% !TeX program = pdflatex
%=====================================================================
% XIV-C.tex
%
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% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-C.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-C.tex
%
% Exact release:
%   - the XIII-C pair and partial-diagonal ambiguity class are frozen;
%   - the eleven branches and 48 labelled matchings are copied;
%   - individual line derivative orders are made explicit;
%   - the first new calculation concerns a retained noncoincident
%     null-related configuration, without cutting off the triangle;
%   - its light-cone defining functions have rank-two differentials
%     on a smooth codimension-four characteristic stratum;
%   - the copied wavefront cancellation is expressed by an explicit
%     positive relation between oriented conormals;
%   - collision-supported changes cannot modify a neighborhood of
%     that configuration;
%   - no uniform cancellation of the complete pair and no admissible
%     characteristic prescription are constructed;
%   - the first unresolved gate is J2.
%
% Ending:
%   Unresolved for the complete coefficient of lambda*hbar.
%
% Principal debugging corrections:
%   - Phi_D^(2) alone is the scalar-linear nine-branch lift.
%     The inherited eleven-branch pair uses
%       O = Phi_D^(2) + A_1(r_g[phi];phi).
%     The full observable is frozen explicitly; no symbol is silently
%     redefined and the scalar-sourced slot is included once.
%   - U_Gamma is the microlocally admissible part of the collision
%     complement, not the entire collision complement.
%   - the product test concerns all graph factors, including internal
%     retarded lines, not only the two cross-lines.
%   - superficial power counting is not a transverse scaling theorem
%     for a distribution that has not been constructed.
%   - the frozen class controls differences of extensions; its
%     contact formulas do not themselves define a reference extension.
%   - the characteristic diagnostic is not an evaluated residue,
%     a nonexistence theorem, or a quantum Ward defect.
%
% XIV-A and XIV-B are recorded only. Their relative antifield complex
% is not an input to the analytic construction.
%
% No R_1^(1), joint renormalized time-ordered product, quantum Ward
% defect, interacting differential, reduced algebra, state, numerical
% loop amplitude, triangle-residue subtraction, or sharp-width limit
% is constructed.
%
% No successful compilation, proof-assistant verification,
% independently executed symbolic computation, independent referee
% review, or journal acceptance is asserted.
%=====================================================================

\documentclass[11pt,a4paper]{article}

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\usepackage{enumitem}
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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{\ER}{E_0^{\mathrm R}}
\newcommand{\EA}{E_0^{\mathrm A}}
\newcommand{\ERm}{E_m^{\mathrm R}}
\newcommand{\EAm}{E_m^{\mathrm A}}
\newcommand{\Pmass}{P_m}
\newcommand{\GR}{G_{\mathrm{act}}^{\mathrm R}}
\newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}}
\newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\NW}{\mathscr N_W}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Csol}{\mathscr C_{\mathrm{sol}}}
\newcommand{\CW}{\mathscr C_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{\TT}{\mathsf T_2}
\newcommand{\Htime}{H_T}
\newcommand{\Rtree}{R_1^{(0)}}
\newcommand{\Rquant}{R_1^{(1)}}
\newcommand{\Cpart}{\mathfrak C_{\mathrm{part}}}
\newcommand{\Ctot}{\mathfrak C_{\mathrm{tot}}}
\newcommand{\Lie}{\mathcal L}
\newcommand{\QH}{Q_H}
\newcommand{\Coll}{\mathcal D_{\Gamma}}
\newcommand{\Admiss}{\mathcal U_{\Gamma}}
\newcommand{\Nchar}{\mathcal N^\circ}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

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

\title{\textbf{Landmark XIV-C}\\[0.4em]
The joint two-line extension of one selected binary pair:\\
noncoincident characteristic configurations,\\
transverse estimates, and the output wavefront gate\\[0.4em]
\large The retained characteristic diagnostic and the exact J2 stop}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
This manuscript continues XIII-C's joint-extension problem on the
unchanged polynomial carrier. It does not continue XIV-B's relative
antifield complex.

We freeze the switched metric--scalar vertex and the complete
displayed second-order coefficient, including its scalar-sourced
slot once. The eleven-branch decomposition, 48 labelled matchings,
partial-diagonal ambiguity class, and elementary parallel-line
extensions are copied. Individual line derivative orders distinguish
those primitive results from the remaining joint problem.

The first new calculation is performed at a retained, noncoincident,
null-collinear configuration. The three light-cone defining functions
have differentials of rank two along a smooth characteristic
stratum of codimension four. Their oriented conormals admit an
explicit positive relation, reproducing the failure of the sufficient
product criterion throughout this stratum. A retained component of
the actual incidence has nonzero coefficient and nonvanishing
differential symbols there. This does not determine whether the
complete tensor and branch sum cancels the configuration.

A separate support lemma shows that the frozen collision contacts
cannot alter any neighborhood disjoint from the collision loci.
They therefore cannot supply an unnamed characteristic prescription.
No uniform cancellation of the complete selected pair, and no
prescription with the required ambiguity and wavefront control, is
constructed. The first unresolved gate remains J2. The subsequent
transverse, gluing, and output obligations are recorded but not
opened.

The complete coefficient of $\lambda\hbar$ remains Unresolved.
No nonexistence theorem, admitted quantum coefficient, residue
subtraction, or Ward defect is asserted.
\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) {XIII-C pair};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Scalar-linear lift};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {XIV-B ghost theory};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The frozen pair is not the XIV-B construction.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{This continuation is XIII-C, not XIV-B. 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-XIV-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) {XIII-C pair};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {XIV-B ghost theory};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The frozen pair is not the XIV-B construction.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{This continuation is XIII-C, not XIV-B. 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-XIV-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) {XIII-C pair};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Scalar-linear lift};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {XIV-B ghost theory};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The frozen pair is not the XIV-B construction.};
\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-XIV-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) {XIII-C 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) {XIV-B ghost theory};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Scalar-linear lift. The frozen pair is not the XIV-B construction.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{XIII-C pair is not XIV-B ghost theory. 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-XIV-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: XIII-C pair};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Scalar-linear lift};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: XIV-B ghost theory};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The frozen pair is not the XIV-B construction.};
\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-XIV-C}
\end{figure}

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


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



\scopebox{Release and first stop}{%
\textbf{Continuation:} XIII-C, not XIV-B.

\smallskip
\textbf{Frozen pair:}
\[
 \left(\Vcal,\Ocal\right),\qquad
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi).
\]
The scalar-linear lift $\PhiD$ alone is not silently identified
with this eleven-branch observable.

\smallskip
\textbf{Copied:} the incidence inventory, $\Cpart$, primitive
parallel-line extensions, tree datum, and inherited support control.

\smallskip
\textbf{Proved here:} the rank-two characteristic geometry,
its positive conormal relation, a nonzero retained component
diagnostic, and the inability of collision-supported changes
to modify a disjoint characteristic neighborhood.

\smallskip
\textbf{First unresolved gate: J2.}
Neither uniform cancellation of the complete pair nor an
admissible prescription through the characteristic set is supplied.

\smallskip
\textbf{Ending for the joint coefficient: Unresolved.}
J3--J5 are not reached. No $\Rquant$ is admitted and no
differential is applied to it.}

\tableofcontents

%=====================================================================
\section{Release criteria and copied prerequisites}
\label{sec:inputs}
%=====================================================================

The three endings apply to the named analytic construction:
\begin{enumerate}
\item \textbf{Constructed}: the joint distribution exists with
the prescribed normalizations, the four successor obligations
are discharged, and its pushed-forward coefficient is admitted.
\item \textbf{Proved obstructed}: a demonstrated incompatibility
excludes every successful construction in a previously frozen
class.
\item \textbf{Unresolved}: a required prescription, estimate,
contact composition, or output theorem has not been established.
\end{enumerate}
Failure of a sufficient product criterion is not the second
ending. A constructed elementary extension is not a joint
extension.

\subsection{The inherited analytic release}

\begin{inputstatement}[XIII-C]
\label{input:XIIC}
For the frozen full displayed coefficient and switched vertex:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the eleven-branch decomposition and all 48 labelled
double cross-contraction matchings are inventoried;
\item $\Ctot$ is a subclass of the widened partial-diagonal
ambiguity class $\Cpart$;
\item elementary parallel-line distributions admit extensions
with their computed scaling-degree bounds;
\item collision exclusion and the $Z_3$ sum-zero bounds do not
establish the complete joint product;
\item distinct null-related vertices admit a nonzero wavefront
cancellation in a retained incidence;
\item the full coefficient of $\lambda\hbar$ remains Unresolved.
\end{enumerate}
These are copied results, not new extension theorems of XIV-C
\cite{XIIIC}.
\end{inputstatement}

The four successor obligations are retained verbatim in substance:
\begin{enumerate}
\item a prescription through the noncoincident characteristic
configurations, or proved cancellation in the selected pair;
\item uniform transverse estimates on the three- and four-vertex
strata and their intersections;
\item compatible partial-diagonal contacts;
\item the output wavefront theorem after proper pushforward.
\end{enumerate}

\begin{inputstatement}[XIII-B, through the supplied XIII-C handoff]
\label{input:XIIIB}
The vertex and the tree datum are admitted. The selected tree
composition is
\[
 \Rtree=-\pair{\Ocal'}{\GR\Vcal'}
 =\pair{\Ocal'_\gamma}{\rg[\phi]}
                   +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]}.
\]
Its properness and compact microcausal kernel are inherited.
The literal normalization $R_1=\Vcal\star_H\Ocal$ is incompatible
with the frozen retarded convention. The quantum joint extension
was not released \cite{XIIIC}.
\end{inputstatement}

Properness controls an existing distributional pushforward.
It does not define an undefined product.

\subsection{XIV-A and XIV-B: recorded, not analytic inputs}

The supplied XIV-A and XIV-B manuscripts discuss a relative
classical master equation, unrestricted classical breaking,
compact relative ghosts and their free-shell boundary, and
evaluation identities on smaller bosonic configuration sets.

None is used to define a propagator, a graph product, a contact,
or an extension in this paper. No Koszul--Tate resolution,
antifield algebra, or quantum interpretation of those
manuscripts is needed here.

Likewise, the auxiliary value $-8I$ and the evaluation identity
on $\mathscr C_{W,\mathrm{bos}}$ do not normalize this quantum
coefficient. They are separate classical statements.

\subsection{Analytic tools and their limited role}

The distribution product criterion is used on domains where
its hypothesis holds. It is sufficient, not a converse
\cite{Hormander,FR}.

Point-extension results control distributions already defined
away from the point with finite scaling degree. Extension along
a submanifold also requires longitudinal and microlocal control
\cite{BF,Dang}. Those tools do not provide the missing joint
hypotheses merely by being cited.

%=====================================================================
\section{J0: the pair and extension contract}
\label{sec:J0}
%=====================================================================

\subsection{Geometry, carrier, and product data}

Fix
\[
 M=\R^4,\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \Pmass=\Box+m^2,
\]
\[
 \MT=[0,T]\times\R^3,\qquad
 \MTo=(0,T)\times\R^3,\qquad 0<\eps<T.
\]
Fix the inherited smooth ramp $\sig$, zero for $t\le0$ and
one for $t\ge\eps$, the compact scalar switch $g$, the observation
$f$, and the frozen translation-invariant Hadamard matrix $H$.

The carrier remains
\[
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                         \Nsol[[\hbar,\kappa]],
 \qquad
 F\star_HG=m\exp(\hbar\Gamma_H)(F\otimes G).
\]
The action Green blocks are
\[
 \GR_{\gamma\gamma}=-\PP\ER,\qquad
 \GR_{\phi\phi}=-\ERm,
 \qquad
 \Htime=H+i\GA.
\]
The metric--scalar blocks vanish. The inverse coefficient is
$-1$, not the differently normalized X-track coefficient $-4$.

We denote by $F_\mu$ the scalar time-ordered numerical line
of mass $\mu$, with the fixed nonzero species fiber factors
separated when convenient. This notation does not change the
frozen Hadamard data.

\subsection{The observable naming correction}

The predecessor notation distinguishes
\[
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD
\]
from the full displayed coefficient
\begin{equation}
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi).
 \label{eq:O}
\end{equation}
The former supplies nine branches; the latter supplies eleven.
The target's instruction to retain the scalar-sourced slot and
the inherited 48-match inventory therefore fixes the pair
$(\Vcal,\Ocal)$, not the nine-branch pair $(\Vcal,\PhiD)$.

This is a naming correction made before the calculation, not
a change of observable after a detector. Throughout this paper,
$\PhiD$ keeps its predecessor meaning.

The interaction is the functional
\begin{equation}
 \Vcal=-\frac12\int g\,\gamma_{\mu\nu}
                              T_0^{\mu\nu}[\phi]\,\dd^4x,
 \qquad
 T_0^{\mu\nu}
 =\partial^\mu\phi\partial^\nu\phi
 -\frac12\eta^{\mu\nu}
                 \bigl((\partial\phi)^2-m^2\phi^2\bigr).
 \label{eq:vertex}
\end{equation}
It is not an Euler source operator.

The inherited filled slots remain
\[
 \rD=\PP\ER\JD=-u_{\mathrm D},
 \qquad
 \rg[\phi]=-\frac12\PP\ER(gT_0[\phi]).
\]
The density-weighted source $\JD$ and its layer coefficients
are those already selected and admitted on the XII/XIII track.
No new Einstein--Hilbert insertion is added.

\subsection{Order and normalization}

The retarded convention is
\begin{equation}
 R_1(\Vcal;\Ocal)
 =\frac{i}{\hbar}
       \bigl(\TT^{\mathrm{ren}}(\Vcal,\Ocal)
                           -\Vcal\star_H\Ocal\bigr).
 \label{eq:R1}
\end{equation}
The requested order is
\[
 R_{\lambda\Vcal}\Ocal
 =\Ocal+\lambda(\Rtree+\hbar\Rquant)
                         +O(\lambda^2,\lambda\hbar^2).
\]
Here $\lambda$ counts one additional vertex insertion. It is
not used to reweight the already filled slots.

The formal two-cross-line expression is
\begin{equation}
 \frac{i}{2}
 \pair{\Vcal''\otimes\Ocal''}
                    {\Htime^{\otimes2}-H^{\otimes2}}.
 \label{eq:formal}
\end{equation}
It remains formal until the complete graph compositions are
defined. The difference is kept grouped; no undefined mixed
intermediate product is declared to exist.

The inherited ordered values are
\[
 \supp\Vcal\text{ strictly later than }\supp\Ocal
       \Longrightarrow R_1=0,
\]
\[
 \supp\Ocal\text{ strictly later than }\supp\Vcal
       \Longrightarrow
       R_1=\frac{i}{\hbar}[\Ocal,\Vcal]_{\star_H}.
\]
These refer to full functional supports.

\subsection{The frozen ambiguity class}

For an inventory graph let $y$ be the interaction vertex.
Connected collision subsets $I$ containing $y$ define
\[
 \Delta_I=\{x_a=x_b:\ a,b\in I\}.
\]
The inherited class includes their relevant intersections and
contracted loci, while observable-only subgraph normalizations
are retained.

Before pushforward, an allowed contact change has the form
\begin{equation}
 D_\Gamma\left[
 w_\Gamma\,\partial_r^\alpha\delta(r_I)
                         S_{\Gamma/I}\right],
 \qquad |\alpha|\le6,
 \label{eq:Cpart}
\end{equation}
subject to XIII-C's conditions: inherited smooth external
weights, retained spectator graph, even parity, ghost number
zero, two residual field slots, compact graph closure, and
unchanged tree and observable-subgraph normalizations.
Only defined products and pushforwards belong to $\Cpart$.

The spectator need not be smooth in tangential variables.
The bound six is the frozen conservative bound; an established
degree can impose a smaller locus-specific bound.

The class controls permitted \emph{differences} of constructions.
It does not by itself select a reference distribution across
the noncoincident characteristic set.

\subsection{A retained diagnostic fixed before extension}

To ensure that the subsequent diagnostic belongs to the retained
graph support, choose once
\[
 \eps<t_*<t_*+2L<T,\qquad L>0,
\]
and the locations
\[
 u_0=(t_*,0,0,0),\quad
 y_0=(t_*+L,L,0,0),\quad
 x_0=(t_*+2L,2L,0,0).
\]
Choose the original observation with $f(x_0)\ne0$ and the
original switch equal to one near these locations, with compact
interior support. All switches and observations are fixed now.

This is an admissible retained instance of the inherited pair.
No cutoff is applied to its numerical triangle, and no
coefficient is chosen after computing a value. The subsequent
diagnostic concerns this fixed instance; no assertion that
every observation retains the same configuration is needed.

\subsection{Negative freeze}

There is no restriction quotient, antifield-dependent observable,
spurion transformation, convolution prescription, new
Einstein--Hilbert vertex, triangle-residue subtraction, or
sharp-width limit. Changing the pair, propagators, or ambiguity
support after J2 would define a new contract.

%=====================================================================
\section{J1: copied branches, line orders, and collision inventory}
\label{sec:J1}
%=====================================================================

This section names the inherited graph data. It constructs no
new extension.

\subsection{Definitions used by the branch display}

Write
\[
 X=X_1[\gamma],\qquad p=p_1[\gamma;\phi],
 \qquad \int_f A=\int fA\,\dd^4x,
\]
where
\[
 X_1^\mu[r]
 =-\ER\partial_\alpha(\sig a_1^{\alpha\mu}[r]),
\]
\[
 p_1[r;\phi]
 =-\ERm\left[
 \partial_\mu(g a_1^{\mu\nu}[r]\partial_\nu\phi)
                         +m^2g b_1[r]\phi\right].
\]
The density coefficients are
\[
 a_1^{\mu\nu}[r]
 =\frac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},
 \qquad b_1[r]=\frac12\tr_\eta r,
\]
\[
 b_2[\gamma]
 =\frac18(\tr_\eta\gamma)^2
                   -\frac14\gamma_{\alpha\beta}\gamma^{\alpha\beta},
\]
\[
 a_2^{\mu\nu}[\gamma]
 =\gamma^\mu{}_\alpha\gamma^{\alpha\nu}
 -\frac12(\tr_\eta\gamma)\gamma^{\mu\nu}
                              +b_2[\gamma]\eta^{\mu\nu}.
\]
Put
\[
 U_\sig[\gamma]q
 =\partial_\alpha(\sig a_1^{\alpha\beta}[\gamma]\partial_\beta q),
\]
\[
 V_{2,g}[\gamma,0]\phi
 =\partial_\mu(g a_2^{\mu\nu}[\gamma]\partial_\nu\phi)
                              +m^2g b_2[\gamma]\phi.
\]

\begin{inputstatement}[Eleven branches, copied]
\label{input:branches}
The full lift \eqref{eq:O} is $\sum_{j=1}^{11}B_j$, with
\begin{align}
 B_1&=-\int_f\ERm(V_{1,g}[\gamma]p),&
 B_2&=-\int_f\ERm(V_{2,g}[\gamma,0]\phi),\\
 B_3&=-\int_fX^\mu\partial_\mu p,&
 B_4&=\int_fX^\nu\partial_\nu X^\mu\partial_\mu\phi,\\
 B_5&=\int_f\ER(U_\sig[\gamma]X^\mu)\partial_\mu\phi,&
 B_6&=\int_f\ER(\partial_\alpha(\sig
                      a_2^{\alpha\mu}[\gamma]))\partial_\mu\phi,\\
 B_7&=\frac12\int_fX^\mu X^\nu\partial_\mu\partial_\nu\phi,&
 B_8&=-\int_fX_1^\mu[\rD]\partial_\mu\phi,\\
 B_9&=\int_fp_1[\rD;\phi],&
 B_{10}&=-\int_fX_1^\mu[\rg]\partial_\mu\phi,\\
 B_{11}&=\int_fp_1[\rg;\phi].
 \label{eq:branches}
\end{align}
The signs, affine slot identity, and inclusion of the
scalar-sourced slot once are inherited from XIII-C \cite{XIIIC}.
\end{inputstatement}

\subsection{Individual line orders}

A displayed divergence is transferred to its incoming numerical
line. Derivatives on unused fields are not transferred to an
unrelated line.

In the following table a superscript on a field slot is its
derivative order, and an edge superscript is the derivative order
on that internal retarded line. The scalar kinetic terms give
the displayed maxima; the mass terms have the corresponding
lower orders.

\begin{center}
\scriptsize
\begin{tabularx}{\linewidth}{@{}c p{.34\linewidth}
                                  p{.34\linewidth} X@{}}
\toprule
Branch & Observable field slots & Internal lines and orders &
Maximum total\\
\midrule
$B_1$ &
$\Gamma^0(u),\Gamma^0(v),\varphi^1(v)$ &
$(x-u)^1,(u-v)^2$ & $4$\\
$B_2$ &
$\Gamma^0(u),\Gamma^0(u),\varphi^1(u)$ &
$(x-u)^1$ & $2$\\
$B_3$ &
$\Gamma^0(u),\Gamma^0(v),\varphi^1(v)$ &
$(x-u)^1,(x-v)^2$ & $4$\\
$B_4$ &
$\Gamma^0(u),\Gamma^0(v),\varphi^1(x)$ &
$(x-u)^1,(x-v)^2$ & $4$\\
$B_5$ &
$\Gamma^0(u),\Gamma^0(v),\varphi^1(x)$ &
$(x-u)^1,(u-v)^2$ & $4$\\
$B_6$ &
$\Gamma^0(u),\Gamma^0(u),\varphi^1(x)$ &
$(x-u)^1$ & $2$\\
$B_7$ &
$\Gamma^0(u),\Gamma^0(v),\varphi^2(x)$ &
$(x-u)^1,(x-v)^1$ & $4$\\
$B_8$ &
$\Gamma^{|\alpha|}(v),\Gamma^{|\beta|}(v),
 \varphi^1(x)$ &
$(x-u)^1,(u-v)^0$ & $4$\\
$B_9$ &
$\Gamma^{|\alpha|}(v),\Gamma^{|\beta|}(v),
 \varphi^1(u)$ &
$(x-u)^1,(u-v)^0$ & $4$\\
$B_{10}$ &
$\varphi^{e_s}(v),\varphi^{e_s}(v),\varphi^1(x)$ &
$(x-u)^1,(u-v)^0$ & $4$\\
$B_{11}$ &
$\varphi^{e_s}(v),\varphi^{e_s}(v),\varphi^1(u)$ &
$(x-u)^1,(u-v)^0$ & $4$\\
\bottomrule
\end{tabularx}
\end{center}

For the gravitational source terms,
$|\alpha|+|\beta|\le2$, including their fixed layer coefficients.
For a stress monomial $e_s=1$ in the kinetic case and $e_s=0$
in the mass case.

The interaction slots at $y$ have orders
\[
 \Gamma^0(y),\qquad
 \varphi^{e_v}(y),\qquad \varphi^{e_v}(y),
 \qquad e_v\in\{0,1\}.
\]
Therefore each cross-line order is fixed by its endpoints:
\begin{equation}
 d_{\gamma\gamma}=d_{\Gamma,\mathrm{obs}},
 \qquad
 d_{\phi\phi}=e_v+d_{\varphi,\mathrm{obs}}.
 \label{eq:cross-orders}
\end{equation}
In $B_7$ the scalar cross-line can have order three. In $B_8$
and $B_9$ the metric cross-line carries the selected source
derivative. For $B_{10}$ and $B_{11}$, each of the two scalar
cross-lines separately uses \eqref{eq:cross-orders}.

These assignments name line orders, not a tensor truncation.
The complete graph has maximum total derivative count six
after inclusion of the interaction.

\subsection{The 48 labelled matchings}

For $j=1,\ldots,9$, label the two observable metric slots
$i=1,2$ and the two interaction scalar slots $a=1,2$.
The matchings
\[
 M_{j;i,a},\qquad i,a\in\{1,2\},
\]
join the interaction metric to the selected observable metric
and scalar slot $a$ to the single observable scalar.
There are four for each branch.

For $j=10,11$, label the three observable scalar slots
$i=1,2,3$. Select an unordered pair $i<k$ and one of its
two bijections to the two interaction scalar slots:
\[
 M_{j;\{i,k\},\pi},\qquad \pi\in S_2.
\]
There are six for each branch. Thus
\[
 9\cdot4+2\cdot6=48.
\]
Coincident slot locations do not erase the slot labels.
Two metric cross-lines are absent because the interaction
has only one metric slot; metric--scalar lines vanish for
the frozen product data.

\subsection{Cycles and collision loci}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}c p{.34\linewidth}X@{}}
\toprule
Branch & Selected locations & Cycle vertices\\
\midrule
$B_1$ & $(u,v)$ or $(v,v)$ &
$\{y,u,v\}$ or $\{y,v\}$\\
$B_2$ & $(u,u)$ & $\{y,u\}$\\
$B_3$ & $(u,v)$ or $(v,v)$ &
$\{y,u,x,v\}$ or $\{y,v\}$\\
$B_4$ & $(u,x)$ or $(v,x)$ &
$\{y,u,x\}$ or $\{y,v,x\}$\\
$B_5$ & $(u,x)$ or $(v,x)$ &
$\{y,u,x\}$ or $\{y,v,u,x\}$\\
$B_6$ & $(u,x)$ & $\{y,u,x\}$\\
$B_7$ & $(u,x)$ or $(v,x)$ &
$\{y,u,x\}$ or $\{y,v,x\}$\\
$B_8$ & $(v,x)$ & $\{y,v,u,x\}$\\
$B_9$ & $(v,u)$ & $\{y,v,u\}$\\
$B_{10}$ & $(v,v)$ or $(v,x)$ &
$\{y,v\}$ or $\{y,v,u,x\}$\\
$B_{11}$ & $(v,v)$ or $(v,u)$ &
$\{y,v\}$ or $\{y,v,u\}$\\
\bottomrule
\end{tabularx}
\end{center}

For a collapsed connected subset $I$, its normal codimension
is $4(|I|-1)$. The inherited superficial count is
$2E_I+D_I$, with numerical internal-edge count $E_I$ and
derivative count $D_I$.

\begin{center}
\small
\begin{tabular}{@{}lcccc@{}}
\toprule
Type & Edges & Codimension & Superficial count &
Count minus codimension\\
\midrule
Parallel pair & $2$ & $4$ & $4+D_I$ & $D_I\le6$\\
Three-cycle & $3$ & $8$ & $6+D_I$ & $D_I-2\le4$\\
Four-cycle & $4$ & $12$ & $8+D_I$ & $D_I-4\le2$\\
Forest, $n$ vertices & $n-1$ & $4(n-1)$ &
$2(n-1)+D_I$ & $D_I-2(n-1)$\\
\bottomrule
\end{tabular}
\end{center}

These are copied power counts, not established scaling degrees
of a presently undefined joint distribution.

\subsection{Primitive extensions already available}

The copied primitive results concern
\[
 t_{\alpha\beta}(r)
   =\partial^\alpha F_\mu(r)\partial^\beta F_\nu(r),
 \qquad r\ne0.
\]
Their degrees satisfy
\[
 \sd(t_{\alpha\beta})\le4+|\alpha|+|\beta|.
\]
The temporal kinetic primitive has degree eight in codimension
four and contact order at most four. These distributions
already admit the stated point extensions \cite{XIIIC}.

Their extension is not an extension of a three- or four-vertex
composition with singular spectators. In particular it does
not settle the noncoincident configuration examined next.

%=====================================================================
\section{J2: the retained characteristic configuration}
\label{sec:J2}
%=====================================================================

\subsection{The admissible set is not the collision complement}

Let $\Coll$ be the relevant union of collision loci in a graph's
vertex space. Define $\Admiss$ to be the subset of its complement
where the full graph product satisfies the distribution product
criterion.

Every line is pulled back to the complete graph variables.
The criterion excludes a selection of line wavefront covectors,
with zero permitted for unused factors, whose sum vanishes at
all vertices. It is not merely a test of the two cross-lines
against each other.

On $\Admiss$ the graph product is defined by the inherited
product calculus. This does not establish equality
\[
 \Admiss=\text{vertex space}\setminus\Coll.
\]
The following calculation identifies the retained failure of
that proposed equality. No triangle cutoff is its starting point.

\subsection{First new calculation: conormals at a retained triple}

For the $B_6$ incidence, the underlying scalar numerical factors
are
\begin{equation}
 E_0^{\mathrm R}(x-u)\,F_0(y-u)\,F_m(y-x).
 \label{eq:triangle}
\end{equation}
Choose
\[
 \ell=(1,1,0,0),\qquad p=\eta(\ell,\cdot).
\]
At the fixed triple in J0,
\[
 y_0-u_0=L\ell,\qquad x_0-u_0=2L\ell.
\]

With the inherited time-ordered orientation, the wavefront
selections at the line endpoints are
\[
 \begin{array}{c|c}
 \text{line}&\text{endpoint covectors}\\ \hline
 E_0^{\mathrm R}(x-u)&(-p,p)\text{ at }(x,u)\\
 F_0(y-u)&(p,-p)\text{ at }(y,u)\\
 F_m(y-x)&(-p,p)\text{ at }(y,x).
 \end{array}
\]
The retarded forward-null singularity permits both conormal
signs. The time-ordered signs are those copied in XIII-C.
A simultaneous reversal of Fourier convention reverses all
three selections and leaves the calculation unchanged.

In the common vertex order $(x,u,y)$ these covectors are
\begin{equation}
 \nu_R=(-p,p,0),\qquad
 \nu_1=(0,-p,p),\qquad
 \nu_2=(p,0,-p).
 \label{eq:nus}
\end{equation}
They are nonzero and
\[
 \nu_R+\nu_1+\nu_2=0.
\]
The original weight $f(x_0)\sig(u_0)g(y_0)$ is nonzero.
None of the vertices coincides with another. This is a
retained configuration of the frozen graph, not a point
removed by collision exclusion.

This selection is the inherited characteristic cancellation.
The new calculation below identifies its geometric rank and
support consequences.

\subsection{A codimension-four characteristic stratum}

Define
\[
 \Nchar=
 \left\{
 (x,u,y):
 \begin{array}{l}
 y=u+\alpha\ell_{\mathbf n},\
 x=u+\beta\ell_{\mathbf n},\\
 \ell_{\mathbf n}=(1,\mathbf n),\
 \mathbf n\in S^2,\
 0<\alpha<\beta
 \end{array}
 \right\},
\]
restricted to the open slab where all three vertices are
interior. This stratum excludes every pairwise collision.

\begin{proposition}[Characteristic geometry]
\label{prop:geometry}
$\Nchar$ is a smooth embedded submanifold of the twelve-dimensional
three-vertex space, of dimension eight and codimension four.

Let
\[
 q_R=(x-u)^2,\qquad q_1=(y-u)^2,\qquad q_2=(x-y)^2,
\]
where the square uses $\eta$. On $\Nchar$,
\begin{equation}
 -\frac1\beta\,\dd q_R+
 \frac1\alpha\,\dd q_1+
 \frac1{\beta-\alpha}\,\dd q_2=0,
 \qquad
 \rank\{\dd q_R,\dd q_1,\dd q_2\}=2.
 \label{eq:rank}
\end{equation}
\end{proposition}

\begin{proof}
The parameterization by $(u,\alpha,\beta,\mathbf n)$ is smooth.
It is injective, with smooth inverse on its image:
\[
 u=u,\quad
 \alpha=y^0-u^0,\quad
 \beta=x^0-u^0,\quad
 \mathbf n=\frac{\mathbf y-\mathbf u}{y^0-u^0}.
\]
Its differential is injective: the $u$ components determine
$\delta u$, the temporal differences determine $\delta\alpha$
and $\delta\beta$, and then the spatial differences determine
the two tangent components of $\delta\mathbf n$.
The dimension is $4+1+1+2=8$.

Writing $p_{\mathbf n}=\eta(\ell_{\mathbf n},\cdot)$ gives
\[
 \dd q_R=2\beta\,p_{\mathbf n}\cdot(\dd x-\dd u),
\]
\[
 \dd q_1=2\alpha\,p_{\mathbf n}\cdot(\dd y-\dd u),
 \qquad
 \dd q_2=2(\beta-\alpha)\,
                         p_{\mathbf n}\cdot(\dd x-\dd y).
\]
These equations give the relation in \eqref{eq:rank}.
The last two differentials are independent because their
nonzero components involve distinct free endpoint variations.
The relation makes the rank exactly two.
\end{proof}

The three light-cone equations consequently do not meet
transversally at this stratum. Their differential rank is
two although the null-collinear stratum has codimension four.
Neither number is the codimension eight of a three-vertex
collision diagonal.

\begin{corollary}[Positive conormal relation]
\label{cor:positive}
At every point of $\Nchar$, the oriented nonzero conormals
\[
 n_R=-\frac{\dd q_R}{2\beta},\qquad
 n_1=\frac{\dd q_1}{2\alpha},\qquad
 n_2=\frac{\dd q_2}{2(\beta-\alpha)}
\]
satisfy $n_R+n_1+n_2=0$ and reproduce
\eqref{eq:nus} with $p=p_{\mathbf n}$.

There is no tangent vector $v$ at that point for which
$n_R(v),n_1(v),n_2(v)$ are all strictly positive.
\end{corollary}

\begin{proof}
The formulas in the proof of Proposition~\ref{prop:geometry}
give the stated covectors and sum. If all three evaluations
were positive, their sum evaluated on $v$ would be positive,
contradicting the zero relation.
\end{proof}

The last assertion only rules out this simultaneous strict
sign separation of the three selected conormals. It is not
a theorem excluding other boundary-value constructions,
finite parts, or cancellation in the complete pair.

\subsection{The derivative symbols do not remove this diagnostic}

The actual $B_6$ graph contains differentiated lines. The
selected temporal term has, up to fixed nonzero constants,
\begin{equation}
 \partial_{u^0}E_0^{\mathrm R}(x-u)\,
 F_0(y-u)\,
 \partial_{x^0}\partial_{y^0}F_m(y-x).
 \label{eq:differentiated}
\end{equation}
Its differential orders are $1,0,2$, exactly the assignments
of J1.

\begin{proposition}[A nonzero retained component]
\label{prop:component}
Before summation over tensor components and branches, the
temporal incidence \eqref{eq:differentiated} has a nonzero
coefficient in the frozen pair. Its line differential symbols
are nonzero at the selections \eqref{eq:nus}.
\end{proposition}

\begin{proof}
For a tensor whose only component is $\gamma_{00}=h$,
the displayed density formula gives
\[
 b_2=-\frac18h^2,\qquad
 a_2^{00}=\frac38h^2.
\]
Thus $B_6$ contains the nonzero monomial
\[
 \frac38\int f(x)\,
 E_0^{\mathrm R}(x-u)\,
 \partial_{u^0}(\sig(u)\gamma_{00}(u)^2)\,
 \partial_{x^0}\phi(x)\,\dd^4u\,\dd^4x.
\]
Transferring the divergence produces a differentiated
retarded line with a nonzero coefficient.

The interaction contains the temporal kinetic monomial
$-\frac14g\,\gamma_{00}(\partial_0\phi)^2$.
One metric cross-contraction is nonzero because
$\mathcal P_{00,00}=1/2$, and one scalar cross-contraction
joins the two displayed temporal scalar derivatives.
The fixed species factors are nonzero.

At \eqref{eq:nus}, $p_0=1$. The temporal retarded derivative
has nonzero symbol, and the two scalar derivatives have
nonzero product symbol. Their principal conormal
singularities are therefore not removed by these
differential operators. Finally, the original external
weight is nonzero at the retained triple fixed in J0.
\end{proof}

This proposition verifies that the diagnostic is not an
unrelated scalar toy graph. It does not prove that the
full $B_6$ tensor sum, or the sum of all eleven branches,
has a nonzero residue. No such residue has been defined.

\subsection{What would count as cancellation}

To discharge J2 by cancellation one would have to construct
or control the complete grouped coefficient and prove that
the dangerous configurations disappear throughout the retained
noncoincident characteristic domain.

A single nonzero component does not disprove cancellation
between components. A single vanishing component does not
prove it. The rank relation above is uniform as geometric
data on $\Nchar$; it is not a uniform theorem about the
summed numerical distribution.

The inherited existence of the $H$ contraction also does not
define the $\Htime$ contraction. The grouped difference
\eqref{eq:formal} must remain grouped until a valid prescription
or cancellation argument is supplied.

%=====================================================================
\section{Collision contacts cannot fill the characteristic gap}
\label{sec:contacts}
%=====================================================================

The following support statement is proved at J2. It is not
a J4 construction or gluing theorem.

\begin{lemma}[A disjoint neighborhood of the retained configuration]
\label{lem:neighborhood}
For the three-vertex graph at $(x_0,u_0,y_0)$, there is an open
neighborhood $V$ whose closure misses every collision locus
of that graph.
\end{lemma}

\begin{proof}
The three locations are pairwise distinct. Their pairwise
Euclidean coordinate distances have a positive minimum.
Choose sufficiently small vertex neighborhoods so that their
closures remain pairwise disjoint. Their product is the
required $V$ and misses every equality of two vertices,
hence every connected collision diagonal.
\end{proof}

\begin{proposition}[Rigidity of collision-supported changes]
\label{prop:rigidity}
Every defined contact change in $\Cpart$ for this graph
restricts to zero on $V$. Consequently any two graph
constructions whose difference belongs to the frozen
$\Cpart$ have the same restriction to $V$.
\end{proposition}

\begin{proof}
Each term \eqref{eq:Cpart} has support in its specified
collision locus. Differentiation does not enlarge support;
nor does multiplication in a defined product. The finite
sum is supported in the collision union. Its restriction
to the disjoint neighborhood $V$ is zero.
\end{proof}

The statement is graphwise before pushforward, where the
ambiguity class was defined. A contact can yield a nonlocal
looking pushed-forward coefficient through its spectator;
that does not change its pre-pushforward collision support.

\begin{corollary}[Characteristic ambiguity is not a partial contact]
\label{cor:charambiguity}
If two proposed prescriptions differ on $V$ by a nonzero
distribution supported on $\Nchar\cap V$, their difference
is not a member of the frozen collision-contact class.
\end{corollary}

\begin{proof}
Every class member vanishes on $V$ by
Proposition~\ref{prop:rigidity}; the proposed difference does not.
\end{proof}

This is a necessary ambiguity check, not nonexistence.
A canonical or otherwise uniquely fixed characteristic
prescription could in principle have no such extra ambiguity.
No theorem establishes that prescription here.

In particular, ``choose the partial contacts'' is not a
definition of the graph on $V$. The contacts can change
collision extensions only after the off-collision graph,
including its characteristic configurations, is defined.

\subsection{Why a cutoff is not substituted for the missing proof}

Removing a neighborhood of $\Nchar$ defines a smaller problem.
A limit restoring that neighborhood needs an actual convergence
or renormalized-limit theorem, its transverse degree, and its
wavefront estimate.

No such limit is introduced. No cutoff triangle is assigned
a value, and no subtraction supported on the characteristic
set is added to $\Cpart$. The class remains the one fixed
in J0.

%=====================================================================
\section{The J2 verdict and the unentered downstream gates}
\label{sec:stop}
%=====================================================================

\begin{theorem}[Exact characteristic-gate verdict]
\label{thm:J2}
For the retained instance of the pair frozen in J0:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the separate-line product criterion fails on the
noncoincident characteristic configuration and its smooth
null-collinear stratum;
\item that failure is not removed by collision exclusion,
line differentiation in the retained temporal component,
or collision-supported contact changes;
\item neither a uniform cancellation theorem for the complete
pair nor a prescription through the characteristic set
with the required ambiguity and wavefront control has
been constructed;
\item no contradiction excluding every possible successful
construction in the frozen class has been proved.
\end{enumerate}
The first unresolved gate is therefore \textbf{J2}, and its
ending is \textbf{Unresolved}.
\end{theorem}

\begin{proof}
The copied wavefront selections and the new
Proposition~\ref{prop:geometry} and
Corollary~\ref{cor:positive} give (i).
Proposition~\ref{prop:component} and
Proposition~\ref{prop:rigidity} give the precise statements
in (ii).

The calculations prove a failure of the sufficient criterion
for the retained separate-line incidence and a restriction
on permissible contact differences. They do not construct
the complete branch sum across this domain, prove its
cancellation, or define a characteristic finite part.
Those missing constructions are exactly (iii).
None of the proved statements is a converse to the
distribution product criterion or a universal incompatibility
of all prescriptions, giving (iv).
\end{proof}

\subsection{J3: the transverse obligation is not reached}

The copied parallel-line degree eight is a normal degree
at a two-vertex collision. It supplies neither a normal
degree along $\Nchar$ nor a uniform degree at the larger
collision strata and their intersections.

A future J3 proof must first have a distribution from J2.
In tubular coordinates $(z,r)$ at a relevant stratum,
it must control normal rescaling $r\mapsto\rho r$
uniformly over compact sets of longitudinal variables $z$.
The rescaled distributions must have controlled test
seminorms and the needed microlocal bounds.

If pointwise slices $t_z$ are used, their existence as
distributional restrictions must also be proved; they are
not automatic. Bounds at one fixed $z$ do not give the
required uniform theorem. The extension literature explicitly
separates these hypotheses from point-extension power counting
\cite{BF,Dang}.

No transverse degree for an undefined characteristic
composition is assigned here. J3 is not opened.

\subsection{J4: the contact and gluing obligation is not reached}

The primitive contacts copied in J1 remain available on
their established domains. A joint construction would have
to multiply them by their singular spectators, impose the
inherited subgraph normalizations, and prove agreement on
every overlap.

No overlap agreement follows from the finite bound
$|\alpha|\le6$. No contact is selected after seeing the
characteristic diagnostic to force a desired value.

Strict full-support ordering prevents an interaction
attachment from coinciding with an observable attachment.
Thus permitted collision contacts must leave those ordered
regions unchanged, as copied from XIII-C. They must also
leave the tree coefficient unchanged because they start
at $\hbar^2$ in the time-ordered product.
These remain normalization requirements, not a completed
gluing proof. J4 is not opened.

\subsection{J5: support control is not output admission}

The inherited compact graph closure and properness remain
support inputs for a support-preserving construction.
They do not turn an unnamed composition into a distribution.

The output target is a compact degree-two kernel with
\[
 \WF(t^{(1)})\subset
 Z_2=
 \{(z_1,p_1;z_2,p_2):p_1+p_2=0\}\setminus0,
\]
under the fixed flat cotangent identifications.
This is a sufficient microcausal bound: a nonzero tuple
lying entirely in one pointed closed causal cone cannot
have zero covector sum.

Assigning balanced covectors separately to edges is not
the output proof. One needs a defined joint extension,
its full graph wavefront theorem, and its actual proper
pushforward. Since J2 is unresolved, none is supplied.
No admission assertion for $\Rquant$ is made. J5 is not opened.

%=====================================================================
\section{J6: recorded boundaries}
\label{sec:J6}
%=====================================================================

The following operations are not performed:
\begin{enumerate}
\item no $s_{\mathrm{ext}}$, $s_\Psi$, or $s_{\mathrm{free}}$
is applied to $\Rquant$;
\item no Hadamard line is contracted against $\JD$;
\item no binary Ward defect is formed;
\item no classical breaking density is identified with a
quantum anomaly of this coefficient;
\item no value $-8I$ or evaluation identity on
$\mathscr C_{W,\mathrm{bos}}$ is used as a subtraction;
\item no triangle residue is identified or subtracted.
\end{enumerate}

The selected vertex has its own admitted local density.
Its appearance in this binary problem is not a quantum
retarded product of the gravitational cubic source.

XIV-A's action identity and XIV-B's relative complex do
not enter the graph construction. Conversely the present
characteristic diagnostic does not evaluate their
classical breaking or observable ledger.

XIII-A's non-ideal restriction theorem also remains
separate: no quotient by $\NW$ is attempted.
The source-switched X-track map and convolution-regulated
coordinate problem are not used. All widths and switches
remain fixed.

%=====================================================================
\section{Decision and successor package}
\label{sec:decision}
%=====================================================================

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.25\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
J0 & Frozen &
Unchanged carrier, full eleven-branch observable, vertex,
fixed product data and width, inherited $\Cpart$.\\[0.35em]
J1 & Copied inventory &
Eleven branches, 48 labelled matchings, individual line
orders, cycle loci, superficial counts, and primitive
extensions.\\[0.35em]
J2 & Unresolved &
Retained characteristic conormal relation and contact
rigidity proved. No complete-pair cancellation or admissible
characteristic prescription constructed.\\[0.35em]
J3 & Not reached &
Uniform transverse and intersection estimates require
the missing J2 distribution.\\[0.35em]
J4 & Not reached &
Spectator compositions, overlap compatibility, and joint
contact gluing not proved.\\[0.35em]
J5 & Not reached &
Support inputs retained; no joint graph pushforward or
output wavefront theorem.\\[0.35em]
J6 & Recorded &
Quantum differential and Ward-defect gate remain unopened;
XIV-A and XIV-B are not analytic inputs.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[XIV-C release]
\label{thm:release}
On the pair and ambiguity class frozen in J0:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the inherited inventory and primitive results remain
valid as copied inputs;
\item the retained noncoincident characteristic configuration
lies on a smooth codimension-four stratum with rank-two
light-cone differentials and an explicit positive conormal
relation;
\item permitted collision contacts cannot modify a
neighborhood of that configuration;
\item these facts do not supply either cancellation of the
complete pair or a joint characteristic prescription;
\item none of XIII-C's four successor obligations is fully
discharged in this release.
\end{enumerate}
The complete coefficient of $\lambda\hbar$ remains
\textbf{Unresolved}, with the first stop at J2.
No admitted $\Rquant$ and no universal extension obstruction
are asserted.
\end{theorem}

\begin{proof}
Input statements~\ref{input:XIIC},
\ref{input:XIIIB}, and \ref{input:branches} establish the
copied scope. Proposition~\ref{prop:geometry},
Corollary~\ref{cor:positive}, and
Proposition~\ref{prop:component} establish the new diagnostic.
Proposition~\ref{prop:rigidity} establishes contact rigidity.
Theorem~\ref{thm:J2} identifies exactly what those statements
do and do not decide. The unentered obligations are recorded
in Section~\ref{sec:stop}.
\end{proof}

\subsection*{What has advanced}

The stop is now geometrically specified: it is not a total
or partial collision, but a nontransversal intersection of
light-cone singularities at separated vertices. Its
codimension and conormal relation are explicit.

The ambiguity issue is also separated. The inherited
collision-contact class cannot serve as an unspecified
choice of extension at that separated configuration.
A future characteristic prescription must state why it
has no forbidden additional ambiguity, or openly change
the contract before using it.

These are analytic diagnostics and necessary conditions.
They are not the missing distribution.

\subsection*{Exact handoff}

A successor receives:
\begin{enumerate}
\item the fixed full pair and the observable naming correction;
\item the copied branches, match labels, and line orders;
\item the retained configuration and characteristic stratum;
\item the rank relation \eqref{eq:rank}, conormal selections,
and nonzero temporal component diagnostic;
\item the support-rigidity theorem for partial contacts;
\item the exact J2 stop and the unchanged four obligations.
\end{enumerate}

To advance beyond J2, the successor must either prove
cancellation for the complete selected pair on the retained
noncoincident domain or construct a prescription there with
the declared normalization, degree, ambiguity, and wavefront
control. Only afterwards can J3--J5 be entered.

It receives no quantum binary coefficient, Ward defect,
interacting differential, reduced algebra, physical
cohomology class, or numerical loop amplitude.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \Ocal=\PhiD+\Aone(\rg[\phi];\phi),\qquad
 \Ctot\subseteq\Cpart,\qquad \Gret=-\PP\ER,\\[2pt]
 \codim\Nchar=4,\qquad
 \rank\{\dd q_R,\dd q_1,\dd q_2\}=2,\\[2pt]
 -\frac{\dd q_R}{2\beta}
 +\frac{\dd q_1}{2\alpha}
 +\frac{\dd q_2}{2(\beta-\alpha)}=0,\\[2pt]
 D\in\Cpart\Longrightarrow D|_V=0
 \quad\text{for }V\cap\Coll=\varnothing,\\[2pt]
 \text{complete characteristic prescription: Unresolved at J2},\\
 \text{complete joint coefficient and admission: Unresolved}.
 \end{gathered}}
\]

A failure of the product test is not nonexistence.
A primitive extension is not the joint coefficient.
An unresolved characteristic configuration remains unresolved.
None of these statements is an interacting BRST algebra.

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

\paragraph{Continuation scope.}
The analytic predecessor is XIII-C. XIV-A and XIV-B are
recorded only and are not hypotheses of any product or
extension theorem here.

\paragraph{Pair correction.}
The full eleven-branch observable is denoted $\Ocal$.
The scalar-linear lift $\PhiD$ retains its original meaning.
The scalar-sourced slot is included once, before any
diagnostic calculation.

\paragraph{Dependency scope.}
The branch inventory, primitive extension statements,
partial-contact class, tree datum, and support control are
copied from the supplied XIII-C handoff. The characteristic
rank calculation, conormal relation, explicit component
check, and contact-support rigidity are proved here.

\paragraph{Unresolved scope.}
No joint characteristic distribution or cancellation theorem
for the complete pair is supplied. J3--J5 are not opened.
The manuscript is an exact analytic stop report, not a
claim of completed renormalization.

\paragraph{Quantum scope.}
No renormalized quantum output, quantum master equation,
Ward defect, state, residue subtraction, interacting
differential, or sharp-width limit is constructed.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
check, independently executed symbolic computation,
independent referee review, or journal acceptance is
asserted. The new calculations are the displayed analytic
ones.

%=====================================================================
\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}.
Primary analytic handoff: branches, matchings, partial contacts,
primitive extensions, characteristic configuration, tree inputs,
properness boundary, and unresolved quantum coefficient.

\bibitem{XIVA}
N.~Cheung,
\emph{Landmark XIV-A: Classical master equation and observable ledger
for one declared Einstein--scalar sheet}.
Supplied manuscript \texttt{XIV-A.tex}.
Recorded only; no analytic product or subtraction is imported.

\bibitem{XIVB}
N.~Cheung,
\emph{Landmark XIV-B: The relative antifield complex on
$\mathfrak F_W$, and its trivial ghost sector on the free shell}.
Supplied manuscript \texttt{XIV-B.tex}.
Recorded only; its relative complex is not an input.

\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.
Sections~5--6: point extensions, transverse scaling, and
microlocal extension along 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.
Extension and wavefront control across embedded submanifolds.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428.
Distribution product criterion, microcausal functionals,
and the distinction between Hadamard and time-ordered products.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
second edition, Springer, 1990.
Distribution products, differential wavefront bounds,
restriction, and pushforward criteria.

\end{thebibliography}

\end{document}