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% Scope:
% One question left open by XI-F: whether the free-shell class of the
% cubic Einstein insertion C_3(f,g) contains a representative in XI-E's
% polynomial microcausal class F_{mc,pol}. Four gates:
%   G1  Cauchy reduction of the onset (the boundary pairing B_Sigma);
%   G2  separation lemma (bulk equation ideal versus the class invariant);
%   G3  wavefront alternative (constructed / obstructed / unresolved);
%   G4  what a representative, or its absence, is not.
%
% Corrections to the production sheet (recorded in the text):
% 1. The class invariant is the trisolution W_u=(E_0 x E_0 x E_m)u,
%    not the Cauchy pairing alone. Bulk equation terms are exactly the
%    kernel of the solution map; the Cauchy pairing is the leading
%    onset coefficient of W_u in counter-propagating null directions.
% 2. The forbidden covector of the class is not a single +-dt in one
%    slot. Pairings with solutions see only null covectors. The
%    conormal +-dt at Sigma is the sum k_1+k_2 of two co-oriented null
%    covectors, one per metric slot; every representative inherits
%    that pair.
% 3. The witness is a pair of exact null plane waves on the gauge-fixed
%    shell, not a Cauchy-data expansion. No restriction theorem to
%    Sigma is used.
% 4. Transverse (Einstein-shell) plane-wave pairs do not detect XI-F's
%    data: the Bianchi identity makes the leading symbol conserved and
%    the total covector 2dt annihilates its 0nu-components. The
%    detecting polarization is pure trace, on the gauge-fixed shell
%    and off the Einstein shell.
% 5. XI-F's one-slot rewriting is the n=1 case of the kernel lemma
%    ker E_mu = (Box+mu^2)E'. It is a corollary, not the two-slot proof.
% 6. The constructed branch is characterized by an iff criterion on
%    WF(W_u) but is not instantiated: the criterion fails for XI-F's
%    data, and for all data whose boundary trace of beta pairs
%    nontrivially with the counter-propagating symbol.
%
% Compile hygiene: no redefinition of predefined macros; no
% theorem-header hacks; all referenced labels are defined.
%
% No ramp on J_3, no moved Cauchy surface, no interacting
% differential, no quantum retarded product, no loop, no state,
% no sharp-width limit, no Einstein-jet repair.
%
% No successful compilation, mechanized verification, independent
% referee review, or journal acceptance is asserted.
%=====================================================================

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\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\PP}{\mathcal P}
\newcommand{\RR}{\mathcal R}
\newcommand{\Lie}{\mathcal L}
\newcommand{\Ldens}{\mathcal L^{\mathrm{dens}}_\xi}
\newcommand{\ga}{\mathfrak g_U}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\Edist}{\mathcal E'}
\newcommand{\ER}{\mathsf E_0^{\mathrm R}}
\newcommand{\EA}{\mathsf E_0^{\mathrm A}}
\newcommand{\ERm}{\mathsf E_m^{\mathrm R}}
\newcommand{\EAm}{\mathsf E_m^{\mathrm A}}
\newcommand{\Ecaus}{\mathsf E_0}
\newcommand{\Ecausm}{\mathsf E_m}
\newcommand{\Emu}{\mathsf E_\mu}
\newcommand{\Ggr}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\Pfree}{P_m}
\newcommand{\Xone}{\mathsf X_1}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Atwo}{\mathcal A_2}
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\newcommand{\Cthree}{\mathcal C_3}
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\newcommand{\Vp}{\widetilde V_+}
\newcommand{\Vm}{\widetilde V_-}
\newcommand{\sxi}{s_\xi}
\newcommand{\zgf}{\zeta_{\mathrm{gf}}}
\newcommand{\Wsol}{\mathsf W}
\newcommand{\Ideal}{\mathcal I_{\mathrm{sol}}}
\newcommand{\qsym}{\mathsf q}
\newcommand{\Bsig}{B_\Sigma}
\newcommand{\cls}[1]{[#1]_{\mathrm{sol}}}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
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    \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
  \textbf{#1}\par\smallskip #2
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}

\title{\textbf{Landmark XI-G}\\[0.4em]
The free-shell class of the cubic Einstein kernel:\\
the onset covector is inherited by every representative}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
Landmark XI-F proved that the frozen kernel $u$ of the cubic Einstein
insertion $\Cthree(f,g)=\Aone[u_3;\phi]$ carries the timelike conormal
$\pm\dd t$ at the initial surface, and left open whether the
free-shell class of $\Cthree$ contains some other representative in
XI-E's polynomial microcausal class $\Fpol$. We decide that question
at fixed regulator width, with $\Ggr=-\PP\ER$ and the frozen
prescription unchanged.

\emph{Separation.} For compact kernels on the ambient Minkowski space,
vanishing on all triples of free solutions is equivalent to membership
in the bulk equation ideal $\Box_1\Edist+\Box_2\Edist+P_{m,3}\Edist$,
and the free-shell class is encoded faithfully by the trisolution
$\Wsol_u=(\Ecaus\otimes\Ecaus\otimes\Ecausm)u$. The class meets $\Fpol$
if and only if $\WF(\Wsol_u)$ avoids $\Vp^{\,3}\cup\Vm^{\,3}$. Pairings
with exact null plane waves are class invariants.

\emph{Obstruction.} For XI-F's admissible observation data, pairing
with the pure-trace counter-propagating plane waves
$\eta\,e^{i\lambda k_a\cdot y}$, $k_1+k_2=2\,\dd t$, gives
$-i\lambda\Lambda_f+O(1)$ with $\Lambda_f\neq0$, whereas every compact
kernel in $\Fpol$ decays rapidly in that direction, because
$(k_1,k_2)$ lies in $\Vp^{\,2}$. Hence $\cls{u}\cap\Fpol=\varnothing$:
every distribution reproducing $\Cthree(f,g)$ on $\Csol$ has a
wavefront point with two co-oriented null covectors, one in each
metric slot and zero in the scalar slot. These are the two halves of
XI-F's $\pm\dd t$. The leading coefficient is the Cauchy pairing
$\Bsig$, computed explicitly on the pure-trace sector. Transverse
(Einstein-shell) pairs are blind to the $00$-component by the Bianchi
identity; the witness lies on the gauge-fixed shell and off the
Einstein shell.

The cubic source is not ramped, the Cauchy surface is not moved, no
interacting differential, retarded quantum product, loop, state, or
sharp-width limit is constructed, and the empty Einstein jet is not
repaired. The classical ledger $-I+I+0=0$ of XI-F is untouched.
\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) {Detecting data};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Excluded free-shell class};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A later admitted insertion};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Changed-source admission does not put the old class back into the algebra.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A class exclusion is not undone by a new source. 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-XI-G}
\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) {Detecting data};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A later admitted insertion};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Changed-source admission does not put the old class back into the algebra.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A class exclusion is not undone by a new source. 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-XI-G}
\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) {Detecting data};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Excluded free-shell class};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A later admitted insertion};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Changed-source admission does not put the old class back into the algebra.};
\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-XI-G}
\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) {Detecting data};
\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) {A later admitted insertion};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Excluded free-shell class. Changed-source admission does not put the old class back into the algebra.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Detecting data is not A later admitted insertion. 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-XI-G}
\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: Detecting data};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Excluded free-shell class};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A later admitted insertion};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Changed-source admission does not put the old class back into the algebra.};
\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-XI-G}
\end{figure}

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


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



\scopebox{Release: four gates, worst ending governs}{%
The question decided here is
\[
 \cls{u}\cap\Fpol\overset{?}{\neq}\varnothing,\qquad
 \cls{u}=\{v:\ v-u\ \text{vanishes on all pairs of free metric
 configurations and every free scalar}\}.
\]
\begin{description}[leftmargin=1.2em,itemsep=2pt]
\item[G1 (Cauchy pairing): \textbf{Constructed}.]
$\Bsig$ is written as a bidifferential form in Cauchy data, printed
on the pure-trace sector, and identified as the leading onset
coefficient of the class invariant.
\item[G2 (separation): \textbf{Constructed}.]
Agreement on the free shell is exactly the bulk equation ideal; the
class is the trisolution $\Wsol_u$; the class meets $\Fpol$ iff
$\WF(\Wsol_u)$ avoids the two causal cones.
\item[G3 (alternative): \textbf{Proved obstructed}.]
For every frozen $g$, every $m$, and XI-F's observation data, every
compact distribution reproducing $\Cthree(f,g)$ on $\Csol$ has
covectors $\pm(k_1,k_2,0)$, $k_1,k_2$ null, co-oriented,
$k_1+k_2=2\,\dd t$. Quantifier: the copied class $\Fpol$, not every
bimodule.
\item[G4: Recorded.] No product, no differential, no loop.
\end{description}
Overall ending: \textbf{Proved obstructed} (Gate 3). The frozen-kernel
theorem of XI-F stands; its covector is inherited, in split form, by
the whole class.}

\tableofcontents

%=====================================================================
\section{Scope, the four gates, and the corrections}
\label{sec:scope}
%=====================================================================

XI-F released two gates for the cubic Einstein response and stopped at
a third \cite{XIF}. Gate F1 proved the frozen representative of
$\Cthree$ obstructed: zero Cauchy data for the second coefficient
together with an unswitched, unramped cubic source produce a sharp
onset $\theta(y^0)$ in a metric slot, and $\pm\dd t\in\WF$ at
$\Sigma$. Gates F2--F3 constructed the classical plateau sum with
ledger $-I+I+0=0$. The on-shell class was left open. This paper
decides it, and nothing else.

\begin{enumerate}[label=\textbf{G\arabic*.}]
\item Restrict both metric slots to the gauge-fixed free shell and
write the boundary pairing produced by $\theta(y^0)$ explicitly in
Cauchy data.
\item Prove that agreement with $u$ on the free shell is a bulk
equation term, identify the class invariant, and show that no bulk
term removes what the shell sees.
\item Either construct $\widetilde u\in\Fpol$ agreeing with $u$ on the
shell, or prove that every reproducing distribution meets a forbidden
covector, or record the missing estimate.
\item Say what the result is not.
\end{enumerate}

Three endings apply to each gate: \emph{Constructed}, \emph{Proved
obstructed}, \emph{Unresolved}. The worst governs the release.
A missing representative is not a no-go; obstruction of the frozen
kernel is not obstruction of the class. The theorems below are stated
with the quantifier written out.

\subsection{Six corrections to the production sheet}

\begin{enumerate}
\item The sheet proposed the Cauchy pairing $\Bsig$ as the remainder
to be tested. The class invariant is the trisolution
$\Wsol_u=(\Ecaus\otimes\Ecaus\otimes\Ecausm)u$; bulk equation terms are
exactly the kernel of the map $u\mapsto\Wsol_u$ on compact kernels
(Lemma~\ref{lem:kernel}). $\Bsig$ is the leading onset coefficient of
$\Wsol_u$ in the counter-propagating null directions
(Proposition~\ref{prop:planewave}), not the invariant itself.
\item The sheet expected the forbidden covector of the class to be
$\pm\dd t$ at $\Sigma$. Pairings with solutions see only null
covectors (Lemma~\ref{lem:transport}). The conormal $\pm\dd t$ is the
sum $k_1+k_2$ of two co-oriented null covectors, one per metric slot;
that pair is what every representative inherits
(Corollary~\ref{cor:covector}).
\item The witness is a pair of exact null plane waves on the
gauge-fixed shell, not a Cauchy-data expansion. The tools are XI-F's
(Definition~8.1.2, Proposition~8.1.3, Theorem~8.2.12 of
\cite{Hormander}) together with Theorems~8.2.4, 8.2.9, 8.2.13 and
8.3.1. No restriction theorem to $\Sigma$ is used.
\item Transverse plane-wave pairs, which lie on the Einstein shell
$A\gamma=0$, do not detect XI-F's data: the Bianchi identity makes the
leading symbol conserved, and the total covector $2\,\dd t$
annihilates its $0\nu$-components (Remark~\ref{rem:einstein}). The
detecting polarization is pure trace, $\epsilon=\eta$, which is on
XI-E's gauge-fixed shell and off the Einstein shell.
\item XI-F's one-slot rewriting of a boundary trace as a bulk
functional is the $n=1$ case of the kernel lemma
$\ker\Emu\cap\Edist=(\Box+\mu^2)\Edist$ (Corollary~\ref{cor:oneslot}).
It is a corollary of G2, not the two-slot proof.
\item The constructed branch of G3 is characterized by an iff
criterion on $\WF(\Wsol_u)$ (Theorem~\ref{thm:criterion}) but is not
instantiated: the criterion fails for XI-F's data
(Theorem~\ref{thm:obstruction}), and for all data whose boundary trace
of $\beta$ pairs nontrivially with the counter-propagating symbol
(Proposition~\ref{prop:general}).
\end{enumerate}

\subsection{Non-imports}

The cubic source $J_3$ is not multiplied by $\sig$ or by $g$. The
metric Cauchy surface stays at $\Sigma=\{t=0\}$. X-E's normalization
$-4\PP\ER$ and X-F's three-tree sum are not used. VII-C's hypotheses
(E1)--(E4) are not the admission interface; XI-E's
Definition of $\Fpol$ is. No reflected residue or finite part is
subtracted. The empty fixed-input Einstein jet
$\Jcal_g(\phi_m)=\varnothing$ remains XI-A's result \cite{XIA}. No
mixed loop is opened: the cubic source is not contracted against a
Hadamard line. No $\widehat s_{\mathrm{int}}$, antibracket, or master
identity is built. No $\eps\to0$ limit is taken.

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

\subsection{Conventions and ambient propagators}

\[
 \MT=[0,T]\times\R^3,\qquad \MTo=(0,T)\times\R^3,\qquad \Sigma=\{t=0\},
 \qquad M=\R^4,
\]
\[
 \eta=\diag(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta,\qquad
 \Pfree=\Box+m^2,\qquad m\geq0 .
\]
Indices are moved with $\eta$; parentheses include $\tfrac12$;
$k\cdot y=\eta_{\mu\nu}k^\mu y^\nu$. A covector $p$ is null if
$\eta^{\mu\nu}p_\mu p_\nu=0$; $\Vp$ ($\Vm$) is the closed cone of
future (past) directed causal covectors with zero included
(XI-E); for a null covector $p\ne0$, $p\in\Vp$ iff $p_0>0$. The ramp
$\sig$, switch $g$, plateau $U$, $\ga$, and the
observation data $f\in C_c^\infty(\MTo)$,
$\Dfset=J^-(\supp f)\cap\MT$, are XI-F's. Trace reversal
$\PP r=r-\tfrac12\eta\tr_\eta r$, $\PP^2=\id$;
$D^\mu r=\partial_\nu(\PP r)^{\nu\mu}$; $(Kv)_{\mu\nu}=\partial_\mu
v_\nu+\partial_\nu v_\mu$; $A=2G_{\mathrm{lin}}$;
\begin{equation}
 A+2D^*D=-\Box\PP,\qquad AK\xi=0,\qquad \Ggr=-\PP\ER .
 \label{eq:algebra}
\end{equation}

For $\mu\in\{0,m\}$ let $\Emu^{\mathrm R},\Emu^{\mathrm A}$ be the
Minkowski retarded and advanced inverses of $\Box+\mu^2$ on $M$; on
compact interior sources they restrict to XI-B's slab inverses
$\ER,\EA,\ERm,\EAm$ \cite{XIB}. Put
\begin{equation}
 \Emu:=\Emu^{\mathrm R}-\Emu^{\mathrm A},
 \qquad
 \Emu^{\mathsf T}=-\Emu,\qquad
 \Emu(x,y)=\mathcal E_\mu(x-y),\qquad
 (\Box+\mu^2)\mathcal E_\mu=0 ,
 \label{eq:Ecaus}
\end{equation}
which is XI-B's causal kernel $\delta_\mu$. The operators
$\Emu^{\mathrm R/A}$ act on distributions with past/future compact
support, satisfy $(\Box+\mu^2)\Emu^{\mathrm R}w=w$ and
$\Emu^{\mathrm R}(\Box+\mu^2)r=r$ for $r$ compactly supported,
$\supp\Emu^{\mathrm R}w\subset J^+(\supp w)$, and the retarded
solution with past-compact support of $(\Box+\mu^2)\Phi=w$ is unique
\cite[Ch.~3]{BGP}. For a distribution $w$ on $M^3$ whose support is
compact in the slot-$i$ direction, $i\in\{1,2,3\}$, the slot operators
$\mathsf E^{\mathrm R}_{\mu,i}$, $\mathsf E^{\mathrm A}_{\mu,i}$,
$\mathsf E_{\mu,i}$ are defined by transposition in slot $i$, for
instance $\pair{\mathsf E_{\mu,i}w}{\varphi}
=-\pair{w}{\mathsf E_{\mu,i}\varphi}$ for $\varphi\in C_c^\infty(M^3)$;
they commute with multiplication by smooth functions of, and
differential operators in, the other slots. We write
$\Ecaus$ for the metric slots and $\Ecausm$ for the scalar slot,
and $\Edist$ for compactly supported distributions on $M^3$ with the
slot type of $u$ below.

\subsection{XI-F's representation of the cubic kernel, copied}

\begin{quote}
$\Cthree(f,g)[\gamma,\phi]=\int_{[0,T]\times\R^3}\beta_{\mu\nu}(y)\,
J_3^{\mu\nu}[\gamma](y)\dd^4y$, with $\beta=\EA(\PP\alpha)$ and
$\supp\beta\subset\Dfset$. The degree-$(2,1)$ kernel has the form
\[
 u(y_1,y_2,x)=\int_{y^0\geq0}\widehat\beta_{\mu\nu}(y,x)\,
 j_3^{\mu\nu}(y;y_1,y_2)\dd^4y .
\]
The factor $\theta(y^0)$ is forced: $\ER$ truncates every source at
$t=0$, and the Einstein--Hilbert density carries neither $g$ nor
$\sig$.
\end{quote}
Here $\alpha=\alpha[f,g,\phi]$ is XI-F's smooth kernel of $\Aone$,
linear in $\phi$, and $j_3$ is the constant-coefficient bilinear local
kernel of $J_3=2\Hcal_2$, so that for all smooth $\gamma_1,\gamma_2,\phi$
on $M$,
\begin{equation}
 \pair{u}{\gamma_1\otimes\gamma_2\otimes\phi}
 =\int_{\{0\le t\le T\}}\beta[f,g,\phi]_{\mu\nu}(y)\,
   J_3^{\mu\nu}(\gamma_1,\gamma_2)(y)\dd^4y,
 \label{eq:upair}
\end{equation}
where $J_3(\cdot,\cdot)$ is the symmetric bilinear form with
$J_3[\gamma]=J_3(\gamma,\gamma)$. The kernel $u$ is compactly
supported on $M^3$, in
$\{y_1=y_2\in\Dfset\cap\{y^0\geq0\}\}\times\Dfset$; no spatial decay
of $\gamma_a$ is needed in \eqref{eq:upair}, exactly as in XI-F's
witness.

\begin{quote}
\textbf{Theorem F1 (XI-F).} For every frozen switch and every mass
there are admissible observation data for which $\pm\dd t$ lies in
the wavefront set of this kernel at $\Sigma$. The kernel leaves
$\Fpol$. The failure is not finite rank and not a loop.
\end{quote}
The data of that theorem, called the \emph{F1 data} below, are:
$t_g:=\min\{t:(t,\mathbf x)\in\supp g\}>0$; $f\in C_c^\infty(\MTo)$,
$f\geq0$, $f\not\equiv0$, $\supp f\subset(0,t_g)\times\R^3$,
$\cos(mt)>0$ on $\supp f$; $\phi=\phi_m$ with $\phi_m=\sin(mt)/m$ for
$m>0$ and $\phi_m=t$ for $m=0$. XI-F's Step~1 computes, for these data,
\begin{equation}
 \beta^{ij}=0,\qquad
 B(s):=\int_{\R^3}\beta^{00}(s,\mathbf x)\dd^3x,\qquad
 \Lambda_f:=B(0)<0,\qquad
 \int_{\R^3}\beta^{0i}(s,\mathbf x)\dd^3x=0,
 \label{eq:F1data}
\end{equation}
the last because $\beta^{0i}=\EA(-\tfrac12\sig\partial_ib^0)$ with
$b^0$ of compact spatial support per slice. $B$ is smooth on $[0,T]$
and vanishes near $T$.

\begin{quote}
\textbf{The open remark (XI-F).} On the free shell, a single-slot
boundary trace equals a bulk functional with a source flat at
$\Sigma$, hence has a smooth representative. The boundary term
carried by $u$ is a bilinear form in the Cauchy data of both metric
slots, weighted by $\beta|_\Sigma$. Whether it has a representative
satisfying the microcausal exclusion after both slots are restricted
to free data is not proved.
\end{quote}

\subsection{XI-E's class and admission, copied}

\begin{definition}[Polynomial microcausal domain]
\label{def:domain}
$\Fpol$ consists of finite graded polynomials
$F(\Phi)=\sum_n\pair{u_n}{\Phi^{\otimes n}}$ on ambient configurations
on $M$, with compact, graded symmetric distributional kernels $u_n$ on
$M^n$ satisfying
\begin{equation}
 \WF(u_n)\cap\bigl(\Vp^{\,n}\cup\Vm^{\,n}\bigr)=\varnothing ,
 \label{eq:mc}
\end{equation}
where a total zero covector is never a wavefront point and finite
field indices do not change the criterion. The free shell $\Csol$ is
\begin{equation}
 -\Box\PP\gamma=0,\qquad b=D\gamma,\qquad \Pfree\phi=0,\qquad
 \Box c=\Box\bar c=0 ,
 \label{eq:shell}
\end{equation}
on $M$; smooth slab solutions extend to $M$ by their Cauchy data.
Admission is $Q_H(F)=[F|_{\Csol}]\in\Ealg$. XI-E admitted $\Bcal_f$
and $\mathcal T$; it did not admit $\Cthree$.
\end{definition}

Since $\PP$ is an invertible constant matrix, $-\Box\PP\gamma=0$ is
$\Box\gamma=0$ componentwise: every $\gamma=\epsilon\,e^{ik\cdot y}$
with constant symmetric $\epsilon$ and null $k$ is a free metric
configuration, for every polarization $\epsilon$.

\subsection{The cubic source and the ledger, retained}

$S_{\mathrm{EH}}=-\frac{2}{\kappa^2}\int\sqrt{-q}R[q]$, $q=\eta+\kappa\gamma$,
$\sqrt{-q}\,G^{\mu\nu}(q)=\kappa\Hcal_1+\kappa^2\Hcal_2+O(\kappa^3)$,
$A\gamma=2\Hcal_1$, $J_3=2\Hcal_2$, with XI-E's finite prescription
for $\Hcal_2$ \cite{XIE}. Two structural facts are used:
$J_3$ is a bilinear differential polynomial with constant
coefficients, every monomial of which carries exactly two
derivatives; and the contracted Bianchi identity
$\nabla_\mu G^{\mu\nu}=0$, written as
$\partial_\mu(\sqrt{-q}G^{\mu\nu})+\sqrt{-q}\,\Gamma^\nu_{\mu\rho}G^{\mu\rho}=0$,
gives at order $\kappa^2$
\begin{equation}
 \partial_\mu J_3^{\mu\nu}[\gamma]=-\Gamma_1{}^\nu_{\mu\rho}[\gamma]\,A^{\mu\rho}\gamma,
 \qquad
 \Gamma_1{}^\rho_{\mu\nu}=\tfrac12\eta^{\rho\sigma}
 (\partial_\mu\gamma_{\nu\sigma}+\partial_\nu\gamma_{\mu\sigma}-\partial_\sigma\gamma_{\mu\nu}),
 \label{eq:bianchi}
\end{equation}
consistent with XI-E's gravitational Noether coefficient.

XI-F's classical ledger is retained and not re-proved:
\[
 u_3[\gamma+K\xi]-u_3[\gamma]
 =\Lie_\xi\gamma+\tfrac12\Lie_\xi K\xi+K\zgf+\PP\ER\Ldens(A\gamma),
 \qquad \zgf=-\ER D\sxi,
\]
and, when $A\gamma=0$ on $U$,
$\Atwothree[\gamma+K\xi]-\Atwothree[\gamma]=\Aone[K\zgf;\phi_0]$, zero
under the causal plateau condition, with witness value $-I+I+0=0$.
Nothing below changes a line of it.

%=====================================================================
\section{The question, written so it cannot collapse}
\label{sec:question}
%=====================================================================

\begin{definition}[Free-shell class]
\label{def:class}
Let $u$ be the frozen kernel of $\Cthree(f,g)$. A compactly supported
distribution $v$ on $M^3$, with two symmetric-tensor slots and one
scalar slot and symmetric in the tensor slots, belongs to $\cls{u}$ if
\[
 \pair{v-u}{\gamma_1\otimes\gamma_2\otimes\phi}=0
 \quad\text{for all smooth }\gamma_1,\gamma_2\text{ with }\Box\gamma_a=0
 \text{ and all smooth }\phi\text{ with }\Pfree\phi=0 .
\]
Real and complex solutions are equivalent here by bilinearity.
We say that $\cls{u}$ \emph{meets $\Fpol$} if some $v\in\cls{u}$
satisfies \eqref{eq:mc} with $n=3$.
\end{definition}

\begin{lemma}[Reduction to one kernel]
\label{lem:reduction}
If $F\in\Fpol$ satisfies $F|_{\Csol}=\Cthree(f,g)|_{\Csol}$, then
$\cls{u}$ meets $\Fpol$. Conversely, if $v\in\cls{u}$ satisfies
\eqref{eq:mc}, then $F=\pair{v}{\gamma\otimes\gamma\otimes\phi}\in\Fpol$
and $F|_{\Csol}=\Cthree(f,g)|_{\Csol}$.
\end{lemma}

\begin{proof}
Restrict $F$ to configurations in $\Csol$ with $c=\bar c=0$, which is
allowed since zero is a free ghost; $\Cthree$ has no ghost dependence,
so the ghost-free part of $F$ already reproduces it. On $\Csol$,
substitute $b=D\gamma$ in every $b$-slot: each kernel becomes
$(\cdots\otimes D^{\mathsf T}\otimes\cdots)u_n$, compact, with
wavefront set not enlarged by a differential operator
\cite[(8.1.11)]{Hormander}, hence still satisfying \eqref{eq:mc}.
The shell is a linear space closed under the independent scalings
$\gamma\mapsto s\gamma$, $\phi\mapsto r\phi$; comparing the
coefficient of $s^2r$ isolates a single symmetric kernel $v$ of slot
type $(\gamma,\gamma,\phi)$ with
$\pair{v}{\gamma\otimes\gamma\otimes\phi}=\Cthree(f,g)[\gamma,\phi]$ on
the shell. Polarization gives $v\in\cls{u}$. The converse is
Definition~\ref{def:domain}.
\end{proof}

The question is therefore
\begin{equation}
 \boxed{\ \cls{u}\cap\Fpol\ \overset{?}{\neq}\ \varnothing\ }
 \label{eq:question}
\end{equation}
Three statements that are \emph{not} this question: $u\notin\Fpol$,
already XI-F's Theorem~F1; the one-slot rewriting of a boundary trace,
true for one slot and not a theorem for two; and the existence of
some larger bimodule containing a singular kernel, which XI-E's
correction leaves open and which this paper does not touch.

%=====================================================================
\section{G1: Cauchy reduction of the onset}
\label{sec:G1}
%=====================================================================

\subsection{The onset family and the boundary pairing}

\begin{definition}[Onset family, boundary pairing]
\label{def:onset}
For smooth $\gamma_1,\gamma_2,\phi$ on $M$ and $0\le s<T$ put
\[
 F_s(\gamma_1,\gamma_2;\phi):=\int_{\{s\le t\le T\}}
 \beta[f,g,\phi]_{\mu\nu}(y)\,J_3^{\mu\nu}(\gamma_1,\gamma_2)(y)\dd^4y,
 \qquad F_0=\pair{u}{\gamma_1\otimes\gamma_2\otimes\phi},
\]
and
\begin{equation}
 \Bsig(\gamma_1,\gamma_2;\phi):=-\frac{\dd}{\dd s}\Big|_{s=0^+}F_s
 =\int_{\R^3}\beta_{\mu\nu}(0,\mathbf x)\,
   J_3^{\mu\nu}(\gamma_1,\gamma_2)(0,\mathbf x)\dd^3x .
 \label{eq:Bsig}
\end{equation}
\end{definition}

\begin{lemma}[Cauchy reduction]
\label{lem:cauchyred}
If $\Box\gamma_1=\Box\gamma_2=0$, then $J_3^{\mu\nu}(\gamma_1,\gamma_2)|_\Sigma
=Q^{\mu\nu}(j_1,j_2)$ is a bidifferential form in the Cauchy data
$j_a=(\gamma_a|_\Sigma,\partial_t\gamma_a|_\Sigma)$, obtained from the jet
expression of $J_3$ by replacing every $\partial_t^2\gamma_a$ by
$\Delta\gamma_a$. Hence $\Bsig=\int_\Sigma\beta|_\Sigma\cdot Q(j_1,j_2)$,
with the scalar slot entering only through the smooth weight
$\beta|_\Sigma=\beta[f,g,\phi]|_\Sigma$, linearly.
\end{lemma}

\begin{proof}
$J_3$ contains at most second derivatives, and on the shell
$\partial_t^2\gamma_a=\Delta\gamma_a$. Mixed derivatives
$\partial_t\partial_i\gamma_a$ are spatial derivatives of the second
datum. The scalar enters $\beta$ linearly through $\alpha[f,g,\phi]$.
\end{proof}

The scalar slot is a spectator: it carries no derivative of $\phi$
that could cancel a metric-slot covector, and in every wavefront
statement below its covector is zero.

\subsection{The pure-trace sector of the cubic source}

The witness of Section~\ref{sec:G3} uses pure-trace configurations
$\gamma=\psi\eta$. Their cubic source is closed-form.

\begin{lemma}[Cubic coefficient on the pure-trace sector]
\label{lem:puretrace}
For smooth $\psi$ and $\gamma=\psi\eta$,
\begin{align}
 \Hcal_1^{\mu\nu}[\psi\eta]&=\eta^{\mu\nu}\Box\psi-\partial^\mu\partial^\nu\psi,
 \label{eq:H1trace}\\
 \Hcal_2^{\mu\nu}[\psi\eta]&=\tfrac32\partial^\mu\psi\partial^\nu\psi
 +\psi\,\partial^\mu\partial^\nu\psi
 -\tfrac34\eta^{\mu\nu}\partial_\rho\psi\partial^\rho\psi
 -\eta^{\mu\nu}\psi\Box\psi .
 \label{eq:H2trace}
\end{align}
Consequently, for the polarized source,
\begin{equation}
 J_3^{\mu\nu}(\psi_1\eta,\psi_2\eta)
 =\tfrac32(\partial^\mu\psi_1\partial^\nu\psi_2+\partial^\mu\psi_2\partial^\nu\psi_1)
 +\psi_1\partial^\mu\partial^\nu\psi_2+\psi_2\partial^\mu\partial^\nu\psi_1
 -\tfrac32\eta^{\mu\nu}\partial_\rho\psi_1\partial^\rho\psi_2
 -\eta^{\mu\nu}(\psi_1\Box\psi_2+\psi_2\Box\psi_1).
 \label{eq:J3trace}
\end{equation}
\end{lemma}

\begin{proof}
$q=\Omega^2\eta$ with $\Omega^2=1+\kappa\psi$ is conformally flat. For
$\tilde g=\Omega^2 g$ in four dimensions with $g$ flat and
$w=\ln\Omega$, the standard conformal transformation of the Ricci
tensor gives
\[
 \tilde R_{\mu\nu}=-2\partial_\mu\partial_\nu w-\eta_{\mu\nu}\Box w
 +2\partial_\mu w\partial_\nu w-2\eta_{\mu\nu}\partial_\rho w\partial^\rho w,
 \qquad
 \tilde R=\Omega^{-2}\bigl(-6\Box w-6\partial_\rho w\partial^\rho w\bigr),
\]
hence
$\tilde G_{\mu\nu}=-2\partial_\mu\partial_\nu w+2\eta_{\mu\nu}\Box w
+2\partial_\mu w\partial_\nu w+\eta_{\mu\nu}\partial_\rho w\partial^\rho w$.
Since $q^{\mu\nu}=\Omega^{-2}\eta^{\mu\nu}$ and $\sqrt{-q}=\Omega^4$,
$\sqrt{-q}\,G^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}\tilde G_{\alpha\beta}$
with no further $\Omega$ factor. Expand
$w=\tfrac12\ln(1+\kappa\psi)=\tfrac\kappa2\psi-\tfrac{\kappa^2}{4}\psi^2+O(\kappa^3)$,
so that
$\partial_\mu w=\tfrac\kappa2\partial_\mu\psi-\tfrac{\kappa^2}2\psi\partial_\mu\psi$ and
$\partial_\mu\partial_\nu w=\tfrac\kappa2\partial_\mu\partial_\nu\psi
-\tfrac{\kappa^2}2(\partial_\mu\psi\partial_\nu\psi+\psi\partial_\mu\partial_\nu\psi)$,
up to $O(\kappa^3)$. The order-$\kappa$ coefficient is
\eqref{eq:H1trace}; the order-$\kappa^2$ coefficient is
$(\partial\psi\partial\psi+\psi\partial\partial\psi)-\eta(|\partial\psi|^2+\psi\Box\psi)
+\tfrac12\partial\psi\partial\psi+\tfrac14\eta|\partial\psi|^2$, which is
\eqref{eq:H2trace}. Polarizing $2\Hcal_2$ gives \eqref{eq:J3trace}.

Two checks. First, by \eqref{eq:algebra}, $\PP(\psi\eta)=-\psi\eta$,
$D^\mu(\psi\eta)=-\partial^\mu\psi$, $D^*D(\psi\eta)=\partial\partial\psi-\tfrac12\eta\Box\psi$,
so $A(\psi\eta)=\Box\psi\,\eta-2\partial\partial\psi+\eta\Box\psi=2\Hcal_1[\psi\eta]$,
as required by $A\gamma=2\Hcal_1$. Second, XI-F's witness $\gamma=t\eta$
gives $q=(1+\kappa t)\eta$, a spatially flat Robertson--Walker metric
in conformal time with $a^2=1+\kappa t$, for which $G_{00}=3(a'/a)^2$;
its $\kappa^2$ coefficient is $\tfrac34$, and \eqref{eq:H2trace} gives
$\Hcal_2^{00}[t\eta]=\tfrac32-\tfrac34=\tfrac34$.
\end{proof}

\begin{corollary}[The Cauchy pairing on the pure-trace sector]
\label{cor:Bsigtrace}
Let $\gamma_a=\psi_a\eta$ with $\Box\psi_a=0$ and Cauchy data
$(\psi_a,\dot\psi_a)|_\Sigma$. Then $\Bsig=\int_{\R^3}
[\beta_{00}J^{00}+2\beta_{0i}J^{0i}+\beta_{ij}J^{ij}](0,\mathbf x)\dd^3x$ with
\begin{align*}
 J^{00}&=\tfrac32\dot\psi_1\dot\psi_2+\tfrac32\nabla\psi_1\cdot\nabla\psi_2
 +\psi_1\Delta\psi_2+\psi_2\Delta\psi_1,\\
 J^{0i}&=-\tfrac32(\dot\psi_1\partial_i\psi_2+\dot\psi_2\partial_i\psi_1)
 -(\psi_1\partial_i\dot\psi_2+\psi_2\partial_i\dot\psi_1),\\
 J^{ij}&=\tfrac32(\partial_i\psi_1\partial_j\psi_2+\partial_i\psi_2\partial_j\psi_1)
 +\psi_1\partial_i\partial_j\psi_2+\psi_2\partial_i\partial_j\psi_1
 +\tfrac32\delta^{ij}(\dot\psi_1\dot\psi_2-\nabla\psi_1\cdot\nabla\psi_2),
\end{align*}
all evaluated at $t=0$. For the F1 data the $\beta_{ij}$ term is
absent.
\end{corollary}

\begin{proof}
Insert $\Box\psi_a=0$ and $\partial_t^2\psi_a=\Delta\psi_a$ into
\eqref{eq:J3trace}, with $\partial^0=\partial_t$, $\partial^i=-\partial_i$,
$\eta^{ij}=-\delta^{ij}$.
\end{proof}

For the spatially homogeneous sector $\gamma_{0\mu}=0$,
$\gamma_{ij}=\gamma_{ij}(t)$, XI-F's Lemma gives
$J_3^{0i}=0$ and $J_3^{00}=q(\dot\gamma,\dot\gamma)$,
$q(a,b)=\tfrac14(\tr a\tr b-a_{ij}b_{ij})$, so there
$\Bsig=\int_\Sigma\beta_{00}\,q(\dot\gamma_1,\dot\gamma_2)$. The full
twenty-component form of $Q$ is XI-E's $\Hcal_2$ with
$\partial_t^2\mapsto\Delta$; the witness needs only the pure-trace
sector printed above.

\subsection{The bilinear symbol and its counter-propagating values}

\begin{definition}[Bilinear symbol]
\label{def:symbol}
For constant symmetric $\epsilon_1,\epsilon_2$ and vectors $k_1,k_2$
let $\qsym^{\mu\nu}(\epsilon_1,k_1;\epsilon_2,k_2)$ be defined by
\[
 J_3^{\mu\nu}\bigl(\epsilon_1e^{ik_1\cdot y},\epsilon_2e^{ik_2\cdot y}\bigr)
 =\qsym^{\mu\nu}(\epsilon_1,k_1;\epsilon_2,k_2)\,e^{i(k_1+k_2)\cdot y}.
\]
\end{definition}
Since every monomial of $J_3$ has exactly two derivatives,
$\qsym$ is real, symmetric under $1\leftrightarrow2$, polynomial, and
homogeneous of degree two in $(k_1,k_2)$ jointly:
$\qsym(\epsilon_1,\lambda k_1;\epsilon_2,\lambda k_2)=\lambda^2\qsym(\epsilon_1,k_1;\epsilon_2,k_2)$.
The operator $Q$ of Lemma~\ref{lem:cauchyred} is $\qsym$ with
$k_a\mapsto-i\partial$ acting on $\gamma_a$, followed by
$\partial_t^2\mapsto\Delta$.

\begin{lemma}[Pure-trace symbol on null pairs]
\label{lem:qvalues}
For null $k_1,k_2$,
\begin{equation}
 \qsym^{\mu\nu}(\eta,k_1;\eta,k_2)
 =-\tfrac32(k_1^\mu k_2^\nu+k_2^\mu k_1^\nu)-(k_1^\mu k_1^\nu+k_2^\mu k_2^\nu)
 +\tfrac32\eta^{\mu\nu}(k_1\cdot k_2),
 \label{eq:qtrace}
\end{equation}
and it satisfies the Bianchi relation
$(k_1+k_2)_\mu\qsym^{\mu\nu}=-(k_1\cdot k_2)(k_1+k_2)^\nu$.
For the counter-propagating pair $k_1^\mu=(1,\mathbf n)$,
$k_2^\mu=(1,-\mathbf n)$, $|\mathbf n|=1$, one has $k_1+k_2=2\,\dd t$
as covectors, $k_1\cdot k_2=2$, and
\begin{equation}
 \qsym^{00}=-2,\qquad \qsym^{0i}=0,\qquad
 \qsym^{ij}=n^in^j-3\delta^{ij}.
 \label{eq:qcounter}
\end{equation}
\end{lemma}

\begin{proof}
Insert $\psi_a=e^{ik_a\cdot y}$ into \eqref{eq:J3trace}:
$\partial^\mu\psi_1\partial^\nu\psi_2=-k_1^\mu k_2^\nu e^{i(k_1+k_2)\cdot y}$,
$\psi_1\partial^\mu\partial^\nu\psi_2=-k_2^\mu k_2^\nu e^{i(k_1+k_2)\cdot y}$,
$\Box\psi_a=0$. This is \eqref{eq:qtrace}. Contracting with
$(k_1+k_2)_\mu$ and using $k_a^2=0$, so that
$(k_1+k_2)\cdot k_a=k_1\cdot k_2=:\sigma$, gives
$-\tfrac32\sigma(k_1+k_2)^\nu-\sigma(k_1+k_2)^\nu+\tfrac32\sigma(k_1+k_2)^\nu
=-\sigma(k_1+k_2)^\nu$. Independently, \eqref{eq:bianchi} polarized
reads $\partial_\mu J_3^{\mu\nu}(\gamma_1,\gamma_2)=-\tfrac12[\Gamma_1{}^\nu_{\mu\rho}[\gamma_1]
A^{\mu\rho}\gamma_2+(1\leftrightarrow2)]$; for $\gamma_a=\psi_a\eta$ with
$\Box\psi_a=0$, $A\gamma_a=-2\partial\partial\psi_a$ and
$\Gamma_1{}^\nu_{\mu\rho}[\psi_1]A^{\mu\rho}\gamma_2=-2\partial_\mu\psi_1\partial^\mu\partial^\nu\psi_2$,
which on plane waves is $2i\sigma k_2^\nu e^{i(k_1+k_2)\cdot y}$; the
right side becomes $-i\sigma(k_1+k_2)^\nu e^{i(k_1+k_2)\cdot y}$ and the
left side $i(k_1+k_2)_\mu\qsym^{\mu\nu}e^{i(k_1+k_2)\cdot y}$. The two
computations agree. For the counter-propagating pair,
$k_1^0k_2^i+k_2^0k_1^i=0$, $k_1^0k_1^i+k_2^0k_2^i=0$,
$k_1^ik_2^j+k_2^ik_1^j=-2n^in^j$, $k_1^ik_1^j+k_2^ik_2^j=2n^in^j$,
and \eqref{eq:qcounter} follows.
\end{proof}

\begin{remark}[Einstein-shell pairs are blind to the $0\nu$ components]
\label{rem:einstein}
If both polarizations are de Donder transverse,
$k_a\cdot\PP\epsilon_a=0$, then $A(\epsilon_ae^{ik_a\cdot y})=0$ by
\eqref{eq:algebra}, and \eqref{eq:bianchi} gives
$(k_1+k_2)_\mu\qsym^{\mu\nu}=0$. For $k_1+k_2=2\,\dd t$ this is
$\qsym^{0\nu}=0$: linearized Einstein pairs see only $\beta_{ij}$,
which vanishes for the F1 data by \eqref{eq:F1data}. The pure-trace
polarization $\epsilon=\eta$ has $k\cdot\PP\eta=-k\neq0$; it lies on
XI-E's gauge-fixed shell $\Box\gamma=0$ and off the Einstein shell.
This is XI-A's distinction between the two shells, met at the level of
the detector.
\end{remark}

Gate~1 ends at \textbf{Constructed}: $\Bsig$ is explicit in Cauchy data
on the sectors used, the scalar slot is a spectator, and
Proposition~\ref{prop:planewave} below identifies $\Bsig$ as the
leading onset coefficient of the class invariant.

%=====================================================================
\section{G2: the separation lemma}
\label{sec:G2}
%=====================================================================

Throughout,
\begin{equation}
 \Ideal:=\Box_1\Edist+\Box_2\Edist+P_{m,3}\Edist,
 \qquad
 \Wsol_w:=\mathsf E_{0,1}\mathsf E_{0,2}\mathsf E_{m,3}\,w
 =(\Ecaus\otimes\Ecaus\otimes\Ecausm)w\quad(w\in\Edist),
 \label{eq:Ideal}
\end{equation}
the \emph{bulk equation ideal} and the \emph{trisolution} of $w$.
The trisolution is a distribution on $M^3$ solving
$\Box_1\Wsol_w=\Box_2\Wsol_w=P_{m,3}\Wsol_w=0$, with support in
$J(K_1)\times J(K_2)\times J(K_3)$, $K_i$ the projection of $\supp w$
to slot $i$ and $J=J^+\cup J^-$; its support is compact in each slot
over every bounded time interval.

\subsection{Solutions and their pairings}

\begin{lemma}[Spatially compact solutions]
\label{lem:spcompact}
Let $\mu\in\{0,m\}$, $P=\Box+\mu^2$, and let $\chi\in C^\infty(\R)$ with
$\chi(t)=0$ for $t\le-T_0$ and $\chi(t)=1$ for $t\ge T_0$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item For $w\in C_c^\infty(M)$, $\Emu w$ is a smooth solution of
$P\Phi=0$ with $\supp\Emu w\subset J(\supp w)$, hence spatially
compact.
\item Every smooth solution $\Phi$ of $P\Phi=0$ with
$\supp\Phi\subset J(K)$, $K$ compact, is $\Phi=\Emu w$ with
$w:=P(\chi\Phi)\in C_c^\infty(M)$.
\item For every compact $K\subset M$ and every smooth solution $\Phi$
there is a spatially compact smooth solution $\Phi_K$ with
$\Phi_K=\Phi$ on a neighbourhood of $K$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) is the support property of $\Emu^{\mathrm R/A}$ and
$P\Emu^{\mathrm R/A}w=w$. (b): $P(\chi\Phi)=[P,\chi]\Phi$ is supported
in $\{|t|\le T_0\}\cap J(K)$, which is compact. $\chi\Phi$ has
past-compact support and solves $P(\chi\Phi)=w$, so by uniqueness of
the retarded solution $\Emu^{\mathrm R}w=\chi\Phi$; likewise
$\Emu^{\mathrm A}w=(\chi-1)\Phi$. Subtracting, $\Emu w=\Phi$.
(c): choose a ball $B\subset\Sigma$ whose domain of dependence
contains a neighbourhood of $K$, and $\vartheta\in C_c^\infty(\R^3)$
with $\vartheta=1$ on $B$. Let $\Phi_K$ solve $P\Phi_K=0$ with Cauchy
data $\vartheta\cdot(\Phi,\partial_t\Phi)|_\Sigma$. By finite
propagation \cite{XA,BGP}, $\Phi_K=\Phi$ on the domain of dependence of
$B$ and $\supp\Phi_K\subset J(\supp\vartheta)$.
\end{proof}

\begin{lemma}[Pairings with solutions are the trisolution]
\label{lem:pairings}
For $w\in\Edist$ the following are equivalent.
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\pair{w}{\gamma_1\otimes\gamma_2\otimes\phi}=0$ for all smooth
solutions $\Box\gamma_1=\Box\gamma_2=0$, $\Pfree\phi=0$ on $M$.
\item The same for all spatially compact smooth solutions.
\item $\Wsol_w=0$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i)$\Rightarrow$(ii) is trivial. (ii)$\Rightarrow$(i): $w$ is
compactly supported, so the pairing depends only on the solutions
near the slot projections $K_i$ of $\supp w$; replace each solution
by the spatially compact one of Lemma~\ref{lem:spcompact}(c).
(ii)$\Leftrightarrow$(iii): by Lemma~\ref{lem:spcompact}(a),(b) the
spatially compact solutions are exactly $\Ecaus w_1$, $\Ecaus w_2$,
$\Ecausm w_3$ with $w_i\in C_c^\infty(M)$, and by three slot
transpositions,
\[
 \pair{w}{\Ecaus w_1\otimes\Ecaus w_2\otimes\Ecausm w_3}
 =(-1)^3\pair{\Wsol_w}{w_1\otimes w_2\otimes w_3}.
\]
Finite sums of products $w_1\otimes w_2\otimes w_3$ are dense in
$C_c^\infty(M^3)$, so (ii) holds iff $\Wsol_w=0$.
\end{proof}

\subsection{The kernel of the solution map is the bulk ideal}

\begin{lemma}[Kernel lemma]
\label{lem:kernel}
Let $\chi_i$ denote the function $\chi$ of Lemma~\ref{lem:spcompact}
in the time variable of slot $i$, with $T_0$ chosen so that
$\supp w\subset\{|t_i|<T_0,\ i=1,2,3\}$. Then for every $w\in\Edist$,
\begin{equation}
 w-\Box_1\Box_2P_{m,3}\bigl[\chi_1\chi_2\chi_3\,\Wsol_w\bigr]\in\Ideal,
 \qquad
 \Box_1\Box_2P_{m,3}\bigl[\chi_1\chi_2\chi_3\,\Wsol_w\bigr]\in\Edist .
 \label{eq:decomp}
\end{equation}
Consequently
\begin{equation}
 \{w\in\Edist:\ \Wsol_w=0\}=\Ideal .
 \label{eq:kernel}
\end{equation}
\end{lemma}

\begin{proof}
\emph{$\Ideal\subset\ker$.} For $r\in\Edist$,
$\mathsf E^{\mathrm R}_{\mu,i}(\Box_i+\mu^2)r=r=\mathsf E^{\mathrm A}_{\mu,i}(\Box_i+\mu^2)r$,
so $\mathsf E_{\mu,i}(\Box_i+\mu^2)r=0$; the other slot operators then
give $\Wsol_{(\Box_i+\mu^2)r}=0$.

\emph{One slot.} For $w$ compactly supported in the slot-$i$
direction, with $P_i$ the slot-$i$ operator ($\Box_i$ or $P_{m,3}$),
write
\[
 \chi_i\mathsf E_iw=\mathsf E_i^{\mathrm R}w-r_i,\qquad
 r_i:=(1-\chi_i)\mathsf E_i^{\mathrm R}w+\chi_i\mathsf E_i^{\mathrm A}w .
\]
The support of $(1-\chi_i)\mathsf E_i^{\mathrm R}w$ lies in
$J^+(\supp w)\cap\{t_i\le T_0\}$ in slot $i$, and in the slot
projections of $\supp w$ otherwise; it is compact. Likewise for
$\chi_i\mathsf E_i^{\mathrm A}w$. Hence
\begin{equation}
 T_iw:=P_i\bigl[\chi_i\mathsf E_iw\bigr]=w-P_ir_i,\qquad
 r_i\ \text{compactly supported},
 \label{eq:Ti}
\end{equation}
so $T_iw\in w+P_i\Edist$, $T_iw$ is compactly supported, and $T_iw$
depends on $w$ only through $\mathsf E_iw$.

\emph{Three slots.} Apply \eqref{eq:Ti} successively:
$w\equiv T_1w\equiv T_2T_1w\equiv T_3T_2T_1w$ modulo $\Ideal$, each
step legitimate because the previous output is compactly supported.
Since $\mathsf E_j$, $\chi_j$ and $P_j$ commute with $\mathsf E_i$,
$\chi_i$ and $P_i$ for $i\ne j$ (they act on different variables, and
$\chi_i\mathsf E_iw$ is compactly supported in every slot other than
$i$, so the slot-$j$ operators are defined on it),
\[
 T_3T_2T_1w=P_{m,3}\bigl[\chi_3\mathsf E_{m,3}\Box_2[\chi_2\mathsf E_{0,2}\Box_1[\chi_1\mathsf E_{0,1}w]]\bigr]
 =\Box_1\Box_2P_{m,3}\bigl[\chi_1\chi_2\chi_3\,\Wsol_w\bigr],
\]
which proves \eqref{eq:decomp}. If $\Wsol_w=0$ the bracket vanishes
and $w\in\Ideal$, proving \eqref{eq:kernel}.
\end{proof}

\begin{corollary}[The one-slot rewriting is the case $n=1$]
\label{cor:oneslot}
$\ker\Emu\cap\Edist(M)=(\Box+\mu^2)\Edist(M)$, and every $w\in\Edist(M)$
agrees on solutions with $(\Box+\mu^2)[\chi\Emu w]$, compactly
supported, with $\WF\subset\WF(\Emu w)$. For a boundary trace
$w=a\,\delta(t)$, $a\in C_c^\infty(\R^3)$, $\Emu w$ is the smooth
solution with Cauchy data $(0,a)$, so the representative is smooth and
flat at $\Sigma$ after the cutoff. This is XI-F's remark. It does not
address two slots, where $\Wsol_u$ is a bisolution in the two metric
variables and its Cauchy data on $\Sigma\times\Sigma$ are not
smooth.
\end{corollary}

\subsection{Covector transport and the criterion}

\begin{lemma}[Covector transport]
\label{lem:transport}
For $w\in\Edist$, every point of $\WF(\Wsol_w)$ has the form
$(z_1,\xi_1;z_2,\xi_2;z_3,\xi_3)$ with
$(y_1,\xi_1;y_2,\xi_2;y_3,\xi_3)\in\WF(w)$ for some $(y_1,y_2,y_3)$,
and each $\xi_i$ null or zero.
\end{lemma}

\begin{proof}
By \eqref{eq:Ecaus}, $\Emu$ is the pullback of $\mathcal E_\mu$ under
the submersion $(z,y)\mapsto z-y$, so by \cite[Theorem~8.2.4]{Hormander}
$\WF(\Emu)\subset\{(z,\xi;y,-\xi):\xi\in\WF_{z-y}(\mathcal E_\mu)\}$,
and by \cite[Theorem~8.3.1]{Hormander} and $(\Box+\mu^2)\mathcal E_\mu=0$
every such $\xi$ is null and nonzero. The kernel of
$w\mapsto\Wsol_w$ on $M^3\times M^3$ is the tensor product
$\Ecaus\otimes\Ecaus\otimes\Ecausm$, whose wavefront set
\cite[Theorem~8.2.9]{Hormander} consists of triples in which each
factor is either in $\WF(\Emu)$ or carries zero covectors in both of
its variables. In particular the $z$-covector is zero exactly when the
$y$-covector is zero in each factor, so the kernel has empty
$\WF_{M^3}$ and $\WF'_{M^3}$, and \cite[Theorem~8.2.13]{Hormander}
gives $\WF(\Wsol_w)\subset\WF'(\Ecaus\otimes\Ecaus\otimes\Ecausm)\circ\WF(w)$,
which is the stated set: the covector triple is transported
unchanged and each entry is null or zero.
\end{proof}

\begin{theorem}[Separation and the admission criterion]
\label{thm:criterion}
Let $u$ be the frozen kernel.
\begin{enumerate}[label=\textup{(\alph*)}]
\item $v\in\cls{u}$ if and only if $v-u\in\Ideal$ if and only if
$\Wsol_v=\Wsol_u$. In particular every pairing of $u$ with a triple
of free solutions is a class invariant, and no bulk equation term
changes it.
\item $\cls{u}$ meets $\Fpol$ if and only if
\begin{equation}
 \WF(\Wsol_u)\cap\bigl(\Vp^{\,3}\cup\Vm^{\,3}\bigr)=\varnothing .
 \label{eq:criterion}
\end{equation}
When \eqref{eq:criterion} holds, an explicit representative is
$\widetilde u=\Box_1\Box_2P_{m,3}[\chi_1\chi_2\chi_3\Wsol_u]$.
\end{enumerate}
\end{theorem}

\begin{proof}
(a) is Lemma~\ref{lem:pairings} applied to $w=v-u$ together with
Lemma~\ref{lem:kernel}. (b) ``Only if'': if $v\in\cls{u}\cap\Fpol$ then
$\Wsol_u=\Wsol_v$ by (a), and by Lemma~\ref{lem:transport} every
covector triple of $\WF(\Wsol_v)$ is a covector triple of $\WF(v)$,
none of which lies in $\Vp^{\,3}\cup\Vm^{\,3}$ by \eqref{eq:mc}.
``If'': $\widetilde u$ is compactly supported and lies in $u+\Ideal$
by \eqref{eq:decomp}, hence in $\cls{u}$ by (a); it is symmetric in
the metric slots because $u$ is and $\chi_1=\chi_2$. Multiplication
by smooth functions and differential operators do not enlarge
wavefront sets \cite[(8.1.11)]{Hormander}, so
$\WF(\widetilde u)\subset\WF(\Wsol_u)$, and \eqref{eq:criterion}
gives \eqref{eq:mc}. Lemma~\ref{lem:reduction} converts
$\widetilde u$ into an element of $\Fpol$ restricting to
$\Cthree(f,g)$ on $\Csol$.
\end{proof}

This is the separation the sheet asked for, stated correctly: the
bulk term is all of $\Ideal$, and nothing is left over inside the
class. What the shell sees is $\Wsol_u$. A remainder appears only when
$u$ is compared with a candidate $\widetilde u$ satisfying
\eqref{eq:mc}; by (a), the comparison is decided by $\Wsol_u$ alone,
and by (b) the candidate exists exactly when \eqref{eq:criterion}
holds.

\subsection{Plane-wave pairings and the Cauchy pairing}

\begin{proposition}[The Cauchy pairing is the leading onset coefficient]
\label{prop:planewave}
Let $k_1^\mu=(1,\mathbf n)$, $k_2^\mu=(1,-\mathbf n)$, $|\mathbf n|=1$,
$\lambda>0$, $\gamma_a^\lambda=\epsilon_ae^{i\lambda k_a\cdot y}$ with
constant symmetric $\epsilon_a$, and $\phi$ smooth. Put
$\bar\beta_{\mu\nu}(t):=\int_{\R^3}\beta[f,g,\phi]_{\mu\nu}(t,\mathbf x)\dd^3x$,
smooth on $[0,T]$ and vanishing near $T$. Then
\begin{equation}
 \pair{u}{\gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi}
 =\lambda^2\,\qsym^{\mu\nu}(\epsilon_1,k_1;\epsilon_2,k_2)\int_0^T\bar\beta_{\mu\nu}(t)e^{2i\lambda t}\dd t
 =\frac{i}{2\lambda}\,\Bsig(\gamma_1^\lambda,\gamma_2^\lambda;\phi)+O(1),
 \label{eq:planewave}
\end{equation}
with $\Bsig(\gamma_1^\lambda,\gamma_2^\lambda;\phi)=\lambda^2\qsym^{\mu\nu}\bar\beta_{\mu\nu}(0)$.
The left side is a class invariant of $\cls{u}$.
\end{proposition}

\begin{proof}
$J_3(\gamma_1^\lambda,\gamma_2^\lambda)=\lambda^2\qsym\,e^{i\lambda(k_1+k_2)\cdot y}
=\lambda^2\qsym\,e^{2i\lambda t}$, independent of $\mathbf x$; insert
in \eqref{eq:upair} and integrate over $\mathbf x$ first. Two
integrations by parts in $t$, using $\bar\beta(T)=\bar\beta'(T)=0$,
give $\int_0^T\bar\beta e^{2i\lambda t}\dd t=-\bar\beta(0)/(2i\lambda)+O(\lambda^{-2})$,
and $-1/(2i)=i/2$. The value of $\Bsig$ on the pair is
\eqref{eq:Bsig} with $J_3|_\Sigma=\lambda^2\qsym$. Invariance is
Theorem~\ref{thm:criterion}(a).
\end{proof}

Gate~2 ends at \textbf{Constructed}: the bulk term and the class
invariant are separated, the invariant is $\Wsol_u$, and
\eqref{eq:planewave} shows that the Cauchy pairing of G1 is what the
invariant carries at leading order in the counter-propagating
directions, divided by the frequency $2\lambda$ of the total covector
$2\lambda\,\dd t$.

%=====================================================================
\section{G3: the wavefront alternative}
\label{sec:G3}
%=====================================================================

The witness is printed before the verdict.

\subsection{The witness}

\begin{proposition}[Pure-trace counter-propagating pairing for the F1 data]
\label{prop:witness}
Let $(f,\phi_m)$ be F1 data, $k_1^\mu=(1,\mathbf n)$, $k_2^\mu=(1,-\mathbf n)$,
and $\gamma_a^\lambda=\eta\,e^{i\lambda k_a\cdot y}$, $\lambda>0$. Both
$\gamma_a^\lambda$ are free metric configurations, $\phi_m$ is a free
scalar, and
\begin{equation}
 \pair{u}{\gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi_m}
 =-2\lambda^2\int_0^TB(t)e^{2i\lambda t}\dd t
 =-i\lambda\Lambda_f+O(1),\qquad
 \Bsig(\gamma_1^\lambda,\gamma_2^\lambda;\phi_m)=-2\lambda^2\Lambda_f\neq0 .
 \label{eq:witnessvalue}
\end{equation}
In particular there is $C$ with
$|\pair{u}{\gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi_m}|\ge\lambda|\Lambda_f|-C$
for all $\lambda>0$.
\end{proposition}

\begin{proof}
$\Box\gamma_a^\lambda=0$ since $k_a$ is null; $\Pfree\phi_m=0$. By
\eqref{eq:F1data}, $\bar\beta_{ij}=0$, $\bar\beta_{0i}=0$ and
$\bar\beta_{00}=B$; by \eqref{eq:qcounter}, $\qsym^{00}=-2$. Insert in
\eqref{eq:planewave}: $-2\lambda^2(-B(0)/(2i\lambda)+O(\lambda^{-2}))
=\lambda B(0)/i+O(1)=-i\lambda\Lambda_f+O(1)$.
\end{proof}

\begin{lemma}[Rapid decay for microcausal kernels]
\label{lem:decay}
Let $v\in\Edist$ satisfy \eqref{eq:mc} with $n=3$, let $\phi_0$ be
smooth on $M$, and put $v_{\phi_0}:=\pair{v}{\cdot\otimes\cdot\otimes\phi_0}$,
a compactly supported tensor-valued distribution on $M^2$. Let
$k_1,k_2$ be nonzero causal covectors, both in $\Vp$ or both in
$\Vm$. Then for all constant $\epsilon_1,\epsilon_2$ and all $N$,
\[
 \pair{v_{\phi_0}}{\epsilon_1e^{i\lambda k_1\cdot y_1}\otimes\epsilon_2e^{i\lambda k_2\cdot y_2}}
 =O(\lambda^{-N}),\qquad\lambda\to+\infty,
\]
uniformly for $(k_1,k_2)$ in a compact subset of $\Vp^{\,2}\setminus0$
or $\Vm^{\,2}\setminus0$.
\end{lemma}

\begin{proof}
By XI-F's smearing lemma \cite[Theorem~8.2.12]{Hormander},
$\WF(v_{\phi_0})\subset\{(y_1,p_1;y_2,p_2):\exists x,\ (y_1,p_1;y_2,p_2;x,0)\in\WF(v)\}$,
so $\WF(v_{\phi_0})$ contains no nonzero point of $\Vp^{\,2}\cup\Vm^{\,2}$,
componentwise in the finite tensor indices. Write
$e^{i\lambda k\cdot y}=e^{i\langle y,\lambda k_\flat\rangle}$ with
$(k_\flat)_\mu=\eta_{\mu\nu}k^\nu$ and the Euclidean pairing
$\langle y,\zeta\rangle=y^\mu\zeta_\mu$. Then the pairing is the
Fourier transform $\widehat{v_{\phi_0}}(\zeta)=\pair{v_{\phi_0}}{e^{-i\langle\cdot,\zeta\rangle}}$
at $\zeta=-\lambda(k_{1\flat},k_{2\flat})$, contracted with
$\epsilon_1\otimes\epsilon_2$. By \cite[Definition~8.1.2,
Proposition~8.1.3]{Hormander}, for a compactly supported distribution
the closed cone $\Sigma(v_{\phi_0})$, the projection of $\WF(v_{\phi_0})$
to the covector, is the complement of the set of directions having a
conic neighbourhood of rapid decrease of the Fourier transform. The
direction $-(k_{1\flat},k_{2\flat})$ lies in $\Vm^{\,2}\setminus0$ or
$\Vp^{\,2}\setminus0$, hence outside $\Sigma(v_{\phi_0})$, and the
compact set of such directions has a conic neighbourhood of rapid
decrease.
\end{proof}

\subsection{The verdict}

\begin{theorem}[The free-shell class of the cubic kernel is obstructed]
\label{thm:obstruction}
Fix $m\ge0$, the frozen $g$ and $\sig$, and F1 data $(f,\phi_m)$. Let
$u$ be the frozen kernel of $\Cthree(f,g)$. Then
\[
 \cls{u}\cap\Fpol=\varnothing .
\]
Equivalently: no $F\in\Fpol$ satisfies $F|_{\Csol}=\Cthree(f,g)|_{\Csol}$,
and $\Cthree(f,g)|_{\Csol}$ is not in the image of XI-E's admission
map $Q_H$ for any choice of representative. Since F1 data exist for
every frozen $g$ and every $m$, no construction admitting
$\Cthree(f,g)$ for all admissible $f$ exists in $\Ealg$.
\end{theorem}

\begin{proof}
Suppose $v\in\cls{u}$ satisfies \eqref{eq:mc}. By
Definition~\ref{def:class}, extended to complex solutions by
bilinearity, $v$ and $u$ have the same pairing with
$\gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi_m$ for the plane
waves of Proposition~\ref{prop:witness}, and this pairing equals
$\pair{v_{\phi_m}}{\gamma_1^\lambda\otimes\gamma_2^\lambda}$. The
covectors $k_{a\flat}=(1,\mp\mathbf n)$ are null with $k_{a,0}=1>0$,
hence both in $\Vp$. Lemma~\ref{lem:decay} gives $O(\lambda^{-1})$;
Proposition~\ref{prop:witness} gives $\ge\lambda|\Lambda_f|-C$ with
$\Lambda_f\ne0$. Contradiction. The equivalent formulations are
Lemma~\ref{lem:reduction}, and the existence of F1 data is XI-F's
Theorem~F1.
\end{proof}

\begin{corollary}[The covector every representative inherits]
\label{cor:covector}
For F1 data, every $v\in\cls{u}$, microcausal or not, has wavefront
points
\[
 (y_1,\pm k_{1\flat};\,y_2,\pm k_{2\flat};\,x,0),\qquad
 k_{1\flat}=(1,-\mathbf n),\quad k_{2\flat}=(1,\mathbf n),\quad
 k_{1\flat}+k_{2\flat}=2\,\dd t,
\]
for some $(y_1,y_2,x)$ and every unit $\mathbf n$: two co-oriented null
covectors, one per metric slot, with opposite spatial momenta and zero
in the scalar slot. Moreover $\WF(\Wsol_u)$ meets $\Vp^{\,3}\cup\Vm^{\,3}$.
\end{corollary}

\begin{proof}
The pairing of $v_{\phi_m}$ with the plane waves does not decay along
the ray $-\lambda(k_{1\flat},k_{2\flat})$, so by
\cite[Proposition~8.1.3]{Hormander} this direction lies in
$\Sigma(v_{\phi_m})$, i.e. some $(y_1,-k_{1\flat};y_2,-k_{2\flat})\in\WF(v_{\phi_m})$,
and by the inclusion in the proof of Lemma~\ref{lem:decay} there is
$x$ with $(y_1,-k_{1\flat};y_2,-k_{2\flat};x,0)\in\WF(v)$. Since
$\beta$ and $\qsym$ are real, the conjugate plane waves give the same
growth, and the $+$ sign. The last statement is
Theorem~\ref{thm:criterion}(b) in contrapositive form.
\end{proof}

XI-F's $\pm\dd t$ was the conormal of the step $\theta(y^0)$ in the
frozen kernel. The class forgets the step and keeps its two null
halves $k_{1\flat}+k_{2\flat}=2\,\dd t$. Representatives in $\Ideal+u$
can move the halves off the diagonal, but by
Theorem~\ref{thm:criterion}(a) they cannot remove them.

\subsection{Scope of the obstruction}

\begin{proposition}[General detection]
\label{prop:general}
Let $(f,g)$ be admissible, $\phi_0$ free, $\beta=\beta[f,g,\phi_0]$,
and for $\mathbf p\in\R^3$ and unit $\mathbf n$ put
\[
 \check\beta_{\mu\nu}(\mathbf p):=\int_{\R^3}\beta_{\mu\nu}(0,\mathbf x)e^{-i\mathbf p\cdot\mathbf x}\dd^3x,
 \qquad
 \mathsf b(\mathbf n,\mathbf p):=-2\check\beta_{00}(\mathbf p)+(n^in^j-3\delta^{ij})\check\beta_{ij}(\mathbf p).
\]
If $\mathsf b\not\equiv0$, then $\cls{u}\cap\Fpol=\varnothing$ for the
kernel $u$ of $\Cthree(f,g)$. In particular, for data of F1 type built
from $\phi_m$, where $\beta_{ij}=0$, the class is obstructed whenever
$\beta_{00}|_\Sigma\not\equiv0$.
\end{proposition}

\begin{proof}
Take $k_1^\mu=(1,\mathbf n)$ and $k_2^{\lambda,\mu}=(\omega_\lambda,-\mathbf n+\mathbf p/\lambda)$
with $\omega_\lambda=|\mathbf n-\mathbf p/\lambda|$, both null and
future; $\gamma_a^\lambda=\eta e^{i\lambda k_a\cdot y}$ are free. Then
$\lambda(k_1+k_2^\lambda)\cdot y=\Omega_\lambda t-\mathbf p\cdot\mathbf x$,
$\Omega_\lambda:=\lambda(1+\omega_\lambda)=2\lambda+O(1)$, and
\[
 \pair{u}{\gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi_0}
 =\lambda^2\qsym^{\mu\nu}(\eta,k_1;\eta,k_2^\lambda)\int_0^T\!\!\int_{\R^3}\beta_{\mu\nu}(t,\mathbf x)e^{i\Omega_\lambda t-i\mathbf p\cdot\mathbf x}\dd^3x\dd t
 =\frac{i\lambda}{2}\,\qsym^{\mu\nu}(\eta,k_1;\eta,k_2)\,\check\beta_{\mu\nu}(\mathbf p)+O(1),
\]
by one integration by parts in $t$, $\lambda^2/\Omega_\lambda\to\lambda/2$,
and continuity of the polynomial $\qsym$ as $k_2^\lambda\to k_2=(1,-\mathbf n)$.
By \eqref{eq:qcounter} the leading coefficient is
$\tfrac{i\lambda}{2}\mathsf b(\mathbf n,\mathbf p)$. If $v\in\cls{u}$
satisfies \eqref{eq:mc}, Lemma~\ref{lem:decay} applies uniformly on a
compact neighbourhood of $(k_{1\flat},k_{2\flat})$ in $\Vp^{\,2}\setminus0$,
which contains $(k_{1\flat},k^\lambda_{2\flat})$ for large $\lambda$,
giving decay; choosing $(\mathbf n,\mathbf p)$ with
$\mathsf b(\mathbf n,\mathbf p)\neq0$ contradicts the growth.
\end{proof}

\begin{remark}[Scope: what is and is not quantified]
\label{rem:scope}
(i) The quantifier of Theorem~\ref{thm:obstruction} is the copied
class $\Fpol$ and the free shell $\Csol$ of XI-E. Nothing is said about
bimodules or algebras with other domains; XI-E's correction stands.
(ii) The theorem obstructs universal admission of $\Cthree(f,g)$ for
all admissible $f$; it also obstructs every individual $f$ of F1 type,
and every $(f,g,\phi_0)$ caught by Proposition~\ref{prop:general}. A
smaller observation class on which $\mathsf b\equiv0$ is not declared
here and not decided; by Theorem~\ref{thm:criterion}(b) its decision
is the cone condition \eqref{eq:criterion}, and the plane-wave test
only certifies the leading onset coefficient. (iii) If both metric
slots are restricted further to the Einstein subshell $A\gamma=0$, the
pure-trace witness is not available and, by Remark~\ref{rem:einstein},
transverse pairs are blind to $\beta_{0\nu}|_\Sigma$; for the F1 data,
where $\beta_{ij}=0$, no plane-wave pair detects the onset on that
subshell. That smaller question is not decided here. XI-E's shell is
the gauge-fixed one, and Theorem~\ref{thm:obstruction} answers the
question as posed.
\end{remark}

\begin{remark}[Mechanism, and the two model changes]
\label{rem:mechanism}
The covector found is the sum of two co-oriented null covectors
producing a timelike total, the pair-creation configuration of a
sharp-time local quadratic functional; the analogy with the
sharp-time Wick square is one of covector type only, as for VII-B in
XI-F, and no kernel is identified. Ramping the source,
$J_3\mapsto\sig J_3$, would replace $\theta(y^0)$ by $\sig$ in
\eqref{eq:upair}; the kernel would then be a local bilinear
distribution on the diagonal $y_1=y_2$ with opposite covectors and
smooth compact weight, in $\Fpol$ outright. Moving the metric Cauchy
surface would move the onset. Both alter XI-A's frozen prescription
and are not done here; Theorem~\ref{thm:criterion}(b) is the test
either successor must pass.
\end{remark}

Gate~3 ends at \textbf{Proved obstructed}. The constructed branch is
characterized by \eqref{eq:criterion} and not instantiated; the
unresolved branch does not arise, because the extension criterion is
discharged by Lemma~\ref{lem:decay}.

%=====================================================================
\section{G4: what the result is not}
\label{sec:G4}
%=====================================================================

\begin{itemize}
\item Theorem~\ref{thm:obstruction} is a statement about compact
distributional kernels and the copied cone exclusion. It is not a
no-go for every bimodule, every renormalized product, or every
extension preserving XI-B's commutators.
\item No quantum retarded product of $J_3$ is formed, no cubic source
is contracted against a Hadamard line, and no loop is evaluated. The
pairings used are classical evaluations on smooth solutions.
\item No $\widehat s_{\mathrm{int}}$, antibracket, or master identity
is constructed. Ghost number zero of $\Cthree$ is known and is not
closure.
\item The classical ledger $-I+I+0=0$ of XI-F is untouched; it is a
coefficient identity on smooth configurations and needs no
representative in $\Ealg$. It is not re-added as $0+I+0$. The
Einstein-shell term $\Aone[\PP\ER\Ldens(A\gamma);\phi_0]$ stays named
and unevaluated when $D^*b\neq0$ on $U$.
\item The onset is the zero-data truncation of an unramped source,
not the ramp layer; no $\eps\to0$ limit is taken, and the two affine
numbers $S_\eps(t)$ and $\eps/2$ are not touched.
\item $\Cthree$ is not identified with $\Bcal_f$, with VII-D's or
IX-E-2's kernels; only the covector type is compared.
\item The empty jet $\Jcal_g(\phi_m)=\varnothing$ is not repaired, and
would not have been repaired by a constructed representative either.
\item No cone, token, or Enter record is imported.
\end{itemize}

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}X X X@{}}
\toprule
Gate & Constructed only if & Obstructed if & Unresolved if\\
\midrule
G1 & $\Bsig$ explicit in Cauchy data & --- & pairing only named\\
G2 & bulk term and remainder separated & --- & one-slot rewrite used as proof\\
G3 & $\widetilde u\in\Fpol$ and agrees on the shell &
every reproducing distribution has a forbidden covector &
pairing nonzero, extension open\\
G4 & exclusions recorded & --- & a product or $\widehat s_{\mathrm{int}}$ is implied\\
\bottomrule
\end{tabularx}
\end{center}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}p{.22\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
G1 & \textbf{Constructed} &
$\Bsig=\int_\Sigma\beta|_\Sigma\cdot Q(j_1,j_2)$ with $Q=\qsym$ on jets,
$\partial_t^2\mapsto\Delta$ (Lemma~\ref{lem:cauchyred}); pure-trace sector
printed (Corollary~\ref{cor:Bsigtrace}); counter-propagating symbol
$\qsym^{00}=-2$, $\qsym^{0i}=0$, $\qsym^{ij}=n^in^j-3\delta^{ij}$ with
Bianchi check (Lemma~\ref{lem:qvalues}); scalar slot a spectator.\\[0.4em]
G2 & \textbf{Constructed} &
$\ker(w\mapsto\Wsol_w)\cap\Edist=\Ideal$ (Lemma~\ref{lem:kernel});
$v\in\cls{u}\iff\Wsol_v=\Wsol_u$; class meets $\Fpol$ iff
\eqref{eq:criterion} (Theorem~\ref{thm:criterion}); plane-wave
pairings are class invariants with leading coefficient
$\tfrac{i}{2\lambda}\Bsig$ (Proposition~\ref{prop:planewave}).\\[0.4em]
G3 & \textbf{Proved obstructed} &
F1 data: pairing $-i\lambda\Lambda_f+O(1)$ versus $O(\lambda^{-\infty})$
for every microcausal kernel; $\cls{u}\cap\Fpol=\varnothing$
(Theorem~\ref{thm:obstruction}); inherited covectors
$\pm(k_{1\flat},k_{2\flat},0)$, $k_{1\flat}+k_{2\flat}=2\,\dd t$
(Corollary~\ref{cor:covector}); general detection by
$\mathsf b(\mathbf n,\mathbf p)\not\equiv0$ (Proposition~\ref{prop:general});
Einstein subshell not decided.\\[0.4em]
G4 & Recorded & No bimodule no-go, no product, no differential, no loop,
no limit, no jet repair.\\
\bottomrule
\end{tabularx}
\end{center}

The worst ending among G1--G3 governs: \textbf{Proved obstructed}, at
Gate~3. The frozen-kernel theorem of XI-F is not reopened; it is
inherited by the class in split form. The classical ledger is not
erased.

\subsection*{Exclusions that persist}

\begin{itemize}
\item No ramp on $J_3$; no switch on the Einstein--Hilbert density;
the metric Cauchy surface is $\Sigma$.
\item $\Ggr=-\PP\ER$; neither $-2\PP\ER$ nor $-4\PP\ER$; X-F's sum is
not used.
\item VII-C's (E1)--(E4) are not the admission interface; no
reflected residue or finite part is subtracted.
\item No state, no Hadamard selection beyond XI-E's product data, no
expectation value, no mixed loop, no sharp limit.
\item $\Jcal_g(\phi_m)=\varnothing$ stands; no Enter record.
\end{itemize}

\subsection*{What a successor receives: the obstructed package}

\begin{enumerate}
\item The class $\cls{u}$, its invariant $\Wsol_u$, and the separation
Theorem~\ref{thm:criterion}: bulk equation terms are exactly $\Ideal$,
and admission into $\Fpol$ is the cone condition \eqref{eq:criterion}.
\item The nonzero Cauchy pairing $\Bsig=-2\lambda^2\Lambda_f$ on the
pure-trace counter-propagating pair, and its identification as the
leading onset coefficient \eqref{eq:planewave}.
\item The covector theorem, quantified to the copied class: every
representative of $\Cthree(f,g)|_{\Csol}$ carries
$\pm(k_{1\flat},k_{2\flat},0)$ with $k_{1\flat}+k_{2\flat}=2\,\dd t$.
\item The detection symbol $\mathsf b(\mathbf n,\mathbf p)$ and the
undecided Einstein-subshell question.
\item The two model changes, ramped cubic source or metric data off
$\Sigma$, each still a change of XI-A's prescription, each to be
tested against \eqref{eq:criterion} and against XI-A's covariance.
\end{enumerate}

\subsection*{What comes after, and only after}

Gate~3 is obstructed, so the next legal paper is a model change: a
ramped cubic source $\sig J_3$, or metric Cauchy data off $\Sigma$, as
a new prescription with its own covariance check against XI-A. Neither
is a repair of XI-F; Remark~\ref{rem:mechanism} records what the first
would do to the kernel, and Theorem~\ref{thm:criterion}(b) records
what either must prove.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 v\in\cls{u}\iff v-u\in\Box_1\Edist+\Box_2\Edist+P_{m,3}\Edist
 \iff\Wsol_v=\Wsol_u;\qquad
 \cls{u}\cap\Fpol\neq\varnothing\iff\WF(\Wsol_u)\cap(\Vp^{\,3}\cup\Vm^{\,3})=\varnothing;\\[2pt]
 \text{F1 data:}\quad
 \pair{u}{\eta e^{i\lambda k_1\cdot y}\otimes\eta e^{i\lambda k_2\cdot y}\otimes\phi_m}
 =-i\lambda\Lambda_f+O(1),\quad k_1+k_2=2\,\dd t,\quad\Lambda_f<0;\\[2pt]
 \cls{u}\cap\Fpol=\varnothing:\ \text{every representative carries }
 \pm(k_{1\flat},k_{2\flat},0),\ k_{1\flat},k_{2\flat}\text{ null, co-oriented.}
 \end{gathered}}
\]
A reader can now see that XI-F's $\pm\dd t$ obstructs one kernel, and
that the free-shell class does not escape it: it inherits the two
null halves of that covector, one in each metric slot, and no bulk
equation term removes them.

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

\paragraph{Proof scope.}
The Cauchy reduction, the pure-trace cubic coefficient, the
counter-propagating symbol and its Bianchi check, the solution-pairing
lemma, the kernel lemma, the covector transport, the admission
criterion, the plane-wave asymptotics, the rapid-decay lemma, the
class obstruction, the inherited covector, and the general detection
statement are proved on smooth configurations and compact
distributional kernels in the displayed conventions. The Einstein
subshell question and smaller observation classes with $\mathsf b\equiv0$
are stated as undecided.

\paragraph{Dependencies.}
X-A supplies finite propagation and the Cauchy facts. XI-A supplies
the signed action and the gauge-fixed versus Einstein distinction.
XI-B supplies the normalization, the causal kernels, and the free
shell. XI-E supplies the algebra, its domain, the admission map, and
the finite cubic coefficient. XI-F supplies the kernel representation,
Theorem~F1 and its data, and the open remark. H\"ormander supplies
Definition~8.1.2, Proposition~8.1.3, (8.1.11), and Theorems~8.2.4,
8.2.9, 8.2.12, 8.2.13, 8.3.1. B\"ar--Ginoux--Pf\"affle supplies the
retarded and advanced inverses on distributions.

\paragraph{Corrections.}
The identity of the class invariant, the split form of the forbidden
covector, the plane-wave witness, the blindness of Einstein-shell
pairs, the status of the one-slot rewriting, and the uninstantiated
constructed branch are corrected relative to the production sheet.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant check,
executed symbolic computation, independent referee review, or journal
acceptance is asserted. All nonzero and zero witnesses are analytic.

%=====================================================================
\begin{thebibliography}{9}
\addcontentsline{toc}{section}{References}

\bibitem{XA}
N.~Cheung,
\emph{Landmark X-A: Classical covariance of one fixed-regulator
switched relational scalar through order $\kappa^2$}.
Supplied manuscript \texttt{X-A.tex}.
Used for finite propagation, the Cauchy lemma, and transposition.

\bibitem{XIA}
N.~Cheung,
\emph{Landmark XI-A: A switched Einstein--scalar action for the
ramp relational scalar}.
Supplied manuscript \texttt{XI-A.tex}.
Used for the signed action, the gauge-fixed versus Einstein
distinction, and the empty fixed-input jet.

\bibitem{XIB}
N.~Cheung,
\emph{Landmark XI-B: The free de Donder multiplet for the ramp
Einstein--scalar sheet}.
Supplied manuscript \texttt{XI-B.tex}.
Used for the normalization $-\PP\ER$, the causal kernels $\delta_\mu$,
and the free shell.

\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 polynomial microcausal domain, the admission map, the
on-shell quotient, and the finite cubic coefficient $\Hcal_2$.

\bibitem{XIF}
N.~Cheung,
\emph{Landmark XI-F: Admission of the cubic Einstein response, and
the plateau sum of the second-order variations}.
Supplied manuscript \texttt{XI-F.tex}.
Used for the kernel representation of $\Cthree$, Theorem~F1 and its
data, the smearing lemma, the homogeneous reduction, the open remark,
and the classical ledger.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
Grundlehren der mathematischen Wissenschaften 256, second edition,
Springer, 1990.
Used for Definition~8.1.2, Proposition~8.1.3, (8.1.11), and
Theorems~8.2.4, 8.2.9, 8.2.12, 8.2.13, 8.3.1.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
ESI Lectures in Mathematics and Physics, European Mathematical
Society, 2007. arXiv:0806.1036.
Used for retarded and advanced inverses on distributions with
past- and future-compact support, uniqueness, and finite propagation.

\end{thebibliography}

\end{document}