% !TeX program = pdflatex
%=====================================================================
% XIV-D.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-D.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-D.tex
%
% Exact release:
%   - the pair, propagators, switches, and ambiguity class of XIV-C
%     remain unchanged;
%   - the complete two-line coefficient is separated into its two
%     nonzero species channels, without expanding H_T into mixed
%     intermediate products;
%   - every branch contribution to the relevant second derivatives
%     is displayed;
%   - on a common attachment/residual-variable neighborhood of the
%     retained configuration, the mixed channel has exactly six
%     potentially surviving branches: B_4,...,B_9;
%   - the other branches are retained globally and proved zero only
%     on the specified local kernel germ;
%   - a full fiber-tensor contraction for B_6 is computed, but it is
%     not called the symbol or residue of the complete branch sum;
%   - the full joint-symbol cancellation test is not established;
%   - no characteristic prescription is constructed.
%
% Ending:
%   Unresolved at D2, within the inherited J2 gate.
%   D3 is not entered on the strength of a branch diagnostic.
%   J3--J5 and the quantum Ward-defect question remain unopened.
%
% Principal debugging corrections:
%   - a distribution cannot be evaluated at a singular configuration
%     merely to decide which branches survive;
%   - graphs with different internal vertex sets must be compared
%     through specified kernel maps, not by setting hidden vertices
%     equal to u_0, y_0, or x_0;
%   - a wavefront-set bound alone does not supply a principal-symbol
%     calculus for the complete mixed graph composition;
%   - cancellation of a leading differential numerator does not
%     control lower orders or establish a distributional product;
%   - a wavefront bound on a whole neighborhood must retain the
%     off-characteristic wavefront set; a pure N^*N bound is not a
%     substitute for it;
%   - degree and conormal control alone need not remove characteristic
%     ambiguities when the transverse degree reaches codimension.
%
% No R_1^(1), characteristic finite part, triangle-residue subtraction,
% new contact class, quantum Ward defect, interacting differential,
% reduced algebra, state, loop value, 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}

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\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

\allowdisplaybreaks
\setlength{\emergencystretch}{3em}
\setlist{leftmargin=*,itemsep=3pt,topsep=5pt}
\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{\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{\GA}{G_{\mathrm{act}}^{\mathrm A}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}}
\newcommand{\Ealg}{\mathscr E_H}
\newcommand{\Vcal}{\mathcal V_g}
\newcommand{\Ocal}{\mathcal O}
\newcommand{\Aone}{\mathcal A_1}
\newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}}
\newcommand{\CD}{\mathcal C_{3,\mathrm D}}
\newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\newcommand{\rD}{r_{\mathrm D}}
\newcommand{\rg}{r_g}
\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{\Nchar}{\mathcal N^\circ}
\newcommand{\Mcal}{\mathcal M}
\newcommand{\Ncal}{\mathcal N}
\newcommand{\Cpol}{\mathcal C}
\newcommand{\Dpol}{\mathcal D}
\newcommand{\Gpol}{\mathcal G}
\newcommand{\Ksum}{K_\Sigma}
\newcommand{\Bform}{\mathcal B_{\mathrm{form}}}
\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-D}\\[0.4em]
Cancellation or prescription on the retained\\
noncoincident characteristic stratum\\[0.4em]
\large Complete retained incidence and the unresolved joint-symbol test}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
This manuscript continues XIV-C's characteristic gate for the
unchanged binary pair. No quantum retarded coefficient was released
there, so no Ward defect is formed here.

We display the complete second-derivative ledger and the two
nonzero species channels of the grouped two-line expression.
A common attachment/residual-variable description makes the
retained local comparison precise. In its mixed channel,
$B_1,B_2,B_3$ vanish as kernel germs by their field-location
supports; the scalar-sector branches vanish in that channel.
The scalar--scalar channel is excluded only on the specified
retained output-location patch. The complete potentially surviving
mixed germ therefore contains $B_4,\ldots,B_9$, including both
gravitational-response legs.

We also compute the full leading fiber-tensor numerator of
$B_6$, rather than its temporal monomial alone. It is nonzero
for suitable residual field jets. This is a branch diagnostic,
not a nonzero symbol section of the complete six-branch germ.

The required common microlocal symbol calculation and its
lower-order control are not established. Neither cancellation
of the complete germ nor failure of that cancellation is proved.
Consequently no characteristic prescription is entered on the
strength of the branch diagnostic. Necessary degree, wavefront,
and ambiguity conditions for a future prescription are separated
from its existence.

The exact ending is Unresolved at D2, within J2. No
characteristic extension, admitted $\Rquant$, quantum Ward defect,
new contact class, or sharp-width limit 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) {XIV-C pair};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {A constructed local step};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {The missing gate};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The local step does not discharge the first missing gate.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The first missing step stays missing. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-C pair};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {The missing gate};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The local step does not discharge the first missing gate.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The first missing step stays missing. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-C pair};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {A constructed local step};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {The missing gate};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The local step does not discharge the first missing gate.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-XIV-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-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) {The missing gate};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {A constructed local step. The local step does not discharge the first missing gate.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{XIV-C pair is not The missing gate. 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-D}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: XIV-C pair};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: A constructed local step};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: The missing gate};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The local step does not discharge the first missing gate.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-XIV-D}
\end{figure}

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


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



\scopebox{Release and first missing step}{%
\textbf{Frozen:} XIV-C's pair, Hadamard matrix, inverse coefficient
$-1$, switches, width, characteristic configuration, and $\Cpart$.

\smallskip
\textbf{Constructed here:} the complete smooth-slot Hessian ledger;
its grouped species decomposition; the common-variable local
incidence comparison; and the reduction of the retained mixed germ
to $B_4,\ldots,B_9$.

\smallskip
\textbf{Additional diagnostic:} the full fiber-tensor leading
numerator of $B_6$ is computed. It is not the full branch sum.

\smallskip
\textbf{First missing step: D2.}
A controlled joint-symbol description of the complete retained
germ, including lower orders, has not been supplied.

\smallskip
\textbf{Ending: Unresolved.}
No full cancellation or nonzero full symbol section is asserted.
D3 is not entered; J3--J5 and the Ward-defect question remain closed.}

\tableofcontents

%=====================================================================
\section{Release criteria and inherited premises}
\label{sec:premises}
%=====================================================================

The three endings concern the characteristic construction:
\begin{enumerate}
\item \textbf{Constructed}: cancellation of the complete retained
germ, with the requisite lower-order control, or one characteristic
prescription with its degree, ambiguity, and wavefront theorem,
discharges the named J2 problem.
\item \textbf{Proved obstructed}: an explicit incompatibility excludes
every prescription in a class fixed before the test.
\item \textbf{Unresolved}: the complete sum is uncontrolled, or a
required prescription or estimate is missing.
\end{enumerate}

A failed sufficient product criterion is not the second ending.
A nonzero numerator of one branch does not establish failure of
cancellation of the complete pair. Even a successful J2 prescription
would not by itself release $\Rquant$.

\begin{inputstatement}[XIV-C's retained problem]
\label{input:XIVC}
The pair is
\[
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD,
\]
with
\[
 \Ctot\subseteq\Cpart,\qquad
 \Gret=-\PP\ER.
\]
The scalar-sourced slot is retained once.

The fixed triple is
\[
 u_0=(t_*,0,0,0),\quad
 y_0=(t_*+L,L,0,0),\quad
 x_0=(t_*+2L,2L,0,0),
\]
where $\eps<t_*<t_*+2L<T$, $f(x_0)\ne0$, and $g=1$ near
these points.

The characteristic stratum is
\[
 \Nchar=
 \left\{
 (x,u,y):
 y=u+\alpha(1,\mathbf n),\
 x=u+\beta(1,\mathbf n),\
 \mathbf n\in S^2,\ 0<\alpha<\beta
 \right\}.
\]
It is smooth of codimension four. For
\[
 q_R=(x-u)^2,\qquad q_1=(y-u)^2,\qquad q_2=(x-y)^2,
\]
the differentials have rank two there and satisfy
\[
 -\frac{\mathrm d q_R}{2\beta}
 +\frac{\mathrm d q_1}{2\alpha}
 +\frac{\mathrm d q_2}{2(\beta-\alpha)}=0.
\]
No tangent vector makes all three selected oriented evaluations
strictly positive.

The temporal $B_6$ monomial has nonzero coefficient and
nonvanishing line differential symbols at the retained triple.
That statement is not a residue of the complete branch sum.

There is a collision-free neighborhood $V$ of the retained
three-vertex configuration. Graphwise collision contacts restrict
to zero there. No characteristic prescription or cancellation of
the complete pair was released, and J3--J5 were not reached
\cite{XIVC}.
\end{inputstatement}

These geometric statements and the temporal monomial diagnostic
are not re-proved.

\begin{inputstatement}[Earlier analytic handoffs]
\label{input:earlier}
XIII-C supplies the eleven branches, 48 labelled matchings,
the partial-diagonal class, and the elementary parallel-line
extensions. XIII-B supplies the tree datum, support control,
retarded convention, and corrected ordered values.
XII-E supplies the completion formulas and their admitted tree
kernels. XII-C supplies the fixed density representative and
actual cubic source \cite{XIIIC,XIIIB,XIIE,XIIC}.
\end{inputstatement}

XIV-A and XIV-B are recorded only. Their relative antifield
complex is not an analytic input. The auxiliary value $-8I$
is not a subtraction or normalization datum.

%=====================================================================
\section{D0: the unchanged contract}
\label{sec:D0}
%=====================================================================

\subsection{Geometry, product data, and ambiguity}

Use
\[
 M=\R^4,\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \Pmass=\Box+m^2,
\]
\[
 \MT=[0,T]\times\R^3,\qquad \MTo=(0,T)\times\R^3.
\]
All widths, observations, switches, and propagators remain those
fixed in XIV-C. The carrier is unchanged:
\[
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                         \Nsol[[\hbar,\kappa]].
\]
The time-ordered line remains
\[
 \Htime=H+i\GA.
\]
Its metric--scalar blocks vanish. No replacement Hadamard
matrix is selected.

The ambiguity class remains the inherited $\Cpart$. Its defined
graphwise members have collision support and the form
\[
 D_\Gamma\left[
 w_\Gamma\,\partial_r^\alpha\delta(r_I)\,
 S_{\Gamma/I}\right],
 \qquad |\alpha|\le6,
\]
with the inherited weights, spectator coefficients, two residual
field slots, support closure, and normalization requirements.

No class of characteristic prescriptions is declared to have been
constructed. In particular, a universal obstruction is not tested
against an unspecified class.

\subsection{The unchanged entries}

Define the polarized stress tensor by
\begin{equation}
 T^{\mu\nu}(\psi_1,\psi_2)
 =
 \frac12\left(
 \partial^\mu\psi_1\partial^\nu\psi_2+
 \partial^\mu\psi_2\partial^\nu\psi_1\right)
 -\frac12\eta^{\mu\nu}
 \left(\partial_\rho\psi_1\partial^\rho\psi_2
                         -m^2\psi_1\psi_2\right).
 \label{eq:Tpolar}
\end{equation}
Thus $T(\phi,\phi)=T_0[\phi]$, and
\[
 \Vcal=-\frac12\int g\,\gamma_{\mu\nu}
                                   T^{\mu\nu}(\phi,\phi)\,\dd^4x.
\]

Put
\[
 a_1^{\mu\nu}[r]
 =\frac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},
 \qquad
 b_1[r]=\frac12\tr_\eta r,
\]
\[
 b_2[\gamma]
 =\frac18(\tr_\eta\gamma)^2
                     -\frac14\gamma_{\rho\sigma}\gamma^{\rho\sigma},
\]
\[
 a_2^{\mu\nu}[\gamma]
 =\gamma^\mu{}_\rho\gamma^{\rho\nu}
 -\frac12(\tr_\eta\gamma)\gamma^{\mu\nu}
                              +b_2[\gamma]\eta^{\mu\nu}.
\]
The first-response operators are
\begin{align}
 X^\mu[r]
 &=-\ER\partial_\alpha(\sig a_1^{\alpha\mu}[r]),\\
 V_{1,g}[r]\psi
 &=\partial_\mu(g a_1^{\mu\nu}[r]\partial_\nu\psi)
                              +m^2g b_1[r]\psi,\\
 p[r;\psi]&=-\ERm(V_{1,g}[r]\psi),\\
 \Aone(r;\psi)
 &=\int f\bigl(p[r;\psi]-X^\mu[r]\partial_\mu\psi\bigr)\,\dd^4x.
 \label{eq:first}
\end{align}
Arguments $(f,g)$ on $\Aone$ are suppressed when fixed.

The inherited filled slots are
\[
 \rD[\gamma]=\PP\ER\JD[\gamma],
 \qquad
 \rg[\phi]=-\frac12\PP\ER\bigl(gT(\phi,\phi)\bigr).
\]
These are the already admitted classical slots, not new quantum
products with $\JD$ as an argument.

For a finite intrinsic description of the fixed source, write
$q(\rho)=\eta+\rho\gamma$ and
\[
 \ell_3[\gamma]
 =-2[\rho^3]\bigl(\sqrt{-\det q(\rho)}\,R[q(\rho)]\bigr),
 \qquad
 \JD[\gamma]=\EL_\gamma(\sig\ell_3[\gamma]).
\]
The curvature representative is the unintegrated one of XII-C,
with
\[
 R_{\mu\nu}
 =\partial_\rho\Gamma^\rho_{\mu\nu}
 -\partial_\nu\Gamma^\rho_{\mu\rho}
 +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
 -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}.
\]
Coefficient extraction is finite at this order. It fixes the same
local quadratic source, including its first and second ramp
derivatives; it changes neither the representative nor the action.

\subsection{The grouped expression}

The convention remains
\[
 R_1(\Vcal;\Ocal)
 =\frac{i}{\hbar}
 \left(\mathsf T_2^{\mathrm{ren}}(\Vcal,\Ocal)
                                      -\Vcal\star_H\Ocal\right).
\]
The requested two-line coefficient is specified formally by
\begin{equation}
 \Bform
 =\frac{i}{2}
 \pair{\Vcal''\otimes\Ocal''}
                  {\Htime^{\otimes2}-H^{\otimes2}}.
 \label{eq:grouped}
\end{equation}
This difference stays grouped. No separate mixed product of $H$
and $\GA$ is assigned an existence statement.

\subsection{Local comparison neighborhoods}

Choose small neighborhoods $A,B,Y$ of $u_0,x_0,y_0$, respectively,
with pairwise disjoint closures and
\[
 B\times A\times Y\subset V
\]
in the vertex order $(x,u,y)$. Choose a neighborhood $Z$ of $u_0$
with $\overline Z\subset A$.

The variables used below are:
\[
 a\in A\quad\hbox{metric attachment},\qquad
 b\in B\quad\hbox{scalar attachment},
\]
\[
 z\in Z\quad\hbox{residual metric variable},\qquad
 y\in Y\quad\hbox{interaction vertex}.
\]
In the retained $B_6$ graph, $(b,a,y)$ is the original triple
$(x,u,y)$ and the local metric vertex enforces $a=z$.

For other branches, $z$ remains an independent field variable.
It is not set equal to $a$. This is a restriction of a kernel
germ to an open set, not an excision of $\Nchar$, a change of
switches, or a cutoff of the triangle.

\subsection{Negative freeze}

There is no characteristic cutoff, triangle-residue subtraction,
new contact class, antifield differential, spurion, convolution
sheet, restriction quotient, new gravitational vertex, or
sharp-width limit.

%=====================================================================
\section{D1: the complete coefficient before joint composition}
\label{sec:D1}
%=====================================================================

The first new calculation concerns the complete second-derivative
sum. It precedes any further branch diagnostic.

\subsection{A polarization of the entire observable}

Let $\Cpol(\gamma_1,\gamma_2;\psi)$ be the symmetric polarization
in the metric arguments of $\Asc(\gamma,0;\psi)$:
\[
 \Cpol(\gamma_1,\gamma_2;\psi)
 =\frac12\left[
 \Asc(\gamma_1+\gamma_2,0;\psi)
 -\Asc(\gamma_1,0;\psi)-\Asc(\gamma_2,0;\psi)\right].
\]
Define symmetric bilinear metric-valued maps
\[
 \Dpol(\gamma_1,\gamma_2)
 =\frac12\left[
 \rD[\gamma_1+\gamma_2]
 -\rD[\gamma_1]-\rD[\gamma_2]\right],
\]
\[
 \Gpol(\psi_1,\psi_2)
 =-\frac12\PP\ER\bigl(gT(\psi_1,\psi_2)\bigr).
\]
Then the complete observable is
\begin{equation}
 \Ocal
 =\Cpol(\gamma,\gamma;\phi)
 +\Aone(\Dpol(\gamma,\gamma);\phi)
 +\Aone(\Gpol(\phi,\phi);\phi).
 \label{eq:polarO}
\end{equation}
This formula includes both legs of each filled slot.

\begin{proposition}[Complete Hessian ledger]
\label{prop:Hessian}
For smooth metric directions $h,h_1,h_2$ and scalar directions
$\psi,\psi_1,\psi_2$,
\begin{align}
 \Vcal''_{\gamma\gamma}[h_1,h_2]&=0,\\
 \Vcal''_{\gamma\phi}[h,\psi]
 &=-\int g\,h_{\mu\nu}T^{\mu\nu}(\phi,\psi)\,\dd^4x,\\
 \Vcal''_{\phi\phi}[\psi_1,\psi_2]
 &=-\int g\,\gamma_{\mu\nu}
                       T^{\mu\nu}(\psi_1,\psi_2)\,\dd^4x,
 \label{eq:VHessian}
\end{align}
and
\begin{align}
 \Ocal''_{\gamma\gamma}[h_1,h_2]
 &=2\Cpol(h_1,h_2;\phi)
                    +2\Aone(\Dpol(h_1,h_2);\phi),\\
 \Ocal''_{\gamma\phi}[h,\psi]
 &=2\Cpol(\gamma,h;\psi)
                    +2\Aone(\Dpol(\gamma,h);\psi),\\
 \Ocal''_{\phi\phi}[\psi_1,\psi_2]
 &=2\Aone(\Gpol(\psi_1,\psi_2);\phi)
   +2\Aone(\Gpol(\phi,\psi_1);\psi_2)
   +2\Aone(\Gpol(\phi,\psi_2);\psi_1).
 \label{eq:OHessian}
\end{align}
The mixed blocks are symmetric under exchanging the derivative
slots.
\end{proposition}

\begin{proof}
The stress tensor \eqref{eq:Tpolar} is symmetric bilinear.
Differentiating $-\frac12\int g\gamma T(\phi,\phi)$ once in
each indicated direction gives \eqref{eq:VHessian}.

Each of $\Cpol$ and $\Dpol$ is symmetric bilinear in its
metric arguments. Each occurrence of $\Aone$ is linear in
both displayed arguments. The first two terms of
\eqref{eq:polarO} are linear in $\phi$, while its last term is
cubic in $\phi$ and independent of $\gamma$.
Applying the product rule gives all three formulas in
\eqref{eq:OHessian}, with the stated factors of two.
\end{proof}

All operations in this proposition concern smooth directions
of existing classical functionals. No characteristic product
is used in its proof.

\subsection{The complete grouped species sum}

For a block-diagonal even line $L$, define
$\Contr_L^{\gamma\phi}$ and $\Contr_L^{\phi\phi}$ to mean the
two-slot contractions using, respectively,
$L_{\gamma\gamma}\otimes L_{\phi\phi}$ and
$L_{\phi\phi}\otimes L_{\phi\phi}$, with the indicated derivative
blocks.

Where the contractions are defined, species expansion of
\eqref{eq:grouped} gives
\begin{align}
 \Bform={}&
 i\,\Contr^{\gamma\phi}
 \left(
 \Vcal''_{\gamma\phi},\Ocal''_{\gamma\phi};
 \Htime_{\gamma\gamma}\otimes\Htime_{\phi\phi}
              -H_{\gamma\gamma}\otimes H_{\phi\phi}\right)
 \nonumber\\
 &+\frac{i}{2}\,\Contr^{\phi\phi}
 \left(
 \Vcal''_{\phi\phi},\Ocal''_{\phi\phi};
 \Htime_{\phi\phi}^{\otimes2}
                          -H_{\phi\phi}^{\otimes2}\right).
 \label{eq:species}
\end{align}
On the unresolved domain this is a formal specification only.

The factor $i$ in the mixed channel includes its two equivalent
slot orderings. There is no two-metric channel because
$\Vcal''_{\gamma\gamma}=0$. No metric--scalar line has been added.

Equation~\eqref{eq:species}, together with
\eqref{eq:VHessian}--\eqref{eq:OHessian}, is the complete
tensor-valued algebraic sum. It does not replace the requested
microlocal composition theorem.

\subsection{Every branch derivative displayed}

Write
\[
 X_\gamma=X[\gamma],\qquad X_h=X[h],
 \qquad p_\gamma=p[\gamma;\psi],\qquad p_h=p[h;\psi],
\]
and
\[
 U[r]q=\partial_\alpha(\sig a_1^{\alpha\beta}[r]\partial_\beta q).
\]
Set
\begin{align}
 \dot b_2[\gamma;h]
 &=\frac14(\tr_\eta\gamma)(\tr_\eta h)
                   -\frac12\gamma_{\rho\sigma}h^{\rho\sigma},\\
 \dot a_2^{\mu\nu}[\gamma;h]
 &=h^\mu{}_\rho\gamma^{\rho\nu}
   +\gamma^\mu{}_\rho h^{\rho\nu}
   -\frac12\left(
      (\tr_\eta h)\gamma^{\mu\nu}
      +(\tr_\eta\gamma)h^{\mu\nu}\right)
                       +\dot b_2[\gamma;h]\eta^{\mu\nu},\\
 \dot V_2[\gamma;h]\psi
 &=\partial_\mu(g\dot a_2^{\mu\nu}[\gamma;h]\partial_\nu\psi)
                      +m^2g\dot b_2[\gamma;h]\psi,\\
 r_h&=\rD'[\gamma]h=2\Dpol(\gamma,h).
 \label{eq:dotcoeff}
\end{align}

Let $\int_f$ denote integration against $f$ over the observation
variable. The mixed branch derivatives
$\Mcal_j[\gamma;h,\psi]=(B_j)''_{\gamma\phi}[h,\psi]$ are
\begin{align}
 \Mcal_1
 &=-\int_f\ERm\left(
 V_{1,g}[h]p_\gamma+V_{1,g}[\gamma]p_h\right),\\
 \Mcal_2
 &=-\int_f\ERm(\dot V_2[\gamma;h]\psi),\\
 \Mcal_3
 &=-\int_f\left(
 X_h^\mu\partial_\mu p_\gamma+
 X_\gamma^\mu\partial_\mu p_h\right),\\
 \Mcal_4
 &=\int_f\left(
 X_h^\nu\partial_\nu X_\gamma^\mu+
 X_\gamma^\nu\partial_\nu X_h^\mu\right)\partial_\mu\psi,\\
 \Mcal_5
 &=\int_f\ER\left(
 U[h]X_\gamma^\mu+U[\gamma]X_h^\mu\right)\partial_\mu\psi,\\
 \Mcal_6
 &=\int_f\ER\left(
 \partial_\alpha(\sig\dot a_2^{\alpha\mu}[\gamma;h])
                                      \right)\partial_\mu\psi,\\
 \Mcal_7
 &=\frac12\int_f
 \left(X_h^\mu X_\gamma^\nu+X_\gamma^\mu X_h^\nu\right)
                                      \partial_\mu\partial_\nu\psi,\\
 \Mcal_8
 &=-\int_fX^\mu[r_h]\partial_\mu\psi,\\
 \Mcal_9
 &=\int_fp[r_h;\psi],\\
 \Mcal_{10}&=0,\qquad \Mcal_{11}=0.
 \label{eq:allmixed}
\end{align}
Thus
\[
 \Ocal''_{\gamma\phi}=\sum_{j=1}^{11}\Mcal_j.
\]

For the scalar--scalar block, put
$\Ncal_j=(B_j)''_{\phi\phi}$. Its complete ledger is
\[
 \Ncal_j=0,\qquad j=1,\ldots,9,
\]
\begin{align}
 \Ncal_{10}[\psi_1,\psi_2]
 =-2\int_f\Bigl(&
 X^\mu[\Gpol(\psi_1,\psi_2)]\partial_\mu\phi
 +X^\mu[\Gpol(\phi,\psi_1)]\partial_\mu\psi_2
 \nonumber\\
 &+X^\mu[\Gpol(\phi,\psi_2)]\partial_\mu\psi_1\Bigr),\\
 \Ncal_{11}[\psi_1,\psi_2]
 =2\int_f\Bigl(&
 p[\Gpol(\psi_1,\psi_2);\phi]
 +p[\Gpol(\phi,\psi_1);\psi_2]
 \nonumber\\
 &+p[\Gpol(\phi,\psi_2);\psi_1]\Bigr).
 \label{eq:allscalar}
\end{align}

\begin{lemma}[Completeness of the branch ledger]
\label{lem:ledger}
Equations~\eqref{eq:allmixed} and \eqref{eq:allscalar} are the
ordinary second derivatives of all eleven inherited branches.
No tensor component, source coefficient, filled-slot leg,
or scalar-slot bijection has been discarded.
\end{lemma}

\begin{proof}
For $B_1,\ldots,B_7$, vary each of the two metric occurrences
in turn, and replace the single scalar by $\psi$. This gives
$\Mcal_1,\ldots,\Mcal_7$, including the symmetric derivative
of $a_2$.

The two metric occurrences in $B_8,B_9$ are inside $\rD$.
Their complete variation is $r_h$, giving $\Mcal_8,\Mcal_9$.
The definition of $\JD$ fixes every coefficient entering $r_h$;
no layer term is separately set to zero.

The last two branches have no metric argument. Their scalar
slot derivatives give \eqref{eq:allscalar}. The first nine
branches have only one scalar occurrence and therefore have
zero second scalar derivative.

For each mixed branch the two observable metric choices and
two interaction scalar choices reproduce its four labelled
matchings. For each scalar-sector branch the three observable
scalar pairs and two bijections reproduce its six matchings.
The count remains $9\cdot4+2\cdot6=48$.
\end{proof}

%=====================================================================
\section{The complete retained local incidence}
\label{sec:localincidence}
%=====================================================================

\subsection{Common kernels, not point values}

Because $\Mcal_j$ is trilinear in $(\gamma,h,\psi)$, let
$K_j$ denote its compact distributional kernel:
\begin{equation}
 \Mcal_j[\gamma;h,\psi]
 =
 \pair{K_j^{ab,cd}(a,b,z)}
       {h_{ab}(a)\,\psi(b)\,\gamma_{cd}(z)}.
 \label{eq:Kj}
\end{equation}
All field and line derivatives are included in this kernel.
Existence of these individual kernels follows from the
inherited tree constructions, not from a two-cross-line
composition.

The complete mixed kernel is
\[
 \Ksum=\sum_{j=1}^{9}K_j.
\]
It lives on the common variables $(a,b,z)$ even when a branch
was originally written with additional internal vertices.
Those vertices have been integrated by its established tree
kernel map.

This comparison does not pull a four-vertex distribution back
to an arbitrarily imposed three-vertex diagonal. Such a
pullback would require its own wavefront test
\cite[Appendix A]{FR}.

\subsection{Branches that are genuinely zero as local germs}

\begin{proposition}[Local support reduction]
\label{prop:reduction}
On $A\times B\times Z$,
\begin{equation}
 K_1=K_2=K_3=0,\qquad
 \Ksum=K_4+K_5+K_6+K_7+K_8+K_9.
 \label{eq:six}
\end{equation}
Moreover $K_6,K_8,K_9$ have support in $\{a=z\}$.
The mixed derivatives of $B_{10},B_{11}$ are identically zero.

These are distributional support statements, not evaluations
of singular kernels at $(u_0,x_0,u_0)$.
\end{proposition}

\begin{proof}
In $B_1$, the scalar and one metric slot share the inner
vertex. Selecting the other metric slot for contraction forces
$b=z$; selecting the metric at the scalar vertex forces $a=b$.
Thus
\[
 \supp K_1\subset\{b=z\}\cup\{a=b\}.
\]
The same slot-location argument applies to $B_3$, giving the
same support inclusion.

All three observable slots in $B_2$ share one vertex, so
\[
 \supp K_2\subset\{a=b=z\}.
\]
Since $\overline A\cap\overline B=\varnothing$ and
$\overline Z\subset A$, these supports miss $A\times B\times Z$.
Differentiating a distribution does not enlarge its support,
so the argument covers all derivative placements.

Both metric slots of $B_6$ share their local vertex.
Both metric slots of $B_8,B_9$ share the source vertex,
including when the source differentiates either slot.
Hence their mixed kernels are supported on $a=z$.

The last assertion follows either directly from the absence
of a metric argument or from \eqref{eq:allmixed}.
\end{proof}

The complete mixed germ is therefore a six-branch sum. In
particular, neither the coordinate gravitational leg $B_8$
nor the scalar gravitational leg $B_9$ may be omitted.

\subsection{The scalar channel on the retained output patch}

In the scalar--scalar channel of \eqref{eq:species}, the
interaction's unused field is its metric at $y$. In the mixed
channel, it is a scalar at $y$ and the residual metric comes
from the observable at $z$.

\begin{lemma}[Channel separation by output locations]
\label{lem:channel}
Retain the interaction vertex in $Y$ and the output metric
variable in $Z$. The scalar--scalar channel has zero kernel
germ on this patch.
\end{lemma}

\begin{proof}
Its residual metric variable equals the interaction vertex.
Its kernel consequently has support in $z=y$, including any
field derivatives. Since $\overline Y\cap\overline Z=\varnothing$,
the restriction is zero.
\end{proof}

This does not delete $B_{10},B_{11}$ from the frozen pair.
Their scalar channel remains part of \eqref{eq:species}
outside the retained output-location patch.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}c p{.31\linewidth}X@{}}
\toprule
Branch & Retained mixed germ & Reason or obligation\\
\midrule
$B_1$ & Zero & Support in $b=z$ or $a=b$.\\
$B_2$ & Zero & Support in $a=b=z$.\\
$B_3$ & Zero & Support in $b=z$ or $a=b$.\\
$B_4$ & Retained & Both metric choices and all tensor components.\\
$B_5$ & Retained & Both internal tree placements retained.\\
$B_6$ & Retained & Local metric diagonal $a=z$.\\
$B_7$ & Retained & Symmetric second scalar derivative retained.\\
$B_8$ & Retained & Full differentiated source; coordinate leg.\\
$B_9$ & Retained & Full differentiated source; scalar leg.\\
$B_{10}$ & Zero in this channel &
Scalar channel retained globally, zero only on the chosen
output-location patch.\\
$B_{11}$ & Zero in this channel &
Same qualification.\\
\bottomrule
\end{tabularx}
\end{center}

There are 24 potentially retained labelled mixed matchings
on this germ: four for each of the six branches. This is not
a count of nonzero tensor monomials.

\subsection{Collision-contact rigidity with attachments retained}

\begin{lemma}[Contact support in the common comparison space]
\label{lem:contact}
For each inventory graph, retain its original two attachment
variables $a,b$ and the interaction vertex $y$.
Every connected collision contact involving $y$ is supported
in
\[
 \{y=a\}\cup\{y=b\}.
\]
Thus every defined inherited contact vanishes on
$Y\times A\times B\times Z$ while these attachment variables
are retained.
\end{lemma}

\begin{proof}
The only edges incident to $y$ are its two cross-lines, whose
observable endpoints are $a$ and $b$. A connected vertex subset
containing $y$ and at least one further vertex must therefore
contain at least one of those endpoints. Collapsing the subset
forces $y=a$ or $y=b$.

Intersections and the corresponding contracted loci retain
that equality when expressed in the original attachment
variables. Differential operators do not enlarge support.
Projection over other internal variables preserves the
displayed equality because $y,a,b$ are not projected out.

The three neighborhoods have disjoint closures, proving
the vanishing.
\end{proof}

After projecting out the attachment variables themselves,
a contact can produce a nonlocal-looking output through its
spectator. No assertion of vanishing after that different
projection is made.

\begin{theorem}[Complete retained algebraic incidence]
\label{thm:incidence}
The full formal coefficient on the retained mixed
attachment/output germ is obtained by contracting
$\Vcal''_{\gamma\phi}$ with the complete kernel
\[
 K_4+K_5+K_6+K_7+K_8+K_9
\]
against the grouped line difference in the first line of
\eqref{eq:species}.

All other branch contributions are zero on this germ for
the reasons proved above. No collision contact defines or
changes this germ while the attachments are retained.
\end{theorem}

\begin{proof}
Use Proposition~\ref{prop:Hessian},
Lemma~\ref{lem:ledger},
Proposition~\ref{prop:reduction},
Lemma~\ref{lem:channel}, and Lemma~\ref{lem:contact}.
\end{proof}

This is a complete incidence statement. It is not yet a
statement that the displayed contraction exists on the
characteristic set.

%=====================================================================
\section{D2: the full joint-symbol test}
\label{sec:D2}
%=====================================================================

\subsection{The exact expression requiring control}

On the retained germ, the required object is
\begin{equation}
 i\,\Contr^{\gamma\phi}\left(
 \Vcal''_{\gamma\phi},
 \sum_{j=4}^{9}K_j;\
 \Htime_{\gamma\gamma}(y,a)
                  \otimes\Htime_{\phi\phi}(y,b)
 -H_{\gamma\gamma}(y,a)\otimes H_{\phi\phi}(y,b)
 \right).
 \label{eq:retainedfull}
\end{equation}
This is the complete sum to be tested, not its $B_6$ summand.

The copied relation is used with $(x,u,y)=(b,a,y)$.
For $\ell=(1,1,0,0)$ and $p=\eta(\ell,\cdot)$, its line
covectors in the order $(b,a,y)$ are
\[
 (-p,p,0),\qquad(0,-p,p),\qquad(p,0,-p).
\]
Their sum is zero. The residual metric variable has zero
covector in this copied product-test selection.

On the microlocally admissible open sets, each contraction
in \eqref{eq:retainedfull} is an ordinary defined graph
composition. At the retained characteristic configurations,
the inherited theorem does not define it.

\subsection{What has not been supplied by the algebraic sum}

The kernels $K_j$ are defined distributions, but a wavefront
inclusion alone does not assign them a common scalar
principal symbol for the composition
\eqref{eq:retainedfull}.

Some kernels include the observable metric diagonal $a=z$.
Others arise from tree compositions with additional internal
vertices. To use a common microlocal symbol calculation one
must specify the corresponding conormal or multiphase
representations, their composition maps, and their orders.
One must then control the complete grouped amplitudes and
the remaining terms. The distribution product and
pushforward criteria do not supply this information merely
from an incidence count \cite{Hormander}.

In particular, the following replacements are not made:
\begin{enumerate}
\item evaluating $K_j$ at a singular point;
\item setting every hidden internal vertex equal to one of
the retained three vertices;
\item adding differential numerators from inequivalent
unidentified phases as if they were one wavefront section;
\item calling a cancellation of highest differential factors
a proof that all lower-order singularities disappear;
\item expanding the grouped line difference into intermediate
products whose existence is unknown.
\end{enumerate}

The first missing analytic step is a controlled common
symbol/composition description of the complete germ
\eqref{eq:retainedfull}. No equality to zero and no nonzero
full symbol section is proved here.

\subsection{A full fiber-tensor check for $B_6$}

The following calculation sharpens the inherited monomial
diagnostic. It does not replace the complete six-branch test.

For a covector $p$, put $n=p^\sharp$ and suppose $p^2=0$.
Let $G_{\mu\nu}$ be the residual metric value at its source
vertex, let $k_\mu$ be the residual scalar derivative at the
interaction, and set
\[
 r_{\mu\nu}=p_\mu k_\nu+p_\nu k_\mu,\qquad
 \lambda=p^\mu k_\mu,\qquad Q=G_{\mu\nu}n^\mu n^\nu.
\]

\begin{proposition}[Full $B_6$ leading fiber numerator]
\label{prop:B6tensor}
The complete density variation satisfies
\begin{equation}
 p_\alpha p_\mu\,
       \dot a_2^{\alpha\mu}[G;r]=Q\lambda.
 \label{eq:B6identity}
\end{equation}
After the metric propagator fiber and the principal kinetic
stress variation are contracted, the corresponding $B_6$
leading differential numerator is a fixed nonzero conventional
factor times
\[
 f(b)\sig(a)g(y)\,Q\lambda.
\]
It is not identically zero as a fiber-tensor polynomial.
\end{proposition}

\begin{proof}
The principal kinetic variation of the stress tensor has
fiber coefficient
\[
 S^{\mu\nu}
 =\frac12(p^\mu k^\nu+p^\nu k^\mu)
                            -\frac12\eta^{\mu\nu}\lambda.
\]
In four dimensions $\tr_\eta S=-\lambda$, so trace reversal
gives
\[
 (\PP S)_{\mu\nu}
 =\frac12(p_\mu k_\nu+p_\nu k_\mu)=\frac12r_{\mu\nu}.
\]
The signs and factors from $\Vcal''_{\gamma\phi}$ and the
fixed metric line are nonzero. The mass term has lower
scalar-line differential order and is not part of this
principal differential numerator.

Because $p^2=0$, the $b_2\eta^{\alpha\mu}$ term makes no
contribution after contraction with $p_\alpha p_\mu$.
Write $\tau=\tr_\eta G$. The remaining quadratic expression is
\[
 p_\alpha p_\mu a_2^{\alpha\mu}[G]
 =\eta(G^\sharp n,G^\sharp n)-\frac{\tau}{2}Q.
\]
For variation $r$,
\[
 r^\sharp n=\lambda n,\qquad
 \tr_\eta r=2\lambda,\qquad r(n,n)=0.
\]
Its directional derivative is therefore
\[
 2\eta(G^\sharp n,\lambda n)
       -\frac{2\lambda}{2}Q-\frac{\tau}{2}\,0
 =2\lambda Q-\lambda Q=\lambda Q,
\]
proving \eqref{eq:B6identity}.

The retarded divergence and observation scalar derivative
supply the two factors of $p$ in \eqref{eq:B6identity}.
The frozen scalar line normalizations and the nonzero
kinetic coefficient supply only fixed nonzero factors.

For example, $G_{00}=1$ with all other components zero,
$p=(1,-1,0,0)$, and $k=(1,0,0,0)$ give $Q=\lambda=1$.
The original weights are nonzero at the retained configuration.
\end{proof}

This proof performs the fiber sum for this leading $B_6$
numerator. It does not evaluate an undefined graph product.
It does not determine the contributions of $K_4,K_5,K_7,K_8,K_9$,
their common microlocal orders, or their lower-order terms.

\begin{corollary}[No full cancellation verdict follows]
\label{cor:branch}
Proposition~\ref{prop:B6tensor} proves neither cancellation
nor failure of cancellation of \eqref{eq:retainedfull}.
It does not justify advancing to D3 on the premise that
complete-pair cancellation has failed.
\end{corollary}

\begin{proof}
The complete retained germ is the six-branch expression
of Theorem~\ref{thm:incidence}. The proposition computes
only one branch's leading differential numerator.
It supplies no sum identity or sum nonvanishing theorem
for that complete expression.
\end{proof}

\begin{theorem}[D2 verdict]
\label{thm:D2}
The complete retained incidence is named by
\eqref{eq:retainedfull}, but its joint-symbol cancellation
test and lower-order control have not been established.
Neither legal decisive D2 result---a complete cancellation
theorem or a nonzero full symbol section---is released.

The first unresolved step is D2, within J2.
\end{theorem}

\begin{proof}
The algebraic and local-support identification is
Theorem~\ref{thm:incidence}. The inherited characteristic
relation does not satisfy the sufficient product criterion.
No common microlocal composition theorem for
\eqref{eq:retainedfull} has been proved.

Proposition~\ref{prop:B6tensor} cannot fill that gap by
Corollary~\ref{cor:branch}. No independent prescription is
constructed below. The missing conclusion is therefore
reported as Unresolved, not as a universal incompatibility.
\end{proof}

%=====================================================================
\section{D3: prescription requirements, not a constructed map}
\label{sec:D3}
%=====================================================================

D3 is not entered as a construction, because no complete
failure-of-cancellation result or independent prescription
has been supplied. This section corrects two possible
misstatements of its requirements.

\subsection{Agreement must be on a defined domain}

A future prescription must specify:
\begin{enumerate}
\item a reference distribution $t^\circ$ on a stated
off-characteristic domain;
\item the exact characteristic locus and graph or common
kernel space on which it is extended;
\item agreement with $t^\circ$ and with the grouped frozen
line prescription;
\item a finite uniform transverse degree;
\item a closed conic wavefront bound and its extension theorem;
\item the normalization selecting a permissible map under
the unchanged ambiguity contract.
\end{enumerate}

It is not enough to say that every separate line is defined.
The reference joint distribution must be defined on the
domain on which agreement is required.

\subsection{The wavefront bound cannot discard inherited singularities}

\begin{lemma}[Necessary restriction compatibility]
\label{lem:WFrestriction}
Let $N$ be a submanifold of an open set $U$, and let
$t\in\mathcal D'(U)$ extend
$t^\circ\in\mathcal D'(U\setminus N)$. Then
\[
 \WF(t)\big|_{U\setminus N}=\WF(t^\circ).
\]
Consequently a bound $\WF(t)\subset N^*N$ on all of $U$
would require $t^\circ$ to be smooth on $U\setminus N$.
\end{lemma}

\begin{proof}
Wavefront sets are local and commute with restriction
to an open set. The conormal bundle $N^*N$ has base points
only in $N$. Hence the proposed bound leaves no wavefront
point outside $N$, which is equivalent to smoothness there.
\end{proof}

Accordingly, ``wavefront contained in the named conormal''
must not be used to erase the inherited off-characteristic
line singularities or observable diagonal singularities.

A minimal-extension target may have the form
\[
 \WF(t)\subset
 \overline{\WF(t^\circ)}\cup N^*N,
\]
but its validity requires additional microlocal hypotheses.
In particular, the conormal landing condition is a hypothesis,
not an automatic consequence of finite scaling degree
\cite[Sections 5--7]{Dang}.

In the common-variable description, pre-existing observable
diagonals such as $a=z$ are also retained. They are not
interaction collision counterterms and are not removed by
calling the new characteristic locus $N$.

No such wavefront theorem is proved for
\eqref{eq:retainedfull}.

\subsection{Degree and wavefront control do not automatically select a map}

The following conditional statement identifies a genuine
ambiguity issue without adding an ambiguity to $\Cpart$.

\begin{proposition}[Characteristic ambiguity at codimension four]
\label{prop:ambiguity}
Let $N$ be a smooth codimension-four submanifold and choose
local coordinates $(s,r)$ with $N=\{r=0\}$, $r\in\R^4$.
Suppose an extension $t$ of a fixed $t^\circ$ has transverse
scaling degree at most $d$, where $d\ge4$.

For every nonzero smooth compactly supported $a(s)$,
\[
 A(s,r)=a(s)\delta^{(4)}(r)
\]
has transverse scaling degree four and wavefront set
contained in $N^*N$. Hence $t+A$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item agrees with $t^\circ$ off $N$;
\item has transverse degree at most $d$;
\item preserves any wavefront bound for $t$ that already
allows all of $N^*N$.
\end{enumerate}
If this characteristic neighborhood is disjoint from the
permitted interaction collision supports, the nonzero
difference $A$ is not a member of $\Cpart$.
\end{proposition}

\begin{proof}
Normal dilation gives
$\delta^{(4)}(\rho r)=\rho^{-4}\delta^{(4)}(r)$.
The smooth tangential coefficient adds no wavefront
covector. Thus the stated degree and conormal bound hold.

The distribution $A$ vanishes off $N$. Scaling degree and
wavefront set of a sum are bounded by the corresponding
maximum and union, proving (i)--(iii).

Every permitted collision change vanishes in the specified
collision-free neighborhood. The nonzero $A$ does not,
so it cannot belong to that class.
\end{proof}

This is a conditional ambiguity diagnostic. The actual
characteristic degree of the retained full coefficient
has not been computed. In particular, the proposition
does not assert that it is at least four.

Under established uniform submanifold extension hypotheses,
degree below codimension can give uniqueness. At or above
codimension, differences are controlled by finite normal
delta derivatives; microlocal conormal control makes their
tangential coefficients smooth
\cite[Theorem 6.9]{BF}\cite[Section 8]{Dang}.
These conclusions presuppose the relevant distribution
and estimates.

No $A$ from Proposition~\ref{prop:ambiguity} is added here.
No characteristic ambiguity class is enlarged. A future
prescription must either establish uniqueness under its
proved bounds or state a normalization that selects one
map without disguising characteristic differences as
members of $\Cpart$.

\subsection{No cutoff or characteristic finite part}

A cutoff excluding $N$, followed by a limit restoring it,
would require its own convergence, transverse-degree,
and wavefront theorem. Such a construction is not introduced.

No finite-part formula, analytic continuation, subtraction
constant, triangle residue, or characteristic contact is
chosen. Collision contacts are not a substitute for these
missing data.

\begin{remark}[Absence is not obstruction]
Failure to supply a prescription does not prove that every
prescription fails. Conversely, specifying only an admissible
difference class does not construct a reference prescription.
Neither inference is made here.
\end{remark}

%=====================================================================
\section{D4: downstream gates and recorded boundaries}
\label{sec:D4}
%=====================================================================

\subsection{J3--J5}

J3 can be opened only for a distribution actually supplied
by D2 or D3. The copied parallel kinetic primitive has
scaling degree eight in collision codimension four.
That is not a degree along the noncoincident characteristic
stratum.

No uniform transverse estimate for the complete retained
germ, its three- or four-vertex collision strata, or their
intersections is proved here.

J4 remains closed: no characteristic distribution has
been glued to the primitive partial contacts, and no
overlap theorem is supplied.

J5 remains closed: inherited properness is support control
for an existing distribution. It does not define
\eqref{eq:retainedfull}, nor does it establish a wavefront
bound for its output.

No admitted degree-two coefficient and no conclusion
about $\Rquant$ in the carrier are written.

\subsection{No quantum Ward defect}

There is no quantum retarded coefficient on which to form
the proposed binary Ward defect. The already constructed
tree datum is not substituted for the missing quantum
coefficient.

No $s_{\mathrm{ext}}$, $s_\Psi$, or $s_0$ is applied to
$\Rquant$. No quantum product having $\JD$ as a separate
argument is constructed.

The classical breaking density from XIV-A is not an anomaly
of this unnamed graph. XIV-B's relative complex supplies
neither the joint product nor a subtraction. The evaluation
identity on $\mathscr C_{W,\mathrm{bos}}$ and the value $-8I$
remain in the non-identification ledger.

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.26\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
D0 & Frozen &
Same pair, source slots, propagators, switches, width,
retained configuration, and $\Cpart$.\\[0.35em]
D1 & Constructed algebraic incidence &
Complete Hessian and branch ledgers. On the specified
retained germ, exactly $B_4,\ldots,B_9$ remain potentially
nonzero. Other branches are zero only on their proved
local domains or species channel.\\[0.35em]
D2 & Unresolved &
The full six-branch joint-symbol test and lower-order
control are missing. The nonzero $B_6$ fiber numerator
is not a nonzero full symbol section.\\[0.35em]
D3 & Not entered as a construction &
No characteristic prescription. Necessary wavefront
and ambiguity requirements are separated from existence.\\[0.35em]
D4 & Recorded &
J3--J5 and the quantum Ward-defect question remain
unopened.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[XIV-D release]
\label{thm:release}
For the unchanged contract of D0:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The full smooth-slot second-derivative sum is
\eqref{eq:VHessian}--\eqref{eq:OHessian}, and the complete
grouped species expression is \eqref{eq:species}.
\item The retained mixed kernel germ is exactly the six-branch
sum \eqref{eq:six}; the local zero statements and channel
separation are proved without evaluating a singular kernel
at a point.
\item The full leading fiber-tensor numerator of $B_6$ obeys
\eqref{eq:B6identity}, but this does not decide cancellation
of \eqref{eq:retainedfull}.
\item No complete cancellation theorem, nonzero full symbol
section, or characteristic prescription is constructed.
\item No incompatibility excluding every possible prescription
under the unchanged contract is proved.
\end{enumerate}
The exact ending is \textbf{Unresolved at D2}, within J2.
No $\Rquant$ or Ward defect is released.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:Hessian} and Lemma~\ref{lem:ledger}
prove the full algebraic ledger.
Proposition~\ref{prop:reduction},
Lemma~\ref{lem:channel}, and
Theorem~\ref{thm:incidence} prove the retained-germ statement.

Proposition~\ref{prop:B6tensor} proves the branch-level
fiber calculation, with its limitation established in
Corollary~\ref{cor:branch}.
Theorem~\ref{thm:D2} identifies the first missing analytic
step. Section~\ref{sec:D3} supplies necessary conditions
but no extension map or universal exclusion theorem.
\end{proof}

\subsection*{What has advanced}

The complete retained incidence is no longer represented
by a temporal monomial or by an unspecified eleven-term
symbol sum. It is a precise common-kernel germ with six
potentially surviving mixed branches and both
gravitational-response legs.

The remaining problem is correspondingly exact: control
the grouped composition of that six-branch kernel with
the two frozen cross-lines on the retained characteristic
relation, including the lower-order terms.

The source and scalar-response branches have not been
declared to cancel or to vanish. The $B_6$ numerator has
not been promoted to a residue. No additional contact has
been chosen to force a value.

\subsection*{Handoff}

A successor receives:
\begin{enumerate}
\item the unchanged pair and ambiguity contract;
\item the full polarized Hessian formulas and species factors;
\item every mixed and scalar branch derivative;
\item the common attachment/residual-variable kernel description;
\item the exact six-branch retained germ and its local zero proofs;
\item the full $B_6$ fiber identity, explicitly limited to its branch;
\item the necessary wavefront and characteristic-ambiguity checks;
\item the unresolved complete joint-symbol test.
\end{enumerate}

To discharge J2, the successor must prove cancellation
of the complete expression \eqref{eq:retainedfull} with
its lower-order control, or construct a prescription for
that expression with its actual transverse degree,
normalization, ambiguity, and wavefront theorem.

It receives neither failure of full cancellation nor
permission to infer it from Proposition~\ref{prop:B6tensor}.
It receives no admitted quantum coefficient.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \Ocal=\PhiD+\Aone(\rg[\phi];\phi),\qquad
 \Gret=-\PP\ER,\qquad
 \Ctot\subseteq\Cpart,\\[2pt]
 \Ksum|_{A\times B\times Z}
   =(K_4+K_5+K_6+K_7+K_8+K_9)|_{A\times B\times Z},\\[2pt]
 p_\alpha p_\mu\dot a_2^{\alpha\mu}[G;p\otimes k+k\otimes p]
   =G(p^\sharp,p^\sharp)(p\cdot k),\\[2pt]
 \text{full joint-symbol cancellation test: Unresolved at D2},\\
 \text{characteristic prescription: not constructed},\\
 \text{J3--J5 and the quantum Ward defect: unopened}.
 \end{gathered}}
\]

A full incidence ledger is not a joint distribution.
A nonzero branch numerator is not a nonzero full symbol section.
A collision contact is not a characteristic prescription.
An undefined quantum coefficient is not an anomaly.

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

\paragraph{Analytic scope.}
This manuscript continues XIV-C's J2 problem. Its new
constructions are the complete smooth-slot derivative
ledger, common-variable support comparison, retained-germ
reduction, and branch-level fiber identity. It does not
claim to solve the joint characteristic extension problem.

\paragraph{Pair scope.}
$\PhiD$ retains its scalar-linear predecessor meaning.
The full observable $\Ocal$ contains the scalar-sourced
slot once. No entry is changed after the retained test.

\paragraph{Locality of zero statements.}
The branches proved zero on the chosen kernel germ remain
in the global pair. The proofs use support and species,
not undefined pointwise evaluations of distributions.

\paragraph{Ambiguity scope.}
$\Cpart$ is unchanged. The conditional characteristic
ambiguity proposition adds no member to it and selects
no reference extension.

\paragraph{Dependency scope.}
XIV-C supplies the characteristic geometry and its
retained monomial diagnostic. XIII-C supplies the
inventory, partial contacts, and primitive extensions.
XIII-B supplies the tree and retarded convention.
XII-E supplies the classical completion and tree kernels.
XII-C supplies the source representative. XIV-A and
XIV-B are not analytic inputs.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, independently executed symbolic computation,
independent referee review, or journal acceptance is
asserted. All new algebraic calculations are displayed.
No numerical residue has been computed.

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

\bibitem{XIVC}
N.~Cheung,
\emph{Landmark XIV-C: The joint two-line extension of one selected
binary pair: noncoincident characteristic configurations,
transverse estimates, and the output wavefront gate}.
Supplied manuscript \texttt{XIV-C.tex}.
Retained configuration, characteristic geometry,
contact rigidity, and the exact J2 stop.

\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}.
Eleven branches, 48 labelled matchings, partial contacts,
primitive extensions, and the unresolved joint problem.

\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, and
ordered-support correction.

\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}.
Completion formulas, affine slot identity, source-slot sign,
and admitted tree kernels.

\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}.
Fixed cubic density representative and actual source.

\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: scaling degree and 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.
Conormal landing conditions, extension wavefront bounds,
and characteristic counterterm structure.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428v2.
Distribution products, pullbacks, and functional domains.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
second edition, Springer, 1990.
Wavefront sets, distribution products, pullbacks,
and pushforward criteria.

\end{thebibliography}

\end{document}