% !TeX program = pdflatex
%=====================================================================
% XIII-E.tex
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% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XIII-E.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIII-E.tex
%
% Exact release:
%   - the selected pair and XIII-C's partial-diagonal class are frozen;
%   - exactly one E1 route is attempted: common-frequency regularization;
%   - the regularized graph functionals are defined and admitted;
%   - convergence on microlocally admissible open sets is established;
%   - independence of auxiliary compact localization is established;
%   - partial-diagonal contacts cannot change an off-collision limit;
%   - the remaining characteristic-set limit is stated precisely;
%   - the joint coefficient has the ending Unresolved at E1.
%
% Important qualifications:
%   - a finite regularization is not a joint distributional prescription;
%   - auxiliary-cutoff independence is not regulator independence;
%   - finite-regulator admission is not admission of an unproved limit;
%   - the relative-coordinate endpoint theorem is conditional;
%   - no obstruction to every extension in C_part is claimed.
%
% No application of the free differential to the quantum coefficient,
% no Ward defect, higher retarded product, general time-ordered family,
% restriction quotient, reduced algebra, interacting differential,
% master equation, state, antifield construction, gravitational
% interaction insertion, or sharp-width limit is supplied.
%
% No successful compilation, proof-assistant verification,
% independently executed symbolic computation, independent referee
% review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
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\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
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\usepackage{enumitem}
\usepackage{tikz}
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\setlength{\emergencystretch}{3em}
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\numberwithin{equation}{section}
\setcounter{tocdepth}{1}

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

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\PP}{\mathcal P}
\newcommand{\RR}{\mathcal R}
\newcommand{\Lie}{\mathcal L}
\newcommand{\ER}{E_0^{\mathrm R}}
\newcommand{\EA}{E_0^{\mathrm A}}
\newcommand{\ERm}{E_m^{\mathrm R}}
\newcommand{\Pmass}{P_m}
\newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\GR}{G_{\mathrm{act}}^{\mathrm R}}
\newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\NW}{\mathscr N_W}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\CW}{\mathscr C_W}
\newcommand{\gW}{\mathfrak g_W}
\newcommand{\Ocal}{\mathcal O}
\newcommand{\Vcal}{\mathcal V_g}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}}
\newcommand{\CD}{\mathcal C_{3,\mathrm D}}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}}
\newcommand{\rD}{r_{\mathrm D}}
\newcommand{\rg}{r_g}
\newcommand{\Bcal}{\mathfrak B}
\newcommand{\Cpart}{\mathfrak C_{\mathrm{part}}}
\newcommand{\Ctot}{\mathfrak C_{\mathrm{tot}}}
\newcommand{\Kclass}{\mathfrak K_{\mathrm{loc,R}}}
\newcommand{\TT}{\mathsf T_2}
\newcommand{\HT}{H_T}
\newcommand{\Rtree}{R_1^{(0)}}
\newcommand{\Rquant}{R_1^{(1)}}
\newcommand{\Edist}{\mathcal E'}
\newcommand{\Ddist}{\mathcal D'}
\newcommand{\Sdist}{\mathcal S'}
\newcommand{\Uadm}{\mathcal U_\Gamma}
\newcommand{\Dcoll}{\mathfrak D_\Gamma}
\newcommand{\Xoff}{X_\Gamma^\circ}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

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\newcommand{\scopebox}[2]{%
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    \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax}
    \textbf{#1}\par\smallskip #2
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}

\title{\textbf{Landmark XIII-E}\\[0.4em]
One joint off-collision prescription for the selected binary pair,\\
or its obstruction in the frozen partial-diagonal class\\[0.4em]
\large A common-frequency prescription attempt,\\
localization independence, and the remaining E1 gate}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We return to the joint-extension problem left open in XIII-C.
The entries are the same switched metric--scalar vertex and the
same admitted second-order composite. The carrier, action,
off-shell lifts, and partial-diagonal ambiguity class are unchanged.

Exactly one route is attempted: a common-frequency regularization
of the complete time-ordered graph factors, with their inherited
tensor coefficients retained. We prove that the finite-regulator
graph functionals are compact microcausal polynomials, that the
regularization converges to the prescribed products on the
microlocally admissible open sets, and that auxiliary compact
localization changes have a vanishing remainder. We also prove that
a partial-diagonal contact cannot change any prescription on the
collision complement.

These results do not establish the remaining limit through the
noncoincident characteristic configurations. The inherited
wavefront cancellation prevents an application of the sufficient
product theorem there, while neither divergence of the complete
pair nor an unavoidable normalization incompatibility is proved.
The earliest unfinished gate is therefore E1.

We formulate the subsequent transverse, overlap, and pushforward
obligations without treating them as completed. A relative-coordinate
lemma identifies a sufficient balanced endpoint once a joint
extension exists; it does not provide existence. The coefficient
linear in $\hbar$ retains the ending Unresolved. No Ward defect,
interacting differential, or restriction quotient is constructed.
\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) {Selected pair};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Common-frequency regularization};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Joint coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The joint coefficient remains unresolved at E1.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{One frequency prescription is not the joint coefficient. 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-XIII-E}
\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) {Selected pair};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Joint coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The joint coefficient remains unresolved at E1.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{One frequency prescription is not the joint coefficient. 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-XIII-E}
\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) {Selected pair};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Common-frequency regularization};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Joint coefficient};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The joint coefficient remains unresolved at E1.};
\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-XIII-E}
\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) {Selected pair};
\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) {Joint coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Common-frequency regularization. The joint coefficient remains unresolved at E1.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Selected pair is not Joint coefficient. 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-XIII-E}
\end{figure}

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


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



\scopebox{Release and the three endings}{%
\textbf{Constructed} means that a joint distribution satisfies all
four successor obligations and has an admitted proper pushforward.

\smallskip
\textbf{Proved obstructed} means that an explicit incompatibility
excludes every extension in the previously frozen class $\Cpart$.

\smallskip
\textbf{Unresolved} means that a required prescription, transverse
estimate, contact compatibility, or output theorem is missing.

\smallskip
\textbf{Results here:} a named common-frequency attempt, admitted
finite-regulator functionals, convergence on the admissible open
sets, auxiliary-localization independence, and the off-collision
rigidity of the permitted contacts.

\smallskip
\textbf{Ending for the requested coefficient: Unresolved at E1.}
The characteristic-set limit and its prescription independence
have not been established. E2--E4 are not released.

\smallskip
A finite regularization is not a completed prescription.
A failed sufficient product criterion is not a nonexistence
theorem. No universal obstruction in $\Cpart$ is claimed.
The free differential is not applied to the absent coefficient.}

\tableofcontents

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

The following are mathematical inputs with their predecessor
scope. They are not new conclusions about the regularization
introduced below.

\subsection{XIII-C's four successor obligations}

\begin{inputstatement}[Successor list]
\label{input:four}
The next extension attempt must supply:
\begin{enumerate}
\item a joint prescription across the noncoincident characteristic
configurations, or a proved cancellation of those configurations
in the complete selected pair;
\item uniform transverse scaling and wavefront estimates for the
three-vertex and four-vertex collision strata and their intersections;
\item compatible extensions with the permitted partial contacts,
rather than only isolated primitive extensions;
\item the full output wavefront estimate after proper pushforward.
\end{enumerate}
If these obligations are met, the successor may release one
$\Rquant$ in the unchanged carrier and only then open the binary
gauge-consistency question at order $\lambda\hbar$ \cite{XIIIC}.
\end{inputstatement}

\begin{inputstatement}[XIII-C's release]
\label{input:C}
Constructed: the incidence inventory, the widened class
$\Ctot\subset\Cpart$, the off-collision products on their
microlocally admissible open sets, and the elementary parallel-line
extensions.

The parallel kinetic primitive has scaling degree $8$ and
codimension $4$; its partial-contact ambiguity has normal derivative
order at most $4$. Sum-zero bounds and collision exclusion do not
establish the joint off-collision product: distinct null-related
vertices can carry a nonzero wavefront cancellation.

The coefficient linear in $\hbar$ is Unresolved. That is a failed
sufficient product criterion, not a nonexistence theorem
\cite{XIIIC}.
\end{inputstatement}

The eleven branches and their labelled incidence matchings are
used as the fixed graph presentation. They are not listed a
second time.

\subsection{XIII-B's tree datum and ordered values}

\begin{inputstatement}[Binary convention]
\label{input:B}
The vertex
\[
 \Vcal=-\frac12\int g\,\gamma_{\mu\nu}
                         T_0^{\mu\nu}[\phi]\,\dd^4x
\]
is admitted. The tree datum
\[
 \Rtree
 =-\pair{\Ocal'}{\GR\Vcal'}
 =\pair{\Ocal'_\gamma}{\rg[\phi]}
                 +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]}
\]
is constructed, proper, and microcausal.

Ordered values of $R_1$ are $0$ or
$\frac{i}{\hbar}[\Ocal,\Vcal]_{\star_H}$.
The literal normalization
$R_1=\Vcal\star_H\Ocal$ is proved obstructed \cite{XIIIB}.
\end{inputstatement}

In particular, the tree datum is not the requested coefficient
linear in $\hbar$.

\subsection{XIII-A and XIII-D}

\begin{inputstatement}[Evaluation boundary]
Under $(\mathrm{UCP}_W)$,
$\Delta_\xi\Ocal|_{\CW}=0$.
The coefficientwise kernel $\NW$ is not a two-sided ideal.
No reduced algebra was handed off.
$\CW$ remains an evaluation set \cite{XIIIA}.
\end{inputstatement}

\begin{inputstatement}[Classical completion boundary]
On the detecting data, every shell-descending action in
$\Kclass$ induces the zero representation, and the pairing is
nonzero on
\[
 \theta=[h\zeta\partial_t,(t-t_*)h\zeta\partial_t].
\]
The ending is Proved obstructed for that class only.
The admitted cochain
$\mathcal C_\xi=-\Bcal_\xi[D\gamma,\phi]$
is not an integrated completion \cite{XIIID}.
\end{inputstatement}

That classical cocycle calculation is not a product obstruction
and supplies no numerical graph distribution.

\subsection{XII-F's full-shell boundary}

The copied witness is retained:
\begin{equation}
 \Delta_\xi\PhiD
 =-8\int f(t,\mathbf x)
       \int_0^t(t-s)^2a(s)\,\dd s\,\dd^4x<0.
 \label{eq:witness}
\end{equation}
Its hypotheses and evaluated sign are those of XII-F
\cite{XIIF}. It is not used to determine a regulator, a contact
coefficient, or an extension normalization.

\subsection{Analytic inputs}

The new arguments use the distribution product criterion,
submersion pullback, differential-operator wavefront bounds, and
proper pushforward. Their microlocal continuity formulation is
the one recalled in \cite[Section~2]{BF} and developed in
\cite[Section~8.2]{Hormander}.

Point extension and extension across a submanifold are distinct.
Scaling and wavefront conditions for the latter are described in
\cite[Sections~5--6]{BF} and \cite{Dang}.
No existence theorem for local entries is imported as an existence
theorem for the present nonlocal pair.

%=====================================================================
\section{E0: the pair, carrier, order, and contact class}
\label{sec:freeze}
%=====================================================================

This freeze precedes the regularized kernel construction.

\subsection{The unchanged sheet}

Use
\[
 M=\R^4,\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \Pmass=\Box+m^2,
\]
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad 0<\eps<T.
\]
The functions $g\in C_c^\infty(\MTo;[0,1])$ and $\sig$ are fixed
backgrounds. All fields are ambient smooth multiplets.
No decay assumption is added.

The carrier is
\begin{equation}
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                         \Nsol[[\hbar,\kappa]].
 \label{eq:carrier}
\end{equation}
There is no quotient by $\NW$.

The action remains XII-C's cubic-only density weighting,
\[
 S_{\mathrm{grav,D}}
 =S_{\mathrm{EH}}+\kappa\int(\sig-1)\ell_3\,\dd^4x.
\]
No Einstein--Hilbert interaction vertex is inserted into the
selected pair \cite{XIIC}.

\subsection{The two displayed lifts}

The only interaction entry is $\Vcal$ of
Input statement~\ref{input:B}. The observable is
\begin{equation}
 \Ocal
 =\PhiD+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD,
 \label{eq:observable}
\end{equation}
with
\begin{equation}
 \rD=\PP\ER\JD=-u_{\mathrm D},
 \qquad
 \rg[\phi]=-\tfrac12\PP\ER(gT_0[\phi]).
 \label{eq:slots}
\end{equation}
The insertion sign and both scalar-completion legs are unchanged
\cite{XIIE}.

The relevant inverse blocks are
\[
 \GR_{\gamma\gamma}=-\PP\ER,\qquad
 \GR_{\phi\phi}=-\ERm.
\]
The frozen translation-invariant product data satisfy
\[
 H-H^{\mathsf T}=i(\GR-\GA),\qquad
 \HT=H+i\GA.
\]
Metric--scalar cross blocks vanish for these frozen data.
All numerical tensor factors and action signs are retained.

The selected expansion is
\begin{equation}
 R_{\lambda\Vcal}\Ocal
 =\Ocal+\lambda(\Rtree+\hbar\Rquant)
                         +O(\lambda^2,\lambda\hbar^2).
 \label{eq:expansion}
\end{equation}
Only the coefficient of $\lambda\hbar$ is sought. There is no
$R_2$, no $\mathsf T_3$, and no claim of completeness of a higher
coupling-order relational coefficient.

The operation remains attached to these off-shell lifts.
A representative-independent operation on all of $\Ealg$ is not
part of the target \cite{XIIIB}.

\subsection{The copied partial-diagonal class}

For a graph $\Gamma$, let $y$ be the interaction vertex.
For each connected subset $I$ containing $y$, with $|I|\ge2$,
\[
 \Delta_I=\{v_a=v_b:\ a,b\in I\}.
\]
Include the intersection and contracted-graph loci specified in
XIII-C. Observable-only subgraphs retain their inherited
normalization.

\begin{definition}[Frozen class $\Cpart$]
\label{def:class}
The ambiguity class is exactly XIII-C's class of well-defined
graphwise contacts
\begin{equation}
 D_\Gamma\!\left[
 w_\Gamma\,\partial_r^\alpha\delta(r_I)\,
                              S_{\Gamma/I}\right],
 \qquad |\alpha|\le6.
 \label{eq:contacts}
\end{equation}
Here $r_I$ is transverse to $\Delta_I$, $S_{\Gamma/I}$ is the
inherited spectator coefficient with the two remaining field slots,
and $D_\Gamma$ is the fixed finite graph differential operator.

The external weights are those built from
$g,\partial g,\sig,\sig',\sig''$ and the already present smooth
graph weights. Contacts are even, have ghost number zero and field
degree two, and remain in the existing compact graph closure.
They leave the tree datum and inherited observable-subgraph
normalizations unchanged.

Only well-defined contact products and pushforwards belong to
the class. The bound six is the copied conservative bound, not
a new power-counting theorem. The parallel kinetic primitive has
the smaller proved normal-order bound four.
\end{definition}

This class includes $y=z$ with $x\ne z$.
It is not replaced by $\Ctot$.
A singular spectator is not declared smooth merely because its
external weight is smooth.

The extension domain is the configuration space of the selected
graphs, not $\CW$. The four obligations are exactly
Input statement~\ref{input:four}.

%=====================================================================
\section{E1: one common-frequency prescription attempt}
\label{sec:E1}
%=====================================================================

\[
 \boxed{\text{Selected E1 move: a common-frequency prescription.}}
\]

No tensor-cancellation theorem and no universal class obstruction
are claimed. The limit attempted here is named explicitly, and its
unproved part is not hidden in the word ``renormalized.''

\subsection{The inherited characteristic configuration}

The triangle identified in XIII-C has numerical structure
\[
 E_0^{\mathrm R}(x-u)\,F_0(y-u)\,F_m(y-x),
\]
with $F_\mu$ the corresponding time-ordered boundary-value line.
The configuration is
\[
 y=u+\ell,\qquad x=u+2\ell,
 \qquad \ell\ne0\text{ future null}.
\]

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
 \node (u) at (0,0) {$u$};
 \node (y) at (3,0) {$y=u+\ell$};
 \node (x) at (6,0) {$x=u+2\ell$};
 \draw (u) -- node[above] {$F_0(y-u)$} (y);
 \draw (y) -- node[above] {$F_m(y-x)$} (x);
 \draw[->] (u) to[bend right=28]
                node[below] {$E_0^{\mathrm R}(x-u)$} (x);
\end{tikzpicture}
\caption{Three vertices on one null line.
The curved edge is the retarded attachment, not a reflected configuration.
The incidence does not prove that the complete tensor sum cancels.}
\label{fig:xiii-e-null-line}
\end{figure}

Figure~\ref{fig:xiii-e-null-line} is the selected configuration.
It is not a cancellation theorem.

XIII-C supplies a nonzero line-covector selection whose sums vanish
at all three vertices. This incidence occurs in the selected pair.
It does not prove that summing the complete tensor coefficients
cannot cancel a singular contribution.

The present attempt retains those coefficients. It neither deletes
the incidence nor adds a subtraction supported on a noncoincident
null locus.

\subsection{Graph variables and the admissible domain}

Let $X_\Gamma=M^{n_\Gamma}$ be the numerical vertex space of one
graph in the copied inventory. Let
\[
 \Dcoll=\bigcup_I\Delta_I,\qquad
 \Xoff=X_\Gamma\setminus\Dcoll.
\]
Observable-only singular subgraphs keep their inherited definition.

Write $k_e(v_a-v_b)$ for each numerical line, including its
specified derivatives and constant fiber maps. The lines are
tempered distributions in relative coordinates. The two new
cross-lines use the full $\HT$ kernels.

Let $\Uadm$ be the open set on which the complete line product
satisfies the H\"ormander sum condition. We use
\[
 U_\Gamma^{\mathrm{good}}=\Uadm\cap\Xoff.
\]
The inherited pinch configuration shows why this set cannot be
identified with the entire collision complement by the supplied
product argument.

\subsection{The fixed regularization}

Fix the Euclidean quadratic form
$|p|_{\mathrm E}^2=\sum_{\mu=0}^3p_\mu^2$ in the given Minkowski
coordinate chart. It is auxiliary regulator data, not a change
of spacetime metric.

For $\tau>0$ define
\begin{equation}
 S_\tau k
 =\mathcal F^{-1}
       \bigl(e^{-\tau|p|_{\mathrm E}^2}\widehat k(p)\bigr)
 =G_\tau*k,
 \qquad
 G_\tau(r)=\frac{e^{-|r|_{\mathrm E}^2/(4\tau)}}{(4\pi\tau)^2}.
 \label{eq:smoothing}
\end{equation}
The parameter $\tau$ is distinct from the fixed ramp width $\eps$.
Only $\tau$ is sent to zero.

Every line uses the same $\tau$. Smooth external weights are not
regularized. Differentiation commutes with $S_\tau$, so the
inherited derivative placement is preserved.

The complete time-ordered graph factor is
\begin{equation}
 A_{\Gamma,\tau}(v)
       =\prod_{e=(a,b)}(S_\tau k_e)(v_a-v_b),
 \label{eq:A-reg}
\end{equation}
with finite tensor sums taken exactly as in the selected pair.
The grouping uses the full time-ordered lines. It does not expand
the desired difference into separately undefined mixed products
of $H$ and $\GA$.

For fixed $\tau$, \eqref{eq:A-reg} is a smooth function.
The attempted off-collision prescription is its weighted
distributional limit on $\Xoff$, followed by the collision
extensions demanded at E2 and E3. Existence on all of $\Xoff$
is an obligation, not a definition by assertion.

\subsection{Compact localization of the regularized graphs}

Gaussian smoothing creates tails, so finite-$\tau$ support is not
the original retarded support. This must be handled explicitly.

Let $K_\Gamma$ be the copied compact closed support set determined
by the graph weights and inherited internal retarded lines.
Choose
\[
 \chi_\Gamma\in C_c^\infty(X_\Gamma),
 \qquad
 \chi_\Gamma=1\text{ near }K_\Gamma.
\]
It is an auxiliary localization, not a contact normalization.

Let $z_1,z_2$ be the two residual field variables, attached to
vertices $v_{a_1},v_{a_2}$. Before pushforward define
\begin{equation}
 \Pi_{\Gamma,\tau}
 =D_\Gamma\!\left[
   \chi_\Gamma(v)w_\Gamma(v)A_{\Gamma,\tau}(v)
   \delta(z_1-v_{a_1})\delta(z_2-v_{a_2})\right].
 \label{eq:Pi}
\end{equation}
The fixed operator $D_\Gamma$ includes the prescribed residual
field derivatives. The attachments are separately labelled even
when they occur at the same vertex.

With the inherited Wick coefficients $c_\Gamma$, define the
regularized time-ordered coefficient kernel
\begin{equation}
 t_\tau^{T,[2]}
       =\frac12\sum_\Gamma c_\Gamma\,
                           \pi_{\Gamma*}\Pi_{\Gamma,\tau}.
 \label{eq:t-reg}
\end{equation}
No new combinatorial multiplier is introduced.

\begin{proposition}[Finite-regulator admission]
\label{prop:finite}
For every $\tau>0$, the kernel \eqref{eq:t-reg} is compact and
defines an even, ghost-number-zero quadratic polynomial.
Its wavefront set is contained in $Z_2$.
\end{proposition}

\begin{proof}
The numerical coefficient in \eqref{eq:Pi} is smooth and compactly
supported in the vertex variables. The two delta attachments are
well defined: each has its own external variable, so a nonzero
covector cancellation between them would require a nonzero
covector in an external variable where no other factor contributes.

Their wavefront covectors are balanced between an external variable
and its attached vertex. Multiplication by the smooth coefficient
and the finite differentiations preserve that bound.

The support is compact because the vertex variables are compact
and the delta attachments place each external variable at a vertex.
The projection is therefore proper. Its wavefront condition sets
all integrated vertex covectors to zero, leaving
$p_1+p_2=0$ in the two external slots.

The sum-zero bound excludes both all-causal cones.
Finite tensor sums and species symmetrization preserve the bound.
The remaining species are even and have ghost number zero
\cite{Hormander}.
\end{proof}

Let $t^{H,[2]}$ be the already defined kernel of the coefficient
of $\hbar^2$ in $\Vcal\star_H\Ocal$. Then
\[
 \mathcal Q_\tau
 =i\,\pair{t_\tau^{T,[2]}-t^{H,[2]}}{\Phi^{\otimes2}}
\]
is an admitted finite-regulator functional. It is not denoted
$\Rquant$. Admission at each positive $\tau$ gives no convergence
or uniform wavefront theorem as $\tau\downarrow0$.

\subsection{Elementary estimates and localization independence}

\begin{lemma}[Smoothing estimates]
\label{lem:smoothing}
For $k\in\Sdist(\R^4)$, a compact set $L$, and any multi-index
$\alpha$, there are constants $C,N$ such that
\[
 \sup_{r\in L}|\partial^\alpha S_\tau k(r)|
                          \le C\tau^{-N},\qquad 0<\tau\le1.
\]
If $L\cap\supp k=\varnothing$, then for every $M$,
\[
 \sup_{r\in L}|\partial^\alpha S_\tau k(r)|
                          =O(\tau^M).
\]
\end{lemma}

\begin{proof}
A tempered distribution is bounded by finitely many Schwartz
seminorms. Apply that bound to
$\partial^\alpha G_\tau(r-\cdot)$.
For $r$ in a fixed compact set, its seminorms grow at most as a
fixed inverse power of $\tau$, proving the first assertion.

For the second, insert a smooth cutoff equal to one near $L$
and supported away from $\supp k$. The pairing with its
near-$L$ part is zero. In the remaining pairing the Gaussian
argument stays a positive distance from zero. Gaussian decay,
including all derivatives and polynomial weights required by the
tempered-distribution seminorm, dominates every power of $\tau$.
This proves the uniform remainder estimate.
\end{proof}

\begin{proposition}[Auxiliary compact-localization independence]
\label{prop:localization}
Replace $\chi_\Gamma$ by another compact smooth function
$\widetilde\chi_\Gamma$ equal to one near $K_\Gamma$.
For every compactly supported graph test and every $M$, the
difference of the resulting pairings in \eqref{eq:Pi} is
$O(\tau^M)$.

Thus auxiliary compact localization cannot change a prospective
distributional limit. This statement does not prove that the
limit exists.
\end{proposition}

\begin{proof}
On the compact support of
$(\chi_\Gamma-\widetilde\chi_\Gamma)w_\Gamma$, the original closed
graph support conditions cannot all hold. Locally, either the
weight is zero or at least one numerical line is evaluated on a
compact set disjoint from its support.

Use a finite cover and a smooth partition of unity. On each
nonzero patch, Lemma~\ref{lem:smoothing} makes that line smaller
than every power of $\tau$, while every other line and its
derivatives has at most polynomial growth in $\tau^{-1}$.
The finite product, including the specified differentiations,
therefore has the same arbitrarily small remainder.

Pairing the delta attachments and their derivatives with a fixed
smooth graph test preserves this estimate. Summing the patches
proves the assertion.
\end{proof}

This eliminates an artificial dependence introduced to restore
compactness at finite regulator. It does not prove independence
of the quadratic form in \eqref{eq:smoothing}, relative damping
rates, or another prescription at the characteristic set.

\subsection{What converges}

\begin{proposition}[Convergence on the admissible open sets]
\label{prop:good-limit}
On every relatively compact open subset of
$U_\Gamma^{\mathrm{good}}$, the regularization
\eqref{eq:A-reg} converges distributionally to the prescribed
complete line product.

The limit there is independent of the smoothing choice among
standard approximate identities satisfying the same local
microlocal approximation estimates.
\end{proposition}

\begin{proof}
Localize every line near the compact set under consideration.
Approximate-identity regularization converges weakly and in the
wavefront-restricted sense outside any slightly enlarged cone
containing the localized wavefront set. The latter assertion
follows from the rapid Fourier decay in those directions;
the separated tails are controlled by
Lemma~\ref{lem:smoothing}.

The sum condition permits such enlarged cones to be chosen
without introducing a zero covector sum on this compact set.
The sequentially continuous product, equivalently the admissible
diagonal pullback of the tensor product, therefore takes the
regularized factors to their canonical product.
This is the microlocal continuity statement recalled in
\cite[Section~2, properties M1--M3]{BF}.

Smooth weights and finite differential operators preserve the
result. The product is canonical on this domain, so two such
regularizations give the same limit there.
\end{proof}

There is also a useful ordered subdomain. If the interaction
attachment $y$ is strictly on the same chronological side of
both cross-line endpoints, their possible nonzero wavefront
covectors at $y$ lie in the same pointed causal cone.
They cannot sum to zero. Removing the cross-lines leaves the
inherited observable tree, where the forest argument rules out a
remaining circulation. Such strictly ordered attachment domains
are therefore covered by the admissible product calculation.

At the copied configuration $u,y,x$, the two cross-line
orientations at $y$ instead permit cancellation. The preceding
proof supplies no uniform estimate there.

\subsection{The exact unfinished E1 statement}

\begin{definition}[Remaining common-frequency obligation]
\label{def:E1-missing}
For the fixed weighted graph family, the remaining obligation is:
\begin{enumerate}[label=\textup{(\roman*)}]
\item establish a distributional limit through every
noncoincident characteristic configuration in $\Xoff$;
\item retain the full tensor and branch summation when deciding
whether singular contributions cancel;
\item show that admissible changes of auxiliary regularization
give the same off-collision prescription, or differ only by the
permitted collision-supported ambiguities after extension;
\item preserve the prescribed values on the strictly ordered
attachment domains.
\end{enumerate}
\end{definition}

No estimate proving (i)--(iii) at the characteristic configuration
has been established here. No test proving divergence of every
prescription has been established either.
The selected move is therefore an \emph{attempt}, not a
Constructed E1 prescription.

%=====================================================================
\section{Contacts cannot repair an unspecified off-collision limit}
\label{sec:rigidity}
%=====================================================================

\begin{lemma}[Off-collision rigidity of $\Cpart$]
\label{lem:rigidity}
Every well-defined contact in $\Cpart$ restricts to zero on
$\Xoff$. Consequently it cannot alter convergence, divergence,
or prescription dependence tested by graph tests supported in
$\Xoff$.
\end{lemma}

\begin{proof}
Each contact is supported on a collision locus $\Delta_I$.
Differentiation does not enlarge support. Its pairing with a test
supported away from the union of those loci is zero.
This remains true for a family of contacts whose coefficients
depend on an auxiliary regulator.
\end{proof}

This lemma separates two questions. Contacts may normalize a
collision extension once the off-collision distribution exists.
They cannot manufacture that distribution at a distinct-vertex
characteristic configuration.

It also gives a precise obstruction test for a future proof:
if a compulsory off-collision normalization were incompatible
with every joint prescription on such an open set, then no member
of $\Cpart$ could repair it. No such universal incompatibility is
proved in this manuscript.

\begin{proposition}[Agreement only on the good set is insufficient]
\label{prop:agreement}
Suppose two candidate distributions agree on
$U_\Gamma^{\mathrm{good}}$.
Their difference is not thereby proved to be a member of
$\Cpart$.

To identify it as a permitted ambiguity, agreement on the full
collision complement and the remaining contact-class conditions
must also be established.
\end{proposition}

\begin{proof}
Agreement on an open set only places the support of the
difference in its complement. That complement can contain
noncoincident characteristic configurations as well as collisions.
A distribution supported on such a characteristic locus need not
be collision-supported.

Membership in $\Cpart$ additionally restricts normal derivative
order, spectator structure, external weights, normalization,
and well-defined composition. Support alone does not verify those
requirements.
\end{proof}

In particular, the inherited sum-zero bound does not turn every
difference between two regularization limits into a permitted
partial contact.

%=====================================================================
\section{E2: transverse estimates, explicitly unfinished}
\label{sec:E2}
%=====================================================================

E1 has not been crossed. No scaling degree is assigned here to an
undefined full triangle or four-vertex distribution.

For a collision subset $I$,
\[
 q_I=\codim\Delta_I=4(|I|-1).
\]
Choose local coordinates $(q,r)$, with $q$ tangential to $\Delta_I$
and $r\in\R^{q_I}$ normal to it. A prospective off-diagonal
distribution $a(q,r)$ would be rescaled by
\[
 a_\rho(q,r)=a(q,\rho r).
\]
In distributional notation this includes the usual
$\rho^{-q_I}$ test-function Jacobian.

The required estimate is not merely a bound for one fixed smooth
value of $q$. It must hold uniformly on compact tangential sets,
with the relevant distributional and wavefront seminorms controlled,
including the approach to the other collision strata.

\begin{definition}[Unfinished transverse-estimate interface]
\label{def:transverse}
After a joint E1 prescription has been obtained, E2 requires
finite transverse degrees for the three-vertex and four-vertex
strata, together with compatible estimates at their intersections.

In each chart, an asserted degree $\omega_I$ must control the
rescaled distribution on bounded families of tests, with the
necessary microlocal control in the tangential variables.
The singular spectators must remain distributions in those
variables; they are not replaced by pointwise smooth coefficients.
\end{definition}

For example, a partial collision $y=z$ has codimension four while
an observation vertex $x$ remains tangential and may satisfy
$x\ne z$. A degree-four point-contact classification at the total
diagonal says nothing by itself about the singular dependence
on this uncollapsed variable.

The inherited primitive statement is retained:
\[
 \sd\bigl((\partial_0^2F_\mu)
                  (\partial_0^2F_\nu)\bigr)=8,\qquad
 \codim\{y=z\}=4.
\]
Its finite contact ambiguity has normal derivative order at most
four \cite{XIIIC}. That primitive extension is not repeated.

The graph expression $2E_I+D_I$ remains a superficial checklist.
It is not substituted for the uniform estimates of
Definition~\ref{def:transverse}.

The available submanifold extension theorems assume an already
defined off-submanifold distribution and appropriate scaling and
microlocal bounds \cite[Sections~5--6]{BF,Dang}.
Here the prerequisite distribution at the characteristic
configuration is missing. Those theorems cannot discharge E1.

\scopebox{E2 status}{%
No uniform transverse theorem for the selected triangles,
four-vertex cycles, or their intersections is released.
E2 is Unresolved and has not been reached by the present E1 attempt.}

%=====================================================================
\section{E3: overlap equations and ordered values}
\label{sec:E3}
%=====================================================================

The isolated primitive extensions are inherited constructions.
The remaining problem is compatible extension of the complete
graphs, not another extension of the same primitive.

Suppose, conditionally, that local extensions $u_i$ have been
constructed on an open cover $\{U_i\}$ of a graph domain.
A permitted adjustment $c_i$ must be a well-defined contact of the
frozen class. Agreement on overlaps requires
\begin{equation}
 (u_i+c_i)|_{U_i\cap U_j}
       =(u_j+c_j)|_{U_i\cap U_j},
 \label{eq:overlap}
\end{equation}
or equivalently
\[
 c_j-c_i=u_i-u_j
 \quad\text{on }U_i\cap U_j.
\]

\begin{lemma}[Conditional gluing]
\label{lem:gluing}
If the local extensions and contacts are defined and satisfy
\eqref{eq:overlap}, they determine a unique distribution on their
union. This statement does not establish existence of contacts
solving \eqref{eq:overlap} in $\Cpart$.
\end{lemma}

\begin{proof}
A smooth locally finite partition of unity subordinate to the
cover defines a distributional pairing by summing the local
pairings. The overlap identities make this pairing independent
of the partition and agree with each local distribution.
Uniqueness follows from localization of every compact test.
\end{proof}

The automatic identity
\[
 (u_i-u_j)+(u_j-u_k)+(u_k-u_i)=0
\]
on a triple overlap is only a consistency condition.
It does not prove solvability within the restricted spectator,
jet, and normal-order class.

Moreover, before E1 is completed, local prescriptions may differ
on noncoincident characteristic sets. Such a difference is not
made contact-supported by the gluing notation.

The ordered values that must survive are
\begin{align}
 \supp\Vcal\succ\supp\Ocal
 &\Longrightarrow R_1=0,\\
 \supp\Ocal\succ\supp\Vcal
 &\Longrightarrow
 R_1=\frac{i}{\hbar}[\Ocal,\Vcal]_{\star_H}.
 \label{eq:ordered}
\end{align}
These are the inherited values for full functional supports,
not a statement using $\supp f$ alone.

Every allowed interaction--observable contact vanishes on a
strictly ordered, disjoint attachment region, by
Lemma~\ref{lem:rigidity}. Thus contacts cannot change its prescribed
off-collision value. This establishes the support part of the
ordered-value audit, not the complete overlap construction.

For the common-frequency attempt, agreement with the canonical
product on the admissible ordered attachment domains follows
from Proposition~\ref{prop:good-limit}.
No exact causal factorization is asserted at finite $\tau$:
Gaussian damping has noncausal tails.

\scopebox{E3 status}{%
The overlap equations and their contact-support restrictions are
explicit. No family of compatible complete-graph extensions
satisfying them has been constructed. E3 is Unresolved.}

%=====================================================================
\section{E4: the conditional endpoint, not applied}
\label{sec:E4}
%=====================================================================

No joint pushforward is performed at this gate because E1--E3
are unfinished. The following lemma identifies one useful
sufficient wavefront input for a future endpoint proof.

\begin{lemma}[Relative-coordinate balance]
\label{lem:relative}
Let $a\in\Ddist(\R^{4(n-1)})$ and define a numerical vertex
distribution by submersion pullback,
\[
 A(v_1,\ldots,v_n)
   =a(v_2-v_1,\ldots,v_n-v_1).
\]
Then
\[
 \WF(A)\subset Z_n.
\]
Multiplication by smooth external weights and finite differential
operators preserves this inclusion.

Furthermore, a distributional limit of unweighted amplitudes
\eqref{eq:A-reg}, when such a limit exists, is invariant under
simultaneous translation of all vertices and has this
relative-coordinate form.
\end{lemma}

\begin{proof}
In coordinates
$s=v_1$ and $r_j=v_j-v_1$, the distribution is
$1_s\otimes a(r)$. Its wavefront covectors have zero $s$ component.
Changing back to vertex coordinates makes that component the sum
of all vertex covectors. This gives the asserted bound by the
submersion pullback theorem \cite{Hormander}.

Smooth multiplication and differentiation do not enlarge the
wavefront set.

Each unweighted regularized amplitude depends only on differences,
so its derivative in every simultaneous-translation direction is
zero. Distributional convergence preserves those equations.
In the coordinates $(s,r)$, a distribution whose $s$ derivatives
all vanish is constant in $s$, hence is $1_s\otimes a(r)$.
\end{proof}

The lemma is not an existence theorem for $a$.
Nor is the relative-coordinate ansatz imposed as an additional
restriction on every extension in $\Cpart$.
It is one sufficient endpoint structure that the selected attempt
would preserve if the needed limits and compatible extensions
were established.

Conditionally, if the complete extended numerical graphs have
the balance of Lemma~\ref{lem:relative}, their residual delta
attachments are compatible with them: a cancellation in the
product would have to vanish at each separate external slot,
forcing the corresponding attachment covectors to vanish.
The compact graph closure then makes the projection proper.

The proper-pushforward wavefront condition sets internal covectors
to zero and leaves a sum-zero bound on the two residual covectors.
This is the conditional endpoint argument, not an application to
an existing joint graph distribution.

The required output would be an even, ghost-number-zero polynomial
of degree two with compact kernel $t^{(1)}$ satisfying
\[
 \WF(t^{(1)})\subset Z_2.
\]
Only after that proof could the selected coefficient
$Q_H\Rquant\in\Ealg$ be released.

The tree coefficient would remain unchanged because the new datum
enters at order $\hbar^2$ in $\TT$ and at order $\hbar$ after the
retarded division. The inverse signs and the inherited branch
overlap cannot be changed at that order.

Properness is available as a support theorem from XIII-B.
It cannot define a missing distribution. Likewise,
Proposition~\ref{prop:finite} cannot be passed to an arbitrary
distributional limit without a uniform or structural wavefront
argument.

\scopebox{E4 status}{%
There is no admitted joint output $\Rquant$ in this release.
The endpoint implication is conditional on the preceding gates.
E4 is Unresolved.}

%=====================================================================
\section{E5: boundaries recorded and the differential left unopened}
\label{sec:boundaries}
%=====================================================================

Four predecessor boundaries remain separate:
\begin{enumerate}
\item XIII-D excludes integrated classical completions in its
parameter-local/one-retarded, shell-descending class on the
detecting data. Its admitted cochain is not an integrated
completion and is not an extension coefficient.
\item XII-F's nonzero witness remains a boundary of unqualified
full-shell invariance. It is not a normalization condition for
the present product.
\item XIII-A's additional vanishing space is not a quantum ideal.
No quotient by $\NW$ is introduced.
\item XIII-B's literal retarded-factorization equality is already
obstructed. Its corrected ordered values are retained.
\end{enumerate}

None of these is a theorem that the selected two-line pair has
no extension.

No action of the free differential on the absent quantum
coefficient is printed. No binary Ward defect is formed.
A Constructed release of the selected pair would remove one
necessary prerequisite for that later question; it would not
automatically construct the binary data with differentiated
entries or prove their compatibility.

No antifields, master equation, interacting differential, state,
or physical cohomology class are added. No reflected-cone
subtraction or triangle residue is used. The ramp width remains
fixed, and no limit $\eps\to0$ is taken.

Evaluation on $\CW$ is subsequent to admission. It neither
defines the graph product nor permits a restriction of its
configuration-space domain. The tree zero-evaluation subclasses
do not imply zeros of a degree-two quantum coefficient.

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.24\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
E0 & Frozen &
Selected pair, unchanged carrier, copied $\Cpart$, and the four
successor obligations are explicit.\\[0.35em]
E1 & Partly Constructed; Unresolved &
One common-frequency attempt; finite-regulator admission,
good-set convergence, and compact-localization independence.
The characteristic-set limit is unfinished.\\[0.35em]
E2 & Unresolved; not reached &
Uniform transverse scaling and microlocal estimates on the
three- and four-vertex strata and intersections are absent.\\[0.35em]
E3 & Unresolved &
Overlap equations and ordered-contact support are specified;
compatible complete-graph extensions are absent.\\[0.35em]
E4 & Conditional endpoint only; Unresolved &
Relative-coordinate balance and the proper-pushforward interface
are identified; no joint output is released.\\[0.35em]
E5 & Recorded &
Classical obstruction, witness, non-ideal theorem, and literal
factorization obstruction remain separate. No differential is
applied to the quantum coefficient.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Exact result of the selected attempt]
\label{thm:release}
For the fixed selected pair and the copied class $\Cpart$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The common-frequency regularization defines compact admitted
quadratic graph functionals at every positive regulator.
\item Its numerical graph products converge to the prescribed
products on the microlocally admissible open sets.
Auxiliary compact-localization changes have a remainder smaller
than every regulator power.
\item Contacts in $\Cpart$ cannot alter a prescription on the
collision complement. Agreement only on the admissible subset
does not prove that two prescriptions differ by permitted contacts.
\item The selected characteristic-set limit, the uniform transverse
estimates, the contact overlaps, and the complete output theorem
have not all been established.
\end{enumerate}
The requested coefficient therefore retains the ending
\textbf{Unresolved}, with E1 the earliest unfinished gate.
No assertion excludes every extension in $\Cpart$.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:finite},
Lemma~\ref{lem:smoothing},
Propositions~\ref{prop:localization} and
\ref{prop:good-limit},
Lemma~\ref{lem:rigidity},
and Proposition~\ref{prop:agreement} prove the affirmative
statements.

Definition~\ref{def:E1-missing} identifies the missing
characteristic-set limit and its prescription independence.
Sections~\ref{sec:E2}--\ref{sec:E4} state the subsequent obligations
without asserting that they have been discharged.
Under the declared release criteria, the absence of those proofs
permits neither Constructed nor Proved obstructed for the full
coefficient.
\end{proof}

The conclusion is deliberately narrower than a joint-extension
theorem. Finite regularization, canonical products on proper
subdomains, and compact support control do not complete that
theorem.

\subsection*{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged pair and contact class;
\item the explicit common-frequency family
\eqref{eq:smoothing}--\eqref{eq:t-reg};
\item the finite-regulator admission and compact-localization
remainder estimate;
\item the canonical limits on the admissible open sets;
\item the theorem that collision contacts cannot repair an
off-collision prescription failure;
\item the precise unfinished characteristic-set obligation;
\item the transverse, overlap, and proper-pushforward interfaces.
\end{enumerate}

It does not receive a named admitted $\Rquant$.
It receives no universal class obstruction and no permission
to form the binary Ward defect.

To continue the same attempt, the successor must prove convergence
and prescription independence on compact tests meeting the
noncoincident characteristic set, with complete tensor sums
retained. Only then can the three- and four-vertex transverse
estimates and compatible partial contacts be addressed as
extensions of an existing distribution.

A different prescription must be named as a different E1 attempt,
while preserving the copied pair and ambiguity class. The
auxiliary-cutoff theorem does not establish equivalence of every
possible prescription.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \text{selected E1 move: common-frequency regularization},\\[2pt]
 \tau>0:
 \quad t_\tau^{T,[2]}\text{ compact and microcausal},\\[2pt]
 U_\Gamma^{\mathrm{good}}:
 \quad A_{\Gamma,\tau}\longrightarrow A_\Gamma^{\mathrm{can}},\\[2pt]
 \Cpart|_{\Xoff}=0,\qquad
 \text{characteristic-set prescription still unfinished},\\[2pt]
 \text{joint coefficient at order }\lambda\hbar:
                         \quad\textbf{Unresolved}.
 \end{gathered}}
\]

The tree datum is not the quantum coefficient.
The sum-zero test does not cross the noncoincident characteristic
set. Exactly one prescription route has been attempted, and its
earliest missing theorem is explicit. Neither the partial results
nor the unresolved interface constitute an interacting BRST
algebra.

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

\paragraph{Dependency scope.}
XIII-C supplies the graph inventory, contact class, primitive
extensions, characteristic cancellation, and four successor
obligations. XIII-B supplies the tree datum, convention, properness,
and ordered values. XIII-A and XIII-D supply boundaries only.
XII-F's witness is copied and not recomputed.

\paragraph{New proof scope.}
The new results concern the common-frequency family, its
finite-regulator kernels, convergence on the admissible sets,
auxiliary compact-localization independence, off-collision contact
rigidity, and a conditional relative-coordinate endpoint.

\paragraph{Unresolved scope.}
No convergence theorem through the full noncoincident
characteristic set is asserted. No superficial degree is promoted
to a scaling theorem for an undefined distribution.
No isolated primitive extension is promoted to a compatible joint
extension.

\paragraph{Normalization scope.}
The selected regulator does not change the frozen action inverse,
the insertion sign, the Wick combinatorics, or the external
backgrounds. Its positive-regulator tails are not called exact
causal factorization. Ordered values are obligations of the limit.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant check,
independently executed symbolic computation, independent referee
review, or journal acceptance is asserted. The manuscript releases
the stated partial proofs and an explicit unresolved joint gate,
not a completed quantum construction.

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

\bibitem{XIIIC}
N.~Cheung,
\emph{Landmark XIII-C: The joint two-line extension of one selected
binary pair, on the unchanged carrier}.
Supplied manuscript \texttt{XIII-C.tex}.
Inventory, partial-diagonal class, primitive extensions,
characteristic cancellation, and four successor obligations.

\bibitem{XIIIB}
N.~Cheung,
\emph{Landmark XIII-B: One binary causal datum for the displayed
coefficient, on the unchanged carrier}.
Supplied manuscript \texttt{XIII-B.tex}.
Tree datum, properness, retarded convention,
ordered values, and unresolved two-line gate.

\bibitem{XIIIA}
N.~Cheung,
\emph{Landmark XIII-A: Classical cancellation on a fixed subclass,
and the obstruction to a coefficientwise restriction quotient}.
Supplied manuscript \texttt{XIII-A.tex}.
Evaluation-only boundary and non-ideal theorem.

\bibitem{XIIID}
N.~Cheung,
\emph{Landmark XIII-D: One bounded classical completion of the
auxiliary pairing, or its obstruction on the unchanged sheet}.
Supplied manuscript \texttt{XIII-D.tex}.
Class-qualified classical obstruction and distinction between
an admitted cochain and an integrated completion.

\bibitem{XIIF}
N.~Cheung,
\emph{Landmark XII-F: The gauge-fixed remainder of
$\Phi^{(2)}_{\mathrm D}$: companion support, the auxiliary pairing,
and what is not a Ward identity}.
Supplied manuscript \texttt{XII-F.tex}.
Retained full-shell witness and its classical scope.

\bibitem{XIIE}
N.~Cheung,
\emph{Landmark XII-E: Assembly of the second-order relational
coefficient after admission of the density-weighted cubic insertion}.
Supplied manuscript \texttt{XII-E.tex}.
Displayed coefficient, insertion sign, and scalar-sourced slot.

\bibitem{XIIC}
N.~Cheung,
\emph{Landmark XII-C: One declared cubic weighting:
actual Euler source, changed Noether identity,
and the coupled-jet question}.
Supplied manuscript \texttt{XII-C.tex}.
Unchanged selected action.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
``Microlocal analysis and interacting quantum field theories:
Renormalization on physical backgrounds,''
\emph{Communications in Mathematical Physics}
\textbf{208} (2000), 623--661,
arXiv:math-ph/9903028.
Microlocal product continuity, scaling degree,
and extension across points and submanifolds.

\bibitem{Dang}
N.~V.~Dang,
``The extension of distributions on manifolds,
a microlocal approach,''
\emph{Annales Henri Poincar\'e}
\textbf{17} (2016), 819--859,
arXiv:1412.2808.
Scaling and wavefront conditions for submanifold extensions.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428.
Functional domains, time ordering, and renormalization;
not an imported joint-extension theorem for the selected pair.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I:
Distribution Theory and Fourier Analysis},
second edition, Springer, 1990.
Submersion pullback, distribution products,
differential-operator wavefront bounds, and proper pushforward.

\end{thebibliography}

\end{document}