% !TeX program = pdflatex
%=====================================================================
% XII-C.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XII-C.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XII-C.tex
%
% Explicit correction of the production specification:
%   Multiplying the entire Einstein--Hilbert density by sigma
%   does NOT preserve the XI-A/XI-B lower-order coefficients.
%   It introduces a flat-background Euler term and changes the
%   quadratic operator.
%
% The single model selected here is therefore:
%   density weighting before variation, at CUBIC ORDER ONLY.
%
% Its gravitational action is
%   S_EH + kappa int (sigma-1) ell_3,
% where ell_3 is the specified, unintegrated cubic coefficient.
% The scalar action and gauge-fixing term remain unchanged.
%
% This correction is stated in the abstract, release box, and C0.
% It is not attributed retroactively to XII-A or XII-B.
%
% Proved here:
%   - a finite prescription for the cubic density representative;
%   - the actual weighted cubic Euler source;
%   - all first- and second-ramp-derivative source terms;
%   - the changed fixed-background Noether identity;
%   - cancellation of apparent third ramp derivatives;
%   - emptiness of the named fixed-input coupled jet;
%   - the vacuum counterexample to universal jet emptiness.
%
% Not constructed:
%   kernel admission, a metric retarded response, wavefront bounds,
%   a response plateau identity, a quantum retarded product,
%   an interacting differential, a state, or a sharp-width limit.
%
% No successful compilation, mechanized verification,
% independent referee review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\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}
\newtheorem{inputstatement}[theorem]{Input statement}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\PP}{\mathcal P}
\newcommand{\RR}{\mathcal R}
\newcommand{\Lie}{\mathcal L}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\Pmass}{P_m}
\newcommand{\Ggr}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\ER}{\mathsf E_0^{\mathrm R}}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Afree}{\mathscr A_0}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Cthree}{\mathcal C_3}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Hcal}{\mathcal H}
\newcommand{\Lag}{\mathfrak L}
\newcommand{\Euler}{\mathsf{EL}}
\newcommand{\Ncal}{\mathcal N}
\newcommand{\Bcal}{\mathcal B}
\newcommand{\Rampdef}{\mathfrak R}
\newcommand{\Jdensity}{J_{3,\mathrm D}}
\newcommand{\Dsource}{\mathcal D_\sig}
\newcommand{\Zsource}{Z_\sig}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\newcommand{\coeff}[2]{\left[#1^{#2}\right]}

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

\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-C}\\[0.4em]
One declared cubic weighting:\\
actual Euler source, changed Noether identity,\\
and the coupled-jet question}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We freeze one density-before-variation prescription at fixed
regulator width. A necessary correction is made explicit:
weighting the entire Einstein--Hilbert density would change the
lower-order sheet and would not retain the XI-B free operator.
To preserve that handoff, only the specified cubic density
coefficient is multiplied by the existing ramp. All other
gravitational coefficients, the switched scalar action, and
the gauge-fixing term remain those of XI-A and XI-B.

A finite, unintegrated prescription fixes the cubic density
representative. Its actual Euler source is computed as
$\sigma_\varepsilon J_3+\mathcal D_{\sigma_\varepsilon}$.
The complete tensor formula displays all first and second
derivatives of the ramp and reproduces XII-B's pure-trace
layer term. The changed fixed-background Noether identity
contains both the scalar-switch defect and the additional
breaking caused by weighting only one gravitational
coefficient. Its apparent third ramp derivatives cancel
algebraically.

For the named free inputs $\phi_m$, the coupled invariant
metric--scalar jet is empty for every switch of the declared
plateau class. This is proved anew: the compatibility condition
has a layer correction, but a nonzero late switching defect
survives where the ramp is already one. Arbitrary leading
vacuum metric data are allowed. The zero-field vacuum remains
a counterexample to universal jet emptiness.

No Green operator is applied. No completed insertion is admitted,
no wavefront theorem or response plateau law is proved, and no
interacting differential 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) {Density before variation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Cubic-order source};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {The full interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Cubic order only. This is not the complete interaction.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The source is cubic order only. 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-C}
\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) {Density before variation};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {The full interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Cubic order only. This is not the complete interaction.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The source is cubic order only. 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-C}
\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) {Density before variation};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Cubic-order source};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {The full interaction};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Cubic order only. This is not the complete interaction.};
\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-C}
\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) {Density before variation};
\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 full interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Cubic-order source. Cubic order only. This is not the complete interaction.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Density before variation is not The full interaction. 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-C}
\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: Density before variation};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Cubic-order source};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: The full interaction};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Cubic order only. This is not the complete interaction.};
\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-C}
\end{figure}

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


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



\scopebox{Release and explicit specification correction}{%
\textbf{Selected model:}
density before variation, at \emph{cubic order only}.
This is the unique weighting used in the manuscript.

\smallskip
\textbf{Correction at C0:}
the production formula weighting the entire Einstein--Hilbert
density cannot also satisfy the requirement that the lower-order
XI-A/XI-B sheet remain unchanged. We therefore replace that
formula by the explicitly declared cubic-only action in
Section~\ref{sec:freeze}. The literal full-density model is not
certified here.

\smallskip
\textbf{Constructed:}
the action freeze, actual cubic source, changed Noether identity,
and successor data.

\smallskip
\textbf{Proved obstructed:}
the fixed-input coupled jet for the family $\phi_m$.
This is an obstruction of that solution fiber, not of an
observable or of XII-A's carrier.

\smallskip
\textbf{Not opened:}
B1--B4, kernel admission, any new metric retarded response,
a response plateau cancellation, a state, or
$\widehat s_{\mathrm{int}}$.}

\tableofcontents

%=====================================================================
\section{Release criteria and the inherited prerequisite sheet}
\label{sec:prerequisites}
%=====================================================================

The three endings remain:
\begin{enumerate}
\item \textbf{Constructed}: the weighting and density representative
are fixed, their actual source and Noether identity are proved,
and the jet question has a named verdict.
\item \textbf{Proved obstructed}: an explicit incompatibility is
proved for the chosen specification or for the named input family.
\item \textbf{Unresolved}: a necessary definition or calculation
is absent.
\end{enumerate}
The action freeze and the solution-fiber verdict are reported
separately. A constructed action need not have a solution for
every prescribed scalar input.

\[
 \boxed{\text{XII-C freezes an action. It does not cross B1.}}
\]

\subsection{Copied prerequisite statements}

The following statements are used with their predecessor scope.
They are not conclusions about the new weighting.

\begin{inputstatement}[XII-A-v2]
The supplied carrier is
\[
 \Ealg
 =
 (\Fpol[[\hbar,\kappa]],\star_H)/
             \Nsol[[\hbar,\kappa]],
\]
with equal equation and action kernels $\Csol$ and a regular
quotient recovering the causal free algebra.
Ordering transport is an isomorphism on the named composite
sector. For detected observations, the original unramped
$\Cthree|_{\Csol}$ is excluded.
Undetected observations, the Einstein subshell, global
hypocontinuity, and enlargement remain undecided.
No quantum retarded product is supplied \cite{XIIA}.
\end{inputstatement}

\begin{inputstatement}[XII-B]
The requested changed-model handoff was incomplete, so B0--B4
were left Unresolved. The proved variational distinction is
\begin{equation}
 \Euler(\sigma\ell_3)
   =\sigma J_3+\mathcal D_\sigma[\ell_3],
 \label{eq:input-weighted}
\end{equation}
with the pure-trace evaluation
\begin{equation}
 \eta_{\mu\nu}\mathcal D_\sigma^{\mu\nu}
       \big|_{\gamma=\psi\eta}
 =-6(\partial_\mu\sigma)\psi\partial^\mu\psi
 \not\equiv0.
 \label{eq:input-trace}
\end{equation}
The simply weighted source is not an Euler derivative on the
unrestricted off-shell class. These results distinguish source
weighting from density weighting; they do not obstruct an action
with its actual layer terms retained \cite{XIIB}.
\end{inputstatement}

\begin{inputstatement}[XI-B]
The inherited inverse normalization is
$\Ggr=-\PP\ER$.
The free rule is $s_0\gamma=\RR c$, with $\RR=K/2$ and
$c=2\xi$ recovering $K\xi$.
The compact plateau residual intersection is zero.
The de Donder layer formula is conditional on $D\gamma=0$,
and propagated support is not layer support \cite{XIB}.
\end{inputstatement}

\begin{inputstatement}[XI-A and XI-G]
XI-A's empty fixed-input Einstein jet concerns its original
unswitched gravitational density.
XI-G's detected unramped shell class remains excluded; its
source-ramping remark is not a completed-kernel admission theorem
\cite{XIA,XIG}.
\end{inputstatement}

Nothing below searches for a representative of the old
$\Cthree$. X-E/X-F's normalization, X-G's reciprocity pairing,
VII-C's word, the IX sharp commutator, and any cone or Enter
readout are not inputs.

The general variational treatment of curvature with a scalar
prefactor and of nondynamical background breaking provides
useful comparison literature \cite{SF,BS}. The model-specific
formulas below are derived in the displayed conventions.

%=====================================================================
\section{C0: one choice and the lower-order consistency correction}
\label{sec:freeze}
%=====================================================================

\[
 \boxed{\text{Choice D: density before variation, at cubic order only.}}
\]

The nonvariational forcing option is not selected.

\subsection{Geometry and fixed external functions}

Use
\[
 M=\R^4,\quad
 \eta=\operatorname{diag}(1,-1,-1,-1),\quad
 \Box=\partial_t^2-\Delta_{\mathbf x},\quad
 \Pmass=\Box+m^2,
\]
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad 0<\eps<T.
\]
All fields are smooth on a neighborhood of the closed slab.
The ramp is exactly the inherited function
\[
 \sig(t)=\chi(t/\eps),\qquad
 \chi(s)=
 \begin{cases}
 0,&s\le0,\\[2pt]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
 1,&s\ge1.
 \end{cases}
\]
The scalar switch satisfies
\[
 g\in C_c^\infty(\MTo;[0,1]),\qquad
 P_g=\Int\{g=1\}\ne\varnothing,
\]
and
\[
 \overline U\subset P_g\cap\{\eps<t<T\},\qquad
 \ga=C_c^\infty(U;TM).
\]
Neither $g$ nor $\sig$ is varied. The class $\ga$ is not
subsequently reduced.

Put
\[
 q_{\mu\nu}=\eta_{\mu\nu}+\kappa\gamma_{\mu\nu},\qquad
 \PP r=r-\tfrac12\eta\,\tr_\eta r,
\]
\[
 D^\nu\gamma=\partial_\mu(\PP\gamma)^{\mu\nu},
 \qquad
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
\]
Parentheses in indexed formulas include $\tfrac12$.
The curvature convention is
\begin{equation}
 R_{\mu\nu}
 =
 \partial_\rho\Gamma^\rho_{\mu\nu}
 -\partial_\nu\Gamma^\rho_{\mu\rho}
 +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
 -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}.
 \label{eq:curvature}
\end{equation}

\subsection{Why the full-density wording must be corrected}

\begin{proposition}[Full weighting changes the free sheet]
\label{prop:fullweight}
Multiplication of the entire signed Einstein--Hilbert density
by $\sig$ does not retain the XI-A/XI-B lower-order model.
Its metric Euler derivative at the flat background contains
\[
 \frac2\kappa
 \left(\eta^{\mu\nu}\Box\sig-\partial^\mu\partial^\nu\sig\right).
\]
This expression is nonzero for the declared ramp.
\end{proposition}

\begin{proof}
The compact-variation calculation proved in
Lemma~\ref{lem:weighted-curvature} below gives the displayed
expression at $q=\eta$.
Its spatial components are
$-2\delta^{ij}\sig''/\kappa$.
The ramp has derivative zero at both endpoints but changes
from zero to one, so $\sig''$ cannot vanish identically.
Thus flat space is not stationary even with zero scalar input.
In particular, the original zero-background free sheet is
not preserved.
\end{proof}

This is a consistency check on a rejected formula, not a second
selected model. Keeping the entire weighted density would require
a different lower-order freeze and a different background audit.

\subsection{The selected formal action}

Let $\ell_3$ be the precise cubic density fixed in
Section~\ref{sec:density}. Define
\begin{equation}
 S_{\mathrm{grav,D}}
 =
 S_{\mathrm{EH}}
 +\kappa\int_{\MT}(\sig-1)\ell_3[\gamma]\dd^4x,
 \qquad
 S_{\mathrm{EH}}
 =-\frac2{\kappa^2}
       \int_{\MT}\sqrt{-q}\,R[q]\dd^4x.
 \label{eq:selected-action}
\end{equation}
The unchanged scalar action is
\begin{equation}
 S_g^{\mathrm{sc}}
 =
 \int_{\MT}
 \bigl((1-g)\Lag_\eta[\phi]+g\Lag_q[\phi]\bigr)\dd^4x,
 \qquad
 \Lag_q[\phi]
 =\tfrac12\sqrt{-q}
       \left(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
                         -m^2\phi^2\right).
 \label{eq:scalar}
\end{equation}
Write
\[
 S_{\mathrm D}=S_{\mathrm{grav,D}}+S_g^{\mathrm{sc}},
 \qquad
 S_{\mathrm{D,gf}}=S_{\mathrm D}+S_{\mathrm{gf}},
 \qquad
 S_{\mathrm{gf}}=\int_{\MT}D_\nu\gamma D^\nu\gamma\dd^4x.
\]

Actions mean formal bulk variational functionals, tested with
compact interior variations. Numerical finiteness on all smooth
configurations is not required. The unweighted linear
Einstein--Hilbert divergence is treated as in XI-A.
The selected cubic representative is not changed by a later
integration by parts.

Equation~\eqref{eq:selected-action} weights only the cubic
coefficient. The quadratic coefficient and all quartic and higher
coefficients remain unweighted. This fixes the higher-order
completion rather than leaving it unspecified.

\begin{proposition}[Unchanged free handoff]
The quadratic gravitational and gauge-fixing coefficients of
$S_{\mathrm{D,gf}}$ are the XI-A/XI-B coefficients. Their metric
operator is
\[
 A+2D^*D=-\Box\PP,\qquad A=2G_{\mathrm{lin}}.
\]
The auxiliary and ghost normalization of XI-B is unchanged.
\end{proposition}

\begin{proof}
The added functional in \eqref{eq:selected-action} is cubic in
$\gamma$. Its first and second derivatives vanish at $\gamma=0$.
It therefore changes neither the free metric Hessian nor the
independent quadratic auxiliary and ghost terms.
\end{proof}

This cubic-only correction is a definition made in XII-C.
It is not attributed retrospectively to XII-A or XII-B.

%=====================================================================
\section{C1: the density representative and actual cubic source}
\label{sec:density}
%=====================================================================

\subsection{A finite unintegrated density prescription}

Write $\tau=\tr_\eta\gamma$ and set
\[
 u_0^{\mu\nu}=\eta^{\mu\nu},\qquad
 u_1^{\mu\nu}=-\gamma^{\mu\nu},\qquad
 u_2^{\mu\nu}=\gamma^\mu{}_\lambda\gamma^{\lambda\nu},
\]
\[
 b_0=1,\qquad b_1=\tfrac12\tau,\qquad
 b_2=\tfrac18\tau^2-\tfrac14\gamma_{\mu\nu}\gamma^{\mu\nu}.
\]
For $n=1,2,3$, define
\begin{align}
 \Gamma_n^\rho{}_{\mu\nu}
 &=
 \tfrac12u_{n-1}^{\rho\lambda}
 \left(\partial_\mu\gamma_{\nu\lambda}
      +\partial_\nu\gamma_{\mu\lambda}
      -\partial_\lambda\gamma_{\mu\nu}\right),\\
 R_{n,\mu\nu}
 &=
 \partial_\rho\Gamma_n^\rho{}_{\mu\nu}
 -\partial_\nu\Gamma_n^\rho{}_{\mu\rho}
 \nonumber\\
 &\quad+
 \sum_{\substack{a+b=n\\a,b\ge1}}
 \left(
 \Gamma_a^\rho{}_{\rho\lambda}\Gamma_b^\lambda{}_{\mu\nu}
 -\Gamma_a^\rho{}_{\nu\lambda}\Gamma_b^\lambda{}_{\mu\rho}
 \right),\\
 r_n&=\sum_{i=0}^{n-1}u_i^{\mu\nu}R_{n-i,\mu\nu}.
 \label{eq:finite-curvature}
\end{align}
The declared representative is
\begin{equation}
 \boxed{\ell_3=-2(r_3+b_1r_2+b_2r_1).}
 \label{eq:ell3}
\end{equation}
It is the coefficient of $\kappa$ in the unintegrated signed
Einstein--Hilbert density. Equivalently, it is $-2$ times the
coefficient of $\kappa^3$ in $\sqrt{-q}R[q]$.

Every term is local, cubic, and of jet order at most two.
The formulas specify the density itself, not only its Euler
equivalence class.

For clarity, the original source can also be evaluated finitely.
Put
\[
 V_{1,\mu\nu}=R_{1,\mu\nu}-\tfrac12\eta_{\mu\nu}r_1,\qquad
 V_{2,\mu\nu}=R_{2,\mu\nu}-\tfrac12\eta_{\mu\nu}r_2
                             -\tfrac12\gamma_{\mu\nu}r_1.
\]
Then
\begin{equation}
 J_3^{\mu\nu}
 =
 2\left(
 V_2^{\mu\nu}+b_1V_1^{\mu\nu}
 -\gamma^{\mu\alpha}V_{1,\alpha}{}^\nu
 -\gamma^{\nu\alpha}V_{1,\alpha}{}^\mu
 \right).
 \label{eq:J3}
\end{equation}
Compact variation of \eqref{eq:ell3} gives
$\Euler_\gamma(\ell_3)=J_3$ with the XI-A sign.

\subsection{The curvature variation needed for the expansion}

\begin{lemma}[Weighted curvature variation]
\label{lem:weighted-curvature}
For a fixed smooth scalar $\sigma$, compact covariant metric
variation gives
\begin{equation}
 \frac{\delta}{\delta\gamma_{\mu\nu}}
 \left(-\frac2{\kappa^2}
        \int\sigma\sqrt{-q}R[q]\dd^4x\right)
 =
 \frac2\kappa\sqrt{-q}
 \left[
 \sigma G^{\mu\nu}
 +q^{\mu\nu}\Box_q\sigma-\nabla^\mu\nabla^\nu\sigma
 \right].
 \label{eq:weighted-full-variation}
\end{equation}
This identity is used for coefficient extraction, not to select
the full-density model.
\end{lemma}

\begin{proof}
For $h_{\mu\nu}=\delta q_{\mu\nu}$, write
$h=q^{\mu\nu}h_{\mu\nu}$ and
$h^{\mu\nu}=q^{\mu\alpha}q^{\nu\beta}h_{\alpha\beta}$.
The Christoffel variation gives
\[
 \delta R=-R^{\mu\nu}h_{\mu\nu}
              +\nabla_\mu\nabla_\nu h^{\mu\nu}-\Box_q h.
\]
Together with $\delta\sqrt{-q}=\frac12\sqrt{-q}\,h$,
two compact integrations by parts give
\[
 \delta\int\sigma\sqrt{-q}R
 =
 \int\sqrt{-q}
 \left[-\sigma G^{\mu\nu}
       +\nabla^\mu\nabla^\nu\sigma-q^{\mu\nu}\Box_q\sigma\right]
 h_{\mu\nu}\dd^4x.
\]
Multiply by $-2/\kappa^2$ and use
$h_{\mu\nu}=\kappa\,\delta\gamma_{\mu\nu}$.
\end{proof}

The coefficient of $\kappa$ on the left is the Euler derivative
of $\sigma\ell_3$. Thus coefficient extraction supplies its
complete layer correction without changing the chosen density.

\subsection{All first and second ramp derivatives}

Define
\[
 C^{\mu\nu\alpha\beta}(q)
 =
 \sqrt{-q}
 \left(q^{\mu\nu}q^{\alpha\beta}
                  -q^{\mu(\alpha}q^{\beta)\nu}\right)
 =C_0+\kappa C_1+\kappa^2C_2+O(\kappa^3).
\]
The three coefficients are explicitly
\begin{align}
 C_0^{\mu\nu\alpha\beta}
 &=\eta^{\mu\nu}\eta^{\alpha\beta}
                         -\eta^{\mu(\alpha}\eta^{\beta)\nu},\\
 C_1^{\mu\nu\alpha\beta}
 &=b_1C_0^{\mu\nu\alpha\beta}
   +u_1^{\mu\nu}\eta^{\alpha\beta}
   +\eta^{\mu\nu}u_1^{\alpha\beta}
 \nonumber\\
 &\quad-u_1^{\mu(\alpha}\eta^{\beta)\nu}
          -\eta^{\mu(\alpha}u_1^{\beta)\nu},\\
 C_2^{\mu\nu\alpha\beta}
 &=b_2C_0^{\mu\nu\alpha\beta}
   +b_1\left(
    u_1^{\mu\nu}\eta^{\alpha\beta}
   +\eta^{\mu\nu}u_1^{\alpha\beta}
   -u_1^{\mu(\alpha}\eta^{\beta)\nu}
   -\eta^{\mu(\alpha}u_1^{\beta)\nu}\right)
 \nonumber\\
 &\quad+u_2^{\mu\nu}\eta^{\alpha\beta}
       +\eta^{\mu\nu}u_2^{\alpha\beta}
       +u_1^{\mu\nu}u_1^{\alpha\beta}
 \nonumber\\
 &\quad-u_2^{\mu(\alpha}\eta^{\beta)\nu}
       -\eta^{\mu(\alpha}u_2^{\beta)\nu}
       -u_1^{\mu(\alpha}u_1^{\beta)\nu}.
 \label{eq:Ccoefficients}
\end{align}
Also define the quadratic tensor
\begin{equation}
 M_2^{\mu\nu\rho}
 =
 C_0^{\mu\nu\alpha\beta}\Gamma_2^\rho{}_{\alpha\beta}
 +C_1^{\mu\nu\alpha\beta}\Gamma_1^\rho{}_{\alpha\beta}.
 \label{eq:M2}
\end{equation}

\begin{theorem}[Actual density-weighted cubic source]
\label{thm:actualsource}
The selected cubic vertex
\[
 V_{3,\mathrm D}[\gamma]
       =\int_{\MT}\sig\ell_3[\gamma]\dd^4x
\]
has Euler source
\begin{equation}
 \boxed{
 \Jdensity^{\mu\nu}
 =\sig J_3^{\mu\nu}+\Dsource^{\mu\nu},\qquad
 \Dsource^{\mu\nu}
 =2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sig
             -2M_2^{\mu\nu\rho}\partial_\rho\sig .
 }
 \label{eq:actualsource}
\end{equation}
For the time-only ramp,
\begin{equation}
 \Dsource^{\mu\nu}
 =2C_2^{\mu\nu00}\sig''
                  -2M_2^{\mu\nu0}\sig'.
 \label{eq:time-layer}
\end{equation}
These are all ramp-derivative terms in the cubic source.
\end{theorem}

\begin{proof}
The extra term in \eqref{eq:weighted-full-variation} is
\[
 \frac2\kappa C^{\mu\nu\alpha\beta}(q)
 \left(
 \partial_\alpha\partial_\beta\sig
            -\Gamma^\rho_{\alpha\beta}(q)\partial_\rho\sig
 \right).
\]
Its coefficient of $\kappa$ is
\eqref{eq:actualsource}, using \eqref{eq:Ccoefficients}
and \eqref{eq:M2}. The Einstein term at that coefficient is
$\sig J_3$. No derivative of the fixed weight beyond order
two occurs.
\end{proof}

The correction is quadratic in $\gamma$ and its derivatives.
It vanishes outside the closed layer $0\le t\le\eps$ and is
smooth and flat at its endpoints. Its nonzero values occur
inside the open layer; its closed support need not avoid the
endpoints. There is no hypersurface distribution.

\subsection{The copied pure-trace check}

XII-B's identity \eqref{eq:input-trace} is a consistency
requirement, not the general source formula.
For $\gamma=\psi\eta$, algebraic substitution gives
$C_1=C_2=0$ and $\Gamma_2=-\psi\Gamma_1$.
Consequently \eqref{eq:actualsource} gives
\begin{equation}
 \Dsource^{\mu\nu}\big|_{\gamma=\psi\eta}
 =
 -\psi\left[
 \eta^{\mu\nu}(\partial_\rho\sig)\partial^\rho\psi
 +(\partial^\mu\sig)\partial^\nu\psi
 +(\partial^\nu\sig)\partial^\mu\psi
 \right].
 \label{eq:full-trace-sector}
\end{equation}
Contracting with $\eta_{\mu\nu}$ reproduces
\eqref{eq:input-trace}. Thus the predecessor's nonzero layer
term is retained.

Density representatives cannot be changed silently.
If $\ell_3$ is replaced by $\ell_3+\partial_\mu B^\mu$,
the weighted source changes by
$-\Euler_\gamma((\partial_\mu\sig)B^\mu)$, as proved in
XII-B. Equation~\eqref{eq:ell3} eliminates that ambiguity.

%=====================================================================
\section{C2: the changed fixed-background Noether identity}
\label{sec:noether}
%=====================================================================

\subsection{Euler operators and the unchanged matter defect}

Define
\[
 e_{\mathrm D}^{\mu\nu}
       =\frac{\delta S_{\mathrm D}}{\delta\gamma_{\mu\nu}},
 \qquad
 E_\phi=\frac{\delta S_{\mathrm D}}{\delta\phi}.
\]
The scalar equation is unchanged. Explicitly,
\begin{equation}
 E_\phi
 =
 -\partial_\mu\left(
 [(1-g)\eta^{\mu\nu}+g\sqrt{-q}\,q^{\mu\nu}]
                         \partial_\nu\phi\right)
 -m^2[(1-g)+g\sqrt{-q}]\,\phi.
 \label{eq:scalar-euler}
\end{equation}

Put $w=\sig-1$ and
\begin{equation}
 \Zsource^{\mu\nu}
   =wJ_3^{\mu\nu}+\Dsource^{\mu\nu}.
 \label{eq:Z}
\end{equation}
The exact formal metric derivative is
\begin{equation}
 e_{\mathrm D}^{\mu\nu}
 =
 \frac2\kappa\sqrt{-q}\,G^{\mu\nu}
 -\frac{\kappa}{2}g\sqrt{-q}\,T_q^{\mu\nu}
 +\kappa\Zsource^{\mu\nu},
 \label{eq:eD}
\end{equation}
where
\[
 T_q^{\mu\nu}
 =q^{\mu\alpha}q^{\nu\beta}\partial_\alpha\phi\partial_\beta\phi
 -\tfrac12q^{\mu\nu}
       (q^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi
                          -m^2\phi^2).
\]
In particular, for coupling-independent fields,
\[
 e_{\mathrm D}
 =A\gamma
   +\kappa\left(\Jdensity-\tfrac12gT_0\right)
   +O(\kappa^2).
\]

For a symmetric contravariant density $v$, set
\begin{align}
 \Ncal_\nu(v)
 &=2\partial_\mu(v^{\mu\rho}q_{\rho\nu})
                           -v^{\mu\rho}\partial_\nu q_{\mu\rho},\\
 N_{0,\nu}(v)
 &=2\eta_{\rho\nu}\partial_\mu v^{\mu\rho},\\
 Q_\nu(v)
 &=2\partial_\mu(v^{\mu\rho}\gamma_{\rho\nu})
                           -v^{\mu\rho}\partial_\nu\gamma_{\mu\rho}.
 \label{eq:Noperators}
\end{align}
Thus $\Ncal_\nu(v)=N_{0,\nu}(v)+\kappa Q_\nu(v)$.

The unchanged scalar-background defect is
\begin{equation}
 \Bcal_{g,\nu}
 =
 (\partial_\mu g)T_0^\mu{}_\nu
 -(1-g)(\Pmass\phi)\partial_\nu\phi
 +(\partial_\nu g)(\Lag_\eta[\phi]-\Lag_q[\phi]).
 \label{eq:matterdefect}
\end{equation}
The derivative-of-switch term remains visible.
There are no derivatives of $g$ in the metric variation of
the minimally coupled scalar density itself.

\subsection{An explicit ramp defect with no third derivatives}

Define
\begin{equation}
 \begin{split}
 \Rampdef_{\sig,\nu}
 ={}&
 2\eta_{\rho\nu}(\partial_\mu\sig)J_3^{\mu\rho}\\
 &+4\eta_{\rho\nu}
 \left[
  (\partial_\mu C_2^{\mu\rho\alpha\beta})
                    \partial_\alpha\partial_\beta\sig
  -(\partial_\mu M_2^{\mu\rho\lambda})\partial_\lambda\sig
  -M_2^{\mu\rho\lambda}
                    \partial_\mu\partial_\lambda\sig
 \right].
 \end{split}
 \label{eq:Rampdef}
\end{equation}
For the declared ramp this is
\begin{equation}
 \begin{split}
 \Rampdef_{\sig,\nu}
 ={}&2\eta_{\rho\nu}J_3^{0\rho}\sig'\\
 &+4\eta_{\rho\nu}
 \left[
   (\partial_\mu C_2^{\mu\rho00})\sig''
   -(\partial_\mu M_2^{\mu\rho0})\sig'
   -M_2^{0\rho0}\sig''
 \right].
 \end{split}
 \label{eq:Rampdef-time}
\end{equation}

\begin{lemma}[Derivative cancellation and cubic divergence]
\label{lem:divergence}
The following identities hold off shell:
\begin{align}
 N_{0,\nu}(J_3)+Q_\nu(A\gamma)&=0,
 \label{eq:original-cubic-noether}\\
 N_{0,\nu}(\Zsource)
 &=-wQ_\nu(A\gamma)+\Rampdef_{\sig,\nu}.
 \label{eq:Zdivergence}
\end{align}
In particular, differentiating the actual source introduces
no surviving third derivative of $\sig$.
\end{lemma}

\begin{proof}
The contracted Bianchi identity, written for
$\sqrt{-q}G^{\mu\nu}$, gives
$\Ncal_\nu(2\sqrt{-q}G/\kappa)=0$.
Its coefficient of $\kappa$ is
\eqref{eq:original-cubic-noether}.

Now differentiate \eqref{eq:Z}. The terms from $wJ_3$ are
\[
 wN_{0,\nu}(J_3)
       +2\eta_{\rho\nu}(\partial_\mu\sig)J_3^{\mu\rho}.
\]
Differentiation of \eqref{eq:actualsource} gives the remaining
terms of \eqref{eq:Rampdef} and an apparent term
\[
 4\eta_{\rho\nu}
 C_2^{\mu\rho\alpha\beta}
               \partial_\mu\partial_\alpha\partial_\beta\sig.
\]
For every symmetric three-covector tensor
$S_{\mu\alpha\beta}$,
\[
 C^{\mu\rho\alpha\beta}(q)S_{\mu\alpha\beta}=0:
\]
the two products of inverse metrics become equal after
permuting the three contracted indices.
Every coefficient $C_r$ has the same property.
The apparent third derivative therefore vanishes.
Substitute \eqref{eq:original-cubic-noether}.
\end{proof}

\subsection{The actual changed identity}

\begin{theorem}[Changed Noether identity]
\label{thm:noether}
For the selected action, off shell,
\begin{equation}
 \boxed{
 \begin{split}
 \Ncal_\nu(e_{\mathrm D})
 =\kappa\bigl[
 &E_\phi\partial_\nu\phi-\Bcal_{g,\nu}
 -wQ_\nu(A\gamma)+\Rampdef_{\sig,\nu}
 +\kappa Q_\nu(\Zsource)
 \bigr].
 \end{split}}
 \label{eq:changed-noether}
\end{equation}
All ramp derivatives are displayed in
\eqref{eq:actualsource} and \eqref{eq:Rampdef}.
The $Q_\nu(\Zsource)$ term is the explicitly defined first-order
operator \eqref{eq:Noperators} applied to the displayed
quadratic source.
\end{theorem}

\begin{proof}
For an arbitrary compact interior $\xi$, use the scaled
metric--scalar variation
\[
 \delta_\xi\gamma=K\xi+\kappa\Lie_\xi\gamma=\Lie_\xi q,
 \qquad
 \delta_\xi\phi=\kappa\xi^\rho\partial_\rho\phi.
\]
Compact integration by parts gives
\[
 \delta_\xi S_{\mathrm D}
 =
 \int\xi^\nu
 \left[-\Ncal_\nu(e_{\mathrm D})
                        +\kappa E_\phi\partial_\nu\phi\right]\dd^4x.
\]

For the unweighted Einstein--Hilbert part, naturality gives
zero variation. Variation of the unchanged switched scalar
action gives
$\kappa\int\xi^\nu\Bcal_{g,\nu}\dd^4x$.
This follows by expanding the flat scalar variation as
\[
 \delta_\xi\Lag_\eta
 =\kappa\left[
 T_0^\mu{}_\nu\partial_\mu\xi^\nu
                   +\partial_\mu(\xi^\mu\Lag_\eta)\right]
\]
and the curved scalar variation as a density divergence,
then integrating the factors $1-g$ and $g$.

The added cubic functional in \eqref{eq:selected-action}
varies as
\[
 \kappa\int
 \Zsource^{\mu\nu}
      (K\xi+\kappa\Lie_\xi\gamma)_{\mu\nu}\dd^4x
 =
 -\kappa\int\xi^\nu
       [N_{0,\nu}(\Zsource)+\kappa Q_\nu(\Zsource)]\dd^4x.
\]
Comparison of the two variation expressions, followed by
Lemma~\ref{lem:divergence}, proves
\eqref{eq:changed-noether}.
\end{proof}

This is not XI-A's identity with a relabeled source.
The additional term $-wQ_\nu(A\gamma)$ is important:
weighting only the cubic coefficient, while leaving the
quadratic coefficient unchanged, produces breaking even
where the weight is a constant different from one.
Thus the new breaking is not exclusively a ramp-derivative
term. Omitting this bulk coefficient mismatch would falsely
treat a cubic-only modification as weighting the complete
natural gravitational density.

For the gauge-fixed action, add the separate contribution
$\Ncal_\nu(e_{\mathrm{gf}})$ to the right, where
\[
 e_{\mathrm{gf}}^{\mu\nu}
 =-2\partial^{(\mu}D^{\nu)}
                      +\eta^{\mu\nu}\partial_\rho D^\rho.
\]
Gauge fixing is not made invariant by the ramp.

\begin{corollary}[Plateau contraction, not a response law]
For $\xi\in\ga$, the new action correction and scalar-switch
breaking vanish in the contracted variational identity.
Thus
\[
 \int\xi^\nu
 \left[-\Ncal_\nu(e_{\mathrm D})
                       +\kappa E_\phi\partial_\nu\phi\right]\dd^4x=0.
\]
\end{corollary}

\begin{proof}
On a neighborhood of $\supp\xi$, one has
$g=\sig=1$ and all their derivatives vanish.
Hence $\Bcal_g$, $w$, $\Dsource$, $\Zsource$, and
$\Rampdef_\sig$ vanish there.
\end{proof}

This local action statement supplies no propagated cancellation.
No source has been evolved, and no plateau ledger is computed.

%=====================================================================
\section{C3: the coupled jet for the selected action}
\label{sec:jet}
%=====================================================================

\begin{definition}[New fixed-input invariant jet]
For a fixed free scalar $\phi_0$, let
$\Jcal_{g,\sig}^{\mathrm D}(\phi_0)$ consist of smooth formal
families
\[
 \gamma(\kappa)=\gamma_0+\kappa\gamma_1+\kappa^2\gamma_2,
 \qquad
 \phi(\kappa)=\phi_0+\kappa\phi_1+\kappa^2\phi_2,
\]
for which the coefficients through $\kappa^2$ of
$e_{\mathrm D}=0$ and $E_\phi=0$ vanish.
Positive-order scalar coefficients have the inherited zero
initial data. Metric Cauchy data are unrestricted.

This is a jet of $S_{\mathrm D}$, not of the gauge-fixed
action. In particular, $\gamma_0$ is not required to vanish.
\end{definition}

\subsection{The recomputed compatibility condition}

\begin{proposition}[Order-one compatibility]
\label{prop:compatibility}
Every element of
$\Jcal_{g,\sig}^{\mathrm D}(\phi_0)$ satisfies
\begin{equation}
 (\partial_\mu g)T_0^\mu{}_\nu[\phi_0]
       =\Rampdef_{\sig,\nu}[\gamma_0].
 \label{eq:new-compatibility}
\end{equation}
For $t>\eps$, this reduces to
\begin{equation}
 (\partial_\mu g)T_0^\mu{}_\nu[\phi_0]=0.
 \label{eq:late-compatibility}
\end{equation}
\end{proposition}

\begin{proof}
The leading metric equation is $A\gamma_0=0$.
The next metric equation is
\[
 A\gamma_1+\Jdensity[\gamma_0]
                          -\tfrac12gT_0[\phi_0]=0.
\]
Apply $N_0$. The linearized Bianchi identity gives
$N_0(A\gamma_1)=0$.
Lemma~\ref{lem:divergence} also gives
\[
 N_{0,\nu}(\Jdensity)
 =-\sig Q_\nu(A\gamma_0)+\Rampdef_{\sig,\nu}
 =\Rampdef_{\sig,\nu}.
\]
Since $\Pmass\phi_0=0$,
$\partial_\mu T_0^\mu{}_\nu[\phi_0]=0$.
The resulting equation is
\eqref{eq:new-compatibility}.
Above the ramp, \eqref{eq:Rampdef-time} vanishes identically.
\end{proof}

The new compatibility condition permits a layer contribution
from leading metric data. It is therefore not obtained by
transferring XI-A's unswitched condition to the whole slab.

\subsection{The named family and a late switching witness}

For every $m\ge0$, set
\[
 \phi_m(t,\mathbf x)=
 \begin{cases}
 \sin(mt)/m,&m>0,\\
 t,&m=0.
 \end{cases}
\]
It is a free scalar and
\[
 T_0^{00}[\phi_m]=\tfrac12,\qquad
 T_0^{0i}[\phi_m]=0.
\]

\begin{lemma}[A falling switch region above the ramp]
\label{lem:late-switch}
There are $c>0$ and a nonempty interior open set
$O\subset\{t>\eps\}$ on which $\partial_tg\le-c$.
\end{lemma}

\begin{proof}
Choose $(t_*,\mathbf x_*)\in U$.
Then $t_*>\eps$ and $g(t_*,\mathbf x_*)=1$.
Compact interior support gives a time
$t_1\in(t_*,T)$ with $g(t_1,\mathbf x_*)=0$.
The mean value theorem yields a point between them where
$\partial_tg<0$. Smoothness gives the claimed open
neighborhood and a strictly positive $c$.
\end{proof}

\begin{theorem}[Empty fixed-input jet for the new action]
\label{thm:empty}
For every mass $m\ge0$ and every switch of the declared
plateau class,
\begin{equation}
 \boxed{\Jcal_{g,\sig}^{\mathrm D}(\phi_m)=\varnothing.}
 \label{eq:empty}
\end{equation}
The obstruction already occurs in the order-one compatibility
condition, independently of the allowed leading metric data.
\end{theorem}

\begin{proof}
For $\nu=0$, equation~\eqref{eq:late-compatibility} becomes
$\frac12\partial_tg=0$ above the ramp.
This contradicts Lemma~\ref{lem:late-switch}.

More explicitly, choose $f\ge0$ supported in $O$ with
$\int f\dd^4x=1$. Then
\[
 \int f\left[
 (\partial_\mu g)T_0^\mu{}_0[\phi_m]
                 -\Rampdef_{\sig,0}[\gamma_0]\right]\dd^4x
 \le-\frac c2<0,
\]
because the ramp defect is zero on $O$.
No leading vacuum metric configuration can alter this
pairing. Higher-order coefficients cannot cancel a nonzero
lower-order compatibility coefficient.
\end{proof}

This proof is new to the selected action.
The layer correction is present in
\eqref{eq:new-compatibility}; it simply cannot cancel the
late switch defect. The witness is in the switching region,
not an asserted nonzero plateau variation of an observable.
The plateau algebra has not been shrunk.

\begin{proposition}[Vacuum and the limits of emptiness]
The unrestricted jet family of the selected action is not
empty. The family with $\gamma=0$ and $\phi=0$ is a solution.
\end{proposition}

\begin{proof}
The flat Einstein tensor and scalar stress vanish.
Both $J_3$ and $\Dsource$ are quadratic in the metric field,
so their values at $\gamma=0$ vanish.
The scalar Euler expression also vanishes.
\end{proof}

This vacuum statement depends on the cubic-only freeze.
It would be false for the flat background of the rejected
full-density formula in Proposition~\ref{prop:fullweight}.

Other scalar inputs are not classified. The necessary
condition \eqref{eq:new-compatibility} is not promoted to a
general sufficiency theorem. Gauge-fixed solvability is a
different equation because $e_{\mathrm{gf}}$ changes the
divergence constraint. No gauge-fixed solution is constructed
here, and none is declared an Einstein-jet repair.

%=====================================================================
\section{C4: successor data without a response}
\label{sec:handoff}
%=====================================================================

The action supplies a local quadratic source and its polarized
form
\[
 \Jdensity(\gamma_1,\gamma_2)
 =
 \tfrac12\left[
 \Jdensity[\gamma_1+\gamma_2]
 -\Jdensity[\gamma_1]-\Jdensity[\gamma_2]\right].
\]
This definition applies equally to the displayed layer terms.
It contains no propagator.

\subsection{An honest classical source-pairing functional}

A successor may prescribe a compact root datum:
a continuous linear map
\[
 a_{f,g}:C^\infty(M)\longrightarrow
 C_c^\infty(M;\operatorname{Sym}^2 T^*M)
\]
whose range is supported in a fixed compact set
\[
 K_a\subset J^-(\supp f)\cap\{t\ge0\}.
\]
Here the range support and the map are \emph{given data}.
They are not claimed to arise from a retarded construction
already performed in this paper.

For such a datum, the classical trilinear source pairing is
\begin{equation}
 \mathcal I_{\mathrm D,a}
       (f,g;\gamma_1,\gamma_2,\phi)
 =
 \int a_{f,g}[\phi]_{\mu\nu}(y)
       \Jdensity^{\mu\nu}(\gamma_1,\gamma_2)(y)\dd^4y.
 \label{eq:source-pairing}
\end{equation}
It is well defined on smooth configurations: the test has
compact support and the source is a local smooth polynomial.

Equation~\eqref{eq:source-pairing} is an actual classical
functional with an independently supplied root datum.
It is not asserted to be the completed insertion of XI-E.
No claim about its distributional kernel or admission is made.

\subsection{The unconstructed completion slot}

The notation $\mathcal C_{3,\mathrm D}(f,g)$ is reserved for
the eventual scalar-completed insertion.
Its intended signature is the inherited completion
$\Aone[f,g](r;\phi)$ with a new metric-response slot $r$.
That slot has not been constructed.

A successor must determine the completed root datum associated
with both the coordinate and scalar-response legs, establish
its support, and then identify the resulting functional.
We do not turn this unconstructed slot into a response formula.

This is a necessary type distinction.
A source pairing is scalar valued; an intermediate compact
test is tensor valued; a metric response is a smooth metric
configuration. Naming one does not construct either of the
others. The action freeze alone cannot supply the compact
root test generated by an unperformed evolution.

\subsection{The inherited inverse is recorded, not applied}

The successor receives the normalization
\[
 \Ggr=-\PP\ER.
\]
No occurrence of this identity in the manuscript is an
application to $\Jdensity$, $\sig J_3$, or any other source.

The successor also receives XII-A's existing carrier and
the unchanged exclusion of the original detected unramped
classes. It does not receive membership of a new insertion
in that carrier.

%=====================================================================
\section{C5: non-identifications and final decision}
\label{sec:decision}
%=====================================================================

The selected source is not XI-F's $J_3$: it is
$\sig J_3+\Dsource$.
It is not the simply source-weighted expression $\sig J_3$.
It is not XII-A's excluded original $\Cthree$ and is not
identified with X-F's mixed map or VII-D's triangle.

No X-track normalization is introduced.
The scalar switch and coordinate ramp are unchanged; neither
has been adjusted to remove a defect.
The original XI-F ledger is not recomputed for this model.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}
                              p{.24\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
C0 & Constructed, with explicit correction &
One cubic-only density weighting; the full-density wording
is rejected because it changes the lower-order sheet.\\[0.4em]
C1 & Constructed &
A fixed density representative and its actual Euler source;
all first and second ramp derivatives retained.\\[0.4em]
C2 & Constructed &
Changed Noether identity, scalar-switch defect, layer defect,
and cubic-only bulk breaking all displayed.\\[0.4em]
C3 & Proved obstructed for the named fiber &
The new fixed-input jet is empty for every $\phi_m$ and
every declared switch; vacuum prevents universal emptiness.\\[0.4em]
C4 & Specified as successor data &
Local source pairing and typed completion slot;
the completed insertion and generated root are not constructed.\\[0.4em]
C5 & Recorded &
No old-class repair, X-track identification, admission,
retarded response, or sharp limit.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Freeze and solution-fiber release]
The selected action is a coherent density-before-variation
cubic modification preserving the XI-A/XI-B free handoff.
Its actual source and changed Noether identity are those of
Theorems~\ref{thm:actualsource} and \ref{thm:noether}.
Its fixed-input family $\phi_m$ has no coupled invariant jet,
by Theorem~\ref{thm:empty}.

The freeze is Constructed.
If a single ending is assigned by the worst-gate convention,
the named fixed-input release is Proved obstructed at C3.
Neither statement is a kernel-admission or observable no-go.
\end{theorem}

\begin{proof}
The action and density are fixed by
\eqref{eq:selected-action} and \eqref{eq:ell3}.
The cubic modification preserves the free Hessian.
The cited source and identity theorems prove its variational
and Noether consistency, while the jet theorem supplies the
specified incompatibility.
No admission or response premise is used in these arguments.
\end{proof}

\subsection*{Exact handoff}

The next paper receives:
\begin{enumerate}
\item the explicit cubic-only action, including the unweighted
higher gravitational coefficients;
\item the density representative \eqref{eq:ell3};
\item the actual source
$\Jdensity=\sig J_3+\Dsource$;
\item the changed identity \eqref{eq:changed-noether};
\item the new jet verdict and its late-switch witness;
\item the unchanged XI-track inverse normalization;
\item XII-A's carrier and the unreopened old exclusion;
\item the classical source-pairing signature and the
unconstructed scalar-completion slot.
\end{enumerate}

It receives no completed-kernel theorem, wavefront estimate,
admission map for this source, response plateau cancellation,
or quantum retarded product.
\textbf{B1--B4 were not opened.}

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}+\kappa\int(\sig-1)\ell_3,\qquad
 \Jdensity=\Euler_\gamma(\sig\ell_3)
                     =\sig J_3+\Dsource,\\[2pt]
 \eta_{\mu\nu}\Dsource^{\mu\nu}\big|_{\gamma=\psi\eta}
 =-6(\partial_\mu\sig)\psi\partial^\mu\psi,\\[2pt]
 \Jcal_{g,\sig}^{\mathrm D}(\phi_m)=\varnothing,
 \qquad
 \text{no metric Green operator has been applied}.
 \end{gathered}}
\]

The nonzero jet defect is not a failure of the polynomial
carrier. Retaining the layer source is not admission.
The changed action is not a repaired representative of the
old cubic class.

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

\paragraph{Specification correction.}
This manuscript selects density weighting at cubic order only.
It explicitly amends the full-density formula in the production
brief because that formula is incompatible with the required
unchanged lower-order handoff. No claim is made that the two
actions are equivalent.

\paragraph{Classical scope.}
All new products and variations are local classical expressions
on smooth configurations. No new quantum product is defined.

\paragraph{Dependency scope.}
XII-B's weighted Euler distinction and pure-trace layer identity
are copied as inputs and used as checks. The new general tensor
source, changed identity, and coupled-jet verdict are proved here.
XI-A's original jet theorem is not used as the new jet verdict.

\paragraph{Boundary scope.}
The cubic representative is fixed before weighting.
All variational proofs use compact interior variations.
No boundary action, boundary charge, or additional variational
principle is constructed.

\paragraph{Unopened work.}
No metric retarded map, kernel admission, wavefront test,
response ledger, interacting differential, antibracket,
ghost loop, state, enlargement, or $\eps\to0$ limit is included.

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

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

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

\bibitem{XIIB}
N.~Cheung,
\emph{Landmark XII-B: Admission and one retarded response of the
ramped cubic Einstein source, at fixed regulator width:
handoff audit, variational consistency, and the unresolved release}.
Supplied manuscript \texttt{XII-B.tex}.
Weighted Euler identity, pure-trace layer term, and
off-shell integrability distinction.

\bibitem{XIA}
N.~Cheung,
\emph{Landmark XI-A: A switched Einstein--scalar action for the
ramp relational scalar: sign conventions, Noether identities,
and an empty fixed-input Einstein jet}.
Supplied manuscript \texttt{XI-A.tex}.
Original signed action, scalar switch, curvature convention,
and original jet quantifier.

\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}.
Free normalization, plateau class, and support distinctions.

\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}.
Inherited scalar-completion signature only;
no new completion is attributed to that manuscript.

\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}.
Closed original obstruction and non-reopening boundary.

\bibitem{SF}
T.~P.~Sotiriou and V.~Faraoni,
``$f(R)$ theories of gravity,''
\emph{Reviews of Modern Physics}
\textbf{82} (2010), 451--497,
arXiv:0805.1726.
Comparison for metric curvature variation and scalar-prefactor
derivative terms; the present coefficient calculation is explicit.

\bibitem{BS}
R.~Bluhm and A.~Sehic,
``Noether identities in gravity theories with nondynamical
backgrounds and explicit spacetime symmetry breaking,''
\emph{Physical Review D}
\textbf{94} (2016), 104034,
arXiv:1610.02892.
Comparison for fixed-background Noether defects;
the selected cubic-only identity is derived here.

\end{thebibliography}

\end{document}