% !TeX program = pdflatex
%=====================================================================
% XIV-E.tex
%
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% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-E.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-E.tex
%
% Exact release:
%   - XIV-D's pair, six-branch germ, propagators, switches, and
%     partial-diagonal ambiguity class remain unchanged;
%   - the complete six-branch smooth-slot expression is regrouped
%     exactly, before any characteristic composition is asserted;
%   - the B_4+B_7 second-scalar-derivative terms cancel after compact
%     integration by parts, with their remaining terms retained;
%   - B_5+B_6+B_8 are combined without discarding either source leg;
%   - B_9 is written with its complete advanced observation root;
%   - the three-vertex characteristic stratum is distinguished from
%     its lifts to the common attachment/residual-variable space;
%   - the elementary sum-of-conormals product phase is shown to be
%     neither nondegenerate nor clean at the retained configuration;
%   - an exact quadratic identity identifies the two directions
%     invisible to the first-order light-cone equations;
%   - no principal symbol of the complete characteristic composition,
%     no full cancellation theorem, and no prescription are asserted.
%
% Ending:
%   Unresolved at E1, within XIV-D's D2 and XIV-C's J2.
%   The first missing step is the common microlocal composition
%   theorem for the complete regrouped six-branch expression.
%   E2 is not entered as a construction; J3--J5 remain unopened.
%
% Principal debugging corrections:
%   - an exact smooth-slot identity is not a characteristic product;
%   - the disappearance of an explicit test-function derivative is
%     not automatically a decrease of microlocal order;
%   - the characteristic locus has codimension four in the
%     three-vertex space, whereas its intersection with the residual
%     metric attachment diagonal has codimension eight in the
%     sixteen-dimensional common space;
%   - the obvious sum of line phases is not an admissible common
%     classical phase at the copied covector cancellation;
%   - failure of that phase construction does not exclude a different
%     construction or cancellation of the complete coefficient;
%   - a characteristic extension must name its ambient space,
%     reference domain, degree, wavefront cone, and normalization;
%   - XIII-E's finite-frequency family is recorded only as an attempt.
%
% No R_1^(1), triangle residue, characteristic counterterm,
% new ambiguity class, Ward defect, antifield differential,
% interacting differential, state, numerical loop amplitude,
% 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}

\usepackage[T1]{fontenc}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\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{\EAm}{E_m^{\mathrm A}}
\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{\HT}{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{\Nbar}{\overline{\mathcal N}_{\mathrm{att}}}
\newcommand{\Nlift}{\widetilde{\mathcal N}_{\mathrm{att}}}
\newcommand{\Dobs}{\mathcal D_{\mathrm{obs}}}
\newcommand{\Mcal}{\mathcal M}
\newcommand{\Mret}{\mathcal M_{\mathrm{ret}}}
\newcommand{\Kret}{K_{\mathrm{ret}}}
\newcommand{\Ucan}{U_{\mathrm{can}}}
\newcommand{\Bform}{\mathcal B_{\mathrm{form}}}
\newcommand{\Eprime}{\mathcal E'}
\newcommand{\Dprime}{\mathcal D'}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}
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\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Contr}{Contr}
\DeclareMathOperator{\EL}{EL}
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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-E}\\[0.4em]
Joint symbol of the six-branch retained germ,\\
or one named characteristic prescription\\[0.4em]
\large Exact operator reduction and the non-clean product-phase stop}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We continue XIV-D's unresolved characteristic composition problem
for the unchanged binary pair. The first calculation concerns the
complete six-branch smooth-slot expression, not the temporal
component of one branch.

An exact regrouping removes the explicit second derivative of the
scalar test from the combined $B_4+B_7$ expression and retains its
remaining first-derivative terms. The coordinate-response branches
$B_5+B_6+B_8$ combine into one divergence with all source coefficients
present. The scalar-response branch $B_9$ is expressed through the
complete advanced observation root. These are identities of
existing tree kernels, not definitions of their joint contraction
with the cross-lines.

We then identify the ambient spaces relevant to a common symbol
calculation. The copied characteristic stratum has codimension four
in the three-vertex space. Its intersection with the residual metric
attachment diagonal has codimension eight in the common
sixteen-dimensional space. The elementary sum of the three selected
line phases and the attachment phase has differential rank six,
rather than seven, while its critical set has codimension eight.
It is neither a nondegenerate nor a clean phase. An exact quadratic
identity exhibits the two transverse directions invisible to the
linearized light-cone equations.

These results do not provide a different common phase, a joint
symbol of the complete coefficient, or lower-order microlocal
control. No complete cancellation or failure-of-cancellation
theorem is obtained, and no characteristic prescription is
constructed. The exact ending is Unresolved at E1, within D2/J2.
J3--J5 and the quantum Ward-defect question remain unopened.
\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) {Six-branch germ};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Exact regrouping};
\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) {An exact regrouping does not close the first missing step.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Regrouping is not the missing step. 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-E}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Six-branch germ};
\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) {An exact regrouping does not close the first missing step.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Regrouping is not the missing step. 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-E}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Six-branch germ};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Exact regrouping};
\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) {An exact regrouping does not close the first missing step.};
\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-E}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Six-branch germ};
\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) {Exact regrouping. An exact regrouping does not close the first missing step.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Six-branch germ 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-E}
\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: Six-branch germ};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Exact regrouping};
\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) {An exact regrouping does not close the first missing step.};
\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-E}
\end{figure}

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


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



\scopebox{Release and first missing step}{%
\textbf{Frozen:} the pair, six-branch germ, Hadamard matrix,
inverse coefficient $-1$, switches, width, and $\Cpart$.

\smallskip
\textbf{Proved here:} an exact regrouping of the entire retained
smooth-slot expression; its complete scalar-response root formula;
the distinction between the two characteristic lifts; and failure
of the elementary sum-of-line-phases construction to be clean.

\smallskip
\textbf{Not proved:} a common principal-symbol calculus for the
grouped characteristic contraction, cancellation of its complete
symbol, control of its lower-order terms, or a characteristic
prescription.

\smallskip
\textbf{First stop: E1, within D2/J2.}
The missing step is the common microlocal composition theorem.
The phase calculation below excludes one proposed shortcut,
not every possible construction.

\smallskip
No $\Rquant$, Ward defect, new contact class, or characteristic
subtraction is released.}

\tableofcontents

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

The three endings apply to the retained characteristic construction:
\begin{enumerate}
\item \textbf{Constructed}: cancellation of the complete retained
germ, with the necessary lower-order control, or one prescription
with proved degree, normalization, ambiguity, and wavefront control.
J3 may then be opened only for that distribution.
\item \textbf{Proved obstructed}: a demonstrated incompatibility
excludes every map in a class fixed before the test.
\item \textbf{Unresolved}: the joint symbol, a required remainder
estimate, or a prescription theorem is missing.
\end{enumerate}

A nonzero $B_6$ numerator is not the second ending. Failure of a
sufficient product test is not a nonexistence theorem. A successful
local characteristic prescription would not, by itself, admit
$\Rquant$ before J3--J5.

\begin{inputstatement}[XIV-D: pair and retained germ]
\label{input:XIVD}
The frozen observable is
\[
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD.
\]
The inverse and ambiguity conventions are
\[
 \Gret=-\PP\ER,\qquad \Ctot\subseteq\Cpart.
\]

Let $A,B,Y$ be the pairwise separated neighborhoods of
$u_0,x_0,y_0$, and let $\overline Z\subset A$ be the residual metric
neighborhood. In the common variables
\[
 a\in A,\qquad b\in B,\qquad z\in Z,\qquad y\in Y,
\]
the mixed observable kernel satisfies
\[
 K_\Sigma|_{A\times B\times Z}
 =(K_4+K_5+K_6+K_7+K_8+K_9)|_{A\times B\times Z}.
\]
The other branches remain in the global pair and vanish only on
the proved germ or species channel. In particular, $K_8$ and $K_9$
retain both gravitational-response legs.

The kernels $K_6,K_8,K_9$ are supported on the observable metric
diagonal $a=z$. The retained mixed germ has 24 potentially surviving
labelled matchings.

The leading fiber identity
\[
 p_\alpha p_\mu\dot a_2^{\alpha\mu}[G;r]=Q\lambda,
 \qquad
 r=p\otimes k+k\otimes p,\quad
 Q=G(p^\sharp,p^\sharp),\quad \lambda=p\cdot k,
\]
is a branch diagnostic. It does not decide the complete sum.

No joint symbol or characteristic prescription was constructed.
The first missing step was D2 \cite{XIVD}.
\end{inputstatement}

\begin{inputstatement}[XIV-C: characteristic geometry]
\label{input:XIVC}
The retained triple is
\[
 u_0=(t_*,0,0,0),\quad
 y_0=(t_*+L,L,0,0),\quad
 x_0=(t_*+2L,2L,0,0),
\]
with $\eps<t_*<t_*+2L<T$, $f(x_0)\ne0$, and $g=1$ near
these points.

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

Collision-supported changes vanish on the retained collision-free
neighborhood, while the original attachment variables are retained
\cite{XIVC,XIVD}.
\end{inputstatement}

Neither copied geometric proposition nor the copied $B_6$ fiber
identity is re-proved.

\begin{inputstatement}[Necessary prescription constraints]
\label{input:constraints}
XIV-D established the following necessary distinctions:
\begin{enumerate}[label=\textup{(\roman*)}]
\item an extension with wavefront set contained only in $N^*N$
on a whole neighborhood would require its off-$N$ reference
distribution to be smooth;
\item in a codimension-four extension problem, degree at least
four and allowance of the full conormal bundle permit tangential
multiples of $\delta^{(4)}(r)$ unless an additional normalization
or uniqueness result removes them;
\item on a collision-free neighborhood, such nonzero
characteristic-supported differences are not members of $\Cpart$.
\end{enumerate}
These constraints do not assert the existence or the actual
degree of the selected characteristic distribution \cite{XIVD}.
\end{inputstatement}

XIII-E's common-frequency family is recorded as an attempt:
its admissible-set convergence does not establish convergence
through this stratum \cite{XIIIE}. No such limit is imported.

XIV-A and XIV-B are not analytic inputs. The value $-8I$ is not
a subtraction, and a classical breaking density is not an anomaly
of the missing graph distribution.

%=====================================================================
\section{E0: the six-branch contract}
\label{sec:E0}
%=====================================================================

\subsection{Geometry and product data}

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 external data remain those fixed in XIV-C and XIV-D. No
observation, switch, or width is chosen after a symbol calculation.

The carrier remains
\[
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                         \Nsol[[\hbar,\kappa]].
\]
The frozen time-ordered line is
\[
 \HT=H+i\GA.
\]
Metric--scalar blocks vanish. The numerical species factors and
the inverse coefficient $-1$ are unchanged.

The interaction is
\[
 \Vcal=-\frac12\int g\,\gamma_{\mu\nu}
                                   T_0^{\mu\nu}[\phi]\,\dd^4x,
\]
where
\[
 T_0^{\mu\nu}[\phi]
 =\partial^\mu\phi\partial^\nu\phi
 -\frac12\eta^{\mu\nu}
                    \bigl((\partial\phi)^2-m^2\phi^2\bigr).
\]
The scalar-sourced slot of $\Ocal$ is included once. It is not
removed because its scalar--scalar channel vanishes on the retained
output-location patch.

\subsection{The grouped characteristic target}

The complete formal two-line expression remains
\[
 \Bform=\frac{i}{2}
 \pair{\Vcal''\otimes\Ocal''}
                       {\HT^{\otimes2}-H^{\otimes2}}.
\]
On the retained mixed germ it is
\begin{equation}
 i\,\Contr^{\gamma\phi}\left(
 \Vcal''_{\gamma\phi},\Kret;\
 \HT_{\gamma\gamma}(y,a)\otimes\HT_{\phi\phi}(y,b)
          -H_{\gamma\gamma}(y,a)\otimes H_{\phi\phi}(y,b)
 \right),
 \qquad
 \Kret=\sum_{j=4}^{9}K_j.
 \label{eq:target}
\end{equation}
The factor $i$ includes the two mixed slot orderings of XIV-D.

This difference stays grouped. Formula~\eqref{eq:target} specifies
the composition to be constructed; it does not assert its existence
on the characteristic set.

\subsection{Ambiguity and negative freeze}

The class $\Cpart$ is unchanged. Its defined graphwise changes
are collision-supported contacts with the inherited spectator,
external-weight, normal-order, support, and normalization
conditions. It controls allowed differences of constructions,
not a reference extension across the separated characteristic set.

There is no cutoff of $\Nchar$, triangle residue, new characteristic
contact class, antifield differential, spurion, replacement
Hadamard matrix, or sharp-width limit. The auxiliary neighborhoods
describe kernel restrictions, not a cutoff of the characteristic
configuration.

No universal class of successful characteristic prescriptions
is asserted to have been constructed. Thus no universal
obstruction is tested against an unspecified class.

%=====================================================================
\section{E1: exact calculation of the complete precomposition sum}
\label{sec:E1}
%=====================================================================

The following calculation concerns all six retained branches.
It is performed before discussing their principal symbols.

\subsection{Operators and complete retained derivatives}

For a smooth metric tensor $r$, put
\[
 a_1^{\mu\nu}[r]
 =\frac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},
 \qquad
 b_1[r]=\frac12\tr_\eta r,
\]
\[
 b_2[G]
 =\frac18(\tr_\eta G)^2-\frac14G_{\rho\sigma}G^{\rho\sigma},
\]
\[
 a_2^{\mu\nu}[G]
 =G^\mu{}_\rho G^{\rho\nu}
 -\frac12(\tr_\eta G)G^{\mu\nu}
                              +b_2[G]\eta^{\mu\nu}.
\]
Define
\begin{align}
 X^\mu[r]
 &=-\ER\partial_\alpha(\sig a_1^{\alpha\mu}[r]),\\
 U[r]v
 &=\partial_\alpha(\sig a_1^{\alpha\beta}[r]\partial_\beta v),\\
 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).
 \label{eq:operators}
\end{align}

Let $G$ be the residual metric direction, $h$ the contracted
metric direction, and $\psi$ the contracted scalar direction.
These are smooth directions of the classical functional, not
values of singular line distributions. Write
\[
 X_G=X[G],\qquad X_h=X[h],
 \qquad r_h=\rD'[G]h=\PP\ER\bigl(\JD'[G]h\bigr).
\]
The source $\JD$ is the unchanged source of XII-C:
\[
 \JD[\gamma]=\EL_\gamma(\sig\ell_3[\gamma]),
 \qquad
 \ell_3[\gamma]
 =-2[\rho^3]\bigl(\sqrt{-\det(\eta+\rho\gamma)}
                              R[\eta+\rho\gamma]\bigr).
\]
The unintegrated curvature representative, and hence every
coefficient of $\JD'[G]h$, is fixed as in XII-C. No source
coefficient is set to zero by the following notation
\cite{XIIC,XIIE}.

The density derivative is
\begin{align}
 \dot b_2[G;h]
 &=\frac14(\tr_\eta G)(\tr_\eta h)
                       -\frac12G_{\rho\sigma}h^{\rho\sigma},\\
 \dot a_2^{\mu\nu}[G;h]
 &=h^\mu{}_\rho G^{\rho\nu}
   +G^\mu{}_\rho h^{\rho\nu}
   -\frac12\left(
       (\tr_\eta h)G^{\mu\nu}
       +(\tr_\eta G)h^{\mu\nu}\right)
                        +\dot b_2[G;h]\eta^{\mu\nu}.
 \label{eq:dotdensity}
\end{align}

The copied mixed branch derivatives are
\begin{align}
 \Mcal_4[G;h,\psi]
 &=\int f\left(
 X_h^\nu\partial_\nu X_G^\mu+
 X_G^\nu\partial_\nu X_h^\mu\right)\partial_\mu\psi\,\dd^4x,\\
 \Mcal_5[G;h,\psi]
 &=\int f\,\ER\left(
 U[h]X_G^\mu+U[G]X_h^\mu\right)\partial_\mu\psi\,\dd^4x,\\
 \Mcal_6[G;h,\psi]
 &=\int f\,\ER\left(
 \partial_\alpha(\sig\dot a_2^{\alpha\mu}[G;h])
                              \right)\partial_\mu\psi\,\dd^4x,\\
 \Mcal_7[G;h,\psi]
 &=\frac12\int f
 \left(X_h^\mu X_G^\nu+X_G^\mu X_h^\nu\right)
                         \partial_\mu\partial_\nu\psi\,\dd^4x,\\
 \Mcal_8[G;h,\psi]
 &=-\int f\,X^\mu[r_h]\partial_\mu\psi\,\dd^4x,\\
 \Mcal_9[G;h,\psi]
 &=\int f\,p[r_h;\psi]\,\dd^4x.
 \label{eq:sixbranches}
\end{align}
They include all metric-slot choices, source derivatives,
and both response legs. Set
\[
 \Mret[G;h,\psi]=\sum_{j=4}^{9}\Mcal_j[G;h,\psi].
\]

\subsection{The first calculation: the entire six-branch sum}

Define the smooth coefficient
\begin{equation}
 C_G^\mu{}_\nu
 =f\,\partial_\nu X_G^\mu
                  -\delta^\mu_\nu\partial_\rho(fX_G^\rho),
 \label{eq:CG}
\end{equation}
and the complete tensor
\begin{equation}
 \begin{split}
 B^{\alpha\mu}[G;h]
 ={}&
 a_1^{\alpha\beta}[h]\partial_\beta X_G^\mu
 +a_1^{\alpha\beta}[G]\partial_\beta X_h^\mu\\
 &+\dot a_2^{\alpha\mu}[G;h]+a_1^{\alpha\mu}[r_h].
 \end{split}
 \label{eq:Btensor}
\end{equation}

\begin{theorem}[Exact regrouping of the complete retained expression]
\label{thm:regroup}
For all smooth directions for which the inherited slab response
formulas are used,
\begin{equation}
 \begin{split}
 \Mret[G;h,\psi]
 ={}&
 \int C_G^\mu{}_\nu X_h^\nu\partial_\mu\psi\,\dd^4x\\
 &+\int f\,\ER\left(
       \partial_\alpha(\sig B^{\alpha\mu}[G;h])
                    \right)\partial_\mu\psi\,\dd^4x\\
 &+\int f\,p[r_h;\psi]\,\dd^4x.
 \end{split}
 \label{eq:fullregroup}
\end{equation}
This identity retains all six branches. It involves no
characteristic composition and no regularization.
\end{theorem}

\begin{proof}
Symmetry of $\partial_\mu\partial_\nu\psi$ gives
\[
 \Mcal_7=\int f X_G^\nu X_h^\mu
                         \partial_\nu\partial_\mu\psi\,\dd^4x.
\]
Compact integration by parts in the second term of $\Mcal_4$
yields
\[
 \begin{split}
 \int fX_G^\nu\partial_\nu X_h^\mu\partial_\mu\psi
 ={}&
 -\int\partial_\nu(fX_G^\nu)X_h^\mu\partial_\mu\psi\\
 &-\int fX_G^\nu X_h^\mu\partial_\nu\partial_\mu\psi.
 \end{split}
\]
The last integral cancels $\Mcal_7$ exactly. The first term
of $\Mcal_4$ and the remaining integral are precisely
\[
 \Mcal_4+\Mcal_7
 =\int C_G^\mu{}_\nu X_h^\nu\partial_\mu\psi\,\dd^4x.
\]

Next, the definition of $U$ expresses $\Mcal_5$ as the retarded
inverse of the divergence of the first two terms in
\eqref{eq:Btensor}. The third term is $\Mcal_6$.
Since
\[
 -X^\mu[r_h]
 =\ER\partial_\alpha(\sig a_1^{\alpha\mu}[r_h]),
\]
the fourth term gives $\Mcal_8$, with its inherited sign.
Linearity of $\ER$ combines these three branches.

Finally $\Mcal_9$ is unchanged. Adding the three groups proves
\eqref{eq:fullregroup}.
\end{proof}

\begin{remark}[The precise cancellation proved]
The theorem proves an exact cancellation of the explicit
second-scalar-test-derivative terms in $B_4+B_7$. It does
not prove cancellation of those two branches, much less
of all six.

It also does not establish that either branch is of lower
microlocal order in a common characteristic calculus.
That conclusion would require the calculus and order
comparison still missing at E1.
\end{remark}

\subsection{The complete scalar-response root}

Let
\[
 F=\EAm f.
\]
The inherited smooth advanced operator makes $F$ smooth.

\begin{proposition}[Exact $B_9$ root formula]
\label{prop:root}
The last term of \eqref{eq:fullregroup} is
\begin{equation}
 \Mcal_9[G;h,\psi]
 =
 \int g\left(
 a_1^{\mu\nu}[r_h]\partial_\mu F\,\partial_\nu\psi
                   -m^2b_1[r_h]F\psi\right)\dd^4x.
 \label{eq:B9root}
\end{equation}
Both the derivative and mass terms are retained.
\end{proposition}

\begin{proof}
Transposition of the inherited zero-data Green operator gives
\[
 \int f\,p[r_h;\psi]
 =-\int F\left(
 \partial_\mu(g a_1^{\mu\nu}[r_h]\partial_\nu\psi)
                  +m^2g b_1[r_h]\psi\right)\dd^4x.
\]
Compactness of $g$ permits integration by parts in the divergence.
This gives \eqref{eq:B9root}. No equation for $\psi$ is assumed.
\end{proof}

The mass term is not deleted merely because it has fewer explicit
scalar derivatives. A prospective remainder estimate must control
it, together with every other lower-order term.

\subsection{Common kernels after a smooth residual test}

For an arbitrary $G\in C_c^\infty(Z;\Sym^2T^*M)$, pair the residual
metric slot of $\Kret$ against $G$. Denote the resulting attachment
kernel by $L_G$:
\[
 \Mret[G;h,\psi]=\pair{L_G(a,b)}{h(a)\psi(b)}.
\]
Testing the residual slot is a distributional pairing. It neither
changes the observable nor sets $z=a$ in branches where that
equality is absent.

\begin{proposition}[Equality of the complete tested kernels]
\label{prop:kernels}
Formula~\eqref{eq:fullregroup}, with \eqref{eq:B9root}, gives the
same attachment distribution $L_G$ as the sum
$\sum_{j=4}^{9}\pair{K_j}{G}$.

For every such $G$, this kernel is compact and satisfies
\[
 \WF(L_G)\subset
 Z_2^{\mathrm{att}}
 =\{(a,\xi;b,\eta):\xi+\eta=0\}\setminus0.
\]
This is an intermediate attachment-kernel bound, not admission
of the final quantum coefficient.
\end{proposition}

\begin{proof}
The exact smooth-slot identity proves equality on tensor-product
tests $h(a)\psi(b)$. Finite sums of these tests determine a
distributional kernel, giving the asserted equality.

Each inherited $K_j$ is a compact tree kernel with the covector
sum-zero bound in $(a,b,z)$. Multiplication by the smooth residual
test and pushforward over its compactly supported variable set
the residual covector to zero. The resulting covectors therefore
satisfy $\xi+\eta=0$. Smooth testing and the finite sum preserve
compactness \cite[Section 8.2]{Hormander}.
\end{proof}

The sum-zero bound does not prove that multiplying $L_G$ by the
two cross-lines is allowed. Their endpoint covectors can cancel
the two nonzero attachment covectors. Nor does testing each fixed
$G$ supply the uniform estimates needed to reconstruct a joint
characteristic distribution in all four common variables.

%=====================================================================
\section{The ambient-space audit for a common symbol}
\label{sec:ambient}
%=====================================================================

\subsection{Two different lifts of the copied stratum}

Let
\[
 W=A\times B\times Z\times Y
\]
in the coordinate order $(a,b,z,y)$, with dimension sixteen.
Let
\[
 P:W\longrightarrow B\times A\times Y,
 \qquad P(a,b,z,y)=(b,a,y).
\]
Define
\[
 \Nbar=P^{-1}(\Nchar),\qquad
 \Dobs=\{a=z\}\cap W,\qquad
 \Nlift=\Nbar\cap\Dobs.
\]

The overline in $\Nbar$ is notation for this full lift, not a
topological closure.

\begin{proposition}[Characteristic and attachment codimensions]
\label{prop:lifts}
Locally at the retained point,
\[
 \codim_W\Nbar=4,\qquad
 \codim_W\Dobs=4,\qquad
 \codim_W\Nlift=8.
\]
The intersection defining $\Nlift$ is transverse.
\end{proposition}

\begin{proof}
The map $P$ is a submersion, so the copied codimension-four
stratum lifts to a codimension-four submanifold with $z$ free.

The equation $a-z=0$ has four independent components.
Every conormal of $\Nbar$ has zero $z$ component, whereas a
conormal to $\Dobs$ has opposite $a$ and $z$ components.
A covector belonging to both conormal spaces must consequently
have its $z$, then its $a$, component zero, and hence is zero.
This proves transversality and the codimension-eight conclusion.
\end{proof}

The codimension-four characteristic ambiguity statement copied
from XIV-D refers to a specified codimension-four extension
problem. It must not be transferred to $\Nlift$ with the same
delta order or degree threshold without identifying the ambient
space.

For example, the numerical three-vertex graph may be extended
first and its observable attachment added later. That is not
automatically the same operation as extending a common kernel
already containing derivatives of $\delta(a-z)$. Their
compatibility requires a theorem.

%=====================================================================
\section{The elementary product phase is not a common symbol calculus}
\label{sec:phase}
%=====================================================================

\subsection{The candidate sum of line and attachment phases}

Consider the smooth homogeneous candidate
\begin{equation}
 \Phi(a,b,z,y;t_R,t_1,t_2,\kappa)
 =-t_Rq_R+t_1q_1+t_2q_2+\kappa_\mu(a^\mu-z^\mu),
 \label{eq:phase}
\end{equation}
where
\[
 (t_R,t_1,t_2)\in(0,\infty)^3,\qquad \kappa\in\R^4.
\]
Its signs encode the copied oriented conormals. A simultaneous
reversal of the Fourier convention reverses the whole expression
and changes none of the rank conclusions.

Formula~\eqref{eq:phase} is the elementary candidate obtained by
adding the selected three line phases and the metric attachment
phase. It is not asserted to represent the full six-branch sum.
Its validity must be tested before amplitudes can be compared
using it.

For a nondegenerate phase, the differentials of its auxiliary
critical equations are independent on the critical set. This is
the standard local criterion for the principal-symbol construction
for Lagrangian distributions \cite[Section 8.2]{Wunsch}.

For the present calculation, call a candidate \emph{clean} if its
auxiliary critical set is smooth and its tangent space equals the
kernel of the differential of those critical equations. This is
the tangent-space requirement needed for clean critical-set
reduction. It is weaker than full independence but still fails
below.

\subsection{The common zero set of the cone equations}

Shrink the original neighborhoods, if necessary, without deleting
the retained configuration, so that
\[
 y^0-a^0>0,\qquad b^0-y^0>0
\]
throughout them.

\begin{lemma}[Local zero locus]
\label{lem:zerolocus}
On these neighborhoods,
\[
 q_R=q_1=q_2=0
 \quad\Longleftrightarrow\quad
 (b,a,y)\in\Nchar.
\]
Consequently the auxiliary critical set of \eqref{eq:phase} is
\[
 C_\Phi=\Nlift\times(0,\infty)^3\times\R^4.
\]
\end{lemma}

\begin{proof}
Set $\xi=b-a$ and $\eta'=y-a$. The equations imply
\[
 \xi^2=(\eta')^2=(\xi-\eta')^2=0,
 \qquad \xi\cdot\eta'=0.
\]
The time-order inequalities make $\xi$ and $\eta'$ nonzero
future null vectors. Write
\[
 \xi=\beta(1,\mathbf n),\qquad
 \eta'=\alpha(1,\mathbf m),
 \qquad \alpha,\beta>0.
\]
Their scalar product is
$\alpha\beta(1-\mathbf n\cdot\mathbf m)$, so it vanishes exactly
when $\mathbf n=\mathbf m$. The second time inequality gives
$\alpha<\beta$. This is the copied parameterization of $\Nchar$.

The auxiliary derivatives of $\Phi$ are
\[
 -q_R,\quad q_1,\quad q_2,\quad a-z.
\]
Their simultaneous vanishing gives the stated critical set.
\end{proof}

This identifies the zero locus; it does not re-prove the copied
smoothness, dimension, or rank of $\Nchar$.

\begin{theorem}[Non-clean elementary product phase]
\label{thm:nonclean}
At the retained characteristic configuration, the seven auxiliary
critical equations of \eqref{eq:phase} have differential rank six.
Their critical set has codimension eight in the
twenty-three-dimensional base/frequency space.

Thus:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the candidate is not nondegenerate;
\item it is not clean under the stated tangent-space criterion;
\item on the ray
\[
 t_R=\frac{c}{2\beta},\qquad
 t_1=\frac{c}{2\alpha},\qquad
 t_2=\frac{c}{2(\beta-\alpha)},\qquad
 \kappa=0,\qquad c>0,
\]
its entire differential vanishes.
\end{enumerate}
It cannot be used as the ordinary common phase defining a
principal symbol of \eqref{eq:target}.
\end{theorem}

\begin{proof}
The three cone differentials have copied rank two. The four
components of $\dd(a-z)$ are independent of them: looking at
their $z$ components forces every coefficient in a proposed
linear relation to vanish. The total rank is therefore six.

By Proposition~\ref{prop:lifts} and
Lemma~\ref{lem:zerolocus}, the critical set has dimension
\[
 \dim C_\Phi=8+7=15
\]
inside a space of dimension $16+7=23$. Its codimension is eight.

The kernel of the differential of the critical equations has
dimension $23-6=17$, whereas the tangent space of $C_\Phi$ has
dimension fifteen. They are unequal. Independence and the
clean tangent-space condition both fail.

On the displayed frequency ray, the base differential is
\[
 c\left(
 -\frac{\dd q_R}{2\beta}
 +\frac{\dd q_1}{2\alpha}
 +\frac{\dd q_2}{2(\beta-\alpha)}\right)=0
\]
by the copied positive relation. The auxiliary differential also
vanishes on $C_\Phi$. Thus $\dd\Phi=0$ there even though the
frequency variables are nonzero.
\end{proof}

\begin{remark}[Scope of this failure]
The theorem excludes this elementary phase shortcut.
It does not prove that every multiphase, resolved, boundary-value,
or other characteristic construction fails.

In particular, an amplitude cancellation could change the
relevant representation. Establishing that cancellation still
requires the complete amplitude calculation and its remainder
estimates; it cannot be inferred from the geometric rank defect.
\end{remark}

\subsection{An exact quadratic identity for the invisible directions}

The rank calculation has a concrete second-order form.

Fix one point of $\Nchar$ with parameters $\alpha,\beta,\mathbf n$.
Choose an orthonormal spatial basis transverse to $\mathbf n$.
Let $v,w\in\R^2$ be transverse spatial displacements, and set
\[
 b-a=\beta\ell_{\mathbf n}+(0,v),\qquad
 y-a=\alpha\ell_{\mathbf n}+(0,w).
\]
Here $(0,v)$ and $(0,w)$ have no time or longitudinal spatial
component.

\begin{proposition}[Exact transverse quadratic defect]
\label{prop:quadratic}
On this affine transverse slice,
\begin{equation}
 -\frac{q_R}{\beta}
 +\frac{q_1}{\alpha}
 +\frac{q_2}{\beta-\alpha}
 =
 -\frac{|\alpha v-\beta w|^2}
         {\alpha\beta(\beta-\alpha)}.
 \label{eq:quadratic}
\end{equation}
There are two independent relative transverse components in
$\alpha v-\beta w$. They are not detected by the first
differentials of the three cone equations at the base point.
\end{proposition}

\begin{proof}
Orthogonality to $\mathbf n$ and nullity of $\ell_{\mathbf n}$
give
\[
 q_R=-|v|^2,\qquad
 q_1=-|w|^2,\qquad
 q_2=-|v-w|^2.
\]
Therefore the left side of \eqref{eq:quadratic} equals
\[
 \frac{|v|^2}{\beta}
 -\frac{|w|^2}{\alpha}
 -\frac{|v-w|^2}{\beta-\alpha}.
\]
Expanding the last square gives exactly the right side.

All three expressions have zero first derivative at $v=w=0$.
The linear map $(v,w)\mapsto\alpha v-\beta w$ has rank two,
which identifies the two relative components invisible to
that linearization.
\end{proof}

Equation~\eqref{eq:quadratic} is a geometric identity, not a
scaling degree of a distribution. Its quadratic behavior does
not determine the transverse degree, convergence, or wavefront
set of the six-branch product.

%=====================================================================
\section{The precise E1 stop}
\label{sec:stop}
%=====================================================================

\subsection{What would be a joint symbol}

To use the notation
\[
 \sigma_{\mathrm{joint}}\!\left(\sum_{j=4}^{9}K_j\right)
\]
for \eqref{eq:target}, a successor must supply a common
microlocal representation and composition theorem for the
grouped coefficient. In particular, it must identify:
\begin{enumerate}
\item the relevant canonical or characteristic relation in the
chosen ambient space;
\item the amplitude orders and the common symbol bundle;
\item the maps accounting for additional internal vertices in
$K_5,K_8,K_9$;
\item the observable attachment diagonal and its derivatives;
\item the complete tensor coefficients and the prescribed grouping
of the cross-line difference;
\item a remainder estimate that controls every term not represented
by the proposed leading section.
\end{enumerate}

The inherited wavefront inclusions do not themselves supply
these data. The ordinary principal-symbol construction requires
more than a wavefront-set inclusion \cite[Section 8]{Wunsch}.

Theorem~\ref{thm:regroup} supplies an exact common operator
expression before characteristic composition.
Theorem~\ref{thm:nonclean} proves that simply adding the selected
line phases does not supply the missing calculus.

Neither theorem assigns the requested joint principal symbol.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}c p{.47\linewidth}X@{}}
\toprule
Branch & Established contribution & Joint-symbol status\\
\midrule
$B_4$ &
Included in the exact $B_4+B_7$ coefficient $C_G$ &
No common characteristic symbol assigned.\\
$B_5$ &
The first two terms of $B^{\alpha\mu}[G;h]$ &
Internal composition and order remain to be controlled.\\
$B_6$ &
$\dot a_2^{\alpha\mu}[G;h]$ and the copied $Q\lambda$ diagnostic &
Not a symbol of the full sum.\\
$B_7$ &
Its explicit second-scalar-derivative term cancels as proved &
Remaining terms retained; no microlocal order claim.\\
$B_8$ &
$a_1^{\alpha\mu}[r_h]$ in the complete coordinate response &
Full differentiated source retained.\\
$B_9$ &
The exact root formula \eqref{eq:B9root} &
Derivative and mass terms both retained.\\
\bottomrule
\end{tabularx}
\end{center}

A vanishing numerator in a particular presentation would have to
be recorded with its remaining terms. It would not authorize
deleting a branch from this table.

\subsection{Reference values on the canonical domain}

Let $\Ucan\subset W$ denote an open domain on which the complete
specified graph products satisfy the inherited product criteria,
with their smooth weights and attachment maps. The grouped
expression there is the canonical reference $t^{\mathrm{can}}$.

No equality
\[
 \Ucan=W\setminus\Nbar
\]
is asserted. Establishing a reference on the whole complement
of a proposed extension locus is itself part of the missing
composition problem.

Consequently an extension of ``the off-characteristic product''
must not be invoked before that product and its precise domain
have been named.

\begin{proposition}[What the present calculation does not decide]
\label{prop:limits}
The proved results establish neither of the decisive E1 conclusions:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a zero joint-symbol section for the complete six-branch
coefficient with controlled remainder;
\item a nonzero joint-symbol section of that complete coefficient.
\end{enumerate}
They also do not demonstrate an incompatibility excluding every
prescription under the frozen contract.
\end{proposition}

\begin{proof}
Theorem~\ref{thm:regroup} is an exact identity for the already
defined smooth-slot tree kernels. It does not compose them
with the cross-lines on the characteristic set.

Theorem~\ref{thm:nonclean} rules out the elementary phase
candidate, not all possible representations.
Proposition~\ref{prop:quadratic} identifies its geometric
degeneracy but supplies no distributional amplitude estimate.

The copied $Q\lambda$ identity concerns only $B_6$.
None of these statements supplies a common joint-symbol
section or a remainder theorem for \eqref{eq:target}.
Nor does any statement quantify over all prescriptions in
a previously fixed candidate class.
\end{proof}

The first unresolved step is therefore \textbf{E1}, within the
inherited D2/J2 gate. It is not E2 on the premise that cancellation
has failed: failure of complete cancellation has not been proved.

%=====================================================================
\section{E2: prescription interface, not entered as a construction}
\label{sec:E2}
%=====================================================================

\subsection{A future map must identify its problem}

No prescription is constructed in this section. Any future map
must name:
\begin{enumerate}
\item its ambient graph or common-kernel space;
\item a distribution $t^\circ$ on a proved reference domain;
\item the exact closed characteristic locus $N$ across which
it is extended;
\item agreement with the grouped frozen prescription on that domain;
\item a finite uniform transverse degree in specified normal
coordinates;
\item a closed conic wavefront bound retaining the inherited
off-characteristic singularities;
\item a normalization selecting a map, or a uniqueness theorem
under the proved degree and wavefront hypotheses.
\end{enumerate}

The numerical stratum $\Nchar$, the full lift $\Nbar$, and the
attachment intersection $\Nlift$ are not interchangeable
choices of $N$. The chosen stage must also account for other
internal compositions present in the complete sum.

\subsection{Degree and wavefront hypotheses}

For a stated codimension-$q$ locus, use normal coordinates
$(s,r)$ with $r\in\R^q$. A finite transverse degree must be
proved uniformly over compact tangential sets with the necessary
microlocal seminorm control.

When their hypotheses hold, submanifold extension theorems
distinguish the degree-below-codimension uniqueness case from
the case requiring normal-jet normalization
\cite[Section 6]{BF}. Those hypotheses are not supplied by a
superficial derivative count.

A prospective minimal wavefront estimate has the form
\[
 \WF(t)\subset
 \overline{\WF(t^\circ)}\cup N^*N,
\]
in the stated ambient space. Such an estimate requires the
appropriate microlocal extension hypotheses, including control
of how the off-$N$ wavefront approaches $N$
\cite[Sections 5--6]{Dang}.

The copied restriction warning remains in force:
$\WF(t)\subset N^*N$ alone cannot retain a nonsmooth reference
outside $N$.

In a common kernel already containing $\delta(a-z)$ or its
derivatives, the inherited attachment singularities must also
be accounted for. No estimate for adding those attachments
after characteristic extension is asserted here.

\subsection{Characteristic ambiguity is not a partial contact}

For a genuine codimension-four problem, the copied ambiguity
diagnostic applies conditionally when the degree is at least
four and the full characteristic conormal is allowed:
\[
 A(s,r)=a(s)\delta^{(4)}(r)
\]
can agree with the reference off the stratum while preserving
the stated degree allowance and conormal bound.

On the retained collision-free neighborhood, a nonzero such
difference is not in $\Cpart$. Hence a future prescription
must not identify it as an inherited collision contact.

For a different extension stage, the relevant codimension
and supported-distribution structure must be recomputed.
For example, $\Nlift$ has codimension eight, while an amplitude
on the three-vertex numerical space still faces codimension four.
No actual degree is assigned to either unresolved distribution.

Under suitable conormal wavefront hypotheses, supported
extension differences have finite normal delta-derivative
expansions with smooth tangential coefficients
\cite[Section 8]{Dang}. This is an ambiguity theorem for an
existing extension problem, not a construction of the selected
reference or extension.

No characteristic-supported term is added here. No normalization
constant is selected, and $\Cpart$ is not enlarged.

\subsection{The recorded Gaussian attempt}

XIII-E's family can be continued only as that named attempt.
To advance it on the present germ, one would need a new
convergence theorem on tests meeting the characteristic set,
with the complete branch sum and uniform wavefront estimates
retained.

Finite-regulator admission, convergence on canonical subdomains,
and independence of auxiliary compact localization do not give
that theorem \cite{XIIIE}. None is claimed here.

No cutoff removing the characteristic stratum and no restoring
limit is introduced as an alternative proof.

%=====================================================================
\section{E3: downstream gates and the Ward-defect boundary}
\label{sec:E3}
%=====================================================================

J3 opens only for a distribution supplied by a successful E1 or
E2 construction. The copied parallel-line degree eight is a
collision degree, not a degree along $\Nchar$.

The exact integrations by parts in
Theorem~\ref{thm:regroup} do not supply uniform transverse
bounds for the unresolved characteristic product, larger
collision strata, or their intersections.

J4 remains closed: no characteristic prescription has been
composed with the singular spectators and compatible partial
contacts.

J5 remains closed: inherited properness applies to an existing
distribution. It does not construct \eqref{eq:target} or prove
the required output wavefront bound.

The intermediate attachment-kernel bound of
Proposition~\ref{prop:kernels} is not the J5 theorem for the
degree-two residual-field output.

No $s_{\mathrm{ext}}$, $s_\Psi$, or $s_0$ is applied to
$\Rquant$. No binary Ward defect is formed. The already
constructed tree coefficient is not substituted for the
missing quantum coefficient.

The classical breaking density from XIV-A and the relative
complex of XIV-B remain separate from this analytic problem.
Neither the value $-8I$ nor a classical evaluation identity
normalizes the graph.

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.25\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
E0 & Frozen &
Same pair, propagators, switches, width, $\Cpart$, and
six-branch germ.\\[0.35em]
E1 & Partial analytic results; Unresolved &
The entire smooth-slot expression is regrouped exactly.
The elementary product phase is proved non-clean.
No joint-symbol section or complete remainder theorem is
constructed.\\[0.35em]
E2 & Not entered as a construction &
No characteristic map, actual transverse degree,
normalization, or extension wavefront theorem.\\[0.35em]
E3 & Recorded &
J3--J5 and the quantum Ward-defect question remain unopened.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[Mathematical release]
\label{thm:release}
For the frozen contract:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The complete six-branch smooth-slot expression equals
\eqref{eq:fullregroup}, with its full scalar-response term
given by \eqref{eq:B9root}.
\item The characteristic lift with the residual metric free has
codimension four, while its intersection with $a=z$ has
codimension eight.
\item The elementary candidate phase \eqref{eq:phase} has rank-six
auxiliary equations and a codimension-eight critical set.
It is neither nondegenerate nor clean.
\item The transverse identity \eqref{eq:quadratic} exhibits the two
relative directions missed by the first-order cone equations.
\end{enumerate}
These conclusions establish an exact operator reduction and a
specific phase-construction failure. They do not establish the
joint characteristic distribution.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:regroup} and Proposition~\ref{prop:root}
give (i). Proposition~\ref{prop:lifts} gives (ii).
Lemma~\ref{lem:zerolocus} and Theorem~\ref{thm:nonclean}
give (iii). Proposition~\ref{prop:quadratic} gives (iv).
The limitation follows from Proposition~\ref{prop:limits}.
\end{proof}

\scopebox{Exact ending}{%
\textbf{Unresolved at E1, within D2/J2.}

\smallskip
The first missing theorem is a controlled common microlocal
composition for the grouped six-branch expression.
Neither full cancellation nor failure of full cancellation has
been established. No characteristic prescription is constructed,
and no universal obstruction is asserted.}

\subsection*{What has advanced}

The full retained sum has a shorter exact precomposition form:
$B_4+B_7$ retain only first scalar-test derivatives after the
proved cancellation, $B_5+B_6+B_8$ appear in one complete
coordinate-response divergence, and $B_9$ has an explicit
complete root formula.

The geometric obstruction to the simplest symbol shortcut is
also precise. The sum of the selected line phases is not an
ordinary clean common phase. Two missing linear constraints
appear quadratically, and the observable attachment changes
the ambient codimension.

These results identify work a common characteristic calculus
must perform. They are not that calculus.

\subsection*{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged pair and six-branch germ;
\item the exact regrouping \eqref{eq:fullregroup};
\item the complete root formula \eqref{eq:B9root};
\item the intermediate tested-kernel equality and balanced
attachment bound;
\item the two characteristic lifts and their codimensions;
\item the non-clean phase theorem and quadratic defect identity;
\item the unchanged degree, ambiguity, and wavefront requirements;
\item the exact E1 stop.
\end{enumerate}

To discharge the retained D2/J2 problem, it must construct a
common representation with a full symbol and controlled
remainder, proving cancellation of the complete grouped
coefficient, or supply a characteristic prescription with
its actual degree, normalization, ambiguity, and wavefront
theorem.

It receives neither a nonzero full symbol section nor permission
to infer one from $Q\lambda$. It receives no $\Rquant$.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \Kret=K_4+K_5+K_6+K_7+K_8+K_9,\\[2pt]
 \Mret
 =\int C_G^\mu{}_\nu X_h^\nu\partial_\mu\psi
  +\int f\,\ER\partial_\alpha(\sig B^{\alpha\mu})
                                      \partial_\mu\psi
  +\int f\,p[r_h;\psi],\\[2pt]
 \codim_W\Nbar=4,\qquad \codim_W\Nlift=8,\\[2pt]
 \rank\dd(\partial_\theta\Phi)=6,\qquad
 \codim C_\Phi=8,\\[2pt]
 \text{common joint symbol and remainder: Unresolved at E1},\\
 \text{characteristic prescription and }\Rquant:
                         \text{not constructed}.
 \end{gathered}}
\]

An exact smooth-slot cancellation is not the full characteristic
cancellation theorem. A branch numerator is not a joint symbol.
A collision contact is not a characteristic extension.
An undefined quantum coefficient is not an anomaly.

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

\paragraph{Analytic scope.}
The new proofs concern the entire retained smooth-slot
regrouping, its root representation, the characteristic
ambient-space distinction, and the elementary phase audit.
No common characteristic principal symbol is asserted.

\paragraph{Dependency scope.}
XIV-D supplies the six-branch germ and derivative ledger.
XIV-C supplies the rank-two characteristic geometry.
XII-C fixes the actual source representative.
XII-E supplies the classical completion and response formulas.
XIII-E is a recorded regulator attempt, not a completed
characteristic prescription.

\paragraph{No branch deletion.}
The exact cancellation of explicit second scalar-test derivatives
does not delete $B_4$ or $B_7$. Both gravitational-response legs
remain present. Terms with fewer explicit derivatives are
retained and are not declared microlocally negligible.

\paragraph{Ambiguity scope.}
The class $\Cpart$ is unchanged. No characteristic-supported
difference is reclassified as a collision contact, and no
normalization is selected after the diagnostic.

\paragraph{Quantum scope.}
No renormalized quantum coefficient, Ward defect, antifield
construction, interacting differential, state, loop value,
or sharp-width limit is supplied.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, independently executed symbolic computation,
independent referee review, or journal acceptance is asserted.
The mathematical calculations claimed in this release are
the displayed analytic proofs.

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

\bibitem{XIVD}
N.~Cheung,
\emph{Landmark XIV-D: Cancellation or prescription on the retained
noncoincident characteristic stratum}.
Supplied manuscript \texttt{XIV-D.tex}.
Complete derivative ledger, six-branch retained germ,
fiber diagnostic, and unresolved D2 composition problem.

\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 characteristic geometry, conormal relation,
collision-contact rigidity, and the J2 stop.

\bibitem{XIIIE}
N.~Cheung,
\emph{Landmark XIII-E: One joint off-collision prescription for
the selected binary pair, or its obstruction in the frozen
partial-diagonal class}.
Supplied manuscript \texttt{XIII-E.tex}.
Common-frequency attempt, admissible-set convergence,
localization independence, and unresolved characteristic limit.

\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, slot sign, response roots,
and admitted classical 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 unintegrated cubic density representative
and actual weighted Euler 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.
Section~6: uniform microlocal scaling 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 estimates,
and supported extension ambiguities.

\bibitem{Wunsch}
J.~Wunsch,
\emph{Microlocal analysis and evolution equations:
Lecture notes from the 2008 CMI/ETH Summer School},
arXiv:0812.3181.
Section~8: conormal and Lagrangian distributions,
nondegenerate phases, and principal-symbol calculus.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I:
Distribution Theory and Fourier Analysis},
second edition, Springer, 1990.
Section~8.2: distribution products, pullbacks,
and pushforward wavefront estimates.

\end{thebibliography}

\end{document}