% !TeX program = pdflatex
%=====================================================================
% XIV-H.tex
%
% Standalone source for pdfLaTeX.
% Build twice:
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-H.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XIV-H.tex
%
% Exact release:
%   - the XIV-G operator route is continued;
%   - the pair, six-branch germ, source representative, inverse -1,
%     switches, width, and partial-diagonal class are unchanged;
%   - the residual metric variable z is retained;
%   - the local source incidence is written before residual testing;
%   - exact kernel maps for K_5, K_8, and K_9 are identified;
%   - the two K_5 placements are represented by submersion pullbacks
%     of tensor products of the existing retarded kernels;
%   - the K_8 internal propagation is evaluated using the existing
%     massless retarded convolution, without a cross-line product;
%   - K_8 and K_9 receive regular conormal representations on the
%     retained separated-endpoint patch;
%   - regular symbol charts for K_5 are constructed away from its
%     internal characteristic-diagonal transition;
%   - the diagonal and flow-out geometries at that transition are
%     distinguished explicitly;
%   - their clean intersection is not called a non-clean matching;
%   - no single ordinary Lagrangian relation contains both full
%     local pieces at that transition;
%   - a uniform internal amplitude calculus there is not constructed.
%
% Ending:
%   Unresolved at H2, at the K_5 internal transition, within
%   G2/F2/E1/D2/J2.
%
% Debugging corrections:
%   1. Defined tree kernels and exact tensor-product formulas are not
%      automatically uniform internal symbol representations.
%   2. A clean intersection of two Lagrangian pieces is not their
%      identification as one smooth Lagrangian relation.
%   3. A missing H2 amplitude theorem is reported at H2, not relabeled
%      as an H3 failure after declaring H2 constructed.
%   4. The simplification of the K_8 intermediate ramp is justified
%      by its actual source incidence and causal support. No hidden
%      vertex is assigned a retained point by hand.
%   5. The internal conormal orders below are not transverse scaling
%      degrees of the grouped quantum coefficient.
%
% No common product phase, excess-bundle construction, factorwise
% cross-line retry, characteristic prescription, new contact class,
% R_1^(1), Ward defect, interacting differential, new vertex,
% numerical loop value, 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}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
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\usepackage{microtype}
\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
\usepackage{array,booktabs,tabularx}
\usepackage{enumitem}
\usepackage[hidelinks]{hyperref}

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\numberwithin{equation}{section}
\setcounter{tocdepth}{1}

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

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\MT}{M_T}
\newcommand{\MTo}{M_T^\circ}
\newcommand{\PP}{\mathcal P}
\newcommand{\ER}{E_0^{\mathrm R}}
\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{\Qsrc}{\mathsf Q}
\newcommand{\Rker}{R}
\newcommand{\Bker}{B}
\newcommand{\Amap}{\mathsf A}
\newcommand{\bmap}{\mathsf b}
\newcommand{\Sk}{S_{\mathrm{resp}}}
\newcommand{\Lresp}{\Lambda_{\mathrm{resp}}}
\newcommand{\Ldiag}{\Lambda_{\mathrm{d}}}
\newcommand{\Lflow}{\Lambda_{\mathrm{f}}}
\newcommand{\pair}[2]{\left\langle #1,#2\right\rangle}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\WF}{WF}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\Contr}{Contr}
\DeclareMathOperator{\EL}{EL}
\DeclareMathOperator{\Diff}{Diff}
\DeclareMathOperator{\id}{id}

\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-H}\\[0.4em]
Internal kernel relations on the operator route\\[0.4em]
\large The two gravitational-response legs,\\
the internal characteristic-diagonal transition,\\
and the first missing amplitude theorem}
\author{Ngo Cheung}
\date{}

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

\begin{abstract}
We continue XIV-G's operator route for the unchanged six-branch
retained germ. The residual metric variable is retained throughout.
No common product phase, excess construction, factorwise
cross-line retry, or characteristic prescription is introduced.

The untested local source is written as a distributional incidence
on its two field slots. This gives exact internal kernel maps for
the coordinate and scalar gravitational-response legs. On the
retained separated-endpoint patch, the coordinate leg reduces to
a derivative of the existing massless retarded convolution. An
explicit calculation evaluates that convolution. Both response
legs consequently admit finite conormal representations with
their source coefficients and scalar-root mass term retained.

The two placements in the remaining internal branch are expressed
as submersion pullbacks of tensor products of the existing
retarded kernels. Their regular separated-cone and
noncharacteristic-diagonal symbol charts are identified.
At the internal characteristic diagonal, however, the diagonal
piece and the limiting flow-out piece are distinct Lagrangians.
Their intersection is clean, but their union is not one smooth
Lagrangian relation. Sectorwise symbols therefore cannot be
identified as a single ordinary symbol there.

A uniform internal amplitude construction across that transition
is not supplied. The first missing gate is H2, rather than H3:
the existence of the tree kernel does not discharge the stronger
internal-symbol requirement. No common characteristic operator
representation of the complete grouped coefficient, joint
section, remainder theorem, or quantum output is released.
The target remains Unresolved within G2/F2/E1/D2/J2.
\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) {Untested source incidence};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Internal formulas for selected kernels};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Admitted quantum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The internal formulas do not admit a quantum coefficient.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Internal kernels are not an admitted coefficient. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-XIV-H}
\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) {Untested source incidence};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Admitted quantum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The internal formulas do not admit a quantum coefficient.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Internal kernels are not an admitted coefficient. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-XIV-H}
\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) {Untested source incidence};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Internal formulas for selected kernels};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Admitted quantum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The internal formulas do not admit a quantum coefficient.};
\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-H}
\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) {Untested source incidence};
\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) {Admitted quantum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Internal formulas for selected kernels. The internal formulas do not admit a quantum coefficient.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Untested source incidence is not Admitted quantum coefficient. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-XIV-H}
\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: Untested source incidence};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Internal formulas for selected kernels};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Admitted quantum coefficient};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The internal formulas do not admit a quantum coefficient.};
\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-H}
\end{figure}

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


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



\scopebox{Exact release and first stop}{%
\textbf{Constructed here:}
the untested source incidence; exact internal kernel formulas for
$K_5,K_8,K_9$; regular conormal representations for $K_8,K_9$;
and regular symbol charts and transition geometry for the two
$K_5$ placements.

\smallskip
\textbf{Not constructed:}
a uniform internal amplitude theorem for the $K_5$
characteristic-diagonal transition. Its defined tensor-product
kernel is not promoted to such a theorem.

\smallskip
\textbf{First stop: Unresolved at H2.}
Declaring H2 complete and then reporting the same missing theorem
at H3 would misstate the dependency.

\smallskip
The clean intersection proved below is not called a non-clean
matching. It is not the tangent defect of XIV-F.

\smallskip
No common characteristic representation, joint section, transverse
degree, prescription, $\Rquant$, or Ward defect is released.}

\tableofcontents

%=====================================================================
\section{H0: copied handoffs and their domains}
\label{sec:H0}
%=====================================================================

Only XIV-G, XIV-F, XIV-E, XIV-D, XIV-C, XIII-C, XIII-B,
XII-E, and XII-C are analytic manuscript inputs.
Their handoffs are used with their stated scope.
No incidence inventory, predecessor phase theorem, or
clean-excess theorem is re-proved.

\subsection{XIV-G's eight handoff items}

\begin{inputstatement}[Successor package, copied from XIV-G]
\label{input:Gpackage}
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}

It receives no full cancellation theorem, no nonzero full
symbol section, and no permission to call the tangent
defect an excess. It receives no $\Rquant$ \cite{XIVG}.
\end{inputstatement}

\begin{inputstatement}[XIV-G's local typing and stop]
\label{input:Gtyping}
For a smooth compact residual test $G$,
\[
 \Qsrc_G=\JD'[G]\in\Diff^{\le2}.
\]
Its local polynomial differential symbol is not the symbol of
either composed gravitational-response leg.

Smooth residual testing can hide attachment-diagonal wavefront
directions. The compact balanced-kernel example in XIV-G proves
that limitation, not a model of the characteristic singularity of
$\Kret$.

The ordinary final product test for the two individually defined
numerical factors in XIV-G fails at the retained opposite
covectors. Their conormal orders
$\mu_U=-\frac12$ and $\mu_V=-1$ belong only to those factors.

\[
 \boxed{\text{Unresolved at G2, within F2/E1/D2/J2.}}
\]

No complete internal symbol relations or common symbol bundle
were supplied \cite{XIVG}.
\end{inputstatement}

Neither factor order is used as a transverse degree below.
The factorwise audit is not repeated.

\subsection{XIV-F's geometry is a boundary, not a relation}

\begin{inputstatement}[Geometric release, copied]
\label{input:F}
On the regular six-constraint stage,
\[
 q_R|_{S_6}
   =\alpha\delta|\mathbf n_1-\mathbf n_2|^2,
 \qquad \beta=\alpha+\delta.
\]
The tangent-defect map is
\[
 \mathfrak d_w(u)
 =\alpha(\delta b-\delta a)_\perp
               -\beta(\delta y-\delta a)_\perp.
\]
Its quotient has rank two. It is not an established excess fiber.

The selected matching has actual dimension fifteen and
linearized matching dimension seventeen. The covector-sum map
satisfies
\[
 \rank\mathrm dJ=14,\qquad
 \dim(\text{local fiber of }J)=1.
\]
Its image is isotropic of dimension fourteen, not Lagrangian
in $T^*W$.

The direct clean-excess-two identification was excluded.
The grouped target remained Unresolved at F2
\cite{XIVF}.
\end{inputstatement}

Neither $S_6$, the matching manifold, nor $J$ is used as an
internal composition relation in this paper.

\subsection{The exact regrouping and complete scalar root}

\begin{inputstatement}[XIV-E's operator identity]
\label{input:E}
With
\[
 X_G=X[G],\qquad X_h=X[h],\qquad
 r_h=\rD'[G]h=\PP\ER(\JD'[G]h),
\]
the complete retained smooth-slot 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
\[
 C_G^\mu{}_\nu
 =f\,\partial_\nu X_G^\mu
       -\delta^\mu_\nu\partial_\rho(fX_G^\rho),
\]
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:B}
\end{equation}
All six branches are retained.

For $F=\EAm f$, the complete scalar-response root is
\begin{equation}
 \int f\,p[r_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:root}
\end{equation}
The derivative and mass terms are both retained.

For each fixed smooth compact $G$, the tested attachment kernel
is compact and satisfies the balanced attachment bound.
This does not give the untested joint symbol or its uniform
estimates \cite{XIVE}.
\end{inputstatement}

\subsection{The retained germ, characteristic locus, and contacts}

\begin{inputstatement}[XIV-D and XIV-C]
\label{input:DC}
The full observable is
\[
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi),
 \qquad
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD.
\]
On the retained mixed patch,
\[
 \Kret=K_4+K_5+K_6+K_7+K_8+K_9.
\]
The kernels $K_6,K_8,K_9$ are supported on the actual local
source or metric attachment incidence $a=z$.

The relation
\[
 p_\alpha p_\mu
 \dot a_2^{\alpha\mu}[G;p\otimes k+k\otimes p]
 =G(p^\sharp,p^\sharp)(p\cdot k)
\]
is limited to the leading differential numerator of $B_6$.
It is not a section of the complete sum \cite{XIVD}.

In the common space
\[
 W=A\times B\times Z\times Y,
 \qquad (a,b,z,y)\text{ in this order},
\]
the test locus is
\[
 N=\Nlift
 =\{a=z\}\cap
 \left\{
 y=a+\alpha\ell_{\mathbf n},\
 b=a+\beta\ell_{\mathbf n},\
 0<\alpha<\beta
 \right\},
\]
where $\ell_{\mathbf n}=(1,\mathbf n)$ and $\mathbf n\in S^2$.
It has codimension eight in the sixteen-dimensional space.

Writing
\[
 q_R=(b-a)^2,\qquad
 q_1=(y-a)^2,\qquad
 q_2=(b-y)^2,
\]
the copied positive relation is
\[
 -\frac{\mathrm dq_R}{2\beta}
 +\frac{\mathrm dq_1}{2\alpha}
 +\frac{\mathrm dq_2}{2(\beta-\alpha)}=0.
\]
It is diagnostic input data, not an output symbol relation.

Defined collision contacts vanish on the retained
collision-free attachment neighborhood \cite{XIVC,XIVD}.
\end{inputstatement}

The neighborhoods $A,B,Y$ have pairwise disjoint closures,
$\overline Z\subset A$, and their closures lie in the open slab.
They are small enough that $b^0>a^0>\eps$ on $B\times A$.
The retained configuration is not removed.

\subsection{Earlier handoffs}

\begin{inputstatement}[XIII-C and XIII-B]
\label{input:BC}
XIII-C supplies four unresolved obligations:
\begin{enumerate}
\item a characteristic prescription or complete-pair cancellation;
\item uniform transverse estimates at the relevant collision
strata and their intersections;
\item compatible partial contacts with their singular spectators;
\item the output wavefront theorem after proper pushforward.
\end{enumerate}
The parallel temporal kinetic primitive has collision scaling
degree eight in codimension four \cite{XIIIC}.

XIII-B supplies the admitted selected tree datum
\[
 \Rtree
 =-\pair{\Ocal'}{G_{\mathrm{act}}^{\mathrm R}\Vcal'},
\]
the retarded convention
\[
 R_1(\Vcal;\Ocal)
 =\frac{i}{\hbar}
  \left(\mathsf T_2^{\mathrm{ren}}(\Vcal,\Ocal)
                             -\Vcal\star_H\Ocal\right),
\]
and the corrected strictly ordered values:
zero or $(i/\hbar)[\Ocal,\Vcal]_{\star_H}$.

Properness is support control for an existing distribution,
not a construction of a missing product \cite{XIIIB}.
\end{inputstatement}

\begin{inputstatement}[XII-E and XII-C]
\label{input:EC}
The response and first-completion formulas are
\[
 \rD=\PP\ER\JD,\qquad
 X^\mu[r]=-\ER\partial_\alpha(\sig a_1^{\alpha\mu}[r]),
\]
\[
 p[r;\psi]=-\ERm(V_{1,g}[r]\psi),
\]
\[
 a_1^{\mu\nu}[r]
 =\frac12\eta^{\mu\nu}\tr_\eta r-r^{\mu\nu},
 \qquad b_1[r]=\frac12\tr_\eta r.
\]
The existing completion kernels are tree kernels
\cite{XIIE}.

The actual source is
\begin{equation}
 \JD
 =\EL_\gamma(\sig\ell_3)
 =\sig J_3+\mathcal D_\sig,
 \qquad
 \mathcal D_\sig^{\mu\nu}
 =2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sig
       -2M_2^{\mu\nu\rho}\partial_\rho\sig .
 \label{eq:actualsource}
\end{equation}
Its fixed unintegrated cubic representative is
\[
 \ell_3[\gamma]
 =-2[\rho^3]
 \left(\sqrt{-\det(\eta+\rho\gamma)}\,R[\eta+\rho\gamma]\right).
\]
The selected weighting is at cubic order only
\cite{XIIC}.
\end{inputstatement}

Every source coefficient below is that of
\eqref{eq:actualsource}. None is replaced by a simply
source-switched expression.

XIV-A and XIV-B are not analytic inputs.
The value $-8I$ is not a normalization.
XIII-E and a resolved paired construction remain unentered.

%=====================================================================
\section{H1: the route, class, and release criteria}
\label{sec:H1}
%=====================================================================

\[
 \boxed{
 \text{Internal relations of }K_5,K_8,K_9,
 \text{ then a common operator representation of the regrouped sum.}}
\]

The carrier, pair, inverse, switches, width, and ambiguity
class remain
\[
 \Ealg=(\Fpol[[\hbar,\kappa]],\star_H)/\Nsol[[\hbar,\kappa]],
 \qquad
 \Gret=-\PP\ER,\qquad
 \Ctot\subseteq\Cpart.
\]
The inverse coefficient is $-1$.

The target is still
\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)\\
 &-H_{\gamma\gamma}(y,a)\otimes H_{\phi\phi}(y,b)
 \Bigr).
 \end{split}
 \label{eq:target}
\end{equation}
The difference stays grouped. No mixed $H$--$G_{\mathrm{act}}$
product is declared.

\begin{definition}[Representation class fixed before the test]
\label{def:class}
The selected class uses the original common space $W$,
with $z$ retained. It permits:
\begin{enumerate}[label=\textup{(\roman*)}]
\item named tensor products, submersion pullbacks, differential
operators, and support-proper pushforwards of the existing
tree kernels;
\item individual regular conormal or canonical-relation charts,
with their orders and amplitudes specified;
\item a composition across intersecting relation charts only
after its matching and uniform amplitude hypotheses have
been proved.
\end{enumerate}
A wavefront inclusion, a union of relation charts, or an exact
distributional tensor-product formula does not by itself
supply the amplitude theorem in \textup{(iii)}.

There is no common product phase and no identification with
XIV-F's excluded matching. The two gravitational-response
legs are kept separate until their own kernel relations
have been written.
\end{definition}

The distinction in item \textup{(iii)} does not prohibit a
future generalized operator theorem. It requires that theorem
to be stated and proved rather than supplied by terminology.

The endings concern only the internal relations and common
operator representation:
\begin{enumerate}
\item \textbf{Constructed}: the necessary internal relation
and amplitude maps exist, and one common representation
is justified on a stated domain.
\item \textbf{Proved obstructed}: a demonstrated incompatibility
excludes every representation in
Definition~\ref{def:class}.
\item \textbf{Unresolved}: an internal amplitude map, matching
hypothesis, or common representation remains missing.
\end{enumerate}

A constructed representation would not by itself discharge
the joint-section, remainder, or quantum-admission gates.

\subsection{The first-missing-gate rule}

H2 is a prerequisite, not a label that may be passed by
recording an existing tree kernel. If its required internal
amplitude theorem is missing, the paper stops there.
H3 cannot be declared reached by renaming that same gap.

No characteristic prescription is opened.
There is no retry of the factorwise product of XIV-G.

%=====================================================================
\section{H2a: the untested local source incidence}
\label{sec:source}
%=====================================================================

\subsection{Conventions}

Use $\eta=\operatorname{diag}(1,-1,-1,-1)$ and
$\Box=\partial_t^2-\Delta_{\mathbf x}$.
Let $\Rker(x-v)$ denote the kernel of $\ER$.
Define constant fiber maps by
\[
 a_1^{\alpha\mu}[r]=\Amap^{\alpha\mu,I}r_I,
 \qquad
 b_1[r]=\bmap^I r_I,
\]
where $I$ ranges over symmetric metric components.
All sums over $I$ use the inherited tensor pairing.

The coordinates $(a,z)$ are the two metric field slots:
$h_I(a)$ is the contracted direction and $G_J(z)$ the
residual direction. They are not replaced by a smooth
residual test in the kernel formulas.

\subsection{A source kernel before testing}

The actual source is a finite sum of local quadratic
monomials. Its full derivative can therefore be written
\begin{equation}
 (\JD'[G]h)^L(v)
 =
 \sum_{\substack{I,J,\alpha,\beta\\|\alpha|+|\beta|\le2}}
 q^{L,I,J}_{\alpha\beta}(v)
          \,\partial^\alpha G_J(v)\,\partial^\beta h_I(v),
 \label{eq:sourcepolynomial}
\end{equation}
where all symmetrization factors are absorbed in $q$.
The smooth coefficients are the fixed coefficients of
\eqref{eq:actualsource}, including the ramp derivatives.

Define the untested incidence kernel
\begin{equation}
 Q^{L,I,J}(v;a,z)
 =
 \sum_{\alpha,\beta}
 q^{L,I,J}_{\alpha\beta}(v)
       \,\partial_v^\alpha\delta^{(4)}(v-z)
       \,\partial_v^\beta\delta^{(4)}(v-a).
 \label{eq:Qincidence}
\end{equation}

\begin{proposition}[The source incidence is regular]
\label{prop:sourceincidence}
The product of deltas in \eqref{eq:Qincidence} is defined.
Its support is $v=a=z$, and its underlying conormal
relation is
\[
 \Lambda_{\mathrm{src}}
 =
 \left\{
 (v,a,z;\nu,\xi,\zeta):
 v=a=z,\ \nu+\xi+\zeta=0
 \right\}\setminus0.
\]
The source kernel has conormal order at most three in
the twelve-dimensional $(v,a,z)$ space.
Its amplitude is the finite polynomial obtained by
applying the displayed derivatives to the two
independent diagonal deltas.

Pairing its $z$ slot against a smooth $G$ recovers
$\Qsrc_G$, but this testing is not used to define the
untested relation.
\end{proposition}

\begin{proof}
The map
\[
 (v,a,z)\longmapsto(v-z,v-a)
\]
is a submersion onto $\R^8$: variations of $z$ and $a$
can prescribe its two outputs independently.
The two-delta product is its pullback of the delta
at the origin in $\R^8$.

Its conormal covectors have the stated zero-sum form.
The defining submanifold has codimension eight in
dimension twelve. With normal amplitude order at most
two, the conormal order is
\[
 2-\frac{12-16}{4}=3.
\]
Lower derivative monomials belong to the corresponding
lower filtered orders.

The pairing formula follows directly from the
distributional evaluation of the derivatives of the
deltas. It reproduces \eqref{eq:sourcepolynomial}.
\end{proof}

Only this actual incidence forces $v=a=z$. No internal
vertex is assigned $u_0$, $y_0$, or $x_0$ by hand.

For operator notation, the incoming $(a,z)$ covectors
may be sign-reversed in the usual primed relation.
Below we use unprimed kernel covectors in $T^*W$.

\subsection{The source response map}

Before composing either gravitational leg, set
\begin{equation}
 T^{I,J}(b;a,z)
 =
 \PP^{I}{}_{L}
 \int \Rker(b-v)\,Q^{L,\cdot,J}(v;a,z)\,\dd^4v,
 \label{eq:responsekernel}
\end{equation}
with the contracted metric index understood in the
source kernel.

This is the untested kernel of
$r_h(b)=\PP\ER(\JD'[G]h)(b)$.
It retains the residual variable.

On the separated patch $b\in B$, $a\in A$, the endpoints
$b,a$ never coincide. Moving the external derivatives
in \eqref{eq:Qincidence} outside the integral writes
\eqref{eq:responsekernel} as a finite sum of derivatives,
of total order at most two, of
\[
 q(a)\,\Rker(b-a)\,\delta^{(4)}(a-z).
 \label{eq:responseseed}
\]
The differentiated coefficients are included in this
identity; they are not omitted remainder terms.

%=====================================================================
\section{H2b: the coordinate gravitational-response leg}
\label{sec:K8}
%=====================================================================

\subsection{Its internal variables and exact map}

The coordinate leg has two internal variables:
$v$ is the actual source vertex and $s$ the intermediate
coordinate-response vertex. Its scalar attachment is $b$.
Its untested kernel is
\begin{equation}
 \begin{split}
 K_8^{I,J}(a,b,z)
 =-\partial_{b^\mu}\Biggl[
 f(b)\int\!\!\int
 \Rker(b-s)\,
 \partial_{s^\alpha}\Bigl(
 \sig(s)\Amap^{\alpha\mu}{}_{L}\PP^{L}{}_{M}
 \Rker(s-v)Q^{M,I,J}(v;a,z)
 \Bigr)\dd^4s\,\dd^4v
 \Biggr].
 \end{split}
 \label{eq:K8map}
\end{equation}
This is the kernel of
\[
 -\int f\,X^\mu[r_h]\partial_\mu\psi,
\]
after the scalar-slot derivative is transferred to the
kernel.

The original maps are therefore:
\[
 (a,z)\longrightarrow v,\qquad
 v\longrightarrow s,\qquad
 s\longrightarrow b.
\]
The first is the regular local incidence of
Proposition~\ref{prop:sourceincidence}.
The other two are the existing retarded maps.

\subsection{A justified intermediate-ramp simplification}

\begin{lemma}[The intermediate coordinate weight on this patch]
\label{lem:weight}
In \eqref{eq:K8map}, replacing the intermediate factor
$\sig(s)$ by one leaves the restricted kernel unchanged.
\end{lemma}

\begin{proof}
The source incidence enforces $v=a=z$, with $a^0>\eps$.
The support of the two retarded kernels enforces
\[
 s\in J^+(a)\cap J^-(b).
\]
Hence $s^0\ge a^0>\eps$, where $\sig(s)=1$ and all its
derivatives vanish.

Differentiation does not enlarge distributional support.
Thus every term containing $\sig(s)-1$ or a derivative
of that factor has empty support on this patch.
\end{proof}

This lemma uses an identified internal vertex and proved
causal support. It does not use the locations of the
retained external triple as substitutes for internal
variables.

The source coefficients $q$ remain those of the actual
source. No simply weighted source is substituted.

\subsection{The existing retarded convolution}

Define $\Bker=\Rker*\Rker$.
For fixed compact endpoint sets the intermediate
integration is support-proper, since its support lies
in compact causal diamonds.

\begin{lemma}[Massless retarded convolution in four dimensions]
\label{lem:convolution}
With the normalization $\Box\ER=\id$,
\begin{equation}
 \Rker(t,\mathbf x)
   =\frac{\Theta(t)}{2\pi}\delta(t^2-|\mathbf x|^2),
 \qquad
 \Bker(t,\mathbf x)
   =\frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2).
 \label{eq:convolution}
\end{equation}
The second equality is an equality of the already
defined retarded convolution distributions.
\end{lemma}

\begin{proof}
The first formula is the retarded massless wave kernel
in the declared normalization.

Let $\rho=|\boldsymbol\xi|$ and take the Fourier
transform only in the spatial variables. The first
kernel has transform
\[
 \widehat{\Rker}(t,\boldsymbol\xi)
       =\Theta(t)\frac{\sin(\rho t)}{\rho}.
\]
Its causal time convolution is
\[
 \Theta(t)\int_0^t
 \frac{\sin(\rho(t-s))}{\rho}
 \frac{\sin(\rho s)}{\rho}\,\dd s
 =
 \Theta(t)\frac{\sin(\rho t)-\rho t\cos(\rho t)}
                         {2\rho^3}.
\]
At $\rho=0$ the continuous value is $\Theta(t)t^3/6$.

The spatial Fourier transform of
$\Theta(t)\mathbf1_{\{|\mathbf x|\le t\}}/(8\pi)$
is the same expression, by radial integration over
the ball of radius $t$.
The formulas define locally tempered distributions,
so equality of their spatial Fourier transforms
proves the result.
\end{proof}

No frequency regulator or characteristic limit is
introduced by this calculation. It concerns the
existing retarded internal propagators only.

\begin{proposition}[Exact reduced coordinate-leg kernel]
\label{prop:K8exact}
On the retained patch, \eqref{eq:K8map} equals
\begin{equation}
 K_8^{I,J}(a,b,z)
 =
 -\partial_{b^\mu}\left[
 f(b)\Amap^{\alpha\mu}{}_{L}\PP^{L}{}_{M}
 \partial_{b^\alpha}
 \int \Bker(b-v)Q^{M,I,J}(v;a,z)\,\dd^4v
 \right].
 \label{eq:K8reduced}
\end{equation}
Since $b\ne a$ and $b^0>a^0$, it is a finite sum of
derivatives of total order at most three of smooth
tensor coefficients times
\[
 \delta((b-a)^2)\delta^{(4)}(a-z).
 \label{eq:K8seed}
\end{proposition}

\begin{proof}
After Lemma~\ref{lem:weight}, constant fiber maps
commute with the scalar propagation.
Causal convolution and differentiation give
\[
 \int \Rker(b-s)
          \partial_{s^\alpha}\Rker(s-v)\,\dd^4s
       =\partial_{b^\alpha}\Bker(b-v),
\]
which proves \eqref{eq:K8reduced}.

On the separated chronological patch,
\[
 \partial_{b^\alpha}\Bker(b-a)
     =\frac{(b-a)_\alpha}{4\pi}\delta((b-a)^2).
\]
The source incidence contributes at most two external
derivatives, and the scalar attachment contributes one
more. Derivatives of the source coefficients are
smooth coefficients of lower terms. This gives the
finite representation \eqref{eq:K8seed}.
\end{proof}

\subsection{Its regular relation and order}

Set
\[
 q=(b-a)^2,\qquad r=a-z,
\]
and define
\begin{equation}
 \Sk=\{q=0,\ r=0\}\subset W,\qquad
 \Lresp=N^*\Sk\setminus0.
 \label{eq:Sresp}
\end{equation}
The $y$ variable is passive, with zero covector.
The five differentials $\mathrm dq,\mathrm dr^0,\ldots,
\mathrm dr^3$ are independent because $b\ne a$ and
the $z$ components isolate the four attachment
differentials.

Consequently $\Sk$ has codimension five, and $\Lresp$
is a sixteen-dimensional Lagrangian kernel relation.
It is not the characteristic test locus $N$, which
has codimension eight.

For the remainder of the paper, the order convention is:
a normal Fourier amplitude of order $d$ along a
codimension-$k$ submanifold of an $n$-dimensional space
has conormal order
\begin{equation}
 \mu=d-\frac{n-2k}{4}.
 \label{eq:order}
\end{equation}
This is the individual conormal convention
\cite[Section 8.1]{Wunsch}.

\begin{corollary}[Coordinate-leg conormal representation]
\label{cor:K8order}
On the retained separated-endpoint patch,
\[
 K_8\in I^{3/2}(W,\Lresp).
\]
The order is an upper filtered order and may decrease
when its complete tensor coefficients cancel.

A full finite amplitude representative is obtained
from \eqref{eq:K8reduced}: apply its finite differential
operators to the normal amplitude of
$\delta(q)\delta^{(4)}(r)$.
No symbol of the grouped quantum coefficient is
assigned.
\end{corollary}

\begin{proof}
The seed has normal amplitude order zero.
The representation in Proposition~\ref{prop:K8exact}
has at most three derivatives. Its normal amplitude
is therefore a finite polynomial of degree at most
three in the five normal frequencies, with smooth
coefficients.
Equation~\eqref{eq:order}, with $n=16,k=5$, gives
$3-(16-10)/4=3/2$.
\end{proof}

\subsection{The actual separated internal flow matching}

For clarity, the interior propagation matching in
\eqref{eq:K8map} can be examined without the source
diagonal. In primed covectors, a separated massless
retarded flow relation is locally
\[
 (x,\xi;v,\xi),\qquad
 \xi^2=0,\quad x=v+\tau\xi^\sharp,
\]
on a fixed sign sheet with the separation future directed.

Two such propagations match by equality of the
intermediate point and covector. On an interior
segment, the matching is parameterized by
\[
 v,\ \xi,\ \tau_1,\ \tau_2,\qquad
 s=v+\tau_2\xi^\sharp,\qquad
 b=s+\tau_1\xi^\sharp .
\]
The matching equations have exactly these linearized
tangent solutions. The projection retains
$v,\xi,\tau_1+\tau_2$ and has a one-dimensional
local fiber.

This is a genuine clean internal propagation matching
on its separated interior chart. Its one-dimensional
fiber comes from splitting an already fixed null
segment. It is unrelated to XIV-F's rank-two defect.
No endpoint symbol integral is inferred from this
interior chart: the exact convolution proof above
already includes those endpoints distributionally.

%=====================================================================
\section{H2c: the scalar gravitational-response leg}
\label{sec:K9}
%=====================================================================

\subsection{The root keeps both terms}

Let $F=\EAm f$, the inherited smooth advanced observation
root. The scalar leg uses the response kernel
\eqref{eq:responsekernel}, with its output at the scalar
attachment $b$. Its exact untested kernel is
\begin{equation}
 \begin{split}
 K_9^{I,J}(a,b,z)
 ={}&
 -\partial_{b^\nu}
 \left[
 g(b)\partial_{b^\mu}F(b)\,
                   \Amap^{\mu\nu}{}_{L}
                   T^{L,I,J}(b;a,z)
 \right]\\
 &-m^2g(b)F(b)\,\bmap_{L}
                   T^{L,I,J}(b;a,z).
 \end{split}
 \label{eq:K9}
\end{equation}

\begin{proposition}[Scalar-leg representation]
\label{prop:K9}
Formula~\eqref{eq:K9} is the complete untested kernel
of the inherited $B_9$ root.
It retains the derivative and mass terms.

On the separated chronological patch,
\[
 K_9\in I^{3/2}(W,\Lresp).
\]
It has a finite conormal amplitude obtained from the
source incidence and the displayed differential
operators. The derivative term has normal amplitude
degree at most three; the mass term has degree at
most two.
\end{proposition}

\begin{proof}
Substitute the untested response kernel into
\eqref{eq:root}. Transferring the derivative of
$\psi(b)$ to the kernel gives the first line of
\eqref{eq:K9}; the second line is its mass term.

On the retained patch,
\[
 \Rker(b-a)=\frac1{2\pi}\delta((b-a)^2),
\]
because $b^0>a^0$ and the endpoints are separated.
Equation~\eqref{eq:responseseed} therefore expresses
the response as at most two derivatives of smooth
coefficients times $\delta(q)\delta^{(4)}(r)$.

The first line of \eqref{eq:K9} adds at most one
derivative. The second adds none. The same regular
conormal coordinates as in \eqref{eq:Sresp} give
the stated orders and amplitudes.
\end{proof}

The equality of the two response-leg relation supports
does not identify their amplitudes. Equations
\eqref{eq:K8reduced} and \eqref{eq:K9} remain separate.

No assertion that the principal amplitude of either
leg is nonzero is needed or made.

%=====================================================================
\section{H2d: the two internal placements in \(K_5\)}
\label{sec:K5}
%=====================================================================

\subsection{The existing tree maps with \(z\) retained}

Define the coordinate-response line
\begin{equation}
 L^{\mu,I}(s,a)
   =\sig(a)\Amap^{\lambda\mu,I}
                   \partial_{a^\lambda}\Rker(s-a).
 \label{eq:L}
\end{equation}
Thus
\[
 X^\mu[h](s)=\int L^{\mu,I}(s,a)h_I(a)\,\dd^4a.
\]

The two terms of $K_5$ have the actual internal maps
\[
 z\longrightarrow s\longrightarrow b,\qquad s=a,
\]
and
\[
 a\longrightarrow s\longrightarrow b,\qquad s=z,
\]
respectively. The equalities $s=a$ and $s=z$ come from
the displayed local metric factor in that placement.
They are not assignments of a singular internal
point to a retained configuration.

\begin{proposition}[Exact two-placement kernel formula]
\label{prop:K5exact}
The untested mixed kernel of $B_5$ is
\begin{equation}
 \begin{split}
 K_5^{I,J}(a,b,z)
 =\partial_{b^\mu}\Bigl[
 &f(b)\sig(a)\,
 \partial_{a^\alpha}\Rker(b-a)\,
 \Amap^{\alpha\beta,I}
                  \partial_{a^\beta}L^{\mu,J}(a,z)\\
 +{}&
 f(b)\sig(z)\,
 \partial_{z^\alpha}\Rker(b-z)\,
 \Amap^{\alpha\beta,J}
                  \partial_{z^\beta}L^{\mu,I}(z,a)
 \Bigr].
 \end{split}
 \label{eq:K5}
\end{equation}
Each placement is a finite differential expression,
of total kernel derivative order at most four, in
the products
\begin{equation}
 \Rker(b-a)\Rker(a-z),
 \qquad
 \Rker(b-z)\Rker(z-a).
 \label{eq:K5products}
\end{equation}
These products are defined, including at $a=z$.
\end{proposition}

\begin{proof}
Start with the two terms
\[
 \int f(b)\Rker(b-s)
 \partial_{s^\alpha}
 \left[
 \sig(s)\left(
 a_1^{\alpha\beta}[h(s)]\partial_{s^\beta}X_G^\mu(s)
 +a_1^{\alpha\beta}[G(s)]\partial_{s^\beta}X_h^\mu(s)
 \right)\right]
 \partial_{b^\mu}\psi(b)\,\dd^4s\,\dd^4b.
\]
Transfer the divergence to the incoming retarded
kernel and the scalar derivative to the final kernel.
The two signs cancel. Insert \eqref{eq:L}, and use the
actual local incidences $s=a$ and $s=z$. This gives
\eqref{eq:K5}.

For the first product in \eqref{eq:K5products}, the
coordinate map
\[
 (a,b,z,y)\longmapsto
       (a,\ b-a,\ a-z,\ y)
\]
is a diffeomorphism onto its image.
The product is therefore the pullback of
$1\otimes\Rker\otimes\Rker\otimes1$.
The second placement uses the analogous diffeomorphism
\[
 (a,b,z,y)\longmapsto
       (z,\ b-z,\ z-a,\ y).
\]
Tensor products of distributions exist, and these
pullbacks are defined. Differentiation and multiplication
by the fixed smooth weights preserve existence.
\end{proof}

This proves the distributional kernel formula. It does
not by itself assign one ordinary symbol class through
the internal cone vertex.

The common $W$ variables are never replaced by the
tested kernel $L_G$ of XIV-E.

\subsection{Separated-cone relations}

On the first placement, define
\[
 q_+=(b-a)^2,\qquad q_{\mathrm{in},+}=(a-z)^2.
\]
When both separations are nonzero and future null,
their defining differentials are independent:
the $b$ and $z$ endpoint components isolate the two
coefficients in a proposed relation.

The regular kernel relation is
\begin{equation}
 \Lambda_{5,+}
 =
 N^*\{q_+=q_{\mathrm{in},+}=0\}\setminus0
 \label{eq:L5plus}
\end{equation}
on that separated chronological sector.

For the second placement use
\[
 q_-=(b-z)^2,\qquad q_{\mathrm{in},-}=(z-a)^2,
\]
and
\[
 \Lambda_{5,-}
 =
 N^*\{q_-=q_{\mathrm{in},-}=0\}\setminus0.
\]

\begin{proposition}[Regular separated-cone orders and amplitudes]
\label{prop:K5flow}
Each placement of $K_5$ has a conormal representation
of upper order one on its corresponding separated
two-cone relation.

The normal amplitude is the finite polynomial obtained
by differentiating
\[
 \frac1{(2\pi)^2}
          \delta(q_\pm)\delta(q_{\mathrm{in},\pm})
\]
and multiplying by the full coefficients in
\eqref{eq:K5}.
This statement includes all lower differential terms
on these regular charts.
\end{proposition}

\begin{proof}
On these sectors both retarded time factors are
constant one. The two defining functions are regular
and independent. The undifferentiated normal amplitude
has degree zero, and the displayed kernel has at most
four derivatives.

The ambient dimension is sixteen and the regular
base codimension is two. Equation~\eqref{eq:order}
gives
\[
 4-\frac{16-4}{4}=1.
\]
The exact polynomial amplitude description follows
by applying the finite differential expression to
the two normal deltas.
\end{proof}

The matching used here is a submersion pullback of a
tensor product. It has no failed product hypothesis.
This is an internal tree calculation, not the
factorwise cross-line audit of XIV-G.

\subsection{The noncharacteristic internal diagonal}

Consider the first placement near $a=z$, with $b-a$
a nonzero future null separation. In the relative
internal variable $r=a-z$, the retarded kernel has
a diagonal operator part. Away from the internal
characteristic covectors
\[
 p(\kappa)=\eta^{\mu\nu}\kappa_\mu\kappa_\nu=0,
\]
its leading Fourier multiplier is $-1/p(\kappa)$,
because the symbol of $\Box$ is $-p(\kappa)$.

This describes the ordinary elliptic diagonal chart
of the fixed retarded kernel. It is not an ellipticity
assertion on its characteristic set.

The corresponding diagonal kernel relation is
\begin{equation}
 \Lambda_{\mathrm d,+}
 =
 \left\{
 (w,\theta\,\mathrm d((b-a)^2)
                 +\kappa_\mu\mathrm d(a^\mu-z^\mu)):
 (b-a)^2=0,\ a=z
 \right\}\setminus0.
 \label{eq:Ldiag}
\end{equation}
The second placement has the analogous formula with
$b-z$ and $z-a$.

\begin{proposition}[Regular noncharacteristic-diagonal chart]
\label{prop:K5diag}
On a conic chart where the outer cone frequency and
the internal diagonal frequency are nonzero and
comparable, and where
\[
 |p(\kappa)|\ge c|\kappa|^2,\qquad c>0,
\]
each $K_5$ placement has an ordinary conormal
representation of upper order $\frac12$.

Its amplitude is obtained from the finite differential
formula \eqref{eq:K5}, the outer delta amplitude,
and the internal classical multiplier
$-1/p(\kappa)$, together with their lower terms.
\end{proposition}

\begin{proof}
The internal multiplier has order $-2$ on the
stated noncharacteristic chart. The complete kernel
differentiates its two elementary factors at most
four times, giving total normal amplitude order at
most two.

The regular diagonal-plus-outer-cone submanifold
has codimension five in dimension sixteen.
Thus
\[
 2-\frac{16-10}{4}=\frac12.
\]
The separation from $p(\kappa)=0$ ensures that the
multiplier and its differentiated coefficients are
classical symbols on this chart.
The finite kernel formula retains their lower
differential terms.
\end{proof}

The frequency comparability is a chart condition for
this order statement, not a cutoff introduced into
the observable. No uniform assertion is inferred
as that chart approaches $p(\kappa)=0$.

%=====================================================================
\section{The internal transition and the first missing theorem}
\label{sec:transition}
%=====================================================================

\subsection{A relation calculation at the cone vertex}

We now keep both the diagonal and the limiting flow-out
of the internal line in the first $K_5$ placement.
This is an internal relation calculation. It is not
the matching excluded in XIV-F.

Work where the outer cone frequency $\theta$ is
nonzero and the internal covector $\kappa$ is
nonzero on one fixed time-oriented null sheet.
Write
\[
 q=(b-a)^2,\qquad r=a-z.
\]

The diagonal Lagrangian piece is
\[
 \Ldiag=
 \left\{
 q=0,\ r=0;\
 \omega=\theta\,\mathrm dq+\kappa_\mu\mathrm dr^\mu
 \right\}.
\]
Here $\kappa$ is unrestricted near the chosen nonzero
null covector.

The separated internal flow-out piece can be
parameterized by
\begin{equation}
 \begin{gathered}
 q=0,\qquad
 z=a-\tau\kappa^\sharp,\qquad
 p(\kappa)=0,\qquad \tau>0,\\
 \omega=\theta\,\mathrm dq
                     +\kappa_\mu(\mathrm da^\mu-\mathrm dz^\mu).
 \end{gathered}
 \label{eq:Lflow}
\end{equation}
Choose the covector sheet so that the separation
$a-z$ is future directed. Its closure has $\tau=0$
as its boundary.

All covectors in these formulas have zero $y$
component. Unlike the balanced testing example,
their outer $b$ covector is nonzero when $\theta\ne0$.

\begin{proposition}[Two pieces, a clean intersection, and no single relation]
\label{prop:branching}
The local diagonal piece $\Ldiag$ and the closure of
the flow-out piece \eqref{eq:Lflow} each have dimension
sixteen. They meet at
\[
 r=0,\qquad q=0,\qquad p(\kappa)=0
\]
in dimension fifteen.

Their tangent intersection is exactly the tangent
space of that intersection. Thus the two pieces
intersect cleanly.

Their tangent sum has dimension seventeen.
Consequently no single smooth sixteen-dimensional
Lagrangian submanifold can contain both full local
pieces at their intersection.
\end{proposition}

\begin{proof}
The parameters for $\Ldiag$ are:
$a$ with four components, the nonzero null separation
$b-a$ with three components, $y$ with four components,
$\theta$ with one component, and $\kappa$ with four
components. Their total is sixteen.

For the flow-out closure, replace the unrestricted
four-dimensional $\kappa$ by a three-dimensional
nonzero null covector and add the parameter $\tau$.
The dimension is again sixteen.

At $\tau=0$ the base incidence is $a=z$.
The further equation $p(\kappa)=0$ gives a
fifteen-dimensional intersection in the diagonal
piece.

A tangent vector to the diagonal piece has
$\delta r=0$, with no constraint on $\delta\kappa$.
A tangent vector to the flow-out closure at $\tau=0$
satisfies
\[
 \delta r=\delta\tau\,\kappa^\sharp,\qquad
 \kappa^\sharp\cdot\delta\kappa=0.
\]
To lie in both tangent spaces it must therefore have
$\delta\tau=0$ and satisfy the linearized null
condition on $\delta\kappa$. These are precisely
the tangent conditions of the fifteen-dimensional
intersection.

The tangent sum has dimension
$16+16-15=17$. A smooth sixteen-dimensional
submanifold containing both pieces would have to
contain this seventeen-dimensional tangent sum,
which is impossible.
\end{proof}

The use of the boundary tangent in this proposition
means the tangent of the smooth extension of the
flow parameterization across $\tau=0$. The
one-sided flow-out has that same tangent space.

\begin{remark}[The exact geometric conclusion]
This is not a non-clean matching. The two pieces
intersect cleanly.

The difficulty is that their clean intersection
does not make their union one ordinary Lagrangian
relation with one classical amplitude chart.
The diagonal symbol chart and the flow-out symbol
chart are different charts with a nontrivial
transition problem.

The number seventeen in the tangent-sum calculation
has no connection to XIV-F's seventeen-dimensional
linearized matching space.
\end{remark}

The second $K_5$ placement has the corresponding
transition with the internal separation reversed.
Neither placement is deleted, and no cancellation
between them is presumed.

\subsection{What the exact tensor-product identity supplies}

Proposition~\ref{prop:K5exact} supplies a defined kernel
at the transition. It also supplies a valid
distributional pullback representation there.

It does not, without a further theorem, supply:
\begin{enumerate}
\item a uniform amplitude space describing the diagonal
and flow-out pieces together;
\item the estimates relating the noncharacteristic
diagonal symbols to the separated flow-out symbols;
\item the uniform parameter dependence needed when
the residual variable $z$ is retained in a subsequent
joint composition.
\end{enumerate}

An intersecting-relation calculus can, in other
problems, encode more than one ordinary Lagrangian
piece. The present paper does not invoke such a
calculus to assert these three conclusions for
the full tensor-valued $K_5$ kernel.

Nor is a resolved paired construction substituted
for the selected route.

\subsection{Why the regular charts do not solve this transition}

The noncharacteristic amplitude contains the
factor $1/p(\kappa)$. Its classical symbol bounds
are not uniform as $p(\kappa)\to0$.
The separated-cone representation uses a different
base incidence, with $\tau>0$.
Proposition~\ref{prop:branching} proves that these
incidences cannot be identified as one smooth
ordinary relation at $\tau=0$.

Thus there is an actual additional amplitude
problem to solve. Merely taking the union of
\eqref{eq:L5plus} and \eqref{eq:Ldiag}, or quoting
a wavefront bound for the existing tree kernel,
does not solve it.

This observation is not a nonexistence theorem
for a suitable amplitude calculus. No universal
claim about all representations in
Definition~\ref{def:class} is made.

\begin{theorem}[Exact H2 verdict]
\label{thm:H2}
The following internal data have been constructed:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the untested regular source incidence;
\item exact kernel maps for $K_8$ and $K_9$, with
regular conormal amplitudes on the retained
separated-endpoint patch;
\item exact tensor-product pullback formulas for
both $K_5$ placements;
\item their regular separated-cone and
noncharacteristic-diagonal symbol charts;
\item the clean diagonal/flow-out intersection
and the failure of a single ordinary Lagrangian
identification there.
\end{enumerate}

A uniform internal amplitude construction across
the $K_5$ characteristic-diagonal transition has
not been supplied. Hence the complete H2
internal-symbol requirement is \textbf{Unresolved}.

The first missing gate is H2, within G2/F2/E1/D2/J2.
H3 is not entered as a characteristic representation
construction.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:sourceincidence} proves (i).
Proposition~\ref{prop:K8exact},
Corollary~\ref{cor:K8order}, and
Proposition~\ref{prop:K9} prove (ii).
Proposition~\ref{prop:K5exact} proves (iii).
Propositions~\ref{prop:K5flow} and
\ref{prop:K5diag} prove (iv).
Proposition~\ref{prop:branching} proves (v).

None of these results constructs the uniform
amplitude map across the identified internal
transition. An exact distributional pullback
formula is not the required symbol-map theorem.
The stated first-missing-gate rule therefore
places the stop at H2.
\end{proof}

\scopebox{Debugged gate assignment}{%
A defined $K_5$ tree kernel is not a missing distribution.
The missing object is its uniform internal amplitude
description for the later joint composition.

\smallskip
It would be incorrect to mark H2 Constructed solely
from that kernel's existence and then report the same
amplitude gap as an H3 failure.

\smallskip
The legal ending is \textbf{Unresolved at H2}.
The narrower single-Lagrangian identification is excluded,
but the full class of allowed representations is not.}

%=====================================================================
\section{H3: common representation not entered}
\label{sec:H3}
%=====================================================================

The exact regrouping \eqref{eq:regroup} continues to
hold for the already defined smooth-slot tree
functionals. The new kernel formulas refine three
pieces of that identity with $z$ retained.

They do not construct one characteristic operator
representation of \eqref{eq:target}.

The pieces remain:
\begin{enumerate}
\item $C_G$ after the exact cancellation of the
explicit second derivative of the scalar test;
\item one divergence of the complete tensor
$B^{\alpha\mu}[G;h]$;
\item the complete scalar root \eqref{eq:root}.
\end{enumerate}
The $B_6$ numerator is not substituted for the second
piece or for the complete sum.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.20\linewidth}
                              p{.39\linewidth}X@{}}
\toprule
Piece & Established internal information & Still required\\
\midrule
$K_5$ &
Exact two-placement tensor pullbacks and regular
relation charts &
Uniform amplitude description at the internal
characteristic diagonal.\\[0.3em]
$K_8$ &
Exact retarded-convolution reduction and regular
conormal representation &
Compatibility with a common representation of all
retained pieces and the grouped cross-lines.\\[0.3em]
$K_9$ &
Complete root, including the mass term, and regular
conormal representation &
The corresponding common-composition theorem.\\[0.3em]
$K_4,K_6,K_7$ &
Inherited exact tree kernels and regrouping &
Their participation in that same justified
common representation.\\
\bottomrule
\end{tabularx}
\end{center}

No ordinary product theorem is asserted for multiplying
these kernels by the grouped cross-line difference
at $N$. The copied XIV-G audit already prevents
an automatic factorwise argument. It is not retried.

A generalized composition would have to be proved
for its own actual relation and amplitude hypotheses.
No such theorem is supplied here.

No equality
\[
 \Ucan=W\setminus\Nbar
\]
is asserted. The pre-existing canonical domain remains
only the domain on which the complete specified
products have already been proved.

%=====================================================================
\section{H4: the joint section and remainder are unentered}
\label{sec:H4}
%=====================================================================

H4 is recorded only. H3 has not been constructed.

The individual conormal orders
\[
 K_8,K_9:\ \mu\le\frac32,\qquad
 K_5\text{ separated cones}:\ \mu\le1,\qquad
 K_5\text{ regular diagonal chart}:\ \mu\le\frac12
\]
are internal kernel orders in their stated charts.
They are not transverse scaling degrees of
$\mathcal T_{\mathrm{form}}$.

If a later paper supplies the common representation,
its joint-section and remainder analysis must control:
\begin{enumerate}
\item lower-order terms of every numerical line;
\item all internal vertices and their actual kernel maps;
\item attachment-diagonal derivatives;
\item the relative transverse directions as geometric
constraints, not integration fibers;
\item every actual source coefficient;
\item both terms of the scalar root;
\item the complete grouped cross-line difference.
\end{enumerate}

No zero or nonzero joint section is assumed.
A common representation, even if constructed,
would not by itself prove cancellation.

\begin{proposition}[No cancellation conclusion]
\label{prop:nocancellation}
The results of this paper prove neither cancellation
nor failure of cancellation of
$\mathcal T_{\mathrm{form}}$.
\end{proposition}

\begin{proof}
The constructed formulas concern internal tree
kernels and regular internal symbol charts.
No joint characteristic composition of the
complete six-branch sum with the cross-lines is
constructed.

The transition calculation concerns the geometry
of two internal relation pieces, not the summed
quantum amplitude. No controlled remainder for
that amplitude is supplied.
\end{proof}

%=====================================================================
\section{H5: the prescription interface remains closed}
\label{sec:H5}
%=====================================================================

No prescription is entered. A future prescription
must name:
\begin{enumerate}
\item its ambient space;
\item a reference distribution on a proved domain;
\item the exact locus across which it extends;
\item agreement with the grouped frozen prescription;
\item a finite uniform transverse degree;
\item a closed conic wavefront bound retaining the
off-locus singularities;
\item a normalization selecting one map or a
proved uniqueness theorem.
\end{enumerate}

The test locus remains
\[
 N=\Nlift,\qquad \codim_WN=8.
\]
It is not converted into a codimension-four
numerical extension problem.

The internal response relation $\Lresp$ does not
change that test locus. It is a relation of a
precomposition kernel, not a characteristic
extension locus for the grouped quantum target.

The prospective bound
\[
 \WF(t)\subset
 \overline{\WF(t^\circ)}\cup N^*N
\]
requires its own hypotheses. A bound by $N^*N$
alone cannot erase nonsmooth reference values
away from $N$.

The Brunetti--Fredenhagen and Dang extension
frameworks remain comparison frameworks only
\cite{BF,Dang}. No theorem from them is applied
to the absent joint characteristic distribution.

XIII-E's family is not imported. No cutoff-and-restore
limit, finite part, characteristic contact, or
new ambiguity class is introduced.

%=====================================================================
\section{H6: downstream gates and non-identifications}
\label{sec:H6}
%=====================================================================

J3 opens only for a distribution produced by a
successful joint-section/remainder construction,
or by a later fully specified prescription.
Neither is supplied.

The parallel-line degree eight remains a collision
degree. It is not a degree along $\Nchar$, $\Nbar$,
or $\Nlift$, and none of the internal conormal
orders above changes that statement.

J4 remains closed: no characteristic distribution
has been glued to the permitted partial contacts
and singular spectators.

J5 remains closed: properness remains support
control for existing kernels. The internal proper
convolution in this paper does not establish the
proper pushforward or output wavefront theorem
for the missing quantum coefficient.

No $\Rquant$ is released.
The tree datum is not substituted for it.

No $s_{\mathrm{ext}}$, $s_\Psi$, or $s_0$ is applied.
No Ward defect is formed.
The classical breaking density and the value
$-8I$ neither define nor normalize this graph.

There is no new vertex, state, numerical loop
value, residue subtraction, or sharp-width limit.
The inverse remains $-1$; neither the X-track
inverse nor a spurion law is introduced.

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

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.07\linewidth}
                              p{.27\linewidth}X@{}}
\toprule
Gate & Ending & Exact scope\\
\midrule
H0 & Frozen &
XIV-G package copied. XIV-F geometry is a boundary,
not a composition relation.\\[0.35em]
H1 & One route named &
Internal-relation route. XIII-E, resolved pairing,
and prescriptions unentered.\\[0.35em]
H2 & Partial construction; Unresolved &
Source incidence and exact maps constructed.
$K_8,K_9$ have regular conormal representations.
$K_5$ has regular charts and explicit transition
geometry, but no uniform internal amplitude map
through that transition.\\[0.35em]
H3 & Not entered &
No common characteristic operator representation
of the regrouped six-branch coefficient.\\[0.35em]
H4 & Recorded only &
No joint section or complete remainder theorem.\\[0.35em]
H5 & Not entered &
No characteristic prescription or transverse degree.\\[0.35em]
H6 & Recorded &
J3--J5 and the Ward question remain unopened.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{theorem}[XIV-H mathematical release]
\label{thm:release}
For the unchanged retained instance:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the local source incidence is defined before
residual testing and retains its complete coefficients;
\item the two gravitational-response legs have
the exact kernel maps
\eqref{eq:K8reduced} and \eqref{eq:K9};
\item on the retained separated-endpoint patch
these legs have regular conormal representations
of upper order $\frac32$;
\item the two $K_5$ placements have the exact
tensor-product pullback formula \eqref{eq:K5};
\item their regular separated-cone and
noncharacteristic-diagonal orders are those
proved above;
\item their internal diagonal and limiting flow-out
pieces intersect cleanly but cannot be identified
as one smooth ordinary Lagrangian relation;
\item the uniform internal amplitude construction
at that transition, and hence the common
characteristic representation, is not supplied.
\end{enumerate}

The ending is \textbf{Unresolved at H2},
within G2/F2/E1/D2/J2.

No universal representation obstruction,
joint-section conclusion, characteristic
prescription, or admitted $\Rquant$ is asserted.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:sourceincidence} proves (i).
Proposition~\ref{prop:K8exact} and
Proposition~\ref{prop:K9} prove (ii).
Corollary~\ref{cor:K8order} and
Proposition~\ref{prop:K9} prove (iii).

Proposition~\ref{prop:K5exact} proves (iv).
Propositions~\ref{prop:K5flow} and
\ref{prop:K5diag} prove (v).
Proposition~\ref{prop:branching} proves (vi).
Theorem~\ref{thm:H2} identifies the missing
amplitude theorem and the correct first stop.
Proposition~\ref{prop:nocancellation} separates
these results from a cancellation verdict.
\end{proof}

\subsection*{What has advanced}

The internal source vertex is no longer hidden
inside a smooth tested operator.
Its two-field incidence is written explicitly.

Both gravitational-response legs have exact
separate kernel formulas. The coordinate leg's
intermediate retarded propagation is evaluated
without touching the quantum cross-lines.
The scalar leg retains its derivative and mass
terms.

The remaining internal issue is also more precise.
For $K_5$, the exact tree kernel exists, but its
regular diagonal and flow-out symbols do not
form one ordinary amplitude chart at the internal
characteristic diagonal. Their clean intersection
is a relation-interface problem, not the excess
identification rejected in XIV-F.

These results do not solve that interface problem
or the complete characteristic contraction.

\subsection*{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged XIV-G package;
\item the untested source incidence and its
regular relation;
\item the exact coordinate-leg convolution map
and its regular conormal representation;
\item the separate complete scalar-leg root map;
\item the exact two-placement $K_5$ tensor
pullbacks;
\item their regular relation charts and orders;
\item the clean diagonal/flow-out intersection
and the exclusion of a single ordinary
Lagrangian identification;
\item the precise missing uniform internal
amplitude theorem and the H2 stop.
\end{enumerate}

It receives no transverse degree of the target,
no common characteristic representation,
no cancellation theorem, and no permission to
call the tangent defect an excess.
It receives no $\Rquant$.

To advance the same route, a successor must first
construct the missing internal amplitude map
without replacing the frozen model or discarding
a placement. It must then justify one common
operator representation of the complete regrouped
sum. Only thereafter may it enter the joint-section
and remainder gates.

\subsection*{Conclusion}

\[
 \boxed{
 \begin{gathered}
 \Kret=K_4+K_5+K_6+K_7+K_8+K_9,\qquad
 \Gret=-\PP\ER,\qquad
 N=\Nlift\subset W,\\[2pt]
 \Rker*\Rker
 =\frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2),\\[2pt]
 K_8,K_9\in I^{3/2}(W,\Lresp)
 \quad\text{on the stated separated patch},\\[2pt]
 K_5\text{ has defined tensor pullbacks and regular
 symbol charts},\\[2pt]
 \text{its uniform internal transition amplitude:
 Unresolved at H2},\\
 \text{common characteristic representation and }\Rquant:
 \text{not constructed}.
 \end{gathered}}
\]

A defined tree kernel is not a uniform internal
symbol theorem. A clean intersection is not one
ordinary relation. An internal conormal order is
not a transverse degree of the quantum target.
An undefined quantum coefficient is not an anomaly.

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

\paragraph{Analytic scope.}
The new proofs concern the untested source incidence,
the three internal kernel maps, an existing retarded
convolution, regular internal symbol charts, and
their characteristic-diagonal transition geometry.
No cross-line characteristic product is constructed.

\paragraph{Gate correction.}
The first missing amplitude theorem is an H2
requirement. The manuscript therefore reports
Unresolved at H2 instead of asserting an unsupported
H2 construction and moving the same gap to H3.

\paragraph{Relation scope.}
The internal response relation and the two
$K_5$ relation pieces are not $J$, the excluded
matching, or $S_6$. Their dimensions and tangent
calculations have separate meanings.
No tangent defect is renamed an excess.

\paragraph{Order scope.}
All orders assigned here belong to specified
internal regular conormal charts. They are upper
filtered orders unless equality is separately
established. No transverse scaling degree of
the grouped target is assigned.

\paragraph{Source and internal-vertex scope.}
All source coefficients remain those of the actual
density-weighted Euler source.
Internal equalities arise only from proved local
incidences. No hidden vertex is assigned a retained
configuration point.

\paragraph{Dependency scope.}
The analytic inputs are exactly those stated in
H0. XIV-A and XIV-B are not hypotheses of any
product, amplitude, or normalization argument.
The inherited relative classical ledger is not
recomputed.

\paragraph{External calculus.}
Individual conormal conventions are used only
after the relevant regular submanifolds and
relations are identified. Defined submersion
pullbacks and proper internal convolutions are
distinguished from the unconstructed uniform
amplitude map. Extension frameworks are not
applied to the missing target.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution,
proof-assistant verification, independently
executed symbolic computation, independent
referee review, or journal acceptance is asserted.

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

\bibitem{XIVG}
N.~Cheung,
\emph{Landmark XIV-G: Operator and pullback composition
of the regrouped retained germ}.
Supplied manuscript \texttt{XIV-G.tex}.
Immediate predecessor: route, local source typing,
residual-testing limitation, factorwise audit,
and exact G2 stop.

\bibitem{XIVF}
N.~Cheung,
\emph{Landmark XIV-F: Paired composition on a proposed
excess bundle}.
Supplied manuscript \texttt{XIV-F.tex}.
Excluded clean-excess identification and geometric
boundaries only.

\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}.
Exact regrouping, complete root, tested-kernel
bound, and characteristic lifts.

\bibitem{XIVD}
N.~Cheung,
\emph{Landmark XIV-D: Cancellation or prescription
on the retained noncoincident characteristic stratum}.
Supplied manuscript \texttt{XIV-D.tex}.
Six-branch germ, actual local incidences,
and branch-limited numerator.

\bibitem{XIVC}
N.~Cheung,
\emph{Landmark XIV-C: The joint two-line extension
of one selected binary pair}.
Supplied manuscript \texttt{XIV-C.tex}.
Retained stratum, positive relation, and
contact rigidity.

\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}.
Four successor obligations and primitive
collision extensions.

\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, retarded 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}.
Response formulas, source-slot sign, roots,
and existing 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 local source.

\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: tensor products, pullbacks,
products, and proper pushforward.

\bibitem{Wunsch}
J.~Wunsch,
\emph{Microlocal analysis and evolution equations:
Lecture notes from the 2008 CMI/ETH Summer School},
arXiv:0812.3181v3, 2023.
Sections~8--9: individual conormal orders,
symbols, canonical relations, and clean
composition conventions.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds
and Quantization},
ESI Lectures in Mathematics and Physics,
European Mathematical Society, 2007,
arXiv:0806.1036.
Causal Green operators and support conventions.
The internal convolution used here is computed
explicitly.

\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.
Recorded extension framework only;
not applied to the missing characteristic 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 only.

\end{thebibliography}

\end{document}