% !TeX program = pdflatex
%=====================================================================
% XIV-G.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-G.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-G.tex
%
% Exact release:
%   - the pair, six-branch germ, source, propagators, switches,
%     regulator width, and partial-diagonal class remain unchanged;
%   - XIV-F's excess correction and F2 stop are retained;
%   - one route is selected: operator/pullback composition from the
%     regrouped expression, without a common product phase;
%   - the characteristic locus and the diagnostic covector relation
%     are named in the common sixteen-dimensional space;
%   - the diagnostic relation is distinguished from a symbol relation;
%   - the differentiated source at a smooth residual test is a
%     differential operator of order at most two;
%   - a compact balanced-kernel example proves that smooth residual
%     testing can hide attachment-diagonal wavefront directions;
%   - two individually defined conormal factors of the retained
%     numerical incidence are examined on their own relations;
%   - their ordinary product/pullback test fails at the retained
%     characteristic covectors, even after legitimate factor grouping;
%   - this excludes an automatic factorwise-composition argument,
%     not every joint-symbol or characteristic construction;
%   - the complete internal symbol relations and the common symbol
%     bundle of the regrouped six-branch coefficient are not supplied.
%
% Ending:
%   Unresolved at G2, within the inherited F2/E1/D2/J2 stop.
%   G3--G5 are not entered as constructions.
%   J3--J5 and the quantum Ward-defect question remain unopened.
%
% Principal debugging corrections:
%   - omitting a common phase does not omit the pullback hypotheses;
%   - a positive conormal cancellation is diagnostic data, not a
%     joint principal-symbol section;
%   - the zero covector is not a point of a punctured symbol relation;
%   - the fourteen-dimensional image from XIV-F is not reused as
%     a Lagrangian relation;
%   - testing the residual metric slot preserves kernel identities
%     but can erase precisely the wavefront directions needed by a
%     later joint composition;
%   - an algebraic differential symbol of the local source is not
%     the composed symbol of either gravitational-response leg;
%   - individual factor symbols do not define multiplication at a
%     characteristic covector cancellation;
%   - the factorwise diagnostic does not decide cancellation of
%     the complete grouped six-branch coefficient.
%
% 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}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=27mm]{geometry}
\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\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{\EAm}{E_m^{\mathrm A}}
\newcommand{\Pmass}{P_m}
\newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}}
\newcommand{\GR}{G_{\mathrm{act}}^{\mathrm R}}
\newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}}
\newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}}
\newcommand{\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{\Fmatch}{\mathscr F_{\mathrm{match}}}
\newcommand{\Defect}{\mathscr D_{\mathrm{tan}}}
\newcommand{\Ssix}{S_6}
\newcommand{\Ucan}{U_{\mathrm{can}}}
\newcommand{\Qsrc}{\mathsf Q}
\newcommand{\Inc}{\mathscr I_{\mathrm{diag}}}
\newcommand{\Lchar}{\Lambda_{\mathrm{char}}}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\sd}{sd}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Contr}{Contr}
\DeclareMathOperator{\EL}{EL}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Int}{int}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Diff}{Diff}

\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-G}\\[0.4em]
Operator and pullback composition of the regrouped retained germ\\[0.4em]
\large Individual symbols, residual testing,\\
and the exact G2 relation stop}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We continue XIV-F's unresolved characteristic problem for the
unchanged binary pair. The selected route works from XIV-E's
regrouped operator expression and uses individual conormal
relations, without constructing a common product phase or
reusing the excluded clean-excess-two matching.

The characteristic test is placed in the common
sixteen-dimensional space, on the intersection with the
observable attachment diagonal. The copied positive conormal
relation specifies a diagnostic cancellation, not a joint symbol
relation. In particular, its zero covector cannot serve as the
base point of an ordinary punctured principal-symbol bundle.

Two operator-level distinctions are proved. First, after a
smooth residual metric test, the differentiated gravitational
source is a differential operator of order at most two. Its
local symbol is well typed, but does not supply the symbols
of the composed response legs. Second, a compact balanced
kernel can have nonzero attachment-diagonal wavefront
directions even though every smooth residual test produces
a smooth attachment kernel.

A separate factorwise audit uses two individually defined
conormal factors of the retained numerical incidence. Their
own relations and orders are named before their symbols are
used. The ordinary two-factor product test fails at the copied
characteristic covectors, even after those factors have been
legitimately grouped. Thus individual symbols do not bypass
the missing pullback hypothesis.

These results do not construct the complete internal symbol
relations, a common symbol bundle, a remainder theorem, or
a characteristic prescription for the grouped six-branch
coefficient. The ending is Unresolved at G2, within the
inherited F2/E1/D2/J2 stop. 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) {Regrouped expression};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Operator and pullback calculus};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A common product phase};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {There is no common product phase in this route.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Pullback calculus has no common phase. 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-G}
\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) {Regrouped expression};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A common product phase};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {There is no common product phase in this route.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Pullback calculus has no common phase. 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-G}
\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) {Regrouped expression};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Operator and pullback calculus};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A common product phase};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {There is no common product phase in this route.};
\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-G}
\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) {Regrouped expression};
\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) {A common product phase};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Operator and pullback calculus. There is no common product phase in this route.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Regrouped expression is not A common product phase. 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-G}
\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: Regrouped expression};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Operator and pullback calculus};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A common product phase};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {There is no common product phase in this route.};
\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-G}
\end{figure}

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


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



\scopebox{Release and first missing gate}{%
\textbf{Selected route:} operator and pullback calculus from the
complete regrouped expression, with no common product phase.

\smallskip
\textbf{Copied boundaries:} the elementary phase is non-clean;
the direct clean-excess-two matching is excluded; the tangent
defect is not an excess fiber; the rank-fourteen image is not
a Lagrangian relation in $T^*W$.

\smallskip
\textbf{Proved here:} the local differential-operator typing of
the tested source; loss of attachment information under smooth
residual testing; and failure of an automatic factorwise
pullback argument at the retained characteristic covectors.

\smallskip
\textbf{First missing gate: G2.}
The complete internal symbol relations and the common relation
and bundle for the grouped coefficient have not been constructed.

\smallskip
\textbf{Ending: Unresolved.}
No joint symbol, complete cancellation, characteristic
prescription, $\Rquant$, or Ward defect is released.}

\tableofcontents

%=====================================================================
\section{G0: copied inputs and the unchanged stop}
\label{sec:G0}
%=====================================================================

Only XIV-F, XIV-E, XIV-D, XIV-C, XIII-C, XIII-B, XII-E, and
XII-C are analytic manuscript inputs. The new proofs below are
conditional on their stated handoffs. They do not verify every
predecessor premise anew.

Quoted passages are copied from the supplied manuscripts.
Displayed formulas are recorded with local labels. No incidence
ledger, elementary phase theorem, clean-matching theorem, or
branch fiber identity is reconstructed.

\subsection{XIV-F: immediate successor package}

\begin{inputstatement}[XIV-F's successor package, copied]
\label{input:Fpackage}
``A successor receives:
\begin{enumerate}
\item the unchanged XIV-E package and grouped target;
\item the selected route and the precise correction to its
excess-two wording;
\item the regular manifold $\Ssix$ and exact formula
$q_R=\alpha\delta|\mathbf n_1-\mathbf n_2|^2$ on it;
\item the non-clean intrinsic conormal matching theorem;
\item the rank-two tangent-defect bundle and its quadratic form;
\item the rank-fourteen projection and one-dimensional
frequency fibers;
\item the persistence of the tangent discrepancy under
smooth frequency reparameterization;
\item the exact F2 stop, with no symbol or prescription released.
\end{enumerate}''

``The successor receives no failure-of-cancellation theorem and no
permission to infer one from $Q\lambda$. It receives no $\Rquant$''
\cite{XIVF}.
\end{inputstatement}

\begin{inputstatement}[XIV-F's geometric release]
\label{input:Fgeometry}
In the common space $W$, the regular manifold is
\[
 \Ssix=\{q_1=q_2=0,\ a=z\},\qquad \dim\Ssix=10.
\]
Its parameterization is
\[
 z=a,\qquad
 y=a+\alpha(1,\mathbf n_1),\qquad
 b=y+\delta(1,\mathbf n_2),
 \qquad \alpha,\delta>0.
\]
The exact formula is
\begin{equation}
 q_R|_{\Ssix}
 =\alpha\delta|\mathbf n_1-\mathbf n_2|^2,
 \qquad \beta=\alpha+\delta.
 \label{eq:Fquadratic}
\end{equation}

The selected conormal matching has
\[
 \dim\Fmatch=15,\qquad
 \dim(\text{linearized matching})=17.
\]
It is smooth and not clean. Its tangent-defect quotient has rank
two and is identified by
\begin{equation}
 \mathfrak d_w(u)
 =\alpha(\delta b-\delta a)_\perp
                -\beta(\delta y-\delta a)_\perp.
 \label{eq:Fdefect}
\end{equation}
The copied quadratic form is
\begin{equation}
 \mathfrak q_w(d_\perp)
 =-\frac{|d_\perp|^2}
         {\alpha\beta(\beta-\alpha)}.
 \label{eq:Fform}
\end{equation}

The covector-sum projection satisfies
\[
 \rank\dd J=14,\qquad
 \dim(\text{local fiber of }J)=1.
\]
Its image is isotropic of dimension fourteen, not Lagrangian
in $T^*W$.

``The two components of $\alpha v-\beta w$ measure a failure
of clean tangency. They are not, merely by that calculation,
an excess-two integration fiber.''

The two-dimensional tangent discrepancy persists under the
specified smooth frequency reparameterizations and local
vertical product reductions.

The ending is ``Unresolved at F2, within the inherited
E1/D2/J2 stop'' \cite{XIVF}.
\end{inputstatement}

These statements are boundaries for the present route. The
matching $\Fmatch$ and the map $J$ are not reused as a
composition relation.

\subsection{XIV-E: regrouping, root, and tested kernels}

\begin{inputstatement}[The exact regrouping]
\label{input:Eregroup}
With
\[
 X_G=X[G],\qquad X_h=X[h],\qquad
 r_h=\rD'[G]h=\PP\ER\bigl(\JD'[G]h\bigr),
\]
the complete retained expression is
\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:regroup}
\end{equation}
where
\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
\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}

``This identity retains all six branches. It involves no
characteristic composition and no regularization.''

For $F=\EAm f$, the complete scalar-response term 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''
\cite{XIVE}.
\end{inputstatement}

\begin{inputstatement}[Tested-kernel equality and bound]
\label{input:Etested}
For $G\in C_c^\infty(Z;\Sym^2T^*M)$,
\[
 \Mret[G;h,\psi]=\pair{L_G(a,b)}{h(a)\psi(b)}.
\]
The kernel $L_G$ is the tested sum of $K_4,\ldots,K_9$,
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.''

``Nor does testing each fixed $G$ supply the uniform estimates
needed to reconstruct a joint characteristic distribution in
all four common variables'' \cite{XIVE}.
\end{inputstatement}

\begin{inputstatement}[Lifts and excluded elementary phase]
\label{input:Elifts}
In
\[
 W=A\times B\times Z\times Y,\qquad
 P(a,b,z,y)=(b,a,y),
\]
define
\[
 \Nbar=P^{-1}(\Nchar),\qquad
 \Dobs=\{a=z\},\qquad
 \Nlift=\Nbar\cap\Dobs.
\]
The copied codimensions are
\[
 \codim_W\Nbar=4,\qquad
 \codim_W\Dobs=4,\qquad
 \codim_W\Nlift=8.
\]

The elementary phase of XIV-E is neither nondegenerate nor
clean. Its auxiliary equations have rank six and its critical
set has codimension eight.

On the copied 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:Equadratic}
\end{equation}

``Equation~\eqref{eq:Equadratic} is a geometric identity,
not a scaling degree of a distribution.''

The ending is ``Unresolved at E1, within D2/J2''
\cite{XIVE}.
\end{inputstatement}

\subsection{XIV-D and XIV-C: the retained germ and conormal relation}

\begin{inputstatement}[Pair, species, and branch limitation]
\label{input:D}
The frozen pair has
\[
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi),\qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD.
\]
On the retained common-variable patch,
\[
 K_\Sigma=K_4+K_5+K_6+K_7+K_8+K_9.
\]
The kernels $K_6,K_8,K_9$ are supported on $a=z$.

``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.''

The inherited mixed species factor is $i$. The
scalar--scalar factor is $i/2$, and that channel vanishes
only on the specified retained output-location patch.

For $p^2=0$ and $r=p\otimes k+k\otimes p$,
\[
 p_\alpha p_\mu\dot a_2^{\alpha\mu}[G;r]
 =G(p^\sharp,p^\sharp)(p\cdot k)=Q\lambda.
\]

``The proposition computes only one branch's leading differential
numerator. It supplies no sum identity or sum nonvanishing theorem
for that complete expression.''

The first unresolved step is D2, within J2 \cite{XIVD}.
\end{inputstatement}

\begin{inputstatement}[Characteristic configuration and rigidity]
\label{input:C}
The fixed retained points are
\[
 u_0=(t_*,0,0,0),\quad
 y_0=(t_*+L,L,0,0),\quad
 x_0=(t_*+2L,2L,0,0),
\]
where
\[
 \eps<t_*<t_*+2L<T,\qquad f(x_0)\ne0,
\]
and the fixed switch is one near these points.

In the order $(b,a,y)=(x,u,y)$,
\[
 \Nchar=
 \left\{
 y=a+\alpha\ell_{\mathbf n},\
 b=a+\beta\ell_{\mathbf n},\
 \ell_{\mathbf n}=(1,\mathbf n),\
 \mathbf n\in S^2,\ 0<\alpha<\beta
 \right\}.
\]
It is smooth of codimension four. With
\[
 q_R=(b-a)^2,\qquad
 q_1=(y-a)^2,\qquad
 q_2=(b-y)^2,
\]
the cone differentials have rank two and satisfy
\begin{equation}
 -\frac{\dd q_R}{2\beta}
 +\frac{\dd q_1}{2\alpha}
 +\frac{\dd q_2}{2(\beta-\alpha)}=0.
 \label{eq:positive}
\end{equation}
The oriented line conormals have the copied positive relation.

``Every defined contact change in $\Cpart$ for this graph
restricts to zero on $V$.''

The first unresolved gate is J2 \cite{XIVC}.
\end{inputstatement}

The neighborhoods $A,B,Y$ have pairwise disjoint closures,
$\overline Z\subset A$, and $B\times A\times Y\subset V$.
They specify restrictions of kernels, not removal of
the retained characteristic configuration.

\subsection{Earlier analytic handoffs}

\begin{inputstatement}[Inventory and four successor obligations]
\label{input:XIIIC}
XIII-C supplies the eleven branches, 48 labelled matchings,
the frozen class $\Cpart$, and the primitive parallel-line
extensions. In particular, the temporal kinetic primitive
has degree eight in collision codimension four.

``The next extension attempt must supply:
\begin{enumerate}
\item a joint prescription across the noncoincident characteristic
configurations, or a proved cancellation of those configurations
in the complete selected pair;
\item uniform transverse scaling and wavefront estimates for the
three-vertex and four-vertex collision strata and their intersections;
\item compatible extensions with the permitted partial contacts,
rather than only isolated primitive extensions;
\item the full output wavefront estimate after proper pushforward.
\end{enumerate}''
The complete coefficient remains Unresolved \cite{XIIIC}.
\end{inputstatement}

\begin{inputstatement}[Tree and retarded convention]
\label{input:XIIIB}
The admitted tree datum is
\[
 \Rtree=-\pair{\Ocal'}{\GR\Vcal'}
 =\pair{\Ocal'_\gamma}{\rg[\phi]}
                  +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]}.
\]
The convention is
\[
 R_1(\Vcal;\Ocal)
 =\frac{i}{\hbar}
 \left(\mathsf T_2^{\mathrm{ren}}(\Vcal,\Ocal)
                                      -\Vcal\star_H\Ocal\right).
\]
Its strictly ordered full-support values are zero or the
scaled commutator $(i/\hbar)[\Ocal,\Vcal]_{\star_H}$.

``Properness cannot repair an undefined distribution product,
and an extension theorem cannot be substituted for properness''
\cite{XIIIB}.
\end{inputstatement}

\begin{inputstatement}[Completion, slots, and roots]
\label{input:XIIE}
The inherited formulas are
\[
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD,\qquad
 \rD=-u_{\mathrm D}=\PP\ER\JD,
\]
\[
 \rg[\phi]=-\frac12\PP\ER(gT_0[\phi]),
\]
\[
 X^\mu[r]=-\ER\partial_\alpha(\sig a_1^{\alpha\mu}[r]),
 \qquad
 p[r;\psi]=-\ERm(V_{1,g}[r]\psi),
\]
where
\[
 a_1^{\mu\nu}[r]
 =\frac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},
 \qquad b_1[r]=\frac12\tr_\eta r,
\]
\[
 V_{1,g}[r]\psi
 =\partial_\mu(g a_1^{\mu\nu}[r]\partial_\nu\psi)
                         +m^2g b_1[r]\psi.
\]
The roots satisfy
\[
 \Aone[f,g](r;\phi)=\int\alpha_{f,g}[\phi]\cdot r\,\dd^4x,
 \qquad
 \beta_{f,g}[\phi]=-\EA(\PP\alpha_{f,g}[\phi]).
\]
The completion and filled-slot kernels are admitted tree kernels
\cite{XIIE}.
\end{inputstatement}

\begin{inputstatement}[Density representative and actual source]
\label{input:XIIC}
For $q(\rho)=\eta+\rho\gamma$,
\[
 \ell_3[\gamma]
 =-2[\rho^3]\bigl(\sqrt{-\det q(\rho)}\,R[q(\rho)]\bigr)
\]
is the fixed unintegrated cubic representative, with
\[
 R_{\mu\nu}
 =\partial_\rho\Gamma^\rho_{\mu\nu}
 -\partial_\nu\Gamma^\rho_{\mu\rho}
 +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
 -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}.
\]
Its actual weighted source is
\begin{equation}
 \begin{split}
 \JD^{\mu\nu}
 &=\EL_\gamma(\sig\ell_3)^{\mu\nu}
   =\sig J_3^{\mu\nu}+\mathcal D_\sig^{\mu\nu},\\
 \mathcal D_\sig^{\mu\nu}
 &=2C_2^{\mu\nu\alpha\beta}
                     \partial_\alpha\partial_\beta\sig
      -2M_2^{\mu\nu\rho}\partial_\rho\sig .
 \end{split}
 \label{eq:source}
\end{equation}
Here $C_2$ is the quadratic coefficient of
\[
 \sqrt{-q}
 \left(q^{\mu\nu}q^{\alpha\beta}
                   -q^{\mu(\alpha}q^{\beta)\nu}\right),
\]
and
\[
 M_2^{\mu\nu\rho}
 =C_0^{\mu\nu\alpha\beta}\Gamma_2^\rho{}_{\alpha\beta}
  +C_1^{\mu\nu\alpha\beta}\Gamma_1^\rho{}_{\alpha\beta}.
\]
The source is density weighted before variation, at cubic
order only \cite{XIIC}.
\end{inputstatement}

No coefficient in \eqref{eq:source} is set to zero.
The retained external points do not specify the locations
of all internal source vertices.

\subsection{Recorded only}

XIII-E's common-frequency family is an attempt; its
finite-regulator and admissible-domain conclusions are not
characteristic convergence here \cite{XIIIE}.

XIV-A and XIV-B are recorded classical boundaries, not
analytic inputs. XIII-A, XIII-D, and XII-F are evaluation,
class, and auxiliary-pairing boundaries, not graph data.
The value $-8I$ is not a normalization.

%=====================================================================
\section{Frozen target and release criteria}
\label{sec:contract}
%=====================================================================

Use
\[
 M=\R^4,\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \Pmass=\Box+m^2.
\]
All external functions, widths, and propagators remain fixed.
The carrier and line conventions are
\[
 \Ealg=
 (\Fpol[[\hbar,\kappa]],\star_H)/
                         \Nsol[[\hbar,\kappa]],
\]
\[
 \Gret=-\PP\ER,\qquad
 \HT=H+i\GA,\qquad
 \Ctot\subseteq\Cpart.
\]
Metric--scalar propagator blocks vanish.

The interaction is
\[
 \Vcal=-\frac12\int g\,\gamma_{\mu\nu}
                                   T_0^{\mu\nu}[\phi]\,\dd^4x.
\]
The observable contains its scalar-sourced slot once:
\[
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi).
\]

The grouped retained target is
\begin{equation}
 \begin{split}
 \mathcal T_{\mathrm{form}}
 =i\,\Contr^{\gamma\phi}\Bigl(
 &\Vcal''_{\gamma\phi},\Kret;\
 \HT_{\gamma\gamma}(y,a)\otimes\HT_{\phi\phi}(y,b)\\
 &\hspace{18mm}
 -H_{\gamma\gamma}(y,a)\otimes H_{\phi\phi}(y,b)
 \Bigr),
 \qquad \Kret=\sum_{j=4}^{9}K_j .
 \end{split}
 \label{eq:target}
\end{equation}
The cross-line difference stays grouped. No intermediate
mixed $H$--$\GA$ product is defined.

The endings, applied only to \eqref{eq:target}, are:
\begin{enumerate}
\item \textbf{Constructed}: a common microlocal representation,
a joint principal-symbol section, and a complete remainder
estimate establish the required cancellation; or one
characteristic prescription has all named domain, degree,
normalization, ambiguity, and wavefront data proved.
\item \textbf{Proved obstructed}: an explicit incompatibility
excludes every map in a previously declared class.
\item \textbf{Unresolved}: the common symbol, remainder,
or prescription theorem is missing.
\end{enumerate}

A vanishing leading section is sufficient only when the remainder
control actually supplies the required distribution. A formal
order improvement does not define an otherwise undefined product.

The elementary phase failure, the excess-route failure, and
$Q\lambda$ are not universal obstruction theorems for this target.

\subsection{Six data still required}

The selected route must identify:
\begin{enumerate}
\item a canonical or characteristic relation in a named ambient space;
\item amplitude orders and a common symbol bundle;
\item internal-vertex maps for $K_5,K_8,K_9$;
\item the attachment diagonal and its derivatives;
\item the complete tensor coefficients and grouped cross-line difference;
\item a remainder estimate controlling every unrepresented term.
\end{enumerate}

Neither inherited wavefront inclusions nor the isotropic
image of $J$ supplies this list.

%=====================================================================
\section{G1: operator and pullback calculus without a common phase}
\label{sec:G1}
%=====================================================================

\[
 \boxed{\text{Selected route: joint-symbol calculus
 from the regrouped operator expression.}}
\]

The construction starts from \eqref{eq:regroup}. Already defined
conormal or Lagrangian factors may be described on their own
relations. Their subsequent operator compositions and pullbacks
must satisfy their own hypotheses.

There is no common sum of line phases. No integration over
$\Defect$ is introduced. Melrose--Uhlmann composition is not
reapplied to $\Fmatch$.

\begin{definition}[The selected route's contract]
\label{def:route}
Before assigning a joint symbol to \eqref{eq:target}, the route
requires:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a named common ambient space and characteristic test locus;
\item valid individual relations, orders, and bundles for each
factor used in the proposed composition;
\item proved internal composition maps for $K_5,K_8,K_9$;
\item a justified pullback or other operator composition for the
complete grouped coefficient;
\item a common symbol identification and a uniform remainder theorem.
\end{enumerate}
The ordinary pullback theorem is used only when its normal-set
condition has been checked. If that condition fails, it supplies
no automatic reference distribution.
\end{definition}

The standard pullback test asks that the wavefront set of the
distribution being pulled back avoid the normal set of the map.
For the diagonal embedding, this becomes the familiar exclusion
of opposite product covectors \cite[Theorems 8.2.4 and 8.2.10]{Hormander}.

Removing a phase construction does not remove this condition.

\subsection{Unentered alternatives}

The following remain unentered:
\begin{enumerate}
\item a resolved or blown-up paired representation, with a new
ambient representation and its own clean-matching theorem;
\item a continuation of XIII-E's common-frequency family, with
a new convergence theorem on tests meeting the stratum.
\end{enumerate}
Neither is substituted after the stop below.

\subsection{Negative freeze}

There is no cutoff of $\Nchar$, restoring limit, characteristic
subtraction, new contact class, or change of propagator.

The tangent defect is not renamed as an excess fiber.
The fourteen-dimensional image of $J$ is not used as a
Lagrangian relation in $T^*W$. No branch numerator is promoted
to a joint section.

%=====================================================================
\section{G2: the chosen locus and the relation audit}
\label{sec:G2}
%=====================================================================

\subsection{One ambient space and one characteristic test locus}

All common-kernel statements take place in
\[
 W=A\times B\times Z\times Y,\qquad \dim W=16,
\]
in the order $(a,b,z,y)$.

For the characteristic test in this paper, choose
\begin{equation}
 N=\Nlift=\Nbar\cap\Dobs.
 \label{eq:chosenN}
\end{equation}
It has codimension eight in $W$. The residual metric variable
is not projected out.

The numerical stratum $\Nchar$ has codimension four in the
three-vertex space. Its full lift $\Nbar$ also has codimension
four, with $z$ free. Neither is substituted for \eqref{eq:chosenN}
in a degree statement. No characteristic degree is assigned
to any of them.

Let
\[
 d=a-z.
\]
At $w\in N$, define the already copied oriented conormals
\[
 n_R=-\frac{\dd q_R}{2\beta},\qquad
 n_1=\frac{\dd q_1}{2\alpha},\qquad
 n_2=\frac{\dd q_2}{2(\beta-\alpha)}.
\]
They satisfy
\[
 n_R+n_1+n_2=0.
\]

\subsection{Diagnostic covectors are not a symbol relation}

Define the diagnostic selection set
\begin{equation}
 \Inc=
 \left\{
 \bigl(w;c\,n_R,c\,n_1,c\,n_2,0\bigr):
          w\in N,\ c>0
 \right\}.
 \label{eq:diagnostic}
\end{equation}
Its entries are four separate cotangent selections: the three
line covectors and a zero attachment covector. It is not a
canonical relation for the complete coefficient.

The natural candidate for a characteristic conormal component,
if such a component were constructed, would be
\[
 \Lchar=N^*N\setminus0.
\]
This merely names a possible target bundle. It does not assert
that the grouped coefficient belongs to a conormal class on it.

\begin{proposition}[Zero-sum reduction does not assign a joint section]
\label{prop:zerosum}
Cotangent addition sends every tuple in \eqref{eq:diagnostic}
to the zero covector of $T^*_wW$.

Accordingly, reducing the diagnostic tuple by its copied
linear dependence does not produce a point of $\Lchar$.
The diagnostic alone cannot define an ordinary principal-symbol
section of the grouped coefficient.
\end{proposition}

\begin{proof}
The sum is
\[
 c(n_R+n_1+n_2)+0=0.
\]
By definition, the zero covector is removed from $\Lchar$.
An ordinary conormal or Lagrangian principal symbol is a
section over its punctured cotangent relation, not a value
assigned at this zero covector.

The calculation specifies a cancellation of input covectors.
It specifies neither membership of an output distribution in
a conormal class nor an amplitude on an output relation.
\end{proof}

This does not exclude a generalized characteristic representation.
It identifies the additional theorem such a representation
would need. The isotropic image from XIV-F cannot fill the
missing symbol relation.

\subsection{The local source has a differential symbol}

The following first operator calculation concerns the full
source, not the $B_6$ numerator.

For a fixed smooth residual metric test
\[
 G\in C_c^\infty(Z;\Sym^2T^*M),
\]
define
\[
 \Qsrc_G h=\JD'[G]h.
\]
This is a metric-density-valued expression.

\begin{proposition}[Typing of the complete tested source]
\label{prop:sourceoperator}
The map $\Qsrc_G$ is a matrix-valued differential operator
of order at most two:
\begin{equation}
 \Qsrc_Gh(s)
 =\sum_{|\nu|\le2}A_\nu[G](s)\,\partial_s^\nu h(s).
 \label{eq:QG}
\end{equation}
Its coefficients are smooth and supported in $\supp G$.

In the Fourier convention where
$\partial^\nu$ has symbol $(i\xi)^\nu$, its degree-two
differential symbol is
\begin{equation}
 q_{G,2}(s,\xi)
 =\sum_{|\nu|=2}A_\nu[G](s)(i\xi)^\nu.
 \label{eq:Qsymbol}
\end{equation}
This is a fiber-valued polynomial on $T^*M$. It is allowed
to vanish and is not asserted to be elliptic.

Its Schwartz kernel is supported on the source diagonal
$s=a$ and is a finite sum
\begin{equation}
 Q_G(s,a)
 =\sum_{|\nu|\le2}A_\nu[G](s)
                         \partial_s^\nu\delta(s-a).
 \label{eq:Qkernel}
\end{equation}
These statements retain every coefficient of the actual
source, including its ramp derivatives.
\end{proposition}

\begin{proof}
The copied quadratic source is a finite sum of local monomials
\[
 c_{\alpha\beta}(s)\,
     \partial^\alpha\gamma(s)\,
     \partial^\beta\gamma(s),
 \qquad |\alpha|+|\beta|\le2,
\]
with constant fiber contractions and the fixed external
ramp coefficients. Its directional derivative at $G$ is the
sum of the two variations of every monomial.

Collecting the derivatives of $h$ gives \eqref{eq:QG}.
Each coefficient contains a derivative of $G$ of order at
most two, multiplied by fixed smooth external data.
It is therefore smooth and supported in $\supp G$.

The symbol \eqref{eq:Qsymbol} is the algebraic highest-degree
differential polynomial of that operator. Lower-degree
coefficients remain in \eqref{eq:QG}; none is omitted.

The kernel identity follows from the distributional formula
for differentiating $h(s)$ through the diagonal delta.
It has exactly the stated support. No propagator or
characteristic multiplication is used.
\end{proof}

\begin{remark}[Local source symbol versus response symbol]
Equation~\eqref{eq:Qsymbol} is a symbol of the local operator
$\Qsrc_G$, on the diagonal operator relation. It is not the
symbol of
\[
 r_h=\PP\ER\Qsrc_Gh,
\]
and still less the common symbol of the coordinate and scalar
response legs after their cross-line composition.

In the untested kernel, the residual metric variable $z$
remains present in the local source incidence. Testing it
against $G$ is not an internal-symbol theorem in all variables.
\end{remark}

\subsection{Smooth residual testing can hide the relevant wavefront}

The complete tested-kernel identity is retained as an input.
The next proposition proves why its regularity conclusions
cannot be strengthened without a uniform joint theorem.

\begin{proposition}[A compact balanced kernel with smooth tested kernels]
\label{prop:testing}
There exists a compact distribution
$T(a,b,z)$ on $A\times B\times Z$ such that
\[
 \WF(T)\subset
 Z_3=\{(a,\xi;b,\eta;z,\zeta):
                         \xi+\eta+\zeta=0\}\setminus0,
\]
and $T$ has nonzero wavefront directions on $a=z$, while
\[
 L_G(a,b)=\pair{T(a,b,z)}{G(z)}
\]
is smooth for every $G\in C_c^\infty(Z)$.
\end{proposition}

\begin{proof}
Choose smooth compact functions $\chi_A,\chi_B,\chi_Z$,
nonzero near the retained $a,b,z$ coordinates, with
$\chi_A\chi_Z$ not identically zero. Set
\begin{equation}
 T(a,b,z)=
 \chi_A(a)\chi_B(b)\chi_Z(z)\delta^{(4)}(a-z).
 \label{eq:testingexample}
\end{equation}
Its support is compact. The conormal directions of the
attachment diagonal are
\[
 (\xi,0,-\xi),\qquad \xi\ne0.
\]
They have zero covector sum. Wherever the smooth weight
is nonzero, the diagonal delta has these nonzero wavefront
directions; multiplication by that nonvanishing weight
does not remove them.

Pairing the residual variable gives
\[
 L_G(a,b)
 =\chi_A(a)\chi_B(b)\chi_Z(a)G(a),
\]
which is smooth for every smooth compact test $G$.
\end{proof}

This is a counterexample to a proposed inference from tested
regularity, not a replacement for $\Kret$. It satisfies the
same relevant compactness and balanced-covector condition.

Smooth tests determine a distributional kernel uniquely.
The proposition does not deny that fact. It shows that the
regularity of every tested kernel can conceal attachment
wavefront directions of the untested kernel. Those directions
are precisely among the data a later joint composition must
control.

Thus the tested equality of XIV-E remains valid, but it
cannot supply the missing common symbol bundle or uniform
characteristic composition estimates by itself.

\subsection{Internal maps remain required}

The source diagonal in \eqref{eq:Qkernel} is a defined local
operator map. The internal response constructions require
more.

For example, the source response and coordinate response
have the exact operator structure
\[
 r_h(s)
 =\PP\int E_0^{\mathrm R}(s-v)\,
                          \Qsrc_Gh(v)\,\dd^4v,
\]
\[
 -X^\mu[r_h](b)
 =\int E_0^{\mathrm R}(b-s)\,
       \partial_{s^\alpha}
       \bigl(\sig(s)a_1^{\alpha\mu}[r_h](s)\bigr)\,\dd^4s.
\]
The internal variables $s,v$ are governed by these kernels
and the prescribed local source incidences. They are not
set equal to $u_0,y_0,x_0$.

The two placements in the $B_5$ part of
\eqref{eq:Btensor}, and the coordinate and scalar response
maps in $B_8,B_9$, must be composed on their own relations.
Their existing tree-kernel pushforwards prove existence
of those classical kernels, not a uniform Lagrangian-symbol
description of their later characteristic composition.

The present paper does not supply:
\begin{enumerate}
\item the complete internal canonical relations with $z$ retained;
\item their microlocal orders and common amplitude identifications;
\item the necessary internal composition and remainder estimates.
\end{enumerate}
These are the first missing data of G2.

The factorwise audit below identifies an additional limitation
of an automatic pullback argument. It does not bypass this
internal stop or enter G3.

%=====================================================================
\section{A factorwise pullback audit within G2}
\label{sec:factor}
%=====================================================================

\subsection{Purpose and scope}

The selected route permits individual symbols only after their
relations, orders, and bundles have been named. The following
audit does that for two elementary factors of the retained
numerical incidence.

It does not assign a symbol to the complete regrouped
coefficient. It does not evaluate a singular graph at a
point and does not add separate branch numerators.

We retain the original temporal numerical factors only for
this pullback diagnostic. The full fiber coefficient and
every other retained branch remain outside its conclusion.

Write $F_\mu$ for the frozen scalar time-ordered numerical
line of mass $\mu$, with its fixed nonzero fiber constants
separated. Consider, on $W$,
\begin{equation}
 \begin{split}
 U(w)&=
 \bigl(\partial_{a^0}E_0^{\mathrm R}(b-a)\bigr)
                              \delta^{(4)}(a-z),\\
 V(w)&=
 F_0(y-a)\,
              \partial_{b^0}\partial_{y^0}F_m(y-b).
 \end{split}
 \label{eq:UV}
\end{equation}
Smooth inherited weights may be included. The statements
about ellipticity below concern the retained open region
where those weights are nonzero.

The two products defining $U$ and $V$ will be justified
individually. Their further multiplication is not assumed.

\subsection{Individual relations and order convention}

Define
\[
 S_{RD}=\{q_R=0,\ a=z\}\subset W,\qquad
 S_{12}=\{q_1=q_2=0\}\subset W.
\]
Near the retained point,
\[
 \codim S_{RD}=5,\qquad \codim S_{12}=2.
\]
The first assertion follows because $\dd q_R$ has no $z$
component and is nonzero. The second follows from independence
of $\dd q_1,\dd q_2$.

Use the conormal order convention in which a normal Fourier
amplitude of order $d$ on a codimension-$k$ submanifold of
an $n$-dimensional space gives conormal order
\begin{equation}
 \mu=d-\frac{n-2k}{4}.
 \label{eq:orderconvention}
\end{equation}
Principal symbols are taken in the usual conormal symbol
quotient, with half-density and Maslov factors and the
fixed numerical fiber factors. Local defining functions
and the fixed coordinate density trivialize those factors
for the calculations here \cite[Section 8.1]{Wunsch}.

The individual relations are:
\[
 \Lambda_U=N^*S_{RD}\setminus0,
\]
and the selected positive sector
\[
 \Lambda_V^+
 =\left\{
 (w,\theta_1\dd q_1+\theta_2\dd q_2):
    w\in S_{12},\ \theta_1,\theta_2>0
 \right\}.
\]
When a joint normal order for $V$ is used, it is understood
microlocally in a sector where both positive frequencies
are nonzero and comparable. No statement about the axes
of that normal sector is needed.

These are relations of the two factors in \eqref{eq:UV},
not the relation of \eqref{eq:target}.

\begin{lemma}[The two preliminary products exist]
\label{lem:UVexist}
Both $U$ and $V$ in \eqref{eq:UV} are defined distributions
on the retained neighborhood.

The relevant conormal directions of $U$ have the form
\[
 \theta_R\dd q_R+\kappa_\mu\dd(a^\mu-z^\mu).
\]
Those of $V$ in the selected sector have the form
\[
 \theta_1\dd q_1+\theta_2\dd q_2,\qquad
 \theta_1,\theta_2>0.
\]
\end{lemma}

\begin{proof}
For $U$, a forbidden product cancellation between the
retarded factor and the attachment delta would have
zero $z$ covector. The attachment covector would therefore
be zero. The retarded covector would then also be zero.
Thus the product criterion passes.

For $V$, the first line depends on $(y,a)$ and the
second on $(y,b)$. In a forbidden cancellation, the
$a$ component forces the first nonzero line selection
to vanish, and the $b$ component forces the second to
vanish. The criterion again passes.

Differentiation preserves the relevant wavefront inclusions.
The pullback and product rules therefore give the stated
directions \cite[Theorems 8.2.4 and 8.2.10]{Hormander}.
\end{proof}

These preliminary products use no matching with the third
line. They do not invoke the excluded fiber product.

\subsection{Individual leading symbols}

At a nonzero future null separation, the massless retarded
line has its nonzero delta-type normal singularity. The
frozen scalar boundary-value lines have nonzero simple-pole
normal parts on the selected conormal sides. The massive
line has the same principal normal pole; its less singular
terms do not change that individual principal symbol.
These are properties of the already frozen numerical lines,
not choices of replacement propagators.

For the present Fourier convention, denote their nonzero
individual leading normal amplitudes by
\[
 c_R,\qquad c_1,\qquad c_2.
\]
The selected time-ordered orientation is the copied one.
A simultaneous reversal of Fourier convention changes the
signs of all selected normal frequencies, not the argument.

\begin{proposition}[Individual conormal symbols at the diagnostic directions]
\label{prop:factorsymbol}
On $\Lambda_U$, near the selected retarded direction,
$U$ has normal amplitude order one and hence conormal
order
\[
 \mu_U=1-\frac{16-10}{4}=-\frac12.
\]
Its leading normal amplitude is a nonzero conventional
factor times
\begin{equation}
 i\theta_R\,\partial_{a^0}q_R\,c_R.
 \label{eq:Usymbol}
\end{equation}

On $\Lambda_V^+$, in the stated positive comparable-frequency
sector, $V$ has normal amplitude order two and hence
conormal order
\[
 \mu_V=2-\frac{16-4}{4}=-1.
\]
Its leading normal amplitude is a nonzero conventional
factor times
\begin{equation}
 (i\theta_2)^2
   \bigl(\partial_{b^0}q_2\bigr)
   \bigl(\partial_{y^0}q_2\bigr)c_1c_2.
 \label{eq:Vsymbol}
\end{equation}

Both individual symbols are nonzero at the copied
characteristic selections. They are not a joint symbol
of their product.
\end{proposition}

\begin{proof}
The normal amplitude of a delta distribution has order
zero. A derivative acting on its normal singularity
multiplies its leading amplitude by
$i\theta_R\partial_{a^0}q_R$. Multiplication by the
attachment delta adds four independent normal variables
with order-zero amplitude. This gives
\eqref{eq:Usymbol} and its order.

For $V$, the two normal defining functions are independent.
In their separate adapted normal coordinates the
simple-pole amplitudes have order zero on their selected
half-rays. The two temporal derivatives of the second
line give the leading degree-two factor in
\eqref{eq:Vsymbol}. Derivatives of amplitudes and the
massive lower normal terms have lower order in this
sector. The usual individual conormal symbol calculus
applies because these normal coordinates are regular
\cite[Section 8]{Wunsch}.

On $N$, with $\delta=\beta-\alpha>0$,
\[
 \partial_{a^0}q_R=-2\beta,\qquad
 \partial_{b^0}q_2=2\delta,\qquad
 \partial_{y^0}q_2=-2\delta.
\]
The selected normal frequencies are nonzero, as are
$c_R,c_1,c_2$. Thus neither leading amplitude vanishes
there.
\end{proof}

No integration over the two tangent-defect directions
occurs in this proof. Each factor is examined on its own
regular conormal relation.

\subsection{The next multiplication fails the ordinary test}

\begin{theorem}[Failure of the automatic two-factor pullback step]
\label{thm:pullbackfailure}
At every retained point $w\in N$ where the diagnostic weights
are nonzero, the two defined factors in \eqref{eq:UV} admit
opposite nonzero wavefront covectors:
\[
 (w,c\,n_R)\in\WF(U),\qquad
 (w,c(n_1+n_2))\in\WF(V),\qquad c>0,
\]
and
\[
 c\,n_R+c(n_1+n_2)=0.
\]
Therefore the ordinary two-factor product criterion does
not define $UV$ at these configurations.

The individually assigned relations and symbols do not
justify this final product step.
\end{theorem}

\begin{proof}
Choose the individual normal frequencies
\[
 \theta_R=-\frac{c}{2\beta},\qquad
 \theta_1=\frac{c}{2\alpha},\qquad
 \theta_2=\frac{c}{2(\beta-\alpha)},
\]
and choose zero attachment covector.

The factor symbols in
Proposition~\ref{prop:factorsymbol} are elliptic at these
conormal selections. Hence the selected directions are
actual individual wavefront directions, not merely
members of an upper bound.

The copied positive relation gives their zero sum.
Thus the wavefront sets of $U$ and $V$ contain opposite
nonzero covectors at the same point. The sufficient
product hypothesis fails
\cite[Theorem 8.2.10]{Hormander}.
\end{proof}

The conclusion is failure of an automatic product
construction, not nonexistence of every possible product
prescription.

\begin{corollary}[What this additional exclusion means]
\label{cor:shortcut}
The following shortcut is excluded for the retained
numerical incidence:
\begin{quote}
Assign symbols to the already defined retarded/attachment
factor and cross-line factor, and then invoke the ordinary
diagonal-pullback theorem without any further characteristic
analysis.
\end{quote}
This exclusion does not establish failure of cancellation
of the complete grouped six-branch coefficient.
\end{corollary}

\begin{proof}
The theorem supplying the proposed final pullback requires
the absent normal-set condition, as
Theorem~\ref{thm:pullbackfailure} shows.

The calculation concerns two elementary numerical factors
of a retained incidence. It neither constructs nor
evaluates the complete tensor and branch sum in
\eqref{eq:target}. It also does not analyze that sum's
lower-order terms. No full cancellation conclusion follows.
\end{proof}

The cross-line factor in the actual target remains grouped
as $\HT^{\otimes2}-H^{\otimes2}$. The preceding audit does
not expand it into mixed $H$--$\GA$ products or replace
it by $V$.

In particular, the audit is not a proof that the complete
grouped coefficient has the displayed wavefront direction.
A cancellation theorem for that coefficient would have
to address the complete sum before claiming such a
direction is present or absent.

%=====================================================================
\section{The exact G2 stop}
\label{sec:stop}
%=====================================================================

\begin{theorem}[Operator-route verdict]
\label{thm:G2}
For the route fixed in Definition~\ref{def:route}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the common ambient space and test locus are named:
$W$ and $N=\Nlift$;
\item the copied positive relation supplies diagnostic input
covectors, not a common symbol relation;
\item the complete tested local source has a valid differential
operator symbol;
\item smooth residual testing alone does not preserve the
attachment wavefront information needed by joint composition;
\item the ordinary factorwise pullback shortcut fails on the
specified retained numerical incidence;
\item the complete internal symbol relations and the common
relation and bundle for \eqref{eq:target} have not been
constructed.
\end{enumerate}
The first unresolved gate is G2, within the inherited
F2/E1/D2/J2 stop.
\end{theorem}

\begin{proof}
The choice \eqref{eq:chosenN} gives (i).
Proposition~\ref{prop:zerosum} gives (ii).
Proposition~\ref{prop:sourceoperator} gives (iii).
Proposition~\ref{prop:testing} gives (iv).
Theorem~\ref{thm:pullbackfailure} and
Corollary~\ref{cor:shortcut} give (v).

The source polynomial and the individual factor symbols
supply none of the missing internal composition and
uniform common-kernel symbol theorems listed in
Section~\ref{sec:G2}. No alternative representation or
characteristic prescription has been constructed.
This is exactly (vi), and it prevents entry into G3.
\end{proof}

\begin{remark}[Scope of the stop]
The selected no-common-phase route is not universally
impossible. What fails is the automatic passage from
individual symbols and inherited wavefront bounds to
the ordinary final pullback.

A representation using cancellations of the full sum,
or a separately proved generalized operator composition,
could in principle advance this route. Neither is
supplied here. No class of all such constructions has
been proved incompatible.
\end{remark}

%=====================================================================
\section{G3 and G4: no joint section and no remainder theorem}
\label{sec:G3G4}
%=====================================================================

G3 is not entered as a construction. The individual symbols
above do not populate a symbol bundle for
\eqref{eq:target}.

The complete regrouped pieces retain the following status:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.19\linewidth}
                              p{.38\linewidth}X@{}}
\toprule
Piece & Copied result & Missing joint data\\
\midrule
$B_4+B_7$ &
The complete first-derivative coefficient $C_G$
after the exact explicit $\partial^2\psi$ cancellation &
Order and symbol of the remaining term in a
common characteristic calculus.\\[0.3em]
$B_5+B_6+B_8$ &
One divergence of the full tensor
$B^{\alpha\mu}[G;h]$ &
Internal relations, composed divergence order,
and common amplitude.\\[0.3em]
$B_6$ inside that tensor &
The diagnostic $Q\lambda$ &
A section of the complete sum; the diagnostic
is not such a section.\\[0.3em]
$B_9$ &
The complete advanced root
\eqref{eq:B9root} &
Its scalar-line composition, including the
derivative and mass contributions.\\
\bottomrule
\end{tabularx}
\end{center}

A loss of an explicit test derivative is not an assigned
microlocal order improvement. A vanishing numerator in one
presentation would have to be recorded with its remaining
terms; it would not delete a branch.

G4 is not entered. Its remainder theorem would have to
control:
\begin{enumerate}
\item lower-order terms of every numerical line;
\item contributions from the internal vertices of $K_5,K_8,K_9$;
\item derivatives of the attachment diagonal;
\item the two relative transverse directions identified by
the copied defect;
\item every retained source and root term.
\end{enumerate}
The tangent defect is a geometric constraint, not an
integration fiber or a characteristic scaling degree.

Neither a zero joint section with adequate remainder
control nor a nonzero full section on a named open
relation is asserted.

\begin{proposition}[No full cancellation verdict]
\label{prop:nocancellation}
The results of this paper establish neither cancellation
nor failure of cancellation of the complete grouped
coefficient \eqref{eq:target}.
\end{proposition}

\begin{proof}
The source symbol belongs to a local differential operator.
The testing example establishes a limitation of a regularity
inference. The factorwise audit establishes failure of the
ordinary final pullback hypothesis for one retained numerical
incidence.

None constructs a common principal-symbol section of the
complete coefficient or its controlled remainder. The
copied $Q\lambda$ identity and the rank of $J$ do not
supply those missing data.
\end{proof}

%=====================================================================
\section{G5: prescription interface not entered}
\label{sec:G5}
%=====================================================================

No nonzero full joint section has been established.
G5 is therefore not entered as a construction.

A future prescription must name all seven data:
\begin{enumerate}
\item its ambient space;
\item a reference distribution on a proved domain;
\item the exact characteristic locus across which it extends;
\item agreement on that domain with the grouped frozen prescription;
\item a finite uniform transverse degree;
\item a closed conic wavefront bound retaining off-locus singularities;
\item a normalization selecting one map, or a uniqueness theorem.
\end{enumerate}

The present choice $N=\Nlift$ is a codimension-eight
common-kernel test stage. It is not silently converted
into a codimension-four numerical extension problem.

The canonical product is known on the inherited admissible
domain $\Ucan$. No equality
\[
 \Ucan=W\setminus\Nbar
\]
is asserted. A proved reference distribution on the required
complement remains part of the construction, not a consequence
of the existence of the individual lines.

The copied extension warning remains:
\[
 \WF(t)\subset N^*N
\]
on a whole neighborhood would require the reference to be
smooth away from $N$. It cannot erase the already existing
off-$N$ singularities. A prospective bound
\[
 \WF(t)\subset
 \overline{\WF(t^\circ)}\cup N^*N
\]
requires its own microlocal extension hypotheses.

For a genuine codimension-four problem with degree allowance
at least four, smooth tangential multiples of
$\delta^{(4)}(r)$ can remain possible unless normalization
or uniqueness excludes them. On the retained collision-free
neighborhood, such nonzero characteristic-supported changes
are not members of $\Cpart$.

The codimension-eight attachment intersection is a separate
extension stage. No degree of the unresolved target is
assigned here.

Brunetti--Fredenhagen's submanifold extension framework and
Dang's microlocal extension framework are recorded only
\cite[Section 6]{BF}\cite[Sections 5--6]{Dang}. Their required
reference and uniform estimates have not been proved for
this target, so no extension theorem from them is applied.

XIII-E's family is not imported as a prescription.
There is no cutoff-and-restore limit. The excluded excess
construction is not reused under a prescription label.

%=====================================================================
\section{G6: downstream gates and the Ward boundary}
\label{sec:G6}
%=====================================================================

J3 opens only for a distribution supplied by a successful
G4 or G5 construction. None is supplied.

The parallel-line degree eight remains a collision degree,
not a degree along $\Nchar$, $\Nbar$, or $\Nlift$.
The copied defect Hessian does not alter that distinction.
The individual conormal orders in
Proposition~\ref{prop:factorsymbol} are likewise not
transverse scaling degrees of the unresolved full product.

J4 remains closed: no characteristic distribution is glued
to singular spectators or compatible partial contacts.

J5 remains closed: inherited properness is support control
for an existing distribution, not a construction of
\eqref{eq:target} or its output wavefront theorem.

No $\Rquant$ is released. The tree coefficient is not
substituted for the missing quantum coefficient.

No $s_{\mathrm{ext}}$, $s_\Psi$, or $s_0$ is applied.
No binary Ward defect is formed. The value $-8I$ and
the classical evaluation identities do not normalize
the graph.

There is no new gravitational interaction vertex,
restriction quotient, state, loop value, characteristic
contact class, triangle residue, or sharp-width limit.

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.27\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
G0 & Frozen &
Copied predecessor packages, unchanged pair, source,
six-branch germ, and ambiguity contract.\\[0.35em]
G1 & One route named &
Operator/pullback calculus from the regrouped expression.
Resolved paired and common-frequency routes unentered.\\[0.35em]
G2 & Partial results; Unresolved &
The common locus is named. Local source typing,
testing limitation, and factorwise pullback failure
are proved. Complete internal relations and a
common symbol relation remain missing.\\[0.35em]
G3 & Not entered &
No joint principal-symbol section or common bundle.\\[0.35em]
G4 & Not entered &
No complete remainder theorem and no full
cancellation or nonvanishing verdict.\\[0.35em]
G5 & Not entered &
No characteristic prescription or actual transverse degree.\\[0.35em]
G6 & Recorded &
J3--J5 and the Ward-defect question remain unopened.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[XIV-G mathematical release]
\label{thm:release}
For the frozen retained instance:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The chosen common space and characteristic test locus are
$W$ and $N=\Nlift$, of dimensions sixteen and eight, respectively.
\item The positive conormal cancellation is diagnostic input data,
not a joint symbol section.
\item For a smooth residual test $G$, the full differentiated
source $\JD'[G]$ is a differential operator of order at most two
with the local symbol \eqref{eq:Qsymbol}.
\item Smooth residual testing can hide nonzero attachment-diagonal
wavefront directions even for compact balanced kernels.
\item Individually defined conormal factors of the retained
numerical incidence still fail the ordinary final product test
at the selected characteristic covectors.
\item No complete internal symbol relation, joint section,
controlled remainder, or characteristic prescription for
\eqref{eq:target} is constructed.
\end{enumerate}
The ending for the grouped target is
\textbf{Unresolved at G2}, within F2/E1/D2/J2.

No universal incompatibility, admitted $\Rquant$, or Ward
defect is asserted.
\end{theorem}

\begin{proof}
The copied lift geometry and \eqref{eq:chosenN} give (i).
Proposition~\ref{prop:zerosum} gives (ii).
Proposition~\ref{prop:sourceoperator} gives (iii).
Proposition~\ref{prop:testing} gives (iv).
Lemma~\ref{lem:UVexist},
Proposition~\ref{prop:factorsymbol}, and
Theorem~\ref{thm:pullbackfailure} give (v).

Theorem~\ref{thm:G2} identifies the first missing common
composition data. Proposition~\ref{prop:nocancellation}
separates the proved diagnostics from a cancellation
verdict. No prescription is supplied in
Section~\ref{sec:G5}, proving (vi) and the stated ending.
\end{proof}

\subsection*{What has advanced}

The selected no-common-phase route has an explicit hypothesis
audit. The local source admits an ordinary differential
symbol, but its composed response symbols have not been
identified.

The information lost under smooth residual testing is also
explicit: attachment-diagonal covectors may disappear from
every tested kernel. Pointwise-in-$G$ regularity does not
replace a joint microlocal theorem.

Finally, two valid preliminary factor products do not make
their final product valid. The retained characteristic
cancellation persists in the ordinary operator/pullback test
without invoking a common phase or an excess bundle.

These results exclude an automatic factorwise shortcut.
They do not exclude every possible no-common-phase
construction of the complete coefficient.

\subsection*{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged XIV-F and XIV-E packages;
\item the selected operator/pullback route and its unentered alternatives;
\item the chosen common ambient space and codimension-eight test locus;
\item the distinction between diagnostic zero-sum covectors and
an output symbol relation;
\item the complete tested-source differential typing and local symbol;
\item the compact balanced-kernel testing counterexample;
\item the individual-factor relations and the failed final
pullback test, explicitly limited to the numerical incidence;
\item the exact G2 stop and the still missing internal
composition and common-symbol data.
\end{enumerate}

To advance this route, the successor must construct the
internal relations and a justified common operator
representation of the complete grouped coefficient.
It must then supply the joint section and remainder, or
one fully specified characteristic prescription.

It receives no full cancellation theorem, no nonzero full
symbol section, and no permission to call the tangent
defect an excess. It receives no $\Rquant$.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \Ocal=\PhiD+\Aone(\rg[\phi];\phi),\qquad
 \Kret=K_4+K_5+K_6+K_7+K_8+K_9,\\[2pt]
 N=\Nlift\subset W,\qquad \codim_WN=8,\\[2pt]
 \Qsrc_G=\JD'[G]\in\Diff^{\le2},\\[2pt]
 \text{smooth tested kernels do not remove joint
 attachment-wavefront obligations},\\[2pt]
 c\,n_R+c(n_1+n_2)=0
 \quad\text{prevents the automatic factorwise product argument},\\[2pt]
 \text{complete common symbol and internal relations:
 Unresolved at G2},\\
 \text{characteristic prescription and }\Rquant:
 \text{not constructed}.
 \end{gathered}}
\]

An individual symbol is not a joint symbol. A residual test
is not a uniform composition theorem. Avoiding a common
phase does not avoid a pullback hypothesis. An undefined
quantum coefficient is not an anomaly.

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

\paragraph{Analytic scope.}
The new proofs concern local source typing, the relation
audit, loss of wavefront information under residual testing,
and the ordinary factorwise pullback test. No joint
characteristic distribution is asserted.

\paragraph{Dependency scope.}
Only XIV-F, XIV-E, XIV-D, XIV-C, XIII-C, XIII-B, XII-E, and
XII-C are analytic manuscript inputs. Their handoffs
retain their predecessor scope. No incidence ledger or
excluded phase or matching is re-proved.

\paragraph{Factor-symbol scope.}
The conormal orders and symbols assigned in the factorwise
audit belong only to its two individually defined factors.
They are not orders, degrees, or symbols of the complete
grouped coefficient. The audit does not expand the
grouped cross-line difference into undefined mixed products.

\paragraph{Source scope.}
The local symbol of $\JD'[G]$ retains the full actual source.
No layer coefficient is discarded. Internal vertices are
governed by their existing kernel maps, not identified with
the retained external points.

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

\paragraph{External calculus.}
Individual regular conormal and differential symbols are
used only after their local relations and orders have been
named. Preliminary products pass the stated product tests.
The failed final test is reported as failure of a sufficient
construction, not nonexistence. No submanifold extension
theorem is applied to the missing target.

\paragraph{Quantum scope.}
No renormalized quantum coefficient, Ward defect,
interacting differential, state, loop value, triangle
residue, 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 release consists of the displayed analytic
proofs and the explicit unresolved gate.

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

\bibitem{XIVF}
N.~Cheung,
\emph{Landmark XIV-F: Paired composition on a proposed excess bundle:
The clean-fiber-product audit, the excess correction,
and the exact F2 stop}.
Supplied manuscript \texttt{XIV-F.tex}.
Immediate predecessor and unchanged geometric boundaries.

\bibitem{XIVE}
N.~Cheung,
\emph{Landmark XIV-E: Joint symbol of the six-branch retained germ,
or one named characteristic prescription}.
Supplied manuscript \texttt{XIV-E.tex}.
Regrouping, root, tested-kernel bound, characteristic lifts,
elementary phase exclusion, and E1 stop.

\bibitem{XIVD}
N.~Cheung,
\emph{Landmark XIV-D: Cancellation or prescription on the retained
noncoincident characteristic stratum}.
Supplied manuscript \texttt{XIV-D.tex}.
Pair, six-branch germ, species split, branch fiber identity,
and D2 stop.

\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}.
Characteristic stratum, positive conormals, contact rigidity,
and 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}.
Inventory, partial contacts, primitive extensions,
and successor obligations.

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

\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, roots, and 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 representative and actual source.

\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}.
Recorded common-frequency attempt only.

\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: pullbacks, tensor products, products,
and wavefront bounds.

\bibitem{Wunsch}
J.~Wunsch,
\emph{Microlocal analysis and evolution equations:
Lecture notes from the 2008 CMI/ETH Summer School},
arXiv:0812.3181, revision v3, 2023.
Section~8: conormal and Lagrangian distributions,
orders, individual symbols, and operator calculus.

\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: recorded submanifold extension framework;
not applied to the unresolved target.

\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.
Recorded microlocal extension and ambiguity framework;
no characteristic prescription is imported.

\end{thebibliography}

\end{document}