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% XI-D.tex
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% Essential corrections:
%
% 1. A classical symbol space is not a quantum algebra extension.
%    No admission theorem is inferred from a wavefront estimate.
%
% 2. XI-B's algebra already imposes the free equations. Its classical
%    symbols therefore live on free configurations. An exact
%    off-shell symbol map reproducing all its linear generators is
%    incompatible with those equations.
%
% 3. The bare dressing B_f is not generally s_0-closed. An explicit
%    witness uses a genuine free ghost with unrestricted Cauchy data.
%    This obstruction occurs before an interacting correction could
%    cancel it in the proposed identity for B_f alone.
%
% 4. The closure of a complete observable F_f+kappa B_f+... is a
%    different coefficient problem. It is not the closure of B_f.
%
% 5. Ghost number zero is verified, but does not imply closure.
%
% 6. An antifield-free vertex does not specify the field component
%    of an interacting BV differential. No antibracket or s_1,s_2
%    is manufactured before quantum admission.
%
% Ending:
%   Quantum admission: Unresolved.
%   Bare-insertion closure: conditionally obstructed, with an
%   explicit free-shell witness for admissible observation tests.
%
% The admission gate remains unfinished, so the overall ending is
% Unresolved, as required by the production sheet.
%
% No state, Hadamard covariance, loop, cubic Einstein contraction,
% interacting BRST class, or sharp-regulator limit is constructed.
%
% No successful compilation, mechanized verification, independent
% referee review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\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{\ga}{\mathfrak g_U}
\newcommand{\Afree}{\mathscr A_0}
\newcommand{\Pfin}{\mathscr P_{\mathrm{fin}}}
\newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}}
\newcommand{\Coff}{\mathscr C_{\mathrm{off}}}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\Esym}{\mathscr 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{\Pfree}{P_m}
\newcommand{\sfree}{s_0}
\newcommand{\Coord}{\mathsf X}
\newcommand{\Xone}{\mathsf X_1}
\newcommand{\Xtwo}{\mathsf X_2}
\newcommand{\Bcal}{\mathcal B}
\newcommand{\Tcal}{\mathcal T}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Atwo}{\mathcal A_2}
\newcommand{\Jcal}{\mathcal J}
\newcommand{\Lag}{\mathfrak L}
\newcommand{\Vone}{\mathsf V_{1,g}}
\newcommand{\Vtwo}{\mathsf V_{2,g}}
\newcommand{\Ueps}{\mathsf U_\eps}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\gh}{gh}

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

\title{\textbf{Landmark XI-D}\\[0.4em]
Admission of the fixed-width graviton tree\\
and one ghost-number-zero identity:\\
the admission interface and a free-order closure obstruction}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
Landmark XI-C constructed a fixed-width classical graviton-tree
symbol, not an admitted quantum insertion. We formulate the
additional admission interface without replacing XI-B's
finite-generator algebra by an unnamed Wick algebra. An essential
distinction is made between exact off-shell representatives and
classical symbols of an algebra that already imposes the free
equations. Requiring the latter to reproduce every linear generator
on all off-shell configurations is inconsistent.

A separate calculation decides a necessary condition for the proposed
closure identity. The bare dressing
$\Bcal_f=-\int f\,\Xone[\gamma]\cdot\partial\phi$ has ghost number
zero, but its free variation is generally nonzero. An explicit
witness satisfies all the free multiplet equations and uses a ghost
with admissible, unrestricted free Cauchy data. Consequently, for
the observation tests in that witness, no admission respecting the
classical free differential can make this bare insertion closed by
adding only positive coupling-order corrections. The inherited
second-order held-fixed-metric defect is also evaluated directly.

No quantum admission map is constructed, and no general
nonexistence theorem for an appropriately formulated quantum
extension is claimed. The overall ending is therefore
\emph{Unresolved}, with a proved conditional obstruction to the
particular bare-insertion closure request. No antibracket,
interacting differential, state, mixed loop, cubic Einstein
contraction, 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) {XI-C classical tree};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Quantum extension};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Admission map};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The overall ending is unresolved. The classical tree remains the XI-C symbol.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The quantum extension is not constructed. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {XI-C classical tree};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Admission map};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The overall ending is unresolved. The classical tree remains the XI-C symbol.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The quantum extension is not constructed. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XI-C classical tree};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Quantum extension};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Admission map};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The overall ending is unresolved. The classical tree remains the XI-C symbol.};
\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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XI-C classical tree};
\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) {Admission map};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Quantum extension. The overall ending is unresolved. The classical tree remains the XI-C symbol.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{XI-C classical tree is not Admission map. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: XI-C classical tree};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Quantum extension};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Admission map};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The overall ending is unresolved. The classical tree remains the XI-C symbol.};
\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-D}
\end{figure}

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


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



\scopebox{Release and legal ending}{%
\textbf{Overall ending: Unresolved.}
The quantum extension, its product, and its admission map are not
constructed. The classical tree remains the XI-C symbol.

Two precise results are proved: an exact off-shell-shadow
interpretation of the unchanged free-shell algebra is incompatible
with its equations; and the bare insertion fails a necessary
free-order closure condition for explicit admissible observation
tests.

The latter is a conditional obstruction to closure, not a theorem
excluding admission alone. No quantum class is released.}

\tableofcontents

%=====================================================================
\section{The question and the three endings}
\label{sec:question}
%=====================================================================

The proposed implication is
\[
 \text{classical tree}
 \longrightarrow
 \text{quantum admission}
 \longrightarrow
 \text{interacting differential}
 \longrightarrow
 \text{closed insertion}.
\]
None of its arrows is supplied merely by the preceding arrow.
In particular, ghost number zero and a compact microcausal kernel
are necessary bookkeeping facts, not closure theorems.

The production sheet permits three endings.

\begin{enumerate}
\item \textbf{Constructed.}
A named quantum extension admits the required symbols, a differential
is defined and proved nilpotent through the declared order, and the
admitted insertion is closed.
\item \textbf{Proved obstructed.}
A stated interface is excluded by an explicit incompatibility.
Its quantifiers must identify whether admission, nilpotence, or
closure is excluded.
\item \textbf{Unresolved.}
An indispensable extension, product, antibracket, or identity remains
unconstructed.
\end{enumerate}

Here admission remains unresolved, so the overall ending is the
third. Nevertheless, one can prove a necessary-condition obstruction
without pretending that an interacting differential has been
constructed. Such a result excludes a specified closure request
\emph{if} admission respects its required classical limit.

This distinction matters because the production sheet asks for the
closure of the \emph{bare dressing} $\Bcal_f$. That is not the same
as closure of a complete perturbative observable whose leading term
is $\int f\phi$. The two coefficient problems have different possible
cancellations. Section~\ref{sec:freeclosure} makes this difference
explicit.

The standard separation between regular polynomial generators,
microcausal functionals, and their quantization is also relevant
\cite{FR}. It does not itself supply the extension required here.
Likewise, a BV construction requires more than names for antifields
\cite{Rejzner}. We use neither reference as a substitute for a
missing construction.

%=====================================================================
\section{Prerequisite and convention sheet}
\label{sec:sheet}
%=====================================================================

\subsection{Geometry, regulator, and fields}

Retain the supplied conventions:
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad
 \Sigma=\{t=0\},
\]
\[
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta,\qquad
 \Pfree=\Box+m^2,\qquad m\geq0.
\]
All coefficient fields are smooth on a neighbourhood of the closed
slab. Parentheses on indices include the factor $1/2$.

Fix $0<\eps<T$ and
\[
 \sig(t)=\chi(t/\eps),\qquad
 \chi(s)=
 \begin{cases}
 0,&s\leq0,\\[2pt]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[7pt]
 1,&s\geq1.
 \end{cases}
\]
Fix
\[
 g\in C_c^\infty(\MTo;[0,1]),\qquad
 P_g=\Int\{g=1\}\neq\varnothing,
\]
and an open relatively compact $U$ with
\[
 \overline U\subset P_g\cap\{\eps<t<T\},\qquad
 \ga=C_c^\infty(U;TM).
\]
The observation test is a real $f\in C_c^\infty(\MTo)$.
The width is never sent to zero.

For a symmetric tensor $r$ define
\[
 \PP r=r-\tfrac12\eta\,\tr_\eta r,\qquad
 D^\mu r=\partial_\nu(\PP r)^{\nu\mu},
\]
\[
 (Kv)_{\mu\nu}=\partial_\mu v_\nu+\partial_\nu v_\mu,
 \qquad \RR=\tfrac12K.
\]
The independent multiplet is
$(\gamma,\phi,c,\bar c,b)$ with parities $(0,0,1,1,0)$ and
ghost numbers $(0,0,1,-1,0)$.

\subsection{What XI-B actually supplies}

XI-B constructs the algebra $\Afree$ generated by compact interior
linear smearings and their finite products, with the free equations
and graded causal commutation relations imposed \cite{XIB}.
Its reduced metric operator and inverse are
\[
 L_\gamma=-\Box\PP,\qquad
 G_{\mathrm{gr}}^{\mathrm R}=-\PP\ER.
\]
The scalar equation inverse is $\ERm$, whereas its action-Hessian
inverse is $-\ERm$.

The copied free differential is
\begin{equation}
 \sfree\gamma=\RR c,\qquad
 \sfree\phi=0,\qquad
 \sfree c=0,\qquad
 \sfree\bar c=ib,\qquad
 \sfree b=0.
 \label{eq:sfree}
\end{equation}
It is nilpotent with $b$ retained and preserves XI-B's free equations
and causal relations. These facts do not admit a new composite.

Write $\Csol$ for smooth multiplet configurations satisfying
\begin{equation}
 -\Box\PP\gamma=0,\qquad
 b=D\gamma,\qquad
 \Pfree\phi=0,\qquad
 \Box c=\Box\bar c=0.
 \label{eq:freeshell}
\end{equation}
These equations are equivalent to XI-B's metric--auxiliary equations
after using $b=D\gamma$. Ghost Cauchy data are unrestricted.
They are not required to vanish at $\Sigma$.

\subsection{The copied XI-C split}

The coordinate operator is
\begin{equation}
 L_\eps r=\Xone[r]
 =-\ER\partial_\alpha(\sig a_1^{\alpha\cdot}[r])
 =\ER\bigl(\sig Dr-\sig'a_1^{0\cdot}[r]\bigr),
 \label{eq:L}
\end{equation}
where
\[
 a_1^{\mu\nu}[r]
 =\tfrac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu}.
\]
The bare insertion and the classical tree are
\begin{align}
 \Bcal_f[\gamma,\phi]
 &=-\int_{\MT}f\,\Xone[\gamma]^\mu\partial_\mu\phi\dd^4x,
 \label{eq:B}\\
 r_g[\phi]
 &=-\tfrac12\PP\ER(gT_0[\phi]),
 \label{eq:rg}\\
 \Tcal(f,g)[\phi]
 &=\int_{\MT}f\,(L_\eps r_g[\phi])^\mu
                         \partial_\mu\phi\dd^4x,
 \label{eq:T}
\end{align}
with
\[
 T_0^{\mu\nu}[\phi]
 =\partial^\mu\phi\partial^\nu\phi
 -\tfrac12\eta^{\mu\nu}
   \bigl(\partial_\rho\phi\partial^\rho\phi-m^2\phi^2\bigr).
\]
The physical one-graviton response of $\Bcal_f$ is
$-\Tcal$, at coupling degree $\kappa^2$.

XI-C proves that these are compact distributional polynomial
symbols satisfying its microcausal exclusion \cite{XIC}.
It does not construct their quantum admission. Its mixed insertion
kernel is not smooth finite-rank. The noncharacteristic diagonal
symbol has a nonzero $00$ component, with the stated numerator
$-1/2$ at $p=\dd t$.

Above the ramp, for the named input
\[
 \phi_m(t)=
 \begin{cases}
 \sin(mt)/m,&m>0,\\
 t,&m=0,
 \end{cases}
\]
the copied response is
\[
 \Xone^0[r_g[\phi_m]]
 =-\tfrac14\ER\ER(\partial_tg)\not\equiv0.
\]
XI-A's empty Einstein jet remains empty \cite{XIA}.
That source obstruction is not a quantum-admission obstruction.

\subsection{Non-inputs}

X-A supplies plateau covariance of the classical coordinate and
switched scalar prescriptions \cite{XA}. It does not supply quantum
closure. X-D is not used.

VII-C's four extension requirements remain unmet: its unchanged
Hadamard Wick embedding, exact sharp admission, specified
normal-topology convergence, and the multiplication/coefficient
continuity package. We neither import its covariance matrix nor
claim that the fixed-width symbols meet those separate interfaces.

%=====================================================================
\section{D1--D2: the admission interface}
\label{sec:admission}
%=====================================================================

\subsection{A source space, not a target algebra}

Let $\Pfin$ be the off-shell polynomial algebra of finite products
of compact smooth linear smearings. Its degree-$n$ kernels are finite
sums of smooth tensor products.

Let $\Pmc$ be the classical space declared in XI-C: finite
polynomials with compact distributional kernels $u_n$ obeying
\begin{equation}
 \WF(u_n)\cap
 \bigl(\widetilde V_+^{\,n}\cup
       \widetilde V_-^{\,n}\bigr)=\varnothing,
 \label{eq:mc}
\end{equation}
where the causal cones include zero covectors and the total zero
covector is excluded from a wavefront set. Include graded field
slots when discussing free variations.

Then
\[
 \Bcal_f\in\Pmc,\qquad
 \Tcal(f,g)\in\Pmc^{(3)}
\]
are classical input statements. Neither $\Pmc[[\hbar]]$ nor its
notation defines a quantum product.

For a possible on-shell formulation define
\[
 \Nsol=\{F\in\Pmc:F|_{\Csol}=0\},\qquad
 \Pmc^{\mathrm{sol}}=\Pmc/\Nsol.
\]
Here equality on free configurations is understood in the graded
sense, including evaluations in finite Grassmann algebras. We do
not identify $\Nsol$ with a generated equation ideal without a
separate proof.

\begin{definition}[On-shell admission interface]
\label{def:interface}
A prospective admission consists of:
\begin{enumerate}[label=\textup{(\alph*)}]
\item a unital quantum algebra $\Esym$ over $\C[[\hbar,\kappa]]$
and an injective homomorphism $j:\Afree[[\kappa]]\to\Esym$;
\item a classical-symbol interpretation of the admitted elements
on $\Csol$, compatible with the old finite-generator symbols;
\item lifts $Q(\Bcal_f)$ and $Q(\Tcal)$ with leading symbols
$[\Bcal_f]$ and $[\Tcal]$ in $\Pmc^{\mathrm{sol}}$;
\item agreement with the old inclusion whenever the symbol is
represented by an old finite product;
\item an explicitly stated topology and continuity requirement
if admission is to be obtained by kernel limits.
\end{enumerate}
This definition names required data; it does not assert their
existence.
\end{definition}

An off-shell version must instead start with an off-shell quantum
presentation and subsequently specify its equation quotient.
Calling the same algebra simultaneously an unchanged free-shell
algebra and an exact algebra of functions on all off-shell
configurations is not a permissible shortcut.

\subsection{An exact off-shell-shadow incompatibility}

\begin{proposition}[The free-shell algebra has no exact off-shell shadow]
\label{prop:shadow}
There is no classical-symbol map on an extension of $\Afree$ which
both preserves its zero elements and sends every old scalar linear
generator $\phi(u)$ to the off-shell functional
$\int u\phi$ on all smooth scalar configurations.
\end{proposition}

\begin{proof}
Take real $w\in C_c^\infty(\MTo)$ with $\Pfree w\neq0$.
Such tests exist. The equation relation in $\Afree$ is
\[
 \phi(\Pfree w)=0.
\]
Every algebra extension preserves this equality, and every symbol
map preserving zero must assign it the zero functional.

The required off-shell interpretation would instead assign
\[
 \phi\longmapsto\int_{\MT}(\Pfree w)\phi\dd^4x.
\]
Evaluate at the smooth configuration $\phi=\Pfree w$.
Its value is
\[
 \int_{\MT}(\Pfree w)^2\dd^4x>0.
\]
This contradicts the zero relation.
\end{proof}

This proposition has deliberately narrow quantifiers. It excludes
an exact off-shell shadow of \emph{all old generators}. It does not
exclude an off-shell representative chosen before an on-shell
quotient, nor admission formulated as
Definition~\ref{def:interface}.

Thus the phrase ``$Q(\Bcal_f)=\Bcal_f$ as a classical functional''
must specify its configuration space. An exact off-shell
representative is meaningful; it cannot be the unrestricted
classical-symbol interpretation of XI-B's already-quotiented
algebra.

\subsection{What a continuity claim would require}

A possible source topology can be specified on kernel stages.
For compact $K_n$ and a closed cone $\Gamma_n$ satisfying
\eqref{eq:mc}, use compactly supported distributions with
$\WF(u_n)\subset\Gamma_n$ and the normal distribution topology.
Besides strong distribution seminorms, its microlocal seminorms
have the form
\[
 p_{N,\psi,V}(u_n)
 =\sup_{\zeta\in V}(1+|\zeta|)^N
       |\widehat{\psi u_n}(\zeta)|,
\]
with $(\supp\psi\times V)\cap\Gamma_n=\varnothing$.
For the insertion, the flat ramp permits the kernel to be viewed
as a compact distribution on the surrounding Minkowski space,
including its possible closed support at $t=0$.

This names a source-side continuity question. It does not name
a quantum target topology or prove continuity of $Q$. The
distributional operation results of \cite{BDH} concern specified
wavefront spaces and specified operations; they are not an
automatic completion theorem for a quantum algebra.

Likewise, the causal commutator on linear fields does not uniquely
define multiplication of all distributional polynomial kernels.
The formal causal-Weyl expansion would require repeated
contractions. Even for local quadratic symbols, its second
contraction involves products whose usual wavefront multiplication
criterion is not satisfied. That observation calls for additional
product data; it is not a universal nonexistence theorem.

No state is required merely to ask this admission question.
Conversely, the absence of a selected state supplies neither the
missing product nor an obstruction to every possible state-independent
extension.

\subsection{Gate verdict}

The finite-rank mismatch proves that XI-B's existing operations
alone do not perform admission. Proposition~\ref{prop:shadow}
settles an inconsistent off-shell interpretation. Neither fact
constructs the on-shell admission of
Definition~\ref{def:interface}, and neither excludes every such
admission.

Accordingly, D1--D2 end at \textbf{Unresolved}. This is the governing
release status.

%=====================================================================
\section{D3--D4: the gate is not crossed}
\label{sec:latergates}
%=====================================================================

Since admission has not been constructed, no antibracket on admitted
quantum composites is defined here. XI-B's antifields remain external
slots. In particular, there is no claimed Jacobi identity for an
unnamed bracket.

There is also no constructed $s_1$ or $s_2$. Saying that $s_1$ is
``the variation of $V_{1,g}$'' leaves an essential ambiguity.
In a canonical BV description, an antifield-free interaction changes
the antifield equations through its Hamiltonian vector field;
it does not, by itself, specify new transformations of the fields.
The necessary antifield-dependent terms and their master identity
must be supplied separately.

The same issue applies more strongly to the observable-dependent
tree $\Tcal(f,g)$. It is an output associated with an observation
test. It is not automatically an interaction Hamiltonian defining
one universal differential.

Consequently no assertion of interacting nilpotence is made.
The gauge-fixed bosonic action's breaking, the auxiliary--ghost
completion, and the invariant action's Noether identity remain
distinct. Omitting the Einstein cubic vertex is still a scope
restriction, not a proof that the missing terms of a master identity
vanish.

The next section only tests a necessary leading coefficient of the
proposed closure equation. It does not bypass these gates.

%=====================================================================
\section{A necessary free-order closure condition}
\label{sec:freeclosure}
%=====================================================================

\subsection{The free variation is an actual classical functional}

\begin{proposition}[Free variation and ghost number]
\label{prop:variation}
The bare insertion has ghost number zero. Its copied free variation,
defined on smooth graded configurations, is
\begin{equation}
 \sfree\Bcal_f
 =-\int_{\MT}f\,
       \bigl(L_\eps\RR c\bigr)^\mu
                         \partial_\mu\phi\dd^4x.
 \label{eq:sB}
\end{equation}
It has ghost number one. Moreover,
\begin{equation}
 \sfree\Tcal(f,g)=0
 \label{eq:sT}
\end{equation}
as a classical identity.
\end{proposition}

\begin{proof}
The coordinate operator is continuous and linear on smooth
tensors at fixed width. Apply the chain rule to \eqref{eq:B},
using \eqref{eq:sfree}. The scalar has zero free variation.
This gives \eqref{eq:sB} and the stated ghost numbers.
The tree depends only on $\phi$, so \eqref{eq:sT} follows.
\end{proof}

The integrals exist by smooth Cauchy theory and compact observation
support. This is a classical calculation, not application of an
unconstructed quantum derivation to an admitted element.

For a free ghost, $D\RR c=\frac12\Box c=0$, so
\begin{equation}
 L_\eps\RR c
 =-\ER\bigl(\sig'a_1^{0\cdot}[\RR c]\bigr).
 \label{eq:ghostlayer}
\end{equation}
The ramp layer therefore remains visible to free ghosts with
nontrivial Cauchy data.

\subsection{An explicit free-shell witness}

\begin{theorem}[The bare insertion is not generally free-closed]
\label{thm:freewitness}
For every $m\geq0$ and every $0<\eps<T$, there are admissible
nonnegative observation tests $f$ and a smooth free multiplet
configuration for which
\[
 \sfree\Bcal_f\neq0.
\]
In particular, this variation is not generally a functional
vanishing on all free configurations.
\end{theorem}

\begin{proof}
Work over the Grassmann algebra $\C[\vartheta]/(\vartheta^2)$
with $\vartheta$ odd and nonzero. Set
\[
 \gamma=0,\qquad b=0,\qquad \bar c=0,\qquad
 c=2\vartheta t\,\partial_t,\qquad \phi=\phi_m.
\]
Every equation in \eqref{eq:freeshell} holds. The ghost is an affine
wave solution, and no ghost initial-data restriction in XI-B
excludes it.

Here
\[
 (\RR c)_{00}=2\vartheta,\qquad
 (\RR c)_{0i}=(\RR c)_{ij}=0,
\]
so
\[
 a_1^{00}[\RR c]=-\vartheta,\qquad
 a_1^{0i}[\RR c]=0.
\]
The spatially constant zero-data wave response is therefore
\[
 (L_\eps\RR c)^0
 =\vartheta\int_0^t(t-s)\sig'(s)\dd s
 =\vartheta S_\eps(t),
 \qquad
 S_\eps(t)=\int_0^t\sig(s)\dd s.
\]
The spatial components vanish. Ramp symmetry gives
\begin{equation}
 S_\eps(t)=t-\eps/2,\qquad t\geq\eps.
 \label{eq:S}
\end{equation}
It follows that
\begin{equation}
 \sfree\Bcal_f
 =-\vartheta\int_{\MT}
       f(t,\mathbf x)S_\eps(t)\cos(mt)\dd^4x,
 \label{eq:witness}
\end{equation}
where $\cos(mt)=1$ when $m=0$.

Choose $t_*\in(\eps,T)$ with $\cos(mt_*)\neq0$, and a small time
interval on which it has a fixed nonzero sign. Choose $f\geq0$,
nonzero, compactly supported in that interval and in a spatial ball.
Since $S_\eps>0$ there, the ordinary coefficient of $\vartheta$
in \eqref{eq:witness} is nonzero.

The switch and plateau may be chosen to contain this observation
region while satisfying the frozen support requirements. They do
not enter \eqref{eq:witness}.
\end{proof}

The theorem supplies explicit allowed tests; it does not assert
nonvanishing for every possible fixed $f$. For a particular test,
\eqref{eq:sB} is the necessary condition to examine. A smearing
can annihilate a nonzero field.

Noncompact smooth ghost data are legitimate here, just as in XI-B.
If compact ghost Cauchy data are desired, choose them equal to the
affine data on a ball containing the initial causal shadow of
$\supp f$. Finite propagation makes the resulting ghost identical
to the displayed one throughout the dependence region, leaving
\eqref{eq:witness} unchanged.

\subsection{The conditional quantum obstruction}

\begin{theorem}[Necessary-condition obstruction to bare closure]
\label{thm:conditional}
Fix an observation test from Theorem~\ref{thm:freewitness}.
Suppose a future admission has the following properties:
\begin{enumerate}[label=\textup{(\alph*)}]
\item the leading classical symbol of $Q(\Bcal_f)$ is
$\Bcal_f|_{\Csol}$;
\item the leading differential on that symbol is the copied
free action \eqref{eq:sB};
\item all further corrections have nonnegative formal powers
of $\hbar$ and strictly positive powers of $\kappa$ in the
differential beyond $\sfree$.
\end{enumerate}
Then it cannot satisfy
\[
 \widehat s\,Q(\Bcal_f)=0
\]
through coupling order $\kappa^2$ and quantum order $\hbar^0$.
The same conclusion holds for the physically weighted bare dressing
$\kappa Q(\Bcal_f)$ through coupling order $\kappa^2$.
\end{theorem}

\begin{proof}
For the unweighted insertion, the coefficient at
$\hbar^0\kappa^0$ of the proposed identity is
$\sfree\Bcal_f|_{\Csol}=0$. Theorem~\ref{thm:freewitness}
contradicts it.

For the weighted insertion, the first coefficient is instead
$\hbar^0\kappa^1$, and it is the same nonzero variation.
Positive-order corrections to the differential cannot alter this
coefficient. Nor can an order-$\kappa^2$ tree correction alter it;
its copied free variation is zero by \eqref{eq:sT}.
\end{proof}

This theorem excludes every prospective admission \emph{combined
with these closure interfaces} for the witness tests. It does not
exclude admission alone. It also does not construct an interacting
differential or claim a failure of its nilpotence.

\subsection{Why a complete observable is a different problem}

Let $F_f=\int f\phi$. For a complete candidate beginning
$F_f+\kappa\Bcal_f$, the first closure coefficient could involve
both
\[
 \sfree\Bcal_f
 \quad\text{and the first interacting action on }F_f.
\]
Theorem~\ref{thm:conditional} does not exclude such a cancellation.
The production sheet's equation for $\Bcal_f$ alone has no leading
$F_f$ term on which that correction can act.

There is a further Cauchy-data distinction. For a compact
plateau parameter $c=2\xi$, X-A's preflight gives
$L_\eps\RR c=c/2$. A proposed scalar co-transformation proportional
to $c\cdot\partial\phi/2$ can then participate in a complete
observable's coefficient cancellation.

The free witness above is not such a plateau parameter.
Its response is $S_\eps(t)$, not $t$, because the coordinate
data are frozen while the free ghost has nonzero initial time
derivative. X-A never asserted covariance for those ghost data.
Thus neither its classical theorem nor ghost number bookkeeping
proves the requested bare closure.

%=====================================================================
\section{The inherited held-fixed-metric defect}
\label{sec:defect}
%=====================================================================

The preceding obstruction is at free order. It must not be confused
with XI-C's second-order background defect. We retain the latter
and reconstruct its classical notation from X-A, rather than
assuming that an unsupplied quantum coefficient exists.

\subsection{Classical scalar completion}

For a smooth tensor $r$ put
\[
 b_1[r]=\tfrac12\tr_\eta r,\qquad
 \Vone[r]u
 =\partial_\mu(ga_1^{\mu\nu}[r]\partial_\nu u)
                  +gm^2b_1[r]u.
\]
For a free scalar $\phi$ define
\begin{equation}
 \Aone[r;\phi]
 =\int f\bigl(-\ERm\Vone[r]\phi
                         -\Xone[r]\cdot\partial\phi\bigr)\dd^4x.
 \label{eq:A1}
\end{equation}
This is the scalar-only first coefficient of X-A.

Let $H$ and $k$ be the first and second metric coefficients.
Use X-A's density coefficient $a_2[H]$ and define
\[
 \Ueps[H]u
 =\partial_\alpha(\sig a_1^{\alpha\beta}[H]\partial_\beta u),
\]
\[
 \Xtwo^\mu[H,k]
 =-\ER\left(
   \Ueps[H]\Xone^\mu[H]
   +\partial_\alpha\bigl(
      \sig(a_1^{\alpha\mu}[k]+a_2^{\alpha\mu}[H])\bigr)\right).
\]
Let $\Vtwo[H,k]$ be X-A's switched second scalar operator and set
\[
 \rho_1=-\ERm\Vone[H]\phi,\qquad
 \rho_2=\ERm\Vone[H]\ERm\Vone[H]\phi
                         -\ERm\Vtwo[H,k]\phi.
\]
The completed classical coefficient is
\begin{align}
 \Atwo^{\mathrm{sc}}[H,k;\phi]
 =\int f\Bigl[
 &\rho_2-\Xone[H]\cdot\partial\rho_1
 +(\Xone[H]\cdot\partial\Xone[H]-\Xtwo[H,k])
                                      \cdot\partial\phi
 \nonumber\\
 &+\tfrac12\Xone^\mu[H]\Xone^\nu[H]
                       \partial_\mu\partial_\nu\phi
 \Bigr]\dd^4x.
 \label{eq:A2}
\end{align}
All these expressions are classical evaluations on actual smooth
configurations.

\begin{lemma}[Dependence on the second metric coefficient]
\label{lem:affine}
For every smooth $r$,
\[
 \Atwo^{\mathrm{sc}}[H,k+r;\phi]
 -\Atwo^{\mathrm{sc}}[H,k;\phi]
 =\Aone[r;\phi].
\]
\end{lemma}

\begin{proof}
The change in $\Xtwo$ is $\Xone[r]$. The change in
$\rho_2$ is $-\ERm\Vone[r]\phi$. No other displayed term
depends on $k$.
\end{proof}

Now set $H=h+\gamma$ and
\[
 \Atwo^{\mathrm{cand}}[h,k,\gamma;\phi]
 =\Atwo^{\mathrm{sc}}[h+\gamma,k;\phi]-\Tcal(f,g)[\phi].
\]
Under the admitted classical transformation
\[
 h'=h+K\xi,\qquad
 k'=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi K\xi,
\]
hold $\gamma$ fixed. Full transformation of $H$ would require
the additional second coefficient $\Lie_\xi\gamma$.
X-A's complete classical intertwiner and
Lemma~\ref{lem:affine} therefore give
\begin{equation}
 \Atwo^{\mathrm{cand}}[h',k',\gamma;\phi]
 -\Atwo^{\mathrm{cand}}[h,k,\gamma;\phi]
 =-\Aone[\Lie_\xi\gamma;\phi].
 \label{eq:forward}
\end{equation}
The tree has no external $h,k,\gamma$ argument and contributes
no variation under this operation. The backward difference has
the opposite sign.

\subsection{The nonzero pairing is retained}

Take $m=0$ and the free configurations
\[
 \gamma=t\eta,\qquad
 \phi=t^2+\tfrac13|\mathbf x|^2.
\]
They satisfy the reduced metric and scalar wave equations.
Set $b=D\gamma$, with ghosts and antighost zero.

Choose nonnegative nonzero
$a\in C_c^\infty((\eps,T))$ and, on a sufficiently large causal
cylinder, $\xi=a(t)\partial_t$. Multiply it by a spatial cutoff
whose derivatives lie outside the observation dependence region.
Choose the plateau to contain its compact support, and observe
after the support of $a$. Finite propagation permits the following
calculation in that cylinder.

For $r=\Lie_\xi\gamma$,
\[
 r_{00}=a+2ta',\qquad r_{ij}=-a\delta_{ij},\qquad r_{0i}=0,
\]
\[
 a_1^{00}[r]=a-ta',\qquad
 a_1^{ij}[r]=-(a+ta')\delta^{ij},\qquad D^0r=ta''.
\]
There is no ramp-layer contribution. Direct zero-data integration
gives
\[
 \Xone^0[r]=ta(t)-2\int_0^t a(s)\dd s,\qquad
 \Xone^i[r]=0.
\]
Also,
\[
 \Vone[r]\phi=-4ta'-2t^2a'',\qquad
 -\ER\Vone[r]\phi
 =2t^2a(t)-4\int_0^t s\,a(s)\dd s.
\]
Consequently
\begin{equation}
 \Aone[\Lie_\xi\gamma;\phi]
 =4\int f(t,\mathbf x)
            \int_0^t(t-s)a(s)\dd s\,\dd^4x>0
 \label{eq:backgroundwitness}
\end{equation}
for nonnegative nonzero observation tests in the stated region.

This is an evaluation on a free multiplet configuration.
The defect is therefore not generally an equation zero.
A cancellation of \eqref{eq:forward} must supply the omitted
co-transformation, not replace $\Lie_\xi\gamma$ by $\RR c$.
Those transformations have different dependence on the metric.

Equation \eqref{eq:backgroundwitness} concerns the completed
second-order classical candidate. Equation \eqref{eq:witness}
concerns the free variation of the bare insertion. Neither is
a quantum Ward identity, and neither can replace the other.

%=====================================================================
\section{Diagnostics and the exclusions that remain}
\label{sec:diagnostics}
%=====================================================================

\paragraph{Scalar reduction.}
Replace the metric by a prescribed classical background $r$
before taking the scalar response. Its first response is
$-\ERm\Vone[r]\phi$, and its completed second coefficient is
\eqref{eq:A2}. Setting that background to zero gives the zero
specialization of those response maps. Setting an external quantum
metric configuration to zero after the graviton response does
not remove $\Tcal$, which has no such external argument.

\paragraph{Pure gauge.}
The correct identities are
\[
 G_{\mathrm{lin}}[K\xi]=0,\qquad D(K\xi)=\Box\xi,
\]
whereas
\[
 (-\Box\PP)(K\xi)=-\PP K(\Box\xi)
\]
need not vanish. A pure-gauge background is not a vanishing test
for a tree sourced by scalar stress.

\paragraph{The empty Einstein jet.}
The response $r_g$ and its divergence exist. Above the ramp,
\[
 D r_g[\phi_0]
 =-\tfrac12\ER\bigl((\partial_\mu g)T_0^{\mu\cdot}[\phi_0]\bigr)
\]
for a free scalar. The named input gives a nonzero source.
The tree propagates this de Donder violation; it does not set it
to zero or restore the ungauge-fixed Einstein equation.

\paragraph{The loop.}
No scalar Hadamard bisolution, covariance matrix, or retarded
quantum product is selected. No mixed-loop expression is multiplied,
extended, or subtracted. Missing loop data are an unresolved stop,
not a no-go theorem for the classical tree.

\paragraph{The gravitational interaction.}
No cubic Einstein vertex is contracted. Its classical role in
nonlinear identities has not disappeared. A future interacting
master identity must account for the appropriate gravitational,
scalar, ghost, and antifield terms.

\paragraph{Boundary and width.}
The initial coordinate data remain frozen at $\Sigma$. No
boundary algebra, boundary charge, reflected-residue subtraction,
or $\eps\to0$ limit is constructed.

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.52\linewidth}X@{}}
\toprule
Gate or statement & Status\\
\midrule
XI-C classical tree and physical sign & Retained:
$R_{\gamma,\mathrm{tree}}=-\Tcal$\\
Compact microcausal source symbols & Retained; not quantum admission\\
Exact off-shell shadow of all XI-B generators &
Proved incompatible with its equation quotient\\
On-shell admission interface & Explicitly formulated\\
Quantum extension, product, and admission map &
Unresolved; no general nonexistence claim\\
Ghost number of $\Bcal_f$ & Zero\\
Free variation of $\Bcal_f$ & Explicit; generally nonzero\\
Free-shell closure witness & Evaluated in
\eqref{eq:witness}\\
Bare closure under a compatible prospective admission &
Conditionally obstructed for the witness tests\\
Antibracket and its Jacobi identity & Not constructed\\
Interacting differential and nilpotence & Not constructed\\
Held-fixed-metric forward defect &
$-\Aone[\Lie_\xi\gamma;\phi]$\\
Inherited positive witness pairing & Retained in
\eqref{eq:backgroundwitness}\\
Mixed loop, state, sharp limit & Not opened\\
Empty fixed-input Einstein jet & Unchanged\\
\bottomrule
\end{tabularx}
\end{center}

The overall ending is \textbf{Unresolved}, because D2 has not been
constructed or excluded in its consistent on-shell formulation.
The paper does not promote a classical symbol to a quantum class.

At the same time, the bare-closure request has a concrete lower-order
problem: for the diagnostic tests, an admission with the correct
free classical action cannot make $\Bcal_f$ closed by positive-order
coupling corrections. This conclusion is independent of a chosen
state and precedes a loop calculation.

A successor therefore receives three distinct tasks:
\begin{enumerate}
\item construct a quantum admission with an explicit distinction
between off-shell representatives and on-shell symbols;
\item define the antibracket, antifield-dependent interaction, and
differential on a named domain;
\item identify the complete observable whose coefficient closure
is intended, instead of requiring closure of its bare dressing
alone.
\end{enumerate}
It also receives both evaluated witnesses. Neither may be erased
by relabelling a background pullback as the free ghost rule.

\subsection*{Conclusion}

The justified release is
\[
 \boxed{
 \begin{gathered}
 \text{quantum admission unresolved},\\
 \text{bare free-order closure condition generally fails},\\
 \text{classical tree, loop boundary, and empty jet remain distinct}.
 \end{gathered}}
\]
A constructed quantum class would require an additional admission
theorem and a different, fully specified closure problem.

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

\paragraph{Proof scope.}
The off-shell-shadow incompatibility, free-shell witness,
necessary-condition obstruction, and classical defect evaluation
are proved in the displayed conventions. The quantum admission
interface is formulated but not solved.

\paragraph{Dependencies.}
The supplied X-A is used for classical Cauchy prescriptions and
plateau intertwiners. XI-A, XI-B, and XI-C supply the signed action,
free multiplet, limited algebraic domain, and classical tree.
The scalar completion is written explicitly, so an unsupplied
X-B-v3 or X-C-v3 theorem is not required as an additional premise.

\paragraph{No quantum certification.}
No admitted composite, interacting BRST cohomology class, state,
renormalized retarded product, or loop extension is claimed.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant check,
executed symbolic computation, independent referee review, or
journal acceptance is asserted. All nonzero 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 the classical prescription, Cauchy facts, and complete
plateau intertwiners.

\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}.
Used for the signed action and Einstein-jet obstruction.

\bibitem{XIB}
N.~Cheung,
\emph{Landmark XI-B: The free de Donder multiplet for the ramp
Einstein--scalar sheet: causal propagators, residual gauge,
and the layer dressing}.
Supplied manuscript \texttt{XI-B.tex}.
Used for the free equations, causal algebra, and differential.

\bibitem{XIC}
N.~Cheung,
\emph{Landmark XI-C: One fixed-regulator graviton tree for the
ramp relational scalar: the retarded symbol, its normalization,
and the boundary of the free-algebra handoff}.
Supplied manuscript \texttt{XI-C.tex}.
Used for the classical tree, kernel bounds, sign, and qualified
admission handoff.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory}.
arXiv:1208.1428v2, 2013.
Background for the distinction between regular polynomial
generators, microcausal functionals, and quantum products.

\bibitem{Rejzner}
K.~Rejzner,
\emph{Batalin--Vilkovisky formalism in locally covariant field theory}.
Ph.D. thesis, arXiv:1111.5130v2, 2013.
Background for separating antifield slots, BV structures, and
quantum differential constructions; no such construction is imported.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
``Continuity of the fundamental operations on distributions having
a specified wave front set (with a counter example by Semyon Alesker),''
\emph{Studia Mathematica} \textbf{232} (2016), 201--226.
arXiv:1409.7662.
Used only to identify the separate normal-topology distributional
continuity question; not as a quantum-completion theorem.

\end{thebibliography}

\end{document}