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% IX-C.tex
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% Release outcome:
% UNRESOLVED GATE, with proved partial-stage, factorization, and
% coefficientwise coherence reductions.
%
% This manuscript does not claim that the missing binary maps exist.
% It also does not turn their absence into a no-go theorem.
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% Analytical dependence:
% The VII-D finite-part theorem is an explicitly stated premise,
% available here through the restatement in IX-B-v2.
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% No proof-assistant verification or independent review is claimed.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\title{\textbf{Landmark IX-C}\\[0.4em]
Actual-functional factorization and coefficientwise coherence\\
for one sharp ordered word}
\author{}
\date{}

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

\begin{abstract}
We formulate the binary coherence problem for the ordered word
\(B_f,W_g,B_{f'}\) on the actual-functional carrier of IX-A.
Both admitted first-stage products are fixed, including the reverse
contraction and its sign. We identify the three one-edge outputs
and give the labelled expressions for the three two-edge outputs,
without assuming that a labelled expression defines a distributional
product. We prove exact algebraic descent criteria for second-stage
coefficient maps and for the calibrated scalar assignment
\(u_\mu^\rho+cr\). The scalar relation tests restrict the real parent
parameter to an empty set, a singleton, or all of \(\R\).
We also prove the coefficientwise equations required for associativity
through order \(\hbar^3\), and distinguish observation compatibility,
involution, presentation independence, and bulk-relative agreement.

The release outcome is an unresolved gate. Neither sharp second
multiplication is constructed, and no incompatibility excluding all
such multiplications is proved. The completed results are the admitted
partial stages and the factorization and coherence reductions.
The numerical triangle calibration remains conditional on the
explicit VII-D analytical premise restated in IX-B-v2.
\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) {One-word coherence};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Necessary tests};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Binary factorization};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A missing map remains an unresolved gate.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Coherence tests are not a factorization. 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-IX-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {One-word coherence};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Binary factorization};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {A missing map remains an unresolved gate.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Coherence tests are not a factorization. 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-IX-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {One-word coherence};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Necessary tests};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Binary factorization};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {A missing map remains an unresolved gate.};
\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-IX-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {One-word coherence};
\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) {Binary factorization};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Necessary tests. A missing map remains an unresolved gate.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{One-word coherence is not Binary factorization. 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-IX-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: One-word coherence};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Necessary tests};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Binary factorization};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {A missing map remains an unresolved gate.};
\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-IX-C}
\end{figure}

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


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



\scopebox{Release claim and proof boundary}{%
This manuscript proves reductions and necessary tests for one-word
binary coherence. It does not claim a completed binary factorization.

A missing map is an unresolved gate, not a no-go theorem.
The unresolved outcome is stated explicitly for both associations.
No ternary numerical finite part is renamed as a binary product.}

\tableofcontents

%=====================================================================
\section{Scope, release outcomes, and interfaces}
\label{sec:scope}
%=====================================================================

The selected word is
\[
A=B_f,\qquad W=W_g,\qquad C=B_{f'}.
\]
The target is a bulk-relative binary factorization of this word
through order \(\hbar^3\), with fully contracted joint-test kernel
\(u_\mu^\rho+cr\), followed by associativity, involution, and
presentation independence on the sector actually constructed.

There are three logically different release outcomes:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \emph{Construction}: the named maps exist and satisfy the
declared interfaces.
\item \emph{Proved obstruction}: an explicit incompatibility excludes
every map satisfying those interfaces.
\item \emph{Unresolved gate}: a required map, descent proof, admission
check, or coherence identity has not been established.
\end{enumerate}
This manuscript has outcome \textup{(iii)}. Its proved results identify
what is already available and what must still be established.

Failure of one candidate excludes that candidate only. Conversely,
the absence of a construction in the supplied inputs does not prove
that construction impossible.

\subsection{The fixed interface}

\begin{requirement}[One-word binary interface]
\label{req:interface}
A proposed construction must specify which of the following clauses
it verifies:
\begin{enumerate}[label=\textbf{C-I\arabic*.}]
\item Every polynomial output belongs to \(\Padm\), including the
metric derivatives and directional descendants required by IX-A.
\item The order-zero coefficient is classical pointwise multiplication.
\item On every specified overlap with the old bulk domain, the
coefficient maps agree with \(\star_H\).
\item The first-stage left coefficient is the locked
\(\mathcal Q_{f,g}\); the right coefficient is its reverse
contraction for the same covariance.
\item The fully contracted joint-test kernel is \(u_\mu^\rho+cr\),
with \(c\in\R\). Declared observation compatibility transports that
same parameter.
\item No subtraction changes an already fixed old bulk product.
\item Unchanged regular approximants are not required to converge to
the sharp insertions in \(\taus\).
\item The maps respect every finite linear relation between actual
presentations of their arguments.
\item Every continuity assertion has a named domain and a proof.
Fixed-support test-family continuity is distinct from continuity or
hypocontinuity on arbitrary carrier stages.
\end{enumerate}
\end{requirement}

Clause C-I8 is the factorization condition. A test-labelled
assignment that fails this clause is not a binary map on \(\Padm\).

Clause C-I9 does not follow from C-I8. Algebraic descent does not
automatically imply continuity in the subspace topology inherited
from \(\taus\).

\subsection{Excluded subjects}

No Einstein equation, backreaction, retarded product, switched
realization, conical identification, or all-graph catalogue is
constructed. No algebra on all of \(\Padm\) is claimed.
The VII-C obstruction is not repaired. IX-D and the separate G2
gate are not opened.

%=====================================================================
\section{Inherited data and the separation from V}
\label{sec:data}
%=====================================================================

\subsection{The actual-functional carrier}

Let
\[
M=\R^{1+3},\qquad M_+=\{t>0\},\qquad X=M_+^3,
\qquad \eta=\operatorname{diag}(-1,1,1,1).
\]
Write \(x=(t,\mathbf x)\), \(y=(s,\mathbf y)\), and
\(x'=(t',\mathbf x')\). Set
\[
\Box=-\partial_t^2+\Delta_{\mathbf x},\qquad
(Ch)^\mu=\partial_\nu h^{\mu\nu}
-\frac12\partial^\mu(\eta^{\alpha\beta}h_{\alpha\beta}),
\qquad L=\Ezero C.
\]
The operator \(\Ezero\) is the zero-value, zero-first-time-derivative
Cauchy inverse on \(\{t=0\}\).

Configurations \((h,\phi)\) are smooth and off shell. The selected
functionals are
\begin{equation}
B_f[h,\phi]=-\int f(x)(Lh)^\mu(x)\partial_\mu\phi(x)\dd^4x,
\qquad
W_g[h]=\int g(y)h_{00}(y)^2\dd^4y.
\label{eq:BW}
\end{equation}

IX-A defines \(\Padm\) as a space of actual finite polynomial
functionals with compact symmetric distributional kernels.
Its admission criterion is
\begin{equation}
\bigl(\Ddir_{\boldsymbol\eta}F\bigr)'_h(\psi)
\in\Slin=\Mlin+\Qlin
\quad
\text{for every smooth \(\psi\) and finite direction list
\(\boldsymbol\eta\)}.
\label{eq:admission}
\end{equation}
Here
\[
q_a=L^*a=C^*(\theta\GA a),\qquad
\Qlin=\{q_a:a\in C_c^\infty(M_+;T^*M)\},
\]
and \(\Mlin\) consists of compact linear metric kernels with no
nonzero nonspacelike wavefront covectors.

The topology \(\taus\) records the strong compact-distribution
coefficients and every intrinsic boundary-trace coordinate
\[
\mathsf b\!\left(
  \bigl(\Ddir_{\boldsymbol\eta}F\bigr)'_h(\psi)
\right).
\]
We inherit the carrier, intrinsic trace, classical closure,
directional closure, and Hausdorff topology from IX-A
\cite[\texttt{def:Padm}, \texttt{lem:charge},
\texttt{thm:closure}, \texttt{def:topology}]{IXA}.
The truncated coefficient module is
\[
\Vmod=\Padm\otimes_\C\C[\hbar]/(\hbar^4),
\]
not a constructed quantum algebra
\cite[\texttt{eq:module}]{IXA}.

\subsection{Covariance and selected coefficients}

The Wick convention is
\[
F\star_HG
=\sum_{j\ge0}\frac{\hbar^j}{j!}
\pair{F^{(j)}}{H^{\otimes j}G^{(j)}}.
\]
The fixed covariance blocks are
\[
H_{hh\,\alpha\beta\rho\sigma}
=P_{\alpha\beta\rho\sigma}\Delta_0^+,\qquad
H_{\phi\phi}=\Delta_m^+,\qquad
H_{h\phi}=H_{\phi h}=0,
\]
with
\[
P_{\alpha\beta\rho\sigma}
=\eta_{\alpha(\rho}\eta_{\sigma)\beta}
-\frac12\eta_{\alpha\beta}\eta_{\rho\sigma}.
\]
We inherit
\[
H_{ww}=0,\qquad w=Lh,
\]
and the mixed component
\begin{equation}
\begin{split}
K(x,y)
={}&\int_{\R^3}\frac{\dd^3p}{(2\pi)^3}
e^{ip\cdot(\mathbf x-\mathbf y)}
\bigg[
\left(-\frac{t}{4|p|}+\frac{i}{8|p|^2}\right)
e^{-i|p|(t-s)}
\\
&\hspace{12em}
-\frac{i}{8|p|^2}e^{i|p|(t+s)}
\bigg].
\end{split}
\label{eq:K}
\end{equation}
These are fixed input identities, not new calculations
\cite[\texttt{lem:tensor}, \texttt{eq:K}]{IXA}
\cite[\texttt{eq:tensor}, \texttt{eq:K}]{IXB}.

IX-B fixes
\begin{equation}
\mathcal Q_{f,g}
=-2\int f(x)g(y)K(x,y)
\,\partial_t\phi(x)h_{00}(y)\dd^4x\dd^4y.
\label{eq:Q}
\end{equation}
Its existence and admission are inherited from IX-A; its sign and
multiplicity are fixed in IX-B
\cite[\texttt{prop:knownoutputs}, item 3]{IXA}
\cite[\texttt{prop:partial}, \texttt{eq:Qpartial}]{IXB}.

\subsection{Explicit analytical premise}

\begin{assumption}[VII-D calibration, as restated in IX-B-v2]
\label{ass:calibration}
For the prescribed radial regulator placement,
\[
T_\eps^\rho(x,y,x')
=2K_\eps(x,y)\overline{K_\eps(x',y)}D_\eps(x-x').
\]
There are distributions \(r,u_\mu^\rho\in\Dtest'(X)\) such that
\[
u_\mu^\rho=U_m+(\kappa_\rho+\log\mu)r,
\]
and, for every compact \(K_0\subset X\),
\begin{equation}
\left|
\pair{T_\eps^\rho-\log((\mu\eps)^{-1})r-u_\mu^\rho}{\Phi}
\right|
\le C_{K_0,\rho,m}\eps(1+|\log\eps|)
p_{K_0,64}(\Phi).
\label{eq:finitepart}
\end{equation}
The estimate includes compacts meeting \(\Rel\cap\Delta_{13}\).

The residue is
\[
r=c_0\Psi_*\nu,\qquad c_0=\frac1{64(2\pi)^3},
\]
where
\[
\Psi(\mathbf z,\mathbf n,t,s,t')
=
\bigl((t,\mathbf z+t\mathbf n),
(s,\mathbf z-s\mathbf n),
(t',\mathbf z+t'\mathbf n)\bigr),
\]
and
\[
\dd\nu=\dd^3\mathbf z\,\dd\Omega(\mathbf n)\,
\dd t\,\dd s\,\dd t'.
\]
Thus \(\Rel=\supp r\).

For the prescribed regulator family,
\[
u_\mu^{\rho_a}-u_\mu^{\rho_b}=c_{ab}r,\qquad c_{ab}\in\R.
\]
The parent distributions satisfy
\[
(u_\mu^\rho)^\dagger=u_\mu^\rho,\qquad r^\dagger=r,
\]
where dagger conjugates and exchanges \(x,x'\).
\end{assumption}

These statements are the analytical premise of VII-D, explicitly
restated in IX-B
\cite[\texttt{ass:VIID}, \texttt{eq:adjoint}]{IXB}
\cite[\texttt{thm:finitepart}, \texttt{thm:conversion};
Section ``Adjoint symmetry'']{VIID}.
No independent verification of VII-D is asserted here.

Fix \(\rho,\mu\). The permitted scalar calibration is
\begin{equation}
\Cal(f,g,f')
=
\pair{u_\mu^\rho+cr}{f\otimes g\otimes f'},
\qquad c\in\R.
\label{eq:Cal}
\end{equation}
This is an imposed parent class. It is not a classification of
all possible graph extensions or binary-product ambiguities.

\subsection{V is a separation source}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}}
\toprule
Object in V & Separation from the selected word\\
\midrule
External \(R_{1,1}\) &
One scalar insertion and a switched external vertex; its output is
a linearly smeared free field.\\
Oscillator coefficient &
A switched mixed bilinear in a finite-mode representation.\\
Newton--wave cubic &
A different composed Wick form, with spatial Newton-transfer
singularities.\\
Selected IX-C word &
Two sharp insertions and one unsmeared quadratic metric observable;
its calibrated triangle has reflected relation \(\Rel\).\\
\bottomrule
\end{tabularx}
\end{center}

These distinctions follow from the definitions in the supplied
V manuscript \cite[Sections ``The external-metric first retarded
product'', ``The one-oscillator mixed insertion'', and
``Contrast: the Newton--wave cubic form'']{V}.
No theorem from V enters a proof below. In particular, its
subcritical extension uniqueness and switched bilinear identity
are not premises for triangle factorization or G2.

%=====================================================================
\section{Both admitted partial stages}
\label{sec:partial}
%=====================================================================

Let \(\mathcal T=C_c^\infty(M_+;\C)\), and define
\[
\Bspace=\Span_\C\{B_f:f\in\mathcal T\},
\qquad
\Wspace=\Span_\C\{W_g:g\in\mathcal T\}.
\]

\begin{lemma}[Injectivity of the generating test maps]
\label{lem:injective}
The maps \(f\mapsto B_f\) and \(g\mapsto W_g\) are injective.
\end{lemma}

\begin{proof}
Injectivity of \(f\mapsto B_f\) is IX-A's result
\cite[\texttt{cor:notbulk}]{IXA}.

Suppose \(W_g=0\). Then \(\int g v^2=0\) for every smooth
scalar function \(v\), by taking \(h_{00}=v\).
Polarization gives \(\int g uv=0\) for every smooth \(u,v\).
Taking \(v=1\) proves that \(g\) is the zero distribution,
hence the zero test.
\end{proof}

\begin{proposition}[Reverse coefficient and its sign]
\label{prop:reverse}
The reverse ordered coefficient is
\begin{equation}
\mathcal Q^R_{g,f'}
=-2\int g(y)f'(x')\overline{K(x',y)}
h_{00}(y)\partial_{t'}\phi(x')\dd^4y\dd^4x'.
\label{eq:QR}
\end{equation}
It belongs to \(\Padm\), and
\begin{equation}
\overline{\mathcal Q_{f,g}}
=\mathcal Q^R_{\bar g,\bar f}.
\label{eq:Qadjoint}
\end{equation}
\end{proposition}

\begin{proof}
The real tensor coefficients in \(H_{hh}\), together with
\(\overline{\Delta_0^+(z)}=\Delta_0^+(-z)\), give reversal
of the mixed covariance by kernel conjugation. The real
zero-Cauchy operator preserves this relation. Therefore the
reverse mixed component is \(\overline{K(x',y)}\).

The two slots of \(h_{00}^2\) supply multiplicity \(2\).
There is one insertion \(B_{f'}\), so its defining minus sign
gives \(-2\). This proves \eqref{eq:QR}.
The formula immediately gives \eqref{eq:Qadjoint}.
Admission follows from IX-A's selected reverse-output result,
or from its conjugation closure applied to \eqref{eq:Qadjoint}
\cite[\texttt{prop:knownoutputs}, item 3;
\texttt{thm:closure}]{IXA}.
\end{proof}

Define actual-functional subspaces
\[
\PLspace=\Span_\C\{B_fW_g\},\qquad
\QLspace=\Span_\C\{\mathcal Q_{f,g}\},
\]
\[
\PRspace=\Span_\C\{W_gB_{f'}\},\qquad
\QRspace=\Span_\C\{\mathcal Q^R_{g,f'}\},
\]
and
\begin{equation}
\Lspace=\PLspace\oplus\QLspace,\qquad
\Rspace=\PRspace\oplus\QRspace.
\label{eq:stages}
\end{equation}
The sums are direct because the two summands are homogeneous of
field degrees four and two. The underlying pointwise spaces
\(\PLspace,\PRspace\) coincide when the test ranges coincide.
We retain the ordered notation to distinguish their roles.
Likewise, their common elements must not be assigned inconsistent
values by any future product.

\begin{theorem}[Admitted first-stage products]
\label{thm:first}
The formulas
\begin{equation}
B_f\star_1W_g=B_fW_g+\hbar\mathcal Q_{f,g},
\qquad
W_g\star_1B_{f'}=W_gB_{f'}+\hbar\mathcal Q^R_{g,f'}
\label{eq:first}
\end{equation}
define bilinear maps on \(\Bspace\times\Wspace\) and
\(\Wspace\times\Bspace\), respectively, with values in \(\Vmod\).
They satisfy classical normalization, the locked first-stage
normalization, and reversal under complex conjugation.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:injective} identifies each element of
\(\Bspace\) and \(\Wspace\) with its unique test. The formulas
are bilinear in those tests and therefore are well defined
on the named actual-functional spaces.

Classical closure places \(B_fW_g\) and \(W_gB_{f'}\) in
\(\Padm\). The selected contractions are admitted by the inherited
result and Proposition~\ref{prop:reverse}. Thus both formal
outputs belong to \(\Vmod\). Their order-zero term is the
pointwise product. Equation~\eqref{eq:Qadjoint} proves reversal.
No additional binary contraction is possible for these pairs:
each sharp insertion has only one metric slot and no middle
scalar slot with which its scalar slot could contract.
\end{proof}

\begin{proposition}[Fixed-support continuity of the first stage]
\label{prop:firstcontinuity}
For fixed compact observation supports, both maps in
\eqref{eq:first} depend jointly continuously on their tests,
with target \(\Vmod\) carrying the inherited topology.
This is a test-family statement, not a quantum hypocontinuity
theorem on \(\Padm\).
\end{proposition}

\begin{proof}
The pointwise terms have the fixed-support test-family continuity
proved in IX-A \cite[\texttt{thm:admission}]{IXA}.

For the contraction term, its compact coefficient kernel is
obtained from \(f(x)g(y)K(x,y)\) by a fixed scalar-slot
derivative and bundle placement. On fixed compact supports,
multiplication of a fixed distribution by varying smooth
coefficients and fixed finite differentiation are continuous
into the strong compact-distribution topology. This follows
directly by applying the distribution to bounded smooth tests
and using its finite-order estimate on the common compact set.

The proof of IX-A's selected-output admission shows that every
first metric derivative of this contraction, at a smooth
configuration and after directional differentiation, is a smooth
compact kernel \cite[\texttt{prop:knownoutputs}, item 3]{IXA}.
Its boundary coordinates therefore vanish. The coefficient
continuity consequently gives continuity in \(\taus\).
Conjugation gives the same result for the reverse contraction.
\end{proof}

Theorem~\ref{thm:first} constructs only the two first stages.
It does not define multiplication of their outputs by another
sharp insertion.

%=====================================================================
\section{The six placements and the nonconstant outputs}
\label{sec:placements}
%=====================================================================

Put
\[
S_{\mu\nu}(x,x')
=\partial_\mu^x\partial_\nu^{x'}\Delta_m^+(x-x').
\]
IX-A admits the scalar-contracted sharp self-product
\begin{equation}
\mathcal T_{f,f'}[h]
=\int f(x)f'(x')S_{\mu\nu}(x,x')
w^\mu(x)w^\nu(x')\dd^4x\dd^4x'.
\label{eq:self}
\end{equation}
This is inherited, not a second-stage closure theorem
\cite[\texttt{prop:knownoutputs}, item 2]{IXA}.

\subsection{The one-edge terms}

Define
\begin{equation}
\Graph_{12}=\mathcal Q_{f,g}B_{f'},\qquad
\Graph_{23}=B_f\mathcal Q^R_{g,f'},\qquad
\Graph_{13}=\mathcal T_{f,f'}W_g.
\label{eq:oneedges}
\end{equation}

\begin{proposition}[Existence and admission of the one-edge outputs]
\label{prop:oneedges}
All three functionals in \eqref{eq:oneedges} exist and belong to
\(\Padm\). They are homogeneous of field degree four.
\end{proposition}

\begin{proof}
Each factor in \eqref{eq:oneedges} is already admitted.
Classical pointwise closure in IX-A gives the products and their
full admission criterion, including directional descendants
\cite[\texttt{thm:closure}]{IXA}.
The degree is the sum of the two residual degrees.
No new singular-variable multiplication is used in this argument.
\end{proof}

\subsection{The two-edge expressions}

The following are labelled candidate expressions:
\begin{align}
\Graph_{12,23}^{\mathrm{lab}}
={}&2\int f(x)g(y)f'(x')
K(x,y)\overline{K(x',y)}
\partial_t\phi(x)\partial_{t'}\phi(x')
\dd^4x\dd^4y\dd^4x',
\label{eq:g1223}\\
\Graph_{12,13}^{\mathrm{lab}}
={}&2\int f(x)g(y)f'(x')
K(x,y)S_{0\nu}(x,x')
h_{00}(y)w^\nu(x')
\dd^4x\dd^4y\dd^4x',
\label{eq:g1213}\\
\Graph_{13,23}^{\mathrm{lab}}
={}&2\int f(x)g(y)f'(x')
\overline{K(x',y)}S_{\mu0}(x,x')
w^\mu(x)h_{00}(y)
\dd^4x\dd^4y\dd^4x'.
\label{eq:g1323}
\end{align}
The superscript ``lab'' means that the expressions identify the
desired contraction placements. It does not assert the existence
of the indicated distributional products.

\begin{proposition}[Signs, multiplicities, and degrees]
\label{prop:placementcount}
At positive cutoff, \eqref{eq:g1223}--\eqref{eq:g1323}, with the
correspondingly regulated kernels and residual fields, are the
three two-edge placements. Each has sign \(+1\) from the two
insertion signs and assignment multiplicity \(2\).
Their residual field degree is two.
The fully contracted placement has coefficient
\[
2K_\eps(x,y)\overline{K_\eps(x',y)}D_\eps(x-x').
\]
\end{proposition}

\begin{proof}
The allowed nonzero edges are the sharp--middle metric edge
\(12\), the scalar edge \(13\), and the middle--sharp metric
edge \(23\). Each choice of two leaves exactly two field slots.

For \(12+23\), both middle slots are used, leaving the two scalar
slots. There are two assignments of the middle slots.
For \(12+13\), either middle slot can be used on edge \(12\);
the remaining middle slot and third-vertex metric slot survive.
The case \(13+23\) is the exchanged case.
Both sharp insertions contribute one minus sign, so all three
coefficients have positive sign.

The inherited tensor cancellation selects the zero vector
component on every sharp--middle metric edge. This gives
\(S_{0\nu}\) and \(S_{\mu0}\) in the latter two placements.
Full contraction uses both metric edges and the scalar edge,
giving the inherited triangle and multiplicity
\cite[\texttt{prop:unique}, \texttt{eq:T}]{IXB}.
\end{proof}

\begin{remark}[Existence and admission are separate]
\label{rem:treeboundary}
For \eqref{eq:g1223}, a compact scalar-only polynomial kernel,
if established, would pass the metric admission test because
all its metric derivatives are zero. The existence of that
kernel must nevertheless be proved.

For \eqref{eq:g1213} and \eqref{eq:g1323}, establishing a compact
distributional kernel is not sufficient: their metric derivatives
must satisfy \eqref{eq:admission}. The supplied calibration theorem
does not provide these graph-specific results.

No claim is made here that these products fail to exist.
Their existence and admission remain unproved in this manuscript.
\end{remark}

%=====================================================================
\section{Actual-functional descent of the second stage}
\label{sec:descent}
%=====================================================================

All tensor products in this section are algebraic tensor products
over \(\C\). No completion or quotient topology is silently introduced.

Let
\[
\mathcal E_L=\mathcal T\otimes\mathcal T,\qquad
\mathcal E_R=\mathcal T\otimes\mathcal T.
\]
Define presentation maps
\begin{align}
p_L(f\otimes g)&=B_fW_g,&
q_L(f\otimes g)&=\mathcal Q_{f,g},
\label{eq:leftpresent}\\
p_R(g\otimes f')&=W_gB_{f'},&
q_R(g\otimes f')&=\mathcal Q^R_{g,f'}.
\label{eq:rightpresent}
\end{align}
Their images are the corresponding actual-functional spaces.
Their kernels contain all finite linear relations between the
displayed presentations.

\begin{theorem}[Algebraic descent criterion]
\label{thm:descent}
Let \(v:E\to V\) be a surjective linear presentation map,
let \(Z,Y\) be vector spaces, and let
\(\lambda:E\otimes Z\to Y\) be linear.
There is a unique bilinear map
\[
\ell:V\times Z\longrightarrow Y
\]
such that
\[
\ell(v(e),z)=\lambda(e\otimes z)
\]
if and only if
\begin{equation}
\lambda\bigl((\kerop v)\otimes Z\bigr)=0.
\label{eq:descentcondition}
\end{equation}
\end{theorem}

\begin{proof}
Necessity follows by evaluating at \(v(e)=0\).
For sufficiency, define the value using any presentation \(e\)
of the first argument. Two presentations differ by an element
of \(\kerop v\), so \eqref{eq:descentcondition} makes the value
independent of the choice. Linearity of \(\lambda\) gives
bilinearity. Surjectivity of \(v\) gives uniqueness.
\end{proof}

For presentation maps on both arguments, the analogous condition
is vanishing on
\[
(\kerop v)\otimes E'
+
E\otimes(\kerop v').
\]
This follows by applying Theorem~\ref{thm:descent} twice.

\subsection{The calibrated scalar relation tests}

Since \(f'\mapsto B_{f'}\) is injective, define
\[
n_c^L:\mathcal E_L\otimes\mathcal T\to\C,
\qquad
n_c^L(f\otimes g\otimes f')=\Cal(f,g,f').
\]
Define similarly
\[
n_c^R:\mathcal T\otimes\mathcal E_R\to\C,
\qquad
n_c^R(f\otimes g\otimes f')=\Cal(f,g,f').
\]

\begin{theorem}[Exact scalar factorization tests]
\label{thm:scalarfactor}
A calibrated bilinear map
\[
\ell_{2,c}:\QLspace\times\Bspace\to\C\,1,
\qquad
\ell_{2,c}(\mathcal Q_{f,g},B_{f'})
=\Cal(f,g,f')\,1,
\]
exists algebraically if and only if
\begin{equation}
n_c^L\bigl((\kerop q_L)\otimes\mathcal T\bigr)=0.
\label{eq:leftscalarcondition}
\end{equation}
The analogous right map
\[
r_{2,c}:\Bspace\times\QRspace\to\C\,1
\]
exists if and only if
\begin{equation}
n_c^R\bigl(\mathcal T\otimes(\kerop q_R)\bigr)=0.
\label{eq:rightscalarcondition}
\end{equation}
When they exist, both maps are unique on these domains.
\end{theorem}

\begin{proof}
Apply Theorem~\ref{thm:descent} to \(q_L\) and \(q_R\),
using injectivity of the sharp test map on the other argument.
Constants belong to \(\Padm\), so the scalar target has no
additional admission issue.
\end{proof}

In particular,
\[
\sum_i\alpha_i\mathcal Q_{f_i,g_i}=0
\]
requires
\[
\sum_i\alpha_i\Cal(f_i,g_i,f')=0
\quad\text{for every }f'.
\]
Theorem~\ref{thm:scalarfactor} is an exact equivalence for
\emph{algebraic scalar descent}. It does not verify the conditions
for the actual kernels \(q_L,q_R\), nor does it construct the
nonconstant maps.

\subsection{The real parameter test}

Let \(\mathcal Z\) be the direct sum of the two relation-test
spaces in \eqref{eq:leftscalarcondition} and
\eqref{eq:rightscalarcondition}. Define linear functionals
\(\alpha,\beta\) on \(\mathcal Z\) by evaluating the corresponding
joint tests against \(u_\mu^\rho\) and \(r\), respectively.
Then both scalar relation tests are equivalent to
\begin{equation}
\alpha(z)+c\beta(z)=0
\quad\text{for every }z\in\mathcal Z.
\label{eq:affine}
\end{equation}

\begin{proposition}[Allowed scalar parameters]
\label{prop:parameters}
The set
\[
\mathcal C_{\mathrm{desc}}
=\{c\in\R:\alpha+c\beta=0\text{ on }\mathcal Z\}
\]
is empty, a singleton, or all of \(\R\).
More precisely:
\begin{enumerate}
\item if \(\beta=0\), it is all of \(\R\) when \(\alpha=0\)
and empty otherwise;
\item if \(\beta\ne0\), choose \(z_0\) with \(\beta(z_0)\ne0\).
The only possible parameter is
\[
c_0=-\frac{\alpha(z_0)}{\beta(z_0)}.
\]
It is allowed exactly when \(c_0\in\R\) and
\(\alpha+c_0\beta=0\) on every relation test.
\end{enumerate}
\end{proposition}

\begin{proof}
If two distinct real parameters solve \eqref{eq:affine},
their difference times \(\beta\) is zero, hence \(\beta=0\);
then \(\alpha=0\). Otherwise there is at most one solution.
The displayed alternatives follow directly.
\end{proof}

An empty set here would prove an incompatibility with the
\emph{imposed parent calibration class}, before any continuity
claim. This manuscript does not establish that the set is empty,
nonempty, or all of \(\R\).

\subsection{Bulk agreement and overlap consistency}

\begin{lemma}[Gluing linear coefficient prescriptions]
\label{lem:gluing}
Let \(D_1,D_2\) be subspaces of a vector space and
\(a_i:D_i\to Y\) linear. A linear map on \(D_1+D_2\) extending
both \(a_i\) exists if and only if
\[
a_1=a_2\quad\text{on }D_1\cap D_2.
\]
When it exists, it is unique.
\end{lemma}

\begin{proof}
Necessity is immediate. For sufficiency, define
\(a(d_1+d_2)=a_1(d_1)+a_2(d_2)\). Any two decompositions differ
by an element of the intersection, on which the maps agree.
\end{proof}

For a proposed coefficient of one partial binary product, relevant
domains inside \(\Padm\otimes\Padm\) include
\[
\Lspace\otimes\Bspace,\qquad
\Bspace\otimes\Rspace,\qquad
\Pmc\otimes\Pmc.
\]
The last domain carries the prescribed old Wick coefficient.
Lemma~\ref{lem:gluing} shows exactly where overlap checks are
required. Calling the left and right maps by different names
does not permit them to disagree on a common actual pair.

The lemma is algebraic. Continuity of a glued map remains a
separate requirement.

%=====================================================================
\section{Second-stage coefficient equations and associativity}
\label{sec:associativity}
%=====================================================================

A candidate construction must supply coefficient maps
\[
L_j:\Lspace\times\Bspace\to\Padm,\qquad
R_j:\Bspace\times\Rspace\to\Padm
\]
on the orders needed below, with \(L_0,R_0\) pointwise
multiplication.

For the selected contraction-degree prescription, no third binary
contraction against a degree-two sharp factor is present.
This is a declared prescription for the candidate sector, not
a consequence excluding arbitrary additional contact terms.
Any departure must be separately formulated.

Write
\[
P_L=AW,\quad Q_L=\mathcal Q_{f,g},
\qquad
P_R=WC,\quad Q_R=\mathcal Q^R_{g,f'}.
\]

\begin{proposition}[Formal coefficient expansion]
\label{prop:expansion}
Under the selected contraction-degree prescription, the two
candidate compositions expand as
\begin{align}
\mathcal A_L
={}&P_LC
+\hbar\bigl(L_1(P_L,C)+Q_LC\bigr)
\nonumber\\
&+\hbar^2\bigl(L_2(P_L,C)+L_1(Q_L,C)\bigr)
+\hbar^3L_2(Q_L,C),
\label{eq:AL}\\
\mathcal A_R
={}&AP_R
+\hbar\bigl(R_1(A,P_R)+AQ_R\bigr)
\nonumber\\
&+\hbar^2\bigl(R_2(A,P_R)+R_1(A,Q_R)\bigr)
+\hbar^3R_2(A,Q_R).
\label{eq:AR}
\end{align}
Their selected field degrees are \(6,4,2,0\), respectively.
\end{proposition}

\begin{proof}
Expand the coefficientwise formal product of
\(P_L+\hbar Q_L\) with \(C\), and of \(A\) with
\(P_R+\hbar Q_R\). Collect equal powers of \(\hbar\).
The degree-two sharp factor permits at most two cross-contractions.
Every contraction removes two field slots, giving the degree ledger.
\end{proof}

\begin{theorem}[Coefficientwise associativity criterion]
\label{thm:coherence}
Assuming the maps in \eqref{eq:AL}--\eqref{eq:AR} are well
defined, the equality
\(\mathcal A_L=\mathcal A_R\bmod\hbar^4\) holds if and only if
the following three identities hold for every permitted
\(f,g,f'\):
\begin{align}
L_1(P_L,C)+Q_LC
&=R_1(A,P_R)+AQ_R,
\label{eq:assoc1}\\
L_2(P_L,C)+L_1(Q_L,C)
&=R_2(A,P_R)+R_1(A,Q_R),
\label{eq:assoc2}\\
L_2(Q_L,C)&=R_2(A,Q_R).
\label{eq:assoc3}
\end{align}
The order-zero identity holds by classical associativity.
\end{theorem}

\begin{proof}
Equality in the truncated module is equality of its four
coefficient coordinates. The zero-order terms coincide because
\(P_LC=(AW)C=A(WC)=AP_R\).
The other three coordinates are exactly
\eqref{eq:assoc1}--\eqref{eq:assoc3}.
\end{proof}

If the two-edge functionals have been established, the intended
placement prescriptions are
\begin{align}
L_1(P_L,C)&=\Graph_{13}+\Graph_{23},&
R_1(A,P_R)&=\Graph_{12}+\Graph_{13},
\label{eq:firstprescription}\\
L_2(P_L,C)&=\Graph_{13,23},&
R_2(A,P_R)&=\Graph_{12,13},
\label{eq:secondprescription}\\
L_1(Q_L,C)&=\Graph_{12,23}+\Graph_{12,13},&
R_1(A,Q_R)&=\Graph_{12,23}+\Graph_{13,23}.
\label{eq:mixedprescription}
\end{align}
These are labelled prescription equations until descent through
\(p_L,q_L,p_R,q_R\), existence, and admission have been proved.

\begin{corollary}[Conditional placement coherence]
\label{cor:placementcoherence}
If the prescriptions \eqref{eq:firstprescription}--\eqref{eq:mixedprescription}
define actual coefficient maps satisfying admission and descent,
and both scalar maps use the same allowed \(c\), then
\eqref{eq:assoc1}--\eqref{eq:assoc3} hold.
\end{corollary}

\begin{proof}
At order one, both sides sum
\(\Graph_{12}+\Graph_{13}+\Graph_{23}\).
At order two, both sides sum
\(\Graph_{12,23}+\Graph_{12,13}+\Graph_{13,23}\).
At order three, both equal \(\Cal(f,g,f')1\).
\end{proof}

\begin{remark}[Why the corollary is not a construction]
The corollary proves coherence of maps \emph{if they exist with
the stated prescriptions}. It does not establish their existence
by defining both associations to be the same graph sum.
That shortcut would omit actual-functional factorization.
\end{remark}

A nonconstant discrepancy in \eqref{eq:assoc1} or
\eqref{eq:assoc2} cannot be repaired by adding \(cr\) to the
constant coefficient. The direct-sum coefficient structure
separates these equations.

%=====================================================================
\section{Observation compatibility, continuity, and one parameter}
\label{sec:observation}
%=====================================================================

\begin{proposition}[Numerical observation descendants]
\label{prop:descendants}
For the fixed scalar calibration \eqref{eq:Cal}, and a fixed
smooth-coefficient differential operator \(D\) on \(X\), the
corresponding numerical descendant is
\[
aD(u_\mu^\rho+cr).
\]
For example,
\[
\Cal(\partial_t^n f,g,f')
=
\pair{(-\partial_{x^0})^n(u_\mu^\rho+cr)}
{f\otimes g\otimes f'}.
\]
Under Assumption~\ref{ass:calibration}, IX-B supplies the associated
strong numerical finite-part estimate with test-order loss
\(64\to64+\ord D\).
\end{proposition}

\begin{proof}
The displayed equality is the definition of distributional
differentiation. Smooth multiplication and differential operators
act linearly, so the parent shift \(cr\) becomes \(caDr\).
The quantified transfer is inherited from IX-B
\cite[\texttt{thm:descendants}, \texttt{cor:conversion}]{IXB}.
\end{proof}

This proposition does not prove that a descended carrier-level
binary map satisfies any additional configuration-derivative
identity. The three operations remain distinct:
\begin{enumerate}
\item differentiation of observation tests or their joint distribution;
\item fixed smooth configuration-direction differentiation in IX-A;
\item exposed functional derivative slots in a new contraction graph.
\end{enumerate}

\begin{proposition}[Fixed-support scalar test continuity]
\label{prop:scalarcontinuity}
For fixed compact supports of \(f,g,f'\), the map
\((f,g,f')\mapsto\Cal(f,g,f')\) is jointly continuous.
The same holds for each fixed observation descendant.
\end{proposition}

\begin{proof}
A distribution has a finite-order estimate on every fixed compact
support. Apply that estimate to \(f\otimes g\otimes f'\).
Leibniz's rule bounds its test seminorms by products of finitely
many smooth seminorms of the three factors.
Fixed differential operators add only finitely many required
test derivatives.
\end{proof}

\begin{remark}[Continuity after descent]
Even if Theorem~\ref{thm:scalarfactor} yields an algebraic map,
Proposition~\ref{prop:scalarcontinuity} proves its continuity
along the named generating test family, not automatically in the
subspace topology of \(\QLspace\times\Bspace\).
No quotient-topology identification has been proved.
No hypocontinuity theorem in \(\taus\) is asserted.
\end{remark}

A general smooth joint multiplier \(a(x,y,x')\) defines a numerical
operation on the joint distribution. It need not encode a binary
operation on the original carrier stages.

%=====================================================================
\section{Involution and quantum presentation independence}
\label{sec:involution}
%=====================================================================

For polynomial functionals, star denotes complex conjugation.
For a joint distribution, dagger is
\[
\pair{U^\dagger}{\Phi}
=
\overline{\pair U{\overline{\Phi\circ\tau}}},
\qquad
\tau(x,y,x')=(x',y,x).
\]

\begin{proposition}[Scalar reversal]
\label{prop:scalarreverse}
For real \(c\),
\[
\overline{\Cal(f,g,f')}
=
\Cal(\bar f',\bar g,\bar f).
\]
If scalar descent holds on the left, the prescription
\[
r_{2,c}(B_f,Q)
=
\bigl(\ell_{2,c}(Q^*,B_f^*)\bigr)^*
\]
defines the corresponding right scalar map on the conjugate
reverse stage, with the required calibration.
\end{proposition}

\begin{proof}
Assumption~\ref{ass:calibration} gives
\((u_\mu^\rho+cr)^\dagger=u_\mu^\rho+cr\).
Pairing this identity with product tests proves the first claim.
Conjugation sends \(\QLspace\) onto \(\QRspace\) by
\eqref{eq:Qadjoint}. Thus the displayed prescription is well
defined whenever the left map is, and its value is the reversed
calibration.
\end{proof}

Parent symmetry does not establish reversal for the nonconstant
second-stage coefficient maps. Those maps would have to satisfy
\[
\bigl(L_j(F,B)\bigr)^*
=
R_j(B^*,F^*)
\]
on their conjugate reverse domains, together with the overlap
conditions of Lemma~\ref{lem:gluing}.

Observation differentiation exchanges first and third observations
under dagger; a descendant is not generally individually
self-adjoint.

\begin{theorem}[Presentation independence after actual descent]
\label{thm:presentation}
Suppose labelled coefficient prescriptions descend to maps on the
named actual-functional spaces and are consistent on their specified
overlaps. Then their values are unchanged by any rewriting that
preserves their arguments as actual off-shell polynomial functionals.
This includes finite linear presentation relations, kernel
permutations, tensor symmetries, polarization identities, and
integration by parts with all boundary terms retained.
\end{theorem}

\begin{proof}
A descended map assigns a value to the actual arguments, not to
their labels. Any equality-preserving rewriting leaves those
arguments unchanged. Equivalently, the difference of two
presentations lies in the relevant kernel of a presentation map
and is annihilated by the descent condition.
Overlap consistency ensures the same statement if the rewriting
moves between two named prescriptions.
\end{proof}

The IX-A identity
\[
C^*(\theta b)
=\theta C^*b+\delta(t)\mathcal B(b|_{\Sigma_0})
\]
is equality of actual kernels
\cite[\texttt{prop:adjoint}, \texttt{thm:presentation}]{IXA}.
Dropping the delta term does not preserve the argument and is
not an admissible presentation change.

Theorem~\ref{thm:presentation} is conditional on descent.
It cannot be used to supply the missing descent proof.

%=====================================================================
\section{Retained obstructions and the unresolved release}
\label{sec:release}
%=====================================================================

\subsection{What remains unchanged}

IX-B's separated witness proves that observation-diagonal
counterterms cannot remove the selected logarithm
\cite[\texttt{thm:diagonal}]{IXB}.
The unchanged regularized coefficient remains divergent on that
witness. The normal-topology extension obstruction of VII-C,
under its declared hypotheses including (E1)--(E4), remains valid
\cite[\texttt{thm:coefficient}, \texttt{ass:extension},
\texttt{cor:noalgebra}]{VIIC}.

IX-A explicitly proves
\[
B_{f,\eps}\not\longrightarrow B_f
\quad\text{in }\taus
\]
for nonzero real \(f\)
\cite[\texttt{thm:IX1}, \texttt{cor:compatibility}]{IXA}.
No opposite continuity requirement is introduced here.

A numerical subtraction from \(T_\eps^\rho\) is not a change
inside the unchanged old product. Subtracting old regular
coefficients would change those products; assigning only a
sharp value would not restore the forbidden normal continuity
\cite[\texttt{rem:subtraction}, Section
``Continuity reconciliation'']{IXB}.

The separate G2 condition is untouched. Where IX-A's stated
switch-gap hypotheses apply, its first-order incompatibility
continues to apply \cite[\texttt{prop:G2}]{IXA}.
No unconditional gauge theorem outside those hypotheses is
inferred.

\subsection{Completed results}

\begin{theorem}[IX-C reduction package and release status]
\label{thm:package}
Relative to the inherited IX-A and IX-B results and the explicit
analytical premise in Assumption~\ref{ass:calibration}, this
manuscript establishes:
\begin{enumerate}[label=\textup{(\roman*)}]
\item both admitted first-stage products, with locked signs and
reverse contraction;
\item actual-functional existence and admission of the three
one-edge outputs;
\item the labelled signs, multiplicities, and residual degrees
of the three two-edge placements;
\item necessary and sufficient algebraic descent tests for a
prescribed labelled coefficient map;
\item necessary and sufficient scalar relation tests for the
calibrated assignment on the named partial-output spaces;
\item the empty/singleton/all-real classification of parameters
passing those scalar relation tests;
\item the three nontrivial coefficient equations for
associativity through order \(\hbar^3\);
\item conditional placement coherence, scalar reversal, and
presentation independence after actual descent.
\end{enumerate}
Neither sharp second multiplication is constructed.
No incompatibility excluding every map satisfying
Requirement~\ref{req:interface} is proved.
The release outcome is therefore an unresolved gate.
\end{theorem}

\begin{proof}
The assertions follow from
Theorem~\ref{thm:first},
Proposition~\ref{prop:oneedges},
Proposition~\ref{prop:placementcount},
Theorem~\ref{thm:descent},
Theorem~\ref{thm:scalarfactor},
Proposition~\ref{prop:parameters},
Theorem~\ref{thm:coherence},
Corollary~\ref{cor:placementcoherence},
Proposition~\ref{prop:scalarreverse}, and
Theorem~\ref{thm:presentation}.

The two-edge existence and admission requirements in
Remark~\ref{rem:treeboundary} have not been discharged.
The scalar relation tests have not been evaluated for the actual
presentation kernels. Consequently the hypotheses needed to
construct the second-stage maps and invoke the conditional
coherence results have not been established.
No violated relation or other universal incompatibility has
been exhibited. The unresolved classification follows.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.49\linewidth}X@{}}
\toprule
Object & Status\\
\midrule
\(B_f\star_1W_g\) &
Constructed on the named first-stage domain\\
\(W_g\star_1B_{f'}\) &
Constructed on the named reverse domain\\
Three one-edge polynomial outputs &
Existence and admission proved from inherited closure\\
Three two-edge labelled expressions &
Identified; sharp existence/admission unresolved here\\
Scalar map on \(\QLspace\times\Bspace\) &
Exact descent test proved; test not discharged\\
Scalar map on \(\Bspace\times\QRspace\) &
Exact descent test proved; test not discharged\\
\((B_f\star_1W_g)\star B_{f'}\) &
Unresolved; second-stage map not constructed\\
\(B_f\star(W_g\star_1B_{f'})\) &
Unresolved; second-stage map not constructed\\
Associativity through order three &
Exact criterion proved; construction hypotheses unresolved\\
Second-stage involution &
Unresolved for nonconstant maps\\
Quantum presentation independence &
Conditional on actual descent; not released as completed\\
VII-C obstruction &
Retained under its original hypotheses\\
IX-D and interacting products &
Not opened\\
G2 &
Untouched; original conditional boundary retained\\
V's three quantum objects &
Separation only; not used in proofs\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{The next proof obligations}

A construction must now do the following:
\begin{enumerate}
\item establish the actual two-edge distributions and their compact
polynomial kernels;
\item check \eqref{eq:admission} for both metric-quadratic outputs,
at every configuration and after the required descendants;
\item prove descent of all second-stage prescriptions through
\(p_L,q_L,p_R,q_R\);
\item evaluate the scalar relation tests and determine whether
\(\mathcal C_{\mathrm{desc}}\) is nonempty;
\item verify overlap agreement with the old bulk coefficients and
between any two prescriptions for the same actual pair;
\item prove the named continuity claims without importing
normal-stage sharp-admission convergence;
\item establish reversal of every nonconstant coefficient map.
\end{enumerate}

An explicit relation with nonzero calibrated value would give a
genuine factorization obstruction for the imposed parent class.
A demonstrated failure of one proposed graph extension would
exclude that candidate only, unless all permitted alternatives
were separately excluded.

\scopebox{Closed conclusion}{%
The admitted partial coefficient has a definite place in the IX-A
carrier. The scalar calibration has an exact relation test for
factoring through that place. Associativity has three separate
nontrivial coefficient equations.

None of those statements supplies the missing second multiplication.
Both sharp associations are explicitly unresolved.
The ternary numerical calibration is retained without being
advertised as a binary product or as an associative algebra.}

%=====================================================================
\appendix
\section{Order, sign, and verification audit}
\label{app:audit}
%=====================================================================

\subsection{Order ledger}

For the selected word,
\[
\begin{array}{c|cccc}
\hbar\text{-order}&0&1&2&3\\ \hline
\text{residual field degree}&6&4&2&0
\end{array}
\]
under the declared contraction-degree prescription.
The fully contracted order-three term is a constant functional,
but the order-two terms are not.

Observation differentiation changes neither the number of
insertions nor the contraction order. It does not supply an
extension theorem for independently opened functional derivative
slots.

\subsection{Sign audit}

Each first-stage contraction contains one insertion sign:
\[
\mathcal Q_{f,g}:\ -2,
\qquad
\mathcal Q^R_{g,f'}:\ -2.
\]
The complete word contains two insertion signs:
\[
(-1)(-1)=+1.
\]
The two middle-slot assignments give multiplicity \(2\), not \(4\).
The partial factor \(2\) is already present when its two remaining
slots are contracted with the last sharp insertion.

\subsection{No hidden equation-of-motion quotient}

All presentation maps act on actual off-shell functionals.
No replacement of a kernel by an equation-of-motion-equivalent
kernel is included. Boundary terms in the sharp adjoint remain
part of the argument.

\subsection{What was and was not verified}

The new proofs are algebraic descent, parameter consistency,
formal coefficient comparison, first-stage reversal, and the
specified inherited-closure consequences.

The parent ultraviolet analysis is the VII-D analytical premise.
The sharp two-edge products, their metric admission, and scalar
descent for the actual presentation kernels have not been verified.

The source is designed for two-pass pdfLaTeX compilation.
Compilation has not been executed in this interface. No successful
build, page count, proof-assistant check, or independent review
is claimed.

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

\paragraph{Nature of the result.}
This is a proof of partial-stage and coherence reductions with an
explicit unresolved release outcome. It is not a completed quantum
factorization theorem.

\paragraph{Source boundary.}
IX-A and IX-B-v2 are supplied sources. VII-D is used only through the
explicit analytical premise restated in IX-B-v2; its full proof is
not supplied or independently certified here. VII-C's stated
obstruction is retained through the source-locked boundary recorded
in IX-B-v2. V is used only to identify different objects.

\paragraph{Quantum release boundary.}
No sharp second multiplication, global quantum algebra, interacting
realization, or gauge-compatible switched construction is claimed.

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

\bibitem{IXA}
N.~Cheung.
\emph{Landmark IX-A:
An off-shell admission domain and boundary-sensitive topology
for sharp Cauchy relational insertions}.
Supplied manuscript, \texttt{IX-A.tex}.
Relevant labels:
\texttt{def:Padm}, \texttt{lem:charge}, \texttt{thm:closure},
\texttt{def:topology}, \texttt{thm:admission},
\texttt{prop:adjoint}, \texttt{lem:tensor},
\texttt{prop:knownoutputs}, \texttt{cor:notbulk},
\texttt{thm:IX1}, \texttt{cor:compatibility},
\texttt{thm:presentation}, \texttt{eq:module},
and \texttt{prop:G2}.

\bibitem{IXB}
\emph{Landmark IX-B:
Derivative-stable triangle calibration and counterterm constraints
for sharp Cauchy insertions}.
Supplied manuscript, \texttt{IX-B-v2.tex}.
Relevant labels:
\texttt{eq:tensor}, \texttt{eq:K}, \texttt{eq:T},
\texttt{ass:VIID}, \texttt{prop:unique},
\texttt{thm:diagonal}, \texttt{thm:descendants},
\texttt{cor:conversion}, \texttt{thm:independence},
\texttt{prop:weights}, \texttt{thm:regularassociation},
\texttt{rem:subtraction}, \texttt{prop:partial},
\texttt{eq:Qpartial}, \texttt{prop:missing},
and \texttt{eq:adjoint}.

\bibitem{VIID}
\emph{Finite part of the regulated Cauchy-reflection triangle:
A strong numerical distributional limit for the VII-C family}.
Companion manuscript, \texttt{VII-D.tex}, identified in IX-B-v2.
The statements used here are explicitly restated in
Assumption~\ref{ass:calibration}.
Relevant source labels reported by IX-B-v2:
\texttt{thm:finitepart}, \texttt{thm:conversion};
also the section ``Adjoint symmetry''.
No independent verification of the companion proof is claimed.

\bibitem{VIIC}
\emph{Landmark VII-C:
A mixed-Wick obstruction to a normal-topology boundary extension
for sharp Cauchy harmonic insertions}.
Companion manuscript, \texttt{VII-C.tex}, identified in IX-B-v2.
Relevant source labels reported by IX-B-v2:
\texttt{thm:coefficient}, \texttt{ass:extension},
and \texttt{cor:noalgebra}.
The obstruction boundary is retained as recorded in the supplied
IX-B-v2 manuscript.

\bibitem{V}
N.~Cheung.
\emph{Admissible currents and first retarded corrections
for a fixed-time scalar response}.
Supplied manuscript, \texttt{V.tex}.
Used only for the separation of its external retarded coefficient,
one-oscillator coefficient, and Newton--wave cubic form from the
selected sharp triangle. No theorem from V enters a proof.

\end{thebibliography}

\end{document}