delta_\xi F$, where $\delta$ is the declared action below.
The functional $F$ is fixed independently of $\xi$.
\item No new Einstein--Hilbert vertex, witness-dependent contact,
additional field, or modification of $\rD$ is permitted.
\end{enumerate}
For a successful member, the directional rule must define an even
derivation of the ordinary polynomial carrier,
\begin{equation}
 \delta_\xi G
 =\pair{G'_\gamma}{Y_\xi^\gamma}
  +\pair{G'_\phi}{Y_\xi^\phi}
  +\pair{G'_b}{Y_\xi^b},
 \label{eq:delta}
\end{equation}
preserve $\Nsol$, and induce a representation with
\begin{equation}
 [\bar\delta_\xi,\bar\delta_\zeta]
                         =\bar\delta_{[\xi,\zeta]}
 \quad\text{on }\Fsol=\Fpol/\Nsol.
 \label{eq:representation}
\end{equation}
Bars denote the inherited classical shell quotient, not a new
quotient by $\NW$.
\end{definition}

The auxiliary component is allowed only to make the multiplet rule
well defined and to test equation preservation. Since $\Ocal$ and
the proposed $F$ have only metric and scalar arguments, it does
not independently change those functionals.

The class is nonempty as a structural class: the zero rule and
zero functional satisfy its bounds and define a representation.
They do not satisfy the nonzero cancellation target.

\begin{remark}[Effect versus integrated functional]
A parameter-indexed family $C_\xi$ is not automatically the
variation of one functional $F$. Calling $C_\xi=-\Bcal_\xi$ a
``counterterm'' does not establish
$\bar\delta_\xi\bar F=-\bar\Bcal_\xi$.
The latter is the integrated interface tested here.
\end{remark}

The only direct slot target is
\begin{equation}
 K_\xi^Y\big|_{\Csol}=-\Bcal_\xi\big|_{\Csol}.
 \label{eq:target}
\end{equation}
An integrated compensating functional must satisfy the corresponding
equation $\bar\delta_\xi\bar F=-\bar\Bcal_\xi$.
No map satisfying only a smooth pointwise assignment is released.

%=====================================================================
\section{D2: representation before the cocycle}
\label{sec:representation}
%=====================================================================

\subsection{Why compact translations do not supply it}

For comparison, define
\[
 T_\xi G=\pair{G'_\gamma}{K\xi}.
\]
These constant directional derivatives commute:
$[T_\xi,T_\zeta]=0$.
They are not the representation in \eqref{eq:representation}.

They also fail equation-ideal preservation. For a compact test $q$,
\[
 E_q(\Phi)=\int q^{\mu\nu}\Box\gamma_{\mu\nu}\,\dd^4x
                                                        \in\Nsol,
\]
but
\[
 T_\xi E_q
 =\int q^{\mu\nu}\Box(K\xi)_{\mu\nu}\,\dd^4x
\]
can be a nonzero constant.

Indeed, if a nonzero compact $\xi$ had $\Box K\xi=0$, Cauchy
uniqueness would give $K\xi=0$. The Killing equation in Minkowski
space then gives $\partial_\mu\partial_\nu\xi_\rho=0$,
so $\xi$ is affine and cannot be nonzero with compact support.
An appropriate compact component test $q$ therefore detects
$\Box K\xi$ \cite[Chapter~3]{BGP}.

The natural tensor pullback rule is not a remedy on the fixed free
sheet. With the convention
\[
 \delta_\xi^{\mathrm{nat}}G
       =-\sum_A\pair{G'_A}{\Lie_\xi\Phi_A},
\]
it has the ordinary vector-field representation sign, but fails
shell descent. At the free configuration $\gamma=\eta$, $b=0$,
its metric variation is $-K\xi$, and the same equation test is
nonzero. The fixed flat wave operator is not preserved by an
arbitrary compact diffeomorphism.

These comparisons do not enter the cocycle calculation.

\subsection{The representation forced by the frozen class}

\begin{lemma}[Causal support of every field component]
\label{lem:causal}
For every structural proposal in $\Kclass$ and every smooth input,
\[
 \supp Y_\xi^A[\Phi]\subseteq J^+(S_\xi),
 \qquad A\in\{\gamma,\phi,b\}.
\]
Every component vanishes on an open time band preceding $S_\xi$.
\end{lemma}

\begin{proof}
The parameter-local terms are supported in $S_\xi$.
The retarded output is supported in its causal future.
Differentiation and multiplication by smooth coefficients do not
enlarge distributional support.
Since $S_\xi$ is compact inside $\{t>\eps\}$, it has a minimum
time strictly above $\eps$.
\end{proof}

\begin{theorem}[Shell descent forces triviality]
\label{thm:triviality}
Suppose a rule in $\Kclass$ defines \eqref{eq:delta} and preserves
$\Nsol$. Then, for every $u\in\Csol$,
\begin{equation}
 Y_\xi^\gamma[u]=Y_\xi^\phi[u]=Y_\xi^b[u]=0.
 \label{eq:Y-zero}
\end{equation}
Consequently its induced action is
\begin{equation}
 \boxed{\bar\delta_\xi=0\quad\text{on }\Fsol.}
 \label{eq:trivial-rep}
\end{equation}
\end{theorem}

\begin{proof}
Apply ideal preservation to all compactly smeared linear equation
functionals. The metric and scalar tests give
\[
 \Box Y_\xi^\gamma[u]=0,\qquad
 \Pmass Y_\xi^\phi[u]=0.
\]
The auxiliary relation gives
\[
 Y_\xi^b[u]=D Y_\xi^\gamma[u].
\]
These identities follow because the compact tests separate smooth
functions and $\delta_\xi E$ vanishes on the shell whenever
$E\in\Nsol$.

By Lemma~\ref{lem:causal}, the metric and scalar variations vanish,
with their first time derivatives, on a Cauchy surface preceding
$S_\xi$. Smooth homogeneous wave uniqueness therefore makes them
zero. The auxiliary identity then gives $Y_\xi^b[u]=0$.
For Grassmann-valued fields the same argument applies to every
coefficient \cite[Chapter~3]{BGP}.

Finally, evaluation of \eqref{eq:delta} on $u$ pairs each derivative
with a zero variation. Thus $\delta_\xi G$ vanishes on $\Csol$ for
every admitted $G$, proving \eqref{eq:trivial-rep}.
\end{proof}

This specifies the representation before any cocycle test:
every successful shell-descending proposal in the frozen class
has the trivial module action. The zero off-shell rule realizes
this representation, so the conclusion is not based on an
unconstructed action.

The theorem uses the simultaneous requirements of causal
zero-past output and free-shell tangency. It does not state that
every retarded response vanishes. A sourced response generally
fails the homogeneous tangent equations.

Nor does it state that every action on free Cauchy data is trivial.
An action with nonzero past data or an unpropagated nonlocal
component lies outside the frozen class.

%=====================================================================
\section{D3: the necessary cocycle and its consequence}
\label{sec:cocycle}
%=====================================================================

Write $\bar\Bcal_\xi$ for the class of the admitted auxiliary row
in $\Fsol$. It is an even linear map from $\gW$ to that vector
space. Parameters and the external functions are not varied.

\begin{lemma}[Necessary cocycle]
\label{lem:cocycle}
For a representation satisfying \eqref{eq:representation}, an
equation
\[
 \bar\delta_\xi\bar G=-\bar\Bcal_\xi
\]
for one fixed functional $G$ requires
\begin{equation}
 \bar\delta_\xi\bar\Bcal_\zeta
 -\bar\delta_\zeta\bar\Bcal_\xi
                         =\bar\Bcal_{[\xi,\zeta]}.
 \label{eq:cocycle}
\end{equation}
For a direct slot proposal, take $G=\Ocal$.
For a compensating-functional proposal, take $G=F$.
\end{lemma}

\begin{proof}
Apply $\bar\delta_\xi$ to the equation with parameter $\zeta$,
subtract the reversed equation, and use
\eqref{eq:representation}. The result is
\[
 -\bar\delta_\xi\bar\Bcal_\zeta
 +\bar\delta_\zeta\bar\Bcal_\xi
 =\bar\delta_{[\xi,\zeta]}\bar G
 =-\bar\Bcal_{[\xi,\zeta]}.
\]
This is the degree-one Chevalley--Eilenberg condition with the
specified representation convention \cite{CE}.
\end{proof}

For the forced action \eqref{eq:trivial-rep}, the condition becomes
\begin{equation}
 \bar\Bcal_{[\xi,\zeta]}=0
 \quad\text{for every }\xi,\zeta\in\gW.
 \label{eq:trivial-cocycle}
\end{equation}
There is no substitution of compact translations into this equation.

\begin{proposition}[Class-wide exclusion criterion]
\label{prop:criterion}
If there are $\xi,\zeta\in\gW$ with
$\bar\Bcal_{[\xi,\zeta]}\ne0$, then no member of $\Kclass$ satisfies
all the representation, descent, and cancellation interfaces.

More directly, any parameter $\theta$ with
$\bar\Bcal_\theta\ne0$ excludes the target
\eqref{eq:target} for every shell-descending rule in the class.
\end{proposition}

\begin{proof}
The first assertion combines Theorem~\ref{thm:triviality} with
Lemma~\ref{lem:cocycle}. For the second, the left side of
\eqref{eq:target} is zero on the shell by
\eqref{eq:Y-zero}, whereas its required right side is nonzero.

Likewise, $\bar\delta_\theta\bar F=0$ for every fixed compensating
functional $F$. Adding one such functional cannot change the
conclusion.
\end{proof}

The criterion excludes successful maps in the whole frozen class,
not just an enumerated ansatz. A proposal whose directional
operator or composed kernel is undefined already fails an interface;
any proposal for which those interfaces are defined is subject to
the same theorem.

The nonzero commutator required by
Proposition~\ref{prop:criterion} is established in
Section~\ref{sec:witness} using the copied witness, not a new
cylinder integration.

%=====================================================================
\section{D4: an admitted cancellation cochain is not a completion}
\label{sec:kernel}
%=====================================================================

A kernel calculation is useful because it isolates what the
obstruction does \emph{not} say.

Let the inherited auxiliary kernel be denoted $b_\xi$, with the
normalization
\[
 \Bcal_\xi[b,\phi]
       =\pair{b_\xi}{b\otimes\phi},
 \qquad
 b_\xi\in\mathcal E'(M^2),\qquad
 \WF(b_\xi)\subset Z_2.
\]
Fiber components are understood. Define the new metric--scalar
cochain
\begin{equation}
 \Ccal_\xi[\gamma,\phi]
       :=-\Bcal_\xi[D\gamma,\phi].
 \label{eq:C-definition}
\end{equation}
This is a functional of the allowed observable slots. It does not
make $b=D\gamma$ an off-shell relation in the inherited carrier.

\begin{proposition}[The cochain kernel and admission]
\label{prop:C-admission}
The family \eqref{eq:C-definition} is even, has ghost number zero,
is linear in $\xi$, and has field degree $(1,1)$ in
$(\gamma,\phi)$. Its unsymmetrized kernel is
\begin{equation}
 c_\xi=-D_y^{\mathsf T}b_\xi,
 \label{eq:C-kernel}
\end{equation}
where $y$ is the auxiliary slot replaced by the metric slot.
After normalized species symmetrization it is compact and satisfies
$\WF(c_\xi)\subset Z_2$. Hence
$\Ccal_\xi\in\Fpol$.

On the complete free shell,
\begin{equation}
 \Ccal_\xi=-\Bcal_\xi.
 \label{eq:C-cancel}
\end{equation}
\end{proposition}

\begin{proof}
The source functional is bilinear and parameter-linear.
Replacing its first smooth argument by $D\gamma$ gives the
transpose formula \eqref{eq:C-kernel}.
A differential operator does not enlarge support or wavefront set.
The finite species symmetrization preserves the sum-zero bound.

A nonzero tuple in either all-future or all-past closed causal
cone cannot have zero covector sum. The bound $Z_2$ therefore
implies the inherited microcausal condition
\cite{Hormander,XIIA}.
The field species are even and have ghost number zero.

On $\Csol$, the equation $b=D\gamma$ holds. Substitution gives
\eqref{eq:C-cancel}.
\end{proof}

This proof uses the actual auxiliary-row kernel. It does not
borrow the wavefront estimate of $\CD$ for a different functional.
No new retarded application is needed; zero such applications is
within the structural bound ``at most one.''

Under \eqref{eq:UCP}, the copied remainder and
\eqref{eq:C-cancel} give the algebraic identity
\[
 \Delta_\xi\Ocal(u)+\Ccal_\xi(u)=0,\qquad u\in\Csol.
\]
That identity is not released as an integrated completion.
The family $\Ccal_\xi$ depends on the parameter. No single
functional whose variation is this family has been constructed.

In fact, for the representation forced by the frozen class,
\[
 \bar\delta_\xi\bar F=0
\]
for every $F$. Thus a nonzero $\bar\Ccal_\xi$ cannot be such a
variation. Its cocycle is
\[
 (d_{\bar\delta}\bar\Ccal)(\xi,\zeta)
             =-\bar\Ccal_{[\xi,\zeta]}
             =\bar\Bcal_{[\xi,\zeta]},
\]
which the next section detects.

\begin{remark}[Why this distinction matters]
If arbitrary parameter-indexed additions were declared to be
completions without an action or integrability requirement,
\eqref{eq:C-definition} would provide a cancellation prescription.
That is a different, weaker interface. The present class requires
a slot action or the variation of one fixed compensating functional.
Its obstruction is not non-admission of the bilinear cochain.
\end{remark}

%=====================================================================
\section{D5: the copied witness on a parameter commutator}
\label{sec:witness}
%=====================================================================

We now specialize the detecting family, after the class and its
representation have been frozen.

Choose the admissible XII-F data with a temporal bump that is a
square. More explicitly, let
\[
 \eps<t_a<t_b<t_c<t_d<T,
 \qquad
 0\ne h\in C_c^\infty((t_a,t_b);\R),
\]
and choose a compact spatial cutoff
$\zeta\in C_c^\infty(\R^3;[0,1])$ equal to one on the dependence
region used by XII-F. Take its temporal bump and spatial
realization to be
\[
 a=h^2,\qquad \zeta_{\mathrm{witness}}=\zeta^2.
\]
These are among XII-F's permitted choices.

Choose $W$ to contain the compact parameter support and choose $g$
to satisfy \eqref{eq:UCP}, as in XIII-A's embedding of the detecting
tuple into fixed restriction data. The switch, observation, region,
and cutoffs are fixed for the entire test. No parameter subclass
is removed afterward.

Define two allowed parameters
\begin{equation}
 \xi=h(t)\zeta(\mathbf x)\partial_t,\qquad
 \eta=(t-t_*)h(t)\zeta(\mathbf x)\partial_t,
 \label{eq:bracket-parameters}
\end{equation}
where $t_*$ is any fixed real number.
Both lie in $\gW$.

\begin{lemma}[The detecting parameter is a commutator]
\label{lem:bracket}
The parameters \eqref{eq:bracket-parameters} satisfy
\begin{equation}
 [\xi,\eta]
       =h(t)^2\zeta(\mathbf x)^2\partial_t
       =:\theta.
 \label{eq:theta}
\end{equation}
\end{lemma}

\begin{proof}
Both vector fields have only a temporal component. Their spatial
cutoff is independent of time. Thus the temporal component of
their bracket is
\[
 h\zeta\,\partial_t\bigl((t-t_*)h\zeta\bigr)
 -(t-t_*)h\zeta\,\partial_t(h\zeta)
 =h^2\zeta^2.
\]
There are no spatial components.
\end{proof}

Use the copied free-shell configuration
\[
 m=0,\qquad
 \psi=t^2+(x^1)^2,\qquad
 \gamma=\psi\eta,\qquad
 \phi=\psi,\qquad
 b=(-2t,2x^1,0,0),\qquad c=\bar c=0,
\]
denoted $u_*$. Its shell membership, compact realization,
causal-plateau hypothesis, and evaluated sign are those of XII-F.

Set
\[
 I=\int f(t,\mathbf x)
          \int_0^t(t-s)^2h(s)^2\,\dd s\,\dd^4x.
\]
The copied witness gives $I>0$ and
\begin{equation}
 \Bcal_\theta(u_*)=-8I<0.
 \label{eq:B-theta}
\end{equation}
No retarded cylinder calculation is repeated.

\begin{theorem}[Cocycle obstruction for the frozen class]
\label{thm:obstruction}
For these detecting fixed data,
\[
 \bar\Bcal_{[\xi,\eta]}\ne0.
\]
Consequently every shell-descending representation in
$\Kclass(f,g,W)$ fails the necessary cocycle condition for the
cancellation target. No member satisfies all declared interfaces.
\end{theorem}

\begin{proof}
The nonzero shell value \eqref{eq:B-theta} proves that
$\Bcal_\theta\notin\Nsol$.
By Theorem~\ref{thm:triviality}, every eligible action induces
$\bar\delta=0$. Therefore
\[
 (d_{\bar\delta}\bar\Bcal)(\xi,\eta)
 =\bar\delta_\xi\bar\Bcal_\eta
  -\bar\delta_\eta\bar\Bcal_\xi
  -\bar\Bcal_{[\xi,\eta]}
 =-\bar\Bcal_\theta\ne0.
\]
At $u_*$ its representative value is $8I>0$.
Apply Proposition~\ref{prop:criterion}.
\end{proof}

\begin{corollary}[What the witness does and does not cancel]
For every shell-descending slot rule in the class,
\[
 K_\theta^Y(u_*)=0,
 \qquad
 K_\theta^Y(u_*)+\Bcal_\theta(u_*)=-8I\ne0.
\]
For the admitted cochain of Section~\ref{sec:kernel},
\[
 \Ccal_\theta(u_*)=8I,
 \qquad
 \Ccal_\theta(u_*)+\Bcal_\theta(u_*)=0.
\]
Nevertheless that cochain is not an integrated compensating
functional in the frozen class.
\end{corollary}

\begin{proof}
The first assertion is \eqref{eq:Y-zero}.
The second is \eqref{eq:C-cancel}.
The lack of an integrated realization follows from
Theorem~\ref{thm:obstruction}, or directly from
$\bar\delta_\theta\bar F=0$.
\end{proof}

The result is observation-qualified. For example, $f=0$ makes
the pairing vanish and removes this detection argument.
We do not assert obstruction for every frozen observation,
every mass, or every conceivable correction class.
For the detecting fixed data, however, the exclusion applies
to every member of the class, not merely to one candidate.

The witness is a classical test of the correction interface.
It is not a binary quantum anomaly.

%=====================================================================
\section{D6: scalar variation, coupling order, and no double count}
\label{sec:bookkeeping}
%=====================================================================

The field $\phi$ is allowed to move in the class definition.
Its contribution to the full target must retain the scalar-sourced
slot. Explicitly,
\begin{equation}
 \begin{split}
 \pair{\Ocal'_\phi}{Y_\xi^\phi}
 ={}&\pair{(\PhiD)'_\phi}{Y_\xi^\phi}\\
 &+\Aone[f,g]\bigl(\rg'_\phi[Y_\xi^\phi];\phi\bigr)
  +\Aone[f,g]\bigl(\rg[\phi];Y_\xi^\phi\bigr),
 \end{split}
 \label{eq:scalar-variation}
\end{equation}
where the inherited derivative is
\[
 \rg'_\phi[Y_\xi^\phi]
 =-\tfrac12\PP\ER
               \bigl(g\,T_0'[\phi][Y_\xi^\phi]\bigr).
\]
This is differentiation of the existing slot, not a new interaction
entry. Its zero metric variation does not authorize deleting either
term in \eqref{eq:scalar-variation}.

For a shell-descending member of the frozen class all three terms
vanish on the shell because $Y_\xi^\phi[u]=0$.
They were not omitted before applying that theorem.

The candidate cochain \eqref{eq:C-definition} has degree two,
whereas $\Ocal$ has degree three. It must not be identified with
$\pair{\Ocal'_\gamma}{r_\xi}$ merely because $r_\xi$ has a plausible
retarded support. For instance, a field-linear $r_\xi$ makes that
slot variation degree three. Homogeneous field scaling distinguishes
it from the bilinear target.

The order of a prospective compensating effect here is
$\kappa^2\hbar^0$. No term is silently added at the first coefficient.
If a future proposal acts on the lower term $\kappa\Aone$, it must
print the additional coupling power producing a $\kappa^2$ effect
and supply its own transformation law.

The filled gravitational slot remains
$\rD=-u_{\mathrm D}$. It is neither modified nor counted a second
time. The admitted parameter-dependent cochain is not inserted
into that slot.

Finally, a finite transformation contains
$\frac12\Lie_\xi K\xi$ in its inherited two-jet law.
A linear parameter substitution is not a construction of that
finite law. The present obstruction already excludes a necessary
infinitesimal integrated interface in the class; it therefore
requires no assertion about an unconstructed finite action.

%=====================================================================
\section{D7: decision and exact successor package}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}
                              p{.25\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
D0 & Recorded and frozen &
Route A with Route D alternative; Route B cited;
one parameter-local/one-retarded class.\\[0.35em]
D1 & Definition completed &
Parity, ghost number, parameter linearity, order, causal support,
and the quadratic kernel requirement are explicit.\\[0.35em]
D2 & Constructed as a necessary representation theorem &
Ideal preservation and zero-past causality force the trivial
action on the complete free-shell quotient.\\[0.35em]
D3 & Proved obstructed for detecting data &
The auxiliary pairing is nonzero on an allowed commutator,
so the required cocycle fails for every induced action in the class.\\[0.35em]
D4 & Cochain admitted; integrated solution excluded &
The new bilinear kernel has the bound $Z_2$.
It is not the variation of one compensating functional
under the forced action.\\[0.35em]
D5 & Class-wide witness failure &
Every eligible slot action gives zero on the copied tuple
and cannot cancel its nonzero pairing.\\[0.35em]
D6 & Recorded &
Scalar-source differentiation retained;
$\rD$ unchanged and no second-slot double count.\\[0.35em]
D7 & Proved obstructed &
One ending for $\Kclass$ on the detecting fixed data;
maps outside that class are undecided.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Classical route release]
\label{thm:release}
On the unchanged cubic-only sheet, for the detecting fixed data of
Section~\ref{sec:witness}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every shell-descending action in $\Kclass(f,g,W)$ induces
the trivial representation on $\Fsol$.
\item The copied auxiliary pairing is not a cocycle for that
representation.
\item No slot action or integrated compensating functional in the
frozen class satisfies the cancellation target on the complete
gauge-fixed free shell.
\item The admitted cochain $\Ccal_\xi=-\Bcal_\xi[D\gamma,\phi]$
provides an algebraic parameter-dependent cancellation, not an
integrated completion in the class.
\end{enumerate}
The ending for the named integrated class is
\textbf{Proved obstructed}. The result claims nothing about
completions outside the parameter-local/one-retarded,
shell-descending class.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:triviality},
Proposition~\ref{prop:C-admission},
Lemma~\ref{lem:bracket},
and Theorem~\ref{thm:obstruction}.
\end{proof}

This obstruction is caused by an explicit incompatibility:
zero-past causal tangency forces zero field variation, whereas
the target pairing has a nonzero class. It is not caused by
the absence of a proposed formula, by a failed wavefront estimate,
or by XIII-C's unconstructed quantum coefficient.

\subsection*{Handoff}

A successor receives:
\begin{enumerate}
\item the exact frozen class, including parameter locality of its
unpropagated components;
\item the theorem forcing its induced shell action to vanish;
\item the explicit commutator parameters and the nonzero
Chevalley--Eilenberg test;
\item the admitted cancellation cochain and its kernel formula,
with the warning that it is not an integrated functional;
\item the observation-qualified obstruction and the unchanged
scalar-slot and coupling bookkeeping.
\end{enumerate}

A proposed successor can leave the class by allowing a nonlocal
action on Cauchy data, nonzero past data, a different domain,
or a different integration interface. It must state that change
before claiming that the present obstruction has been avoided.
No such construction is supplied here.

It receives no reduced algebra, completed quantum binary datum,
interacting differential, or physical cohomology class.
The class exclusion supplies no permission to differentiate the
unconstructed $\Rquant$.

\subsection*{The retained quantum gate}

The next quantum paper still begins at XIII-C's four obligations:
\begin{enumerate}
\item a prescription across the noncoincident null-related
configurations;
\item transverse estimates on the three- and four-vertex strata;
\item compatible partial-diagonal contacts;
\item the output wavefront estimate after pushforward.
\end{enumerate}
Those obligations remain Unresolved. The present classical
cocycle calculation is not the binary quantum gauge-consistency
defect.

Compactly supported free plateau ghosts cannot replace the
parameters in this proof. A homogeneous ghost with compact
time support has zero Cauchy data in an earlier band and therefore
vanishes by uniqueness. Pairing against that zero is not a
nontrivial cancellation identity.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \text{parameter-local/one-retarded output}
 +\text{ free-shell descent}
 \Longrightarrow \bar\delta_\xi=0,\\[2pt]
 \theta=[h\zeta\partial_t,(t-t_*)h\zeta\partial_t]
                  =h^2\zeta^2\partial_t,\\[2pt]
 \Bcal_\theta(u_*)=-8I<0,\qquad
 (d_{\bar\delta}\bar\Bcal)(\xi,\eta)
                  =-\bar\Bcal_\theta\ne0,\\[2pt]
 \Kclass\text{ on the detecting fixed data: }
                         \textbf{Proved obstructed},\\[2pt]
 \Ccal_\theta(u_*)=8I
 \quad\text{is an admitted cochain value, not an integrated completion.}
 \end{gathered}}
\]

$\CW$ remains an evaluation set. The quantum two-line coefficient
remains unresolved. Exactly one bounded integrated classical class
has been excluded, with its support and representation assumptions
visible.

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

\paragraph{Class scope.}
The tested class is a precise parameter-local/one-retarded
realization of the production envelope. Parameter locality of
unpropagated components is an additional explicit ansatz, frozen
before the witness test. No universal exclusion of nonlocal
Cauchy-data actions is claimed.

\paragraph{Obstruction scope.}
The theorem concerns integrated cancellations admitting a
shell-descending representation in this class.
It does not deny existence of parameter-dependent bilinear
cancellation cochains; one such cochain is constructed explicitly.

\paragraph{Dependency scope.}
The predecessor commutator, ordering counterexample, tree datum,
quantum stop, assembly sign, and witness value are copied.
The causal tangency theorem, representation identification,
commutator realization of the detecting parameter, and cochain
kernel calculation are proved here.

\paragraph{Classical scope.}
All new actions and cocycles are even and classical.
No application of $s_0$ to a quantum binary coefficient,
interacting differential, master equation, state, antifield,
Einstein--Hilbert interaction insertion, or sharp-width limit
is supplied.

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

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

\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}.
Evaluation-only route, restriction-ideal obstruction,
ordering counterexample, and conditional completion interface.

\bibitem{XIIIB}
N.~Cheung,
\emph{Landmark XIII-B: One binary causal datum for the displayed
coefficient, on the unchanged carrier}.
Supplied manuscript \texttt{XIII-B.tex}.
Constructed tree datum, corrected ordered values,
and unresolved two-line extension.

\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}.
Partial-diagonal class, primitive extensions,
characteristic cancellation, and retained quantum stop.

\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-row kernel, compact witness family,
copied evaluated value, and nonzero carrier class.

\bibitem{XIIE}
N.~Cheung,
\emph{Landmark XII-E: Assembly of the second-order relational
coefficient after admission of the density-weighted cubic insertion}.
Supplied manuscript \texttt{XII-E.tex}.
Assembly sign, affine slot identity,
and scalar-sourced slot.

\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}.
Unchanged selected action and fixed-input jet boundary.

\bibitem{XIIA}
N.~Cheung,
\emph{Landmark XII-A: One graded polynomial carrier
for the finite-ramp multiplet}.
Supplied manuscript \texttt{XII-A-v2.tex}.
Polynomial carrier, complete shell, null ideal,
and frozen free differential.

\bibitem{XIB}
N.~Cheung,
\emph{Landmark XI-B: The free de Donder multiplet
for the ramp Einstein--scalar sheet}.
Predecessor normalization input, as stated in the supplied
XII-A, XII-E, and XIII-A manuscripts.
Inverse coefficient and residual-gauge distinction.

\bibitem{CE}
C.~Chevalley and S.~Eilenberg,
``Cohomology theory of Lie groups and Lie algebras,''
\emph{Transactions of the American Mathematical Society}
\textbf{63} (1948), 85--124.
Representation-dependent Lie algebra cocycles.

\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 uniqueness and causal Green operators.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
second edition, Springer, 1990.
Differential-operator wavefront bounds
used in the new bilinear kernel calculation.

\end{thebibliography}

\end{document}