% !TeX program = pdflatex
%=====================================================================
% XII-B.tex
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% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XII-B.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XII-B.tex
%
% B0 stop-rule edition.
%
% Supplied dependencies:
%   XII-A-v2.tex, XI-B.tex, XI-E.tex, XI-G.tex.
%
% Audit result:
%   XII-A-v2 constructs the polynomial carrier and excludes the
%   detected classes of the original, unramped cubic response.
%   It does not freeze a density-ramped cubic model, compute its
%   Noether defect, or decide its coupled-jet question.
%
% Consequently:
%   - the requested changed-model handoff is not available;
%   - no new cubic retarded response is defined here;
%   - B1--B4 are not claimed constructed;
%   - the overall ending is Unresolved.
%
% Mathematical debugging:
%   - source weighting and density weighting are distinguished;
%   - the weighted Euler derivative is computed;
%   - an explicit pure-trace Einstein--Hilbert calculation shows
%     that density weighting produces a nonzero layer term;
%   - a mixed-second-variation test shows that sigma J_3 is not
%     itself an Euler source on the unrestricted off-shell class;
%   - the proposed response functional and a compact root test
%     are distinguished by their types.
%
% No enlargement, state, loop, interacting differential,
% sharp-width limit, contact prescription, or X-track merger.
%
% No successful compilation, mechanized verification,
% independent referee review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
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\usepackage{lmodern}
\usepackage[margin=27mm]{geometry}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

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

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\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{\Ggr}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\ER}{\mathsf E_0^{\mathrm R}}
\newcommand{\EA}{\mathsf E_0^{\mathrm A}}
\newcommand{\Pmass}{P_m}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Afree}{\mathscr A_0}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Bcal}{\mathcal B}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Cthree}{\mathcal C_3}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Hcal}{\mathcal H}
\newcommand{\Vp}{\widetilde V_+}
\newcommand{\Vm}{\widetilde V_-}
\newcommand{\Euler}{\mathsf{EL}}
\newcommand{\Dtot}{\mathsf D}
\newcommand{\Edist}{\mathcal E'}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

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

\title{\textbf{Landmark XII-B}\\[0.4em]
Admission and one retarded response\\
of the ramped cubic Einstein source,\\
at fixed regulator width\\[0.5em]
\large Handoff audit, variational consistency,\\
and the unresolved release}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
This manuscript applies the stated stop rules to the proposed
ramped-cubic construction. The supplied XII-A-v2 establishes a
graded polynomial carrier, ordering transport, and an exclusion
for the original cubic response. It does not provide the required
changed-model freeze, its new Noether defect, or a decision about
the changed coupled jet. The handoff gate therefore ends
Unresolved, and no new cubic retarded response is defined.

A separate consistency audit identifies a mathematical issue in
the proposed specification. Multiplication of a cubic source by
a ramp is not the same operation as multiplication of its
Lagrangian density before variation. We derive the weighted
Euler formula and compute the discrepancy explicitly on the
pure-trace Einstein--Hilbert sector. It contains a nonzero
ramp-derivative term. A mixed-second-variation test further shows
that the simply weighted source cannot be the Euler derivative
of an action on the unrestricted off-shell class.

These results neither reopen XI-G nor obstruct a correctly
specified density-ramped model. They identify the information
and convention changes required before the completed-kernel,
admission, response, and plateau gates can be evaluated.
No state, loop, interacting differential, contact subtraction,
or sharp-width limit is introduced.
\end{abstract}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Source weighting};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Density weighting};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {The B0 stop};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The two weightings stay distinct. The B0 stop still applies.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Source weighting is not density weighting. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-XII-B}
\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) {Source weighting};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {The B0 stop};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The two weightings stay distinct. The B0 stop still applies.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Source weighting is not density weighting. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-XII-B}
\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) {Source weighting};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Density weighting};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {The B0 stop};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The two weightings stay distinct. The B0 stop still applies.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-XII-B}
\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) {Source weighting};
\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) {The B0 stop};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Density weighting. The two weightings stay distinct. The B0 stop still applies.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Source weighting is not The B0 stop. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-XII-B}
\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: Source weighting};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Density weighting};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: The B0 stop};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The two weightings stay distinct. The B0 stop still applies.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-XII-B}
\end{figure}

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


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



\scopebox{Release: the B0 stop rule applies}{%
\textbf{Constructed here:} the handoff audit; the distinction
between source weighting and density weighting; the weighted
Euler identity; an explicit Einstein--Hilbert layer discrepancy;
and a variational integrability test.

\smallskip
\textbf{Overall ending: Unresolved.}
The supplied XII-A-v2 does not contain the changed-model handoff
required by the production specification. Accordingly this
manuscript does not define the new cubic retarded response and
does not claim B1--B4 constructed.

\smallskip
\textbf{Not inferred:} a forbidden wavefront tuple for a correctly
specified new insertion, a plateau obstruction for that insertion,
or nonexistence of a density-ramped model.

\smallskip
XI-G's original class obstruction remains closed.
Missing handoff data are not a physical no-go theorem.}

\tableofcontents

%=====================================================================
\section{The three endings and the non-reopening rule}
\label{sec:release}
%=====================================================================

The intended affirmative target is admission of a
scalar-completed, ramped cubic Einstein insertion and construction
of one classical retarded response with a proved plateau law.
The three endings are retained:
\begin{enumerate}
\item \textbf{Constructed}: the changed model is frozen, its
completed kernel is microcausal, the named response is defined
on its stated domain, and its plateau identity is proved.
\item \textbf{Proved obstructed}: a forbidden tuple is exhibited
for the named completed kernel, or a nonzero interior defect is
proved for the named plateau identity.
\item \textbf{Unresolved}: the prerequisite model, completed
kernel, or response map has not been specified sufficiently to
prove either alternative.
\end{enumerate}

The production specification contains an additional dependency
rule: XII-B consumes the changed-model freeze from XII-A; it does
not supply that freeze itself. If the required XII-A statements
are absent, XII-B stops and does not write the new response
formula. That rule governs the present release.

XI-G proves, for its detected observation classes, that the
original free-shell cubic class has no representative in
$\Fpol$ \cite{XIG}. The inherited split null covectors are part
of that predecessor result. Nothing below searches for another
representative of the original $\Cthree$.

A changed source would be a different model. Its admission cannot
be deduced merely by negating the original obstruction, and its
plateau law cannot be obtained by relabeling the original
response. Likewise, absence of a new construction does not
extend XI-G's obstruction to every possible changed model.

%=====================================================================
\section{B0: audit of the supplied XII-A handoff}
\label{sec:audit}
%=====================================================================

The supplied predecessor is \texttt{XII-A-v2.tex}, cited here as
XII-A \cite{XIIA}. Its actual release is a polynomial-carrier
construction for the frozen XI-track multiplet. It includes:
\begin{itemize}
\item the separation of equation geometry from action
normalization;
\item a regular quantum presentation and a shell quotient;
\item a smoothing-based descent argument;
\item ordering transport and the named composite sector;
\item delimited continuity statements;
\item robustness of the detected observation class;
\item a regular ambient--slab identification.
\end{itemize}
These are relevant algebraic prerequisites. They are not a
definition of a new cubic action.

\subsection{The required sentences and their actual status}

The production specification requires the following handoff.
We distinguish requirements from statements actually supplied.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}X
                              p{.27\linewidth}@{}}
\toprule
 & Required XII-A statement & Supplied status\\
\midrule
H1 &
The cubic model is changed before response formation, with
a specified ramped density and its actual Euler source. &
Not supplied. XII-A retains the frozen XI-F/XI-G cubic
functional.\\[0.4em]
H2 &
The inverse remains $-\PP\ER$, with no X-track rescaling. &
Supplied at the free normalization level.\\[0.4em]
H3 &
The changed cubic Noether identity and all ramp-derivative
terms are displayed. &
Not supplied for a changed cubic density.\\[0.4em]
H4 &
The new map is not identified with the X-track
source-switched mixed map. &
No new cubic map is defined, so the required changed-model
comparison is not supplied.\\[0.4em]
H5 &
The effect, if any, of the changed density on the fixed-input
coupled jet is stated and justified. &
Not supplied. A disclaimer of jet repair is not a calculation
for a new action.\\
\bottomrule
\end{tabularx}
\end{center}

In particular, XII-A's exclusion section identifies its
$\Cthree$ as the frozen XI-F/XI-G functional. It does not replace
that functional by a density-ramped one. Its declarations and
proof ledger do not contain a changed cubic Noether theorem.

The normalization requirement H2 passes. The conjunction
H1--H5 does not.

\begin{theorem}[B0 audit]
\label{thm:B0}
Relative to the supplied XII-A-v2 and the stated production
stop rule, the changed-model handoff is incomplete.
Gate B0 ends Unresolved. Consequently no new cubic retarded
response, completed-kernel admission, or plateau theorem is
released in this manuscript.
\end{theorem}

\begin{proof}
An affirmative B0 ending requires all the changed-model
obligations, not only the inherited free inverse.
The supplied text contains neither the changed density/source
freeze nor its new Noether identity and coupled-jet decision.
The production rule explicitly makes these omissions a stop.
Thus the dependent construction is not authorized.
This is a dependency conclusion, not a wavefront or plateau
nonexistence argument.
\end{proof}

The proposed handoff also requires a mathematical clarification
before it can be supplied consistently: its phrases
``multiply the source'' and ``multiply the density before
variation'' do not specify the same operation.
Sections~\ref{sec:weighted}--\ref{sec:trace} prove this point.

%=====================================================================
\section{Inherited sheet and the absent changed object}
\label{sec:sheet}
%=====================================================================

The inherited geometry is
\[
 M=\R^4,\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \Pmass=\Box+m^2.
\]
Fix $0<\eps<T$ and
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3.
\]
The ramp is the one supplied in XI-B and XI-E:
\[
 \sig(t)=\chi(t/\eps),\qquad
 \chi(s)=
 \begin{cases}
 0,&s\le0,\\[2pt]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
 1,&s\ge1.
 \end{cases}
\]
It is smooth and flat at both endpoints.
Moreover $\sig'(t)>0$ for $0<t<\eps$. Indeed, on $(0,1)$,
\[
 \frac{\dd}{\dd s}\log\frac{\chi(s)}{1-\chi(s)}
 =\frac1{s^2}+\frac1{(1-s)^2}>0.
\]

The switch and plateau satisfy
\[
 g\in C_c^\infty(\MTo;[0,1]),\qquad
 P_g=\Int\{g=1\}\ne\varnothing,
\]
\[
 \overline U\subset P_g\cap\{\eps<t<T\},\qquad
 \ga=C_c^\infty(U;TM).
\]
This class is not reduced after the audit.

Trace reversal and the differential normalization are
\[
 \PP r=r-\tfrac12\eta\,\tr_\eta r,\qquad
 \RR=\tfrac12K,\qquad
 s_0\gamma=\RR c,\qquad
 \Ggr=-\PP\ER .
\]
Parentheses include $\tfrac12$. The equation inverse and action
inverse remain distinct as in XI-B \cite{XIB}.
The factor $-4$ is not imported.

The original cubic source is
\[
 J_3[\gamma]=2\Hcal_2[\gamma],
\]
with the curvature convention and Einstein--Hilbert action
\[
 S_{\mathrm{EH}}[q]
 =-\frac{2}{\kappa^2}\int\sqrt{-q}\,R[q]\dd^4x,
 \qquad q=\eta+\kappa\gamma.
\]
These data specify the original local coefficient, not its
new weighted variational completion.

The scalar completion $\Aone$ and the coordinate ramp are those
of XI-E \cite{XIE}. The new completed insertion would need both
its scalar-response leg and its coordinate leg. Neither is
discarded merely because a local metric source has a smooth
weight. No completed new kernel is defined here, because B0
has stopped.

%=====================================================================
\section{Density weighting is not source weighting}
\label{sec:weighted}
%=====================================================================

This section is a variational consistency calculation, not a
construction of the prohibited new retarded response.

\subsection{A local Euler convention}

Let $\psi^a$ denote finitely many even field components, and
let $\ell(x,j^r\psi)$ be a local density of finite jet order.
For a multi-index $I$, write
\[
 \psi_I^a=\partial^I\psi^a,\qquad
 \Dtot_I=\Dtot_1^{I_1}\cdots\Dtot_4^{I_4}.
\]
Total derivatives act on local jet expressions; when applied to
a function of $x$ alone, they are ordinary derivatives.

Compact variations define the Euler expression
\begin{equation}
 \Euler_a(\ell)
 =\sum_{|I|\le r}(-1)^{|I|}
       \Dtot_I\left(\frac{\partial\ell}{\partial\psi_I^a}\right).
 \label{eq:Euler}
\end{equation}
Thus
\[
 \delta\int\ell\,\dd^4x
 =\int \Euler_a(\ell)\,\delta\psi^a\,\dd^4x
\]
for compact variations. This is the ordinary local variational
calculus; see also \cite{KO}. No global convergence of an
unlocalized action on arbitrary configurations is needed.

\begin{lemma}[Weighted Euler expression]
\label{lem:weighted}
For a fixed smooth scalar function $\sigma$,
\begin{equation}
 \Euler_a(\sigma\ell)
 =\sigma\,\Euler_a(\ell)+\mathcal D_{\sigma,a}[\ell],
 \label{eq:weightedEuler}
\end{equation}
where
\begin{equation}
 \mathcal D_{\sigma,a}[\ell]
 =
 \sum_{|I|\le r}
 \ \sum_{\substack{0<J\le I}}
 (-1)^{|I|}\binom{I}{J}
 (\partial^J\sigma)\,
 \Dtot_{I-J}
       \left(\frac{\partial\ell}{\partial\psi_I^a}\right).
 \label{eq:layerEuler}
\end{equation}
For a time-dependent ramp, all terms in
$\mathcal D_{\sigma,a}$ contain a positive-order time derivative
of $\sigma$.
\end{lemma}

\begin{proof}
Since $\sigma$ is independent of the field jets,
\[
 \frac{\partial(\sigma\ell)}{\partial\psi_I^a}
 =\sigma\frac{\partial\ell}{\partial\psi_I^a}.
\]
Apply the multi-index Leibniz rule to every term of
\eqref{eq:Euler}. The terms with $J=0$ give
$\sigma\Euler_a(\ell)$. The others give
\eqref{eq:layerEuler}. Spatial derivatives of a time-only
weight vanish.
\end{proof}

For a first-order density this becomes
\begin{equation}
 \mathcal D_{\sigma,a}[\ell]
 =-(\partial_\mu\sigma)
       \frac{\partial\ell}{\partial(\partial_\mu\psi^a)}.
 \label{eq:firstorderlayer}
\end{equation}
For a second-order density, first and second derivatives of
the ramp may occur.

With the fixed $\sig$, these layer terms vanish outside the
closed layer $0\le t\le\eps$ and are smooth and flat at its
endpoints. Their nonzero values lie in $0<t<\eps$.
They are not sharp-surface distributions.

\subsection{Two different specifications}

Let $\ell_3$ be a cubic density normalized so that
$\Euler(\ell_3)=J_3$.
There are then two distinct prescriptions:
\begin{enumerate}[label=\textup{(\alph*)}]
\item \emph{Source weighting}: prescribe the local expression
$\sig J_3$ as the new forcing term.
\item \emph{Density weighting}: prescribe $\sig\ell_3$ before
variation, whose Euler expression is
$\sig J_3+\mathcal D_\sig[\ell_3]$.
\end{enumerate}
The production sheet asks for a density-ramped vertex but also
writes a response driven only by $\sig J_3$.
That combination needs justification; it is not a consequence
of multiplication by a smooth ramp.

The next section shows that the omitted term is genuinely
nonzero for the displayed Einstein--Hilbert convention.
It is not merely a possible correction for unrelated models.

\begin{lemma}[Boundary representatives matter after weighting]
\label{lem:boundary}
If $\ell_3'=\ell_3+\partial_\mu B^\mu$, then
\[
 \Euler(\ell_3')=\Euler(\ell_3),
\]
but
\begin{equation}
 \Euler(\sig\ell_3')-\Euler(\sig\ell_3)
 =-\Euler\bigl((\partial_\mu\sig)B^\mu\bigr).
 \label{eq:boundaryrepresentative}
\end{equation}
\end{lemma}

\begin{proof}
Euler variation annihilates a total divergence under compact
variations. Use
\[
 \sig\,\partial_\mu B^\mu
 =\partial_\mu(\sig B^\mu)-(\partial_\mu\sig)B^\mu
\]
and apply that fact.
\end{proof}

Thus an unweighted source alone does not identify a
density-ramped vertex. A density representative, or an explicit
boundary convention giving the same information, must be named.
A silent integration by parts after the ramp is introduced
can change the model.

%=====================================================================
\section{An explicit Einstein--Hilbert layer discrepancy}
\label{sec:trace}
%=====================================================================

The following calculation uses the pure-trace sector already
displayed in XI-G, but concerns density variation rather than
XI-G's plane-wave obstruction.

\subsection{The cubic density on the pure-trace sector}

Set
\[
 \gamma_{\mu\nu}=\psi\eta_{\mu\nu},\qquad
 q_{\mu\nu}=(1+\kappa\psi)\eta_{\mu\nu}=e^{2w}\eta_{\mu\nu},
 \qquad
 w=\tfrac12\log(1+\kappa\psi).
\]
All expressions are formal near $\kappa=0$.

\begin{lemma}[Pure-trace cubic density]
\label{lem:tracedensity}
For the displayed Einstein--Hilbert density, its coefficient of
$\kappa^1$ on this sector is
\begin{equation}
 \ell_3[\psi\eta]
 =3\psi\,\partial_\mu\psi\,\partial^\mu\psi .
 \label{eq:tracedensity}
\end{equation}
Its Euler derivative is
\begin{equation}
 \Euler_\psi(\ell_3[\psi\eta])
 =-3\,\partial_\mu\psi\,\partial^\mu\psi-6\psi\Box\psi
 =\eta_{\mu\nu}J_3^{\mu\nu}[\psi\eta].
 \label{eq:traceEuler}
\end{equation}
\end{lemma}

\begin{proof}
Computing the Christoffel symbols of $e^{2w}\eta$ gives
\[
 \sqrt{-q}\,R[q]
 =-6e^{2w}
       \left(\Box w+\partial_\mu w\,\partial^\mu w\right).
\]
With $e^{2w}=1+\kappa\psi$, this is
\[
 -3\kappa\Box\psi
 +\frac{3\kappa^2}{2(1+\kappa\psi)}
           \partial_\mu\psi\,\partial^\mu\psi .
\]
Therefore the signed density is exactly
\[
 -\frac2{\kappa^2}\sqrt{-q}\,R[q]
 =\frac6\kappa\Box\psi
  -\frac3{1+\kappa\psi}
        \partial_\mu\psi\,\partial^\mu\psi .
\]
Its coefficient of $\kappa$ is \eqref{eq:tracedensity}.

Direct variation gives
\[
 3(\partial\psi)^2
 -\partial_\mu(6\psi\partial^\mu\psi)
 =-3(\partial\psi)^2-6\psi\Box\psi.
\]
Variation along $\delta\gamma_{\mu\nu}
=\eta_{\mu\nu}\delta\psi$ contracts the full cubic Euler source
with $\eta_{\mu\nu}$, proving the last equality.
It agrees with the pure-trace coefficient in XI-G \cite{XIG}.
\end{proof}

\begin{proposition}[Nonzero layer term]
\label{prop:tracelayer}
Density weighting gives
\begin{equation}
 \Euler_\psi(\sig\ell_3[\psi\eta])
 =
 \sig\,\eta_{\mu\nu}J_3^{\mu\nu}[\psi\eta]
 -6(\partial_\mu\sig)\psi\partial^\mu\psi.
 \label{eq:tracelayer}
\end{equation}
The second term is not identically zero on the unrestricted
off-shell configuration class.
\end{proposition}

\begin{proof}
Apply \eqref{eq:firstorderlayer} to
\eqref{eq:tracedensity}.
To detect the discrepancy, choose a spacetime box with
$0<t<\eps$ and a smooth $\psi$ equal to $t$ near that box.
Let $h\ge0$ be a nonzero compact variation supported inside it.
The discrepancy pairing is
\[
 -6\int \sig'(t)\,t\,h(t,\mathbf x)\dd^4x<0.
\]
The strict sign follows from $t>0$ and $\sig'>0$ there.
All variations are compact and supported away from the layer
endpoints, so no boundary convention enters this pairing.
\end{proof}

This is a defect in the identification of the proposed
\emph{vertex source}, not a computed B4 plateau defect.
The test $h$ is a compact metric variation used to audit an
Euler derivative; it is not a replacement for the stipulated
plateau parameter class $\ga$.

\subsection{Simply weighting the source is not variational}

The preceding discrepancy is stronger than the failure of one
chosen density representative.

\begin{proposition}[Off-shell variational integrability]
\label{prop:integrability}
For nonconstant $\sig$, the full source $\sig J_3$ cannot be
the Euler derivative of a twice differentiable action on an
off-shell neighborhood containing the unrestricted pure-trace
configurations and compact metric variations.
\end{proposition}

\begin{proof}
If such an action existed, restrict it to
$\gamma=\psi\eta$. Its first derivative would be
\[
 F_\sig[\psi]
 =-3\sig(\partial\psi)^2-6\sig\psi\Box\psi.
\]
Symmetry of mixed second variations requires
\[
 \int h\,F_\sig'[\psi]k\,\dd^4x
 =
 \int k\,F_\sig'[\psi]h\,\dd^4x
\]
for every pair of compact variations $h,k$.

Choose $\psi=1$ on a neighborhood of their supports.
There
\[
 F_\sig'[1]k=-6\sig\Box k.
\]
Compact integration by parts gives
\begin{equation}
 \begin{split}
 &\int h\,F_\sig'[1]k\,\dd^4x
       -\int k\,F_\sig'[1]h\,\dd^4x\\
 &\qquad
 =6\int(\partial_\mu\sig)
       \bigl(h\partial^\mu k-k\partial^\mu h\bigr)\dd^4x.
 \end{split}
 \label{eq:Hessianantisymmetry}
\end{equation}
Take a nonzero real $h$ supported strictly inside the layer and
put $k=th$. Both variations are compact, and
\[
 h\partial_t(th)-th\partial_t h=h^2.
\]
Since the ramp depends only on time,
\eqref{eq:Hessianantisymmetry} equals
\[
 6\int\sig'(t)h^2\dd^4x>0.
\]
This contradicts symmetry of the Hessian.
\end{proof}

Consequently the simultaneous demands
``an action vertex varied before response'' and
``Euler source exactly $\sig J_3$ on unrestricted off-shell
tests'' are not interchangeable specifications.
Adding an unnoticed total divergence cannot evade
Proposition~\ref{prop:integrability}.

A source-weighted forcing prescription may still be considered
as a nonvariational model. A density-ramped action may also be
considered, with its actual layer terms included. Neither
possibility is ruled out. Choosing between them, and freezing
the relevant identities, is outside this stopped construction.

%=====================================================================
\section{Dependent gates: what is not established}
\label{sec:dependent}
%=====================================================================

\subsection{B1: the completed kernel}

The relevant comparison criterion from XI-G is
\[
 \text{a compact free-shell class meets }\Fpol
 \quad\Longleftrightarrow\quad
 \WF(W_u)\cap(\Vp^{\,3}\cup\Vm^{\,3})=\varnothing.
\]
Its role is comparison after a new compact kernel has been
defined. It does not define that kernel.

For a corrected density-ramped model, the completed kernel
must include:
\begin{enumerate}
\item both metric slots of the actual cubic Euler source,
including its ramp-derivative terms;
\item the metric retarded leg with the XI-track normalization;
\item the coordinate leg with its own divergence-form ramp;
\item the scalar-response part of $\Aone$, including the switch
and massive advanced scalar leg after transposition;
\item every support and properness condition used in the
composition.
\end{enumerate}
Checking only a smooth weight on the local metric diagonal
would leave the scalar completion untested.

No new $u^\sigma$ or $W_{u^\sigma}$ is defined here.
Accordingly no wavefront inclusion, admission criterion
evaluation, or forbidden tuple for that new object is asserted.

\subsection{B2: admission and the old algebra}

XII-A supplies an inherited regular-algebra embedding into
its named polynomial shell carrier:
\[
 j_H:\Afree(M)[[\kappa]]\hookrightarrow\Ealg.
\]
This is the old embedding, not admission of an unnamed new
functional. Its Hadamard matrix is product data, not a selected
state.

The existence of the old carrier does not decide which changed
cubic action is being used. Nor does it imply membership of the
completed insertion before B1 is discharged.
Thus B2 has an available carrier prerequisite but no new
admission conclusion.

\subsection{B3: a response functional is not a test function}

There is also a type issue to resolve in a future B3 statement.
For the original quadratic insertion $\Bcal_f$, its metric
derivative at a smooth scalar configuration is a linear
covector in the metric slot, with scalar field degree one.
A cubic vertex has a quadratic Euler source in the metric.

Their single rooted contraction is therefore a scalar-valued
functional of field degree $(2,1)$, not an element of
$C_c^\infty(M)$.
This agrees with the distinction between classical response
functionals and their functional derivative kernels
\cite{FR}.

A smooth compact test can instead be an intermediate rooted
coefficient after fixing the smooth fields and transposing
the propagator, possibly with source localization.
Those are different types. A complete B3 theorem must state:
\begin{itemize}
\item the domain of fields and external tests;
\item whether its codomain is a test space, a polynomial
functional space, or a continuous dual pairing;
\item which intermediate coefficient is claimed compact;
\item the exact support statement for that coefficient.
\end{itemize}

This observation does not prove that the eventual map leaves
a test space. The map and test-space identification are not
yet fixed. Under the B0 stop rule, no new cubic retarded
formula is supplied to fill that gap.

\subsection{B4: local layer support is not a plateau cancellation}

If a correctly specified cubic density has a layer correction,
that correction is locally supported where derivatives of
$\sig$ are nonzero. Retarded propagation of a layer-supported
source need not be layer-supported, as the explicit XI-B layer
diagnostics show \cite{XIB}.

Consequently a support statement about a local Noether
breaking term is not, by itself, a theorem that its completed
response vanishes on the plateau.
Conversely, compact plateau parameters do not justify
discarding a term before its variation and propagation have
been calculated.

The stipulated class remains
$\ga=C_c^\infty(U;TM)$.
No contact term is selected, no layer term is deleted, and
this class is not shrunk.
No new B4 variation has been computed because the new vertex
has not passed B0.

The nonzero pairing in
Proposition~\ref{prop:tracelayer} is not relabeled as a
nonzero plateau pairing. It establishes a source-definition
mismatch only. Ending Proved obstructed for B4 would require
a separate, correctly typed interior witness for the actual
completed new response.

%=====================================================================
\section{B5: exclusions, the jet, and the classical ledger}
\label{sec:exclusions}
%=====================================================================

The following boundaries remain explicit.

\paragraph{The X-track comparison.}
No X-E or X-F inverse, density switch, or mixed-tree prescription
is imported. No field map is identified with $k_{\mathrm{mix}}$.
The reciprocity constraint mentioned in the production brief
is a constraint on a proposed identification, not a permission
to remove ramp derivatives from a new action.

The X-G manuscript is not among the supplied dependencies.
Its numerical reciprocity witness is therefore not rederived
or used as a premise of any theorem here.

\paragraph{The original coupled jet.}
The original empty fixed-input Einstein jet
$\Jcal_g(\phi_m)=\varnothing$ is retained as the predecessor
result for its original action, as cited by XI-B and XI-E.
A new action requires its own coupled equation and conservation
analysis.

The statement ``no jet repair is claimed'' in XII-A is not a
decision about a model that XII-A never defines. This manuscript
does not turn it into one. The changed jet is unresolved.

\paragraph{The original ledger.}
The classical ledger $-I+I+0=0$ belongs to the original
XI-F response and its stated coefficient comparison.
It is not recomputed here, not promoted to a quantum relation,
and not used as the plateau law of an undefined changed
response.

\paragraph{Unopened quantum constructions.}
No second graviton line, fish graph, triangle subtraction,
$\mathsf T_2$, reflected residue, ghost loop, state, or
expectation value is formed.
No interacting differential, master equation, or interacting
BRST class is claimed.

\paragraph{Fixed width.}
The ramp width remains $\eps>0$ throughout.
No limiting argument is used to suppress the layer terms.
Smoothness at fixed width does not identify the new model with
a sharp insertion.

\paragraph{No additional readout.}
No X-Enter record, cone token, conical current, or independent
readout is assigned to the kernel. None is a premise of this
audit.

%=====================================================================
\section{Decision table and exact handoff}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}
                              p{.22\linewidth}X@{}}
\toprule
Gate & Ending & Exact reason\\
\midrule
B0 & \textbf{Unresolved} &
The supplied XII-A-v2 does not freeze the changed cubic
density, its actual Euler source, new Noether identity,
and changed-jet status.\\[0.4em]
B1 & \textbf{Unresolved} &
The completed new three-slot kernel is not defined; no
wavefront verdict is inferred from a local weight.\\[0.4em]
B2 & \textbf{Unresolved} &
The inherited carrier and regular embedding are available,
but the new insertion has no admission proof.\\[0.4em]
B3 & \textbf{Unresolved} &
No new vertex response is defined; the functional/test
codomain distinction also needs specification.\\[0.4em]
B4 & \textbf{Unresolved} &
No new completed variation or plateau witness is evaluated.
The layer discrepancy proved here is not a plateau verdict.\\[0.4em]
B5 & Recorded &
Non-identifications and exclusions are explicit; no loop,
state, interacting differential, or sharp limit is introduced.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Final release]
\label{thm:final}
For the supplied dependency set, the requested affirmative
XII-B release is not established. Its ending is Unresolved
at B0, with the dependent gates left unresolved.

The audit additionally proves that a density-ramped cubic
Einstein action and a forcing prescription whose source is
exactly $\sig J_3$ are different specifications on the
unrestricted off-shell class.
This does not obstruct a correctly defined new action or a
separately defined source-weighted model.
\end{theorem}

\begin{proof}
The B0 conclusion is Theorem~\ref{thm:B0}.
The distinction is
Lemma~\ref{lem:weighted},
Proposition~\ref{prop:tracelayer}, and
Proposition~\ref{prop:integrability}.
No theorem here exhibits a forbidden completed-kernel tuple
or an interior defect of a specified new plateau identity.
Therefore neither of the stated grounds for a new
Proved obstructed release has been established.
\end{proof}

\subsection*{What must be supplied before the construction resumes}

A consistent updated handoff must:
\begin{enumerate}
\item choose a density-ramped action or an independently
source-weighted forcing prescription;
\item if it chooses an action, name the cubic density
representative and retain its actual Euler source, including
the terms generated by differentiating the ramp;
\item keep the XI-track free inverse and state exactly which
quadratic and lower-order coefficients remain unchanged;
\item derive the changed Noether identity before propagating
or completing it;
\item state the changed coupled-jet conclusion, or amend the
handoff contract to permit that question to remain unresolved;
\item define the full scalar-completed insertion;
\item specify the response codomain and the intermediate
compact test to which its support theorem applies.
\end{enumerate}

These are missing data, not a newly supplied freeze.
They must not be inserted retroactively into XII-A while
continuing to cite the old version as if it contained them.

\subsection*{What XII-C does and does not receive}

At this stage XII-C receives an audit and the exact
variational consistency calculations.
It does not receive an admitted ramped insertion, a new
retarded response, or a proved new plateau law.
Opening those objects remains conditional on a corrected
handoff and completion of B1--B4.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \text{Supplied XII-A handoff incomplete}
 \quad\Longrightarrow\quad
 \text{XII-B stops at B0: Unresolved},\\[3pt]
 \Euler(\sig\ell_3)
 =\sig J_3+\mathcal D_\sig[\ell_3],\\[3pt]
 \eta_{\mu\nu}\mathcal D_\sig^{\mu\nu}
 \big|_{\gamma=\psi\eta}
 =-6(\partial_\mu\sig)\psi\partial^\mu\psi
 \not\equiv0.
 \end{gathered}}
\]
The old XI-G question stays closed.
The density/source mismatch is displayed rather than hidden.
No new response is written before its prerequisites are supplied.

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

\paragraph{Proof scope.}
The proofs given here concern the dependency audit, local
Euler variation, the explicit pure-trace Einstein--Hilbert
coefficient, and variational integrability.
They do not constitute the proposed admission and retarded
response construction.

\paragraph{Predecessor scope.}
XI-B, XI-E, XI-G, and XII-A-v2 are used in their supplied
forms. Their inherited statements are distinguished from
the calculations proved here.
The changed-model freeze is not attributed to a predecessor
that does not contain it.

\paragraph{No-go scope.}
The integrability result concerns the source
$\sig J_3$ as an Euler derivative on the unrestricted
off-shell class. It is not a no-go for density-ramped
actions with their actual layer terms, nonvariational
forcing prescriptions, larger algebras, or future
admission constructions.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
check, independent referee review, or journal acceptance is
asserted. The source uses standard theorem environments and
defines all internal reference targets.

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

\bibitem{XIIA}
N.~Cheung,
\emph{Landmark XII-A: One graded polynomial carrier for the
finite-ramp multiplet, with ordering-stable composites and
a robust observation exclusion}.
Supplied manuscript \texttt{XII-A-v2.tex}.
Used for the actual carrier handoff and the B0 audit.

\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}.
Used for the free inverse, ghost normalization, plateau
class, ramp, and distinction between layer source support
and propagated support.

\bibitem{XIE}
N.~Cheung,
\emph{Landmark XI-E: On-shell admission of the fixed-width
tree, a localized first-order identity, and the held-fixed
defect}.
Supplied manuscript \texttt{XI-E.tex}.
Used for the scalar completion, coordinate prescription,
polynomial carrier, and distinction between product data
and a state.

\bibitem{XIG}
N.~Cheung,
\emph{Landmark XI-G: The free-shell class of the cubic
Einstein kernel: the onset covector is inherited by every
representative}.
Supplied manuscript \texttt{XI-G.tex}.
Used for the closed original obstruction, the class
criterion, and comparison with the pure-trace cubic
coefficient. Its plane-wave proof is not repeated.

\bibitem{KO}
I.~A.~Kogan and P.~J.~Olver,
``Invariant Euler--Lagrange equations and the invariant
variational bicomplex,''
\emph{Acta Applicandae Mathematicae}
\textbf{76} (2003), 137--193.
Background for local Euler operators and variational
integration by parts. The weighted identities used here
are proved explicitly.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428.
Background for the distinction between classical response
functionals, functional derivative kernels, and test
functions. No quantum retarded product is imported.

\end{thebibliography}

\end{document}