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% Em01-B-v3.tex
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% Revision B.4:
%   - complete indexed vertex and branch--vertex inventories;
%   - global finite-order internal-frequency estimates, including
%     kappa=0 and low frequencies;
%   - localized microlocal canonical-product convergence;
%   - exact eventual independence of auxiliary cutoffs;
%   - a proved polynomial smoothing-growth estimate for the complete
%     retained contraction on tests meeting the characteristic locus;
%   - separate fixed-kernel and continuous-map limit criteria;
%   - explicit treatment of auxiliary-choice comparison and gluing;
%   - degree-four reconstruction obstruction and corrected projection.
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% It proves moderate growth, not existence of a characteristic limit.
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% referee review, or quantum admission is claimed.
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\hypersetup{
  pdftitle={Em01-B: Intrinsic Regularization of the Joint Characteristic Map},
  pdfauthor={}
}

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\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\N}{\mathbb N}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma_\eps}
\newcommand{\Test}{\mathcal D}
\newcommand{\Dist}{\mathcal D'}
\newcommand{\PP}{\mathcal P}
\newcommand{\ER}{E_0^{\mathrm R}}
\newcommand{\EAm}{E_m^{\mathrm A}}
\newcommand{\Rker}{R}
\newcommand{\Bker}{B}
\newcommand{\Amap}{\mathsf A}
\newcommand{\bmap}{\mathsf b}
\newcommand{\JD}{\mathcal J_{\mathrm D}}
\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_{\mathrm D}^{(2)}}
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    \textbf{#1}\par\smallskip #2
  \end{minipage}}
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}

\title{\textbf{Em01-B}\\[0.4em]
Intrinsic Regularization of the Joint Characteristic Map\\[0.4em]
\large Descent, complete-contraction growth estimates,\\
conditional limits, and output geometry}
\author{}
\date{}

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

\begin{abstract}
We study the complete retained XIV Einstein--scalar
contraction on the space of actual reconstructed internal
kernels. An explicit mixed-vertex formula and finite
branch--vertex inventory define its smooth-cross-line
regularization on arbitrary joint tests. Every regularized
member respects every reconstruction identity.

The internal retarded-frequency estimates are extended from
nonzero pole charts to compact frequency sets containing the
origin. Localized microlocal approximation proves convergence
to ordinary products on their product-admissible domains.
A separate estimate controls the complete regularized
contraction, including tests meeting the characteristic
locus: its distributional order is at most five and its
smoothing growth is polynomial, with an explicit exponent
in terms of the crossed distribution's local order. This is
a moderate-growth estimate, not a smoothing-uniform bound
or a characteristic-limit theorem.

A degree-four null coefficient array excludes a specified
highest-degree reconstruction rule, while preserving its
associated-graded class. Fixed-kernel and continuous-map
limit criteria distinguish convergence of the named sum
from a construction on the whole reconstructed space.
Auxiliary cutoff locality, regularization comparisons,
characteristic-conormal projection, and extension ambiguity
are treated separately.

No characteristic limit, normalized quantum coefficient,
or quantum admission for the complete frozen graph is
asserted. The submission consists of the stated partial
construction and estimates.
\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) {Retained response kernel};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Positive-scale contractions};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Limit on the aligned locus};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The growth bound does not evaluate the limit on tests that meet the locus.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The scale family is not the characteristic limit. 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-Em01-B}
\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) {Retained response kernel};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Limit on the aligned locus};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The growth bound does not evaluate the limit on tests that meet the locus.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The scale family is not the characteristic limit. 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-Em01-B}
\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) {Retained response kernel};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Positive-scale contractions};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Limit on the aligned locus};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The growth bound does not evaluate the limit on tests that meet the locus.};
\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-Em01-B}
\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) {Retained response kernel};
\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) {Limit on the aligned locus};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Positive-scale contractions. The growth bound does not evaluate the limit on tests that meet the locus.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Retained response kernel is not Limit on the aligned locus. 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-Em01-B}
\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: Retained response kernel};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Positive-scale contractions};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Limit on the aligned locus};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The growth bound does not evaluate the limit on tests that meet the locus.};
\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-Em01-B}
\end{figure}

Figure~\ref{fig:split-Em01-B} records one view of the same separation. Figure~\ref{fig:chain-Em01-B} records one view of the same separation. Figure~\ref{fig:pair-Em01-B} records one view of the same separation. Figure~\ref{fig:gate-Em01-B} records one view of the same separation. The drawings add no hypothesis.


Figure~\ref{fig:scope-Em01-B} records that separation.
The drawing adds no hypothesis.



\scopebox{Scope fixed for this paper}{%
All retained internal branches and vertex terms are included.
The construction is intrinsic to the reconstructed kernel,
not to an amplitude presentation.

\smallskip
The complete-family estimate allows polynomial growth as the
smoothing parameter tends to zero. It does not establish
convergence through the characteristic locus.

\smallskip
Existence for fixed auxiliary choices, independence of those
choices, global compatibility, normalization, and quantum
admission are separate obligations.}

\tableofcontents

%=====================================================================
\section{Main results, dependencies, and conventions}
\label{sec:introduction}
%=====================================================================

\subsection{The mathematical object}

The immediate object is a local family
\[
 \Fmap_\delta:\Kint\longrightarrow
             \Dist(U;V_{\mathrm{out}}),\qquad \delta>0,
\]
obtained by smoothing the grouped crossed distribution and
keeping the internal kernel exact. Its value on the named
kernel \(\Kret\) is the complete retained contraction, not
the full globally assembled quantum coefficient.

The new conclusions are summarized below.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}Y@{}}
\toprule
Result & Exact dependency\\
\midrule
Complete regularized map and descent &
Defined internal distributions, the explicit vertex,
and smooth crossed coefficients.\\[0.3em]
Internal-frequency estimates &
The fixed retarded cone kernel and its Fourier transform;
no crossed-line product.\\[0.3em]
Ordinary-product convergence &
Localized angular separation and normal-topology
operation theorems.\\[0.3em]
Complete-contraction growth bound &
Order-zero internal seeds, at most four internal
derivatives, and finite order of the compact crossed
distribution.\\[0.3em]
Characteristic-limit criteria &
Additional weak convergence or summable Cauchy estimates;
for a map theorem, uniform input and Fourier estimates.\\[0.3em]
Characteristic projection and ambiguity &
The explicit characteristic submanifold and compact
distributional pushforward.\\
\bottomrule
\end{tabularx}
\end{center}

The complete supplied XIV-H/I/J sources fix the retained
instance \cite{XIVH,XIVI,XIVJ}. Their internal formulas
are reproduced below. Their identification with the
designated observable Hessian, global contact inventory,
and global canonical-domain construction remain inherited
inputs. The new distributional theorems are valid for
every fixed smooth source-coefficient array satisfying
the stated finite differential form.

The internal-module conclusions are checked within
XIV-I's named finite module. No equivalence with a general
intersecting-Lagrangian calculus is asserted. Source-history
details are placed in Appendix~\ref{app:evidence}.

\subsection{Distributional conventions}

Distributions and tests use a complex-bilinear pairing.
Tensor components are taken in fixed finite-dimensional
bases. Local coordinate-density conversions are included
in smooth coefficients.

For a compact \(L\) and a test supported in \(L\), write
\[
 \|\varphi\|_{r}
 =\max_{|\alpha|\le r}\sup|\partial^\alpha\varphi|.
\]
For tensor-valued tests take the maximum over components.

The strong distribution topology is generated by
\[
 p_{\mathcal B}(u)=\sup_{\varphi\in\mathcal B}|u(\varphi)|,
\]
where \(\mathcal B\) is a bounded test set. Such a set has
common compact support and bounded derivatives of every order.

For a closed cone \(\Gamma\subset T^*U\setminus0\), the
normal topology on \(\Dist_\Gamma(U)\) additionally uses
\begin{equation}
 q_{r,\chi,V}(u)
 =\sup_{\xi\in V}(1+|\xi|)^r
                    |\widehat{\chi u}(\xi)|,
 \label{eq:normal-seminorm}
\end{equation}
where \((\supp\chi\times V)\cap\Gamma=\varnothing\).
The Fourier convention is
\[
 \widehat u(\xi)=\int e^{-ix\cdot\xi}u(x)\dd x,
\]
with inverse factor \((2\pi)^{-\dim x}\).

We use normal-topology pullback continuity, fixed-factor
product continuity under the nonopposition condition, and
proper-support pushforward continuity
\cite[Proposition 5.1 and Theorems 6.1, 6.3]{BDH}.
Their wavefront criteria are the standard distributional
calculus \cite[Section 8.2]{Hormander}.

%=====================================================================
\section{Frozen expression and complete contraction}
\label{sec:expression}
%=====================================================================

\subsection{Spacetime, normalization, and geometry}

Let
\[
 M=\R^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x}.
\]
Indices are raised and lowered with \(\eta\).
Trace reversal is
\[
 (\PP r)_{\mu\nu}
 =r_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta r,
 \qquad \PP^2=\id.
\]

Retain
\begin{equation}
 G_{\mathrm{gr}}^{\mathrm R}=-\PP\ER,\qquad
 r_{\mathrm D}=\PP\ER\JD,\qquad
 \rg[\phi]=-\frac12\PP\ER(gT_0[\phi]).
 \label{eq:inverse}
\end{equation}
The frozen pair is
\begin{equation}
 \Vcal=-\frac12\int g\gamma_{\mu\nu}
                         T_0^{\mu\nu}[\phi]\dd^4x,
 \qquad
 \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi),
 \label{eq:pair}
\end{equation}
where
\[
 \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD.
\]
The inverse and insertion coefficients remain \(-1\),
and the scalar-sourced slot occurs once.

The width \(\eps>0\), observations, switches, masses,
component pairings, and partial-diagonal class \(\Part\)
remain fixed. The smooth switch \(\sig\) equals one after
the frozen onset region.

Work on a sufficiently small patch
\[
 W=A\times B\times Z\times Y,\qquad w=(a,b,z,y),
\]
with separated closures of \(A,B,Y\), \(\overline Z\subset A\),
and
\[
 b^0>y^0>a^0>\eps.
\]
The outer separations \(b-a,b-z\) are nonzero and future
directed. The residual variable \(z\) is retained.

Put
\[
 q_R=(b-a)^2,\qquad q_1=(y-a)^2,\qquad q_2=(b-y)^2,
\]
and
\begin{equation}
 \begin{split}
 N=\{(a,b,z,y):\quad
 &z=a,\quad y=a+\alpha\ell_{\mathbf n},\\
 &b=a+\beta\ell_{\mathbf n},\quad
 0<\alpha<\beta,\quad
 \ell_{\mathbf n}=(1,\mathbf n),\quad \mathbf n\in S^2\}.
 \end{split}
 \label{eq:N}
\end{equation}
This is a smooth submanifold of dimension eight in
the sixteen-dimensional \(W\).

The inherited canonical domain \(\Ucan\) is not identified
with \(W\setminus N\) or any larger characteristic complement.

\subsection{Internal source and branch formulas}

Let \(I,J,L,M\) denote component indices in the fixed
symmetric-tensor bases. Define constant maps by
\[
 a_1^{\alpha\mu}[r]=\Amap^{\alpha\mu,I}r_I,\qquad
 b_1[r]=\bmap^I r_I,
\]
and
\[
 L^{\mu,I}(s,a)
 =\sig(a)\Amap^{\lambda\mu,I}
                    \partial_{a^\lambda}\Rker(s-a).
\]
Write
\[
 \dot a_2^{\alpha\mu}[G;h]
 =\mathsf D^{\alpha\mu,I,J}h_I G_J.
\]

The designated local source derivative has coefficients
\(q_{\alpha\beta}^{L,I,J}\in C^\infty\) with
\(|\alpha|+|\beta|\le2\), and kernel
\begin{equation}
 Q^{L,I,J}(v;a,z)
 =\sum_{|\alpha|+|\beta|\le2}
 q_{\alpha\beta}^{L,I,J}(v)
 \partial_v^\alpha\delta^{(4)}(v-z)
 \partial_v^\beta\delta^{(4)}(v-a).
 \label{eq:Q}
\end{equation}
All coefficients of the designated source are retained.
The analytic results below require its finite local form,
not an unprinted numerical simplification.

The fixed retarded kernels are
\begin{equation}
 \Rker(t,\mathbf x)
 =\frac{\Theta(t)}{2\pi}\delta(t^2-|\mathbf x|^2),
 \qquad
 \Bker=\Rker*\Rker
 =\frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2).
 \label{eq:R}
\end{equation}
Set
\[
 T^{L,I,J}(b;a,z)
 =\PP^L{}_M\int\Rker(b-v)Q^{M,I,J}(v;a,z)\dd^4v,
 \qquad F=\EAm f.
\]

The complete retained sum is
\begin{equation}
 \Kret=K_{47}+K_{5,+}+K_{5,-}+K_6+K_8+K_9,
 \qquad K_{47}=K_4+K_7,
 \label{eq:Kret}
\end{equation}
with
\begin{align}
 K_{47}^{I,J}
 &=-\partial_{b^\mu}\Bigl[
 \bigl(
 f(b)\partial_{b^\nu}L^{\mu,J}(b,z)
 -\delta^\mu_\nu\partial_{b^\rho}
                     (f(b)L^{\rho,J}(b,z))
 \bigr)L^{\nu,I}(b,a)\Bigr],
 \label{eq:K47}\\
 K_{5,+}^{I,J}
 &=\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)\Bigr],
 \label{eq:K5plus}\\
 K_{5,-}^{I,J}
 &=\partial_{b^\mu}\Bigl[
 f(b)\sig(z)\partial_{z^\alpha}\Rker(b-z)
 \Amap^{\alpha\beta,J}
                  \partial_{z^\beta}L^{\mu,I}(z,a)\Bigr],
 \label{eq:K5minus}\\
 K_6^{I,J}
 &=\partial_{b^\mu}\Bigl[
 f(b)\sig(a)\mathsf D^{\alpha\mu,I,J}
 \partial_{a^\alpha}\Rker(b-a)\delta^{(4)}(a-z)\Bigr],
 \label{eq:K6}\\
 K_8^{I,J}
 &=-\partial_{b^\mu}\Bigl[
 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\Bigr],
 \label{eq:K8}\\
 K_9^{I,J}
 &=-\partial_{b^\nu}\Bigl[
 g(b)\partial_{b^\mu}F(b)
 \Amap^{\mu\nu}{}_L T^{L,I,J}(b;a,z)\Bigr]
 \notag\\
 &\hspace{1.5em}
 -m^2g(b)F(b)\bmap_L T^{L,I,J}(b;a,z).
 \label{eq:K9}
\end{align}
These formulas, with the fixed coefficient arrays, make the
retained input independently specified as a distributional
expression. Their physical identification remains the
designated inherited identity.

\subsection{Basis pairing and stress polarization}

Choose a basis \(e_I\) of covariant symmetric tensors and
its dual basis. All component contractions below use this
basis--dual-basis pairing. Off-diagonal symmetric components
therefore introduce no additional ad hoc factor of two.

For signature \(+---\), define
\begin{align}
 T_{0,\mu\nu}[\phi]
 &=\partial_\mu\phi\,\partial_\nu\phi
 -\frac12\eta_{\mu\nu}
   \bigl(\partial^\lambda\phi\,\partial_\lambda\phi-m^2\phi^2\bigr),
 \label{eq:stress}\\
 T_{\mu\nu}(\phi,\psi)
 &=\partial_{(\mu}\phi\,\partial_{\nu)}\psi
 -\frac12\eta_{\mu\nu}
   \bigl(\partial^\lambda\phi\,\partial_\lambda\psi-m^2\phi\psi\bigr).
 \label{eq:polarization}
\end{align}
Then \(T_0[\phi]=T(\phi,\phi)\), and differentiation
of \(T_0\) gives \(2T(\phi,\psi)\). Hence
\begin{equation}
 \Vcal''_{\gamma\phi}[h,\psi]
 =-\int g(y)h^{\mu\nu}(y)
                         T_{\mu\nu}(\phi,\psi)(y)\dd^4y.
 \label{eq:Hessian}
\end{equation}

The grouped crossed distribution has components
\begin{equation}
 C_I^{\mu\nu}(x_1,x_2)
 =
 H_{T,\gamma}^{\mu\nu}{}_I(x_1)\otimes H_{T,\phi}(x_2)
 -
 H_{\gamma}^{\mu\nu}{}_I(x_1)\otimes H_\phi(x_2).
 \label{eq:C}
\end{equation}
Both terms are defined tensor products. Their difference
is retained as a single array. No mixed product or
alternative covariance is introduced.

\subsection{Explicit vertex and its arbitrary-test action}

Define the linear submersion
\[
 \mathfrak f(w)=(y-a,y-b).
\]
The symbol \(\partial_{2,\mu}\) differentiates only the
second relative variable; \(\partial_2^\nu
=\eta^{\nu\lambda}\partial_{2,\lambda}\).

For smooth \(C\), put
\[
 Z_I^\nu(C)
 =\mathfrak f^*\left(
 \partial_{2,\mu}C_I^{\mu\nu}
 -\frac12\eta_{\mu\lambda}\partial_2^\nu C_I^{\mu\lambda}
 \right),
 \qquad
 M_I(C)=\mathfrak f^*(\eta_{\mu\nu}C_I^{\mu\nu}).
\]

\begin{definition}[Complete vertex operation]
\label{def:vertex}
For \(K=(K^{I,J})\in\Dist(W;V_{\mathrm{int}})\) and smooth \(C\),
define
\begin{equation}
 \Vertex(K,C)^J
 =
 \partial_{y^\nu}\left[g(y)\sum_IK^{I,J}Z_I^\nu(C)\right]
 -
 \frac{m^2}{2}g(y)\sum_IK^{I,J}M_I(C).
 \label{eq:vertex}
\end{equation}
For every compact joint dual test \(\Phi=(\Phi_J)\), this means
\begin{equation}
 \pair{\Vertex(K,C)}{\Phi}
 =
 -\sum_{I,J}\pair{K^{I,J}}{
 g(y)\left[
 Z_I^\nu(C)\partial_{y^\nu}\Phi_J
 +\frac{m^2}{2}M_I(C)\Phi_J\right]}.
 \label{eq:vertex-test}
\end{equation}
\end{definition}

\begin{lemma}[Vertex normalization]
\label{lem:vertex}
Equation \eqref{eq:vertex} is the full contraction of
\eqref{eq:Hessian}, with the remaining scalar derivative
transferred to the output kernel. It includes the kinetic,
trace-kinetic, and mass terms.
\end{lemma}

\begin{proof}
Symmetry of the metric tensor converts the symmetrized kinetic
term into
\(h^{\mu\nu}\partial_\nu\phi\,\partial_\mu\psi\).
Contracting the crossed \(\psi\) produces
\(\partial_{2,\mu}C_I^{\mu\nu}\).
The trace kinetic term produces
\(-\eta_{\mu\lambda}\partial_2^\nu C_I^{\mu\lambda}/2\).
These are the two contributions to \(Z_I^\nu\).

The polarized mass term is
\(m^2\eta_{\mu\nu}\phi\psi/2\). The leading minus sign
of the Hessian gives the negative mass coefficient in
\eqref{eq:vertex}. Integration by parts in the remaining
scalar slot gives the output divergence and
\eqref{eq:vertex-test}.
The differentiation of the quadratic stress has already
used the factor \(1/2\) in the interaction, so there is
no additional polarization factor.
\end{proof}

The output divergence is a total distributional derivative.
It retains derivatives of \(g(y)\), both relative-variable
contributions to a total \(y\)-derivative of the cross-lines,
and any \(y\)-dependence of \(K\). Appendix~\ref{app:inventory}
lists every vertex operation and every internal placement.

%=====================================================================
\section{Internal reconstruction and all-region frequency estimates}
\label{sec:internal}
%=====================================================================

\subsection{Defined seeds and finite reconstruction}

\begin{lemma}[Internal seeds]
\label{lem:seeds}
The seeds underlying \(K_{47}\), \(K_{5,+}\), and \(K_{5,-}\)
are defined tensor-product pullbacks, including at \(a=z\).
The source incidence \eqref{eq:Q} is defined.

On the retained patch, the supported sector
\[
 \Ksup=K_6+K_8+K_9,\qquad
 \Sresp=\{q_R=0,\ a=z\},
\]
is a finite sum of at most three derivatives of smooth
coefficients times \(\delta(q_R)\delta^{(4)}(a-z)\).
\end{lemma}

\begin{proof}
The coordinate maps
\[
 (a,b,z,y)\mapsto(b,b-a,b-z,y),
\]
\[
 (a,b,z,y)\mapsto(a,b-a,a-z,y),\qquad
 (a,b,z,y)\mapsto(z,b-z,z-a,y)
\]
are invertible linear maps. They identify the three seeds
with tensor products of existing retarded distributions.

The map \((v,a,z)\mapsto(v-z,v-a)\) is a submersion.
Its pullback of the delta at the origin defines the
source incidence before differentiation.

The convolution in \eqref{eq:R} is support-proper in the
intermediate variable on compact endpoint sets. Its spatial
transform is
\[
 \Theta(t)
 \frac{\sin(\rho t)-\rho t\cos(\rho t)}{2\rho^3},
\]
with value \(\Theta(t)t^3/6\) at \(\rho=0\), obtained by
causal time convolution. The displayed formula for
\(\Bker\) has the same spatial transform.

On the separated chronological patch,
\[
 \Rker(b-a)=\frac1{2\pi}\delta(q_R),\qquad
 \partial_{b^\alpha}\Bker(b-a)
 =\frac{(b-a)_\alpha}{4\pi}\delta(q_R).
\]
The source contributes at most two derivatives and the
output divergence at most one. Thus \(K_8,K_9\) have
the asserted form; \(K_6\) needs at most two derivatives.
\end{proof}

Choose adapted charts for the two \(K_5\) placements,
\[
 (u_\pm,\zeta_\pm,q_\pm,r_\pm,y),
\]
with
\[
 (q_+,r_+)=((b-a)^2,a-z),\qquad
 (q_-,r_-)=((b-z)^2,z-a).
\]
Here \(u_+=a,u_-=z\), and \(\zeta_\pm\) complete the
outer coordinates. Also use response coordinates
\[
 (s,n),\qquad n=(q_R,a-z)\in\R^5.
\]

\begin{definition}[Presentation and reconstructed spaces]
\label{def:presentation}
For \(0\le d\le4\), the \(K_5\) array spaces are
\[
 P_{5,\pm,d}
 =
 \prod_{j+|\beta|\le d}C^\infty(W;V_{\mathrm{int}}),
\]
with reconstruction
\begin{equation}
 \Rec_{5,\pm,d}(c)
 =
 \sum_{j+|\beta|\le d}
 c_{\pm,j\beta}(w)\delta^{(j)}(q_\pm)
                         \partial_{r_\pm}^\beta\Rker(r_\pm).
 \label{eq:Rec5}
\end{equation}
For \(K_{47}\), use arrays multiplying derivatives of order
at most four of
\(\Rker(b-a)\Rker(b-z)\).
For \(K_6\), use arrays multiplying
\(\partial_n^\alpha\delta^{(5)}(n)\), \(|\alpha|\le2\).
For \(K_8,K_9\), use separate such arrays with
\(|\alpha|\le3\).

Their finite direct sum is \(\Pres\).
The reconstruction map \(\Rec\) sums the blocks, and
\[
 \Kint=\im\Rec\subset\Dist(W;V_{\mathrm{int}})
\]
carries the quotient topology transported from
\(\Pres/\ker\Rec\).
All array spaces use compact-set \(C^\infty\) seminorms.
\end{definition}

The fixed kernels are elements of these spaces after finite
chain-rule and Leibniz expansion. The spaces permit arbitrary
smooth arrays in the specified module; this does not make
every array a new physical source.

\begin{lemma}[Order-four reconstruction estimate]
\label{lem:reconstruction}
For every compact \(L\Subset W\), there are a continuous
coefficient seminorm \(p_{L,4}\) and a constant \(C_L\) such that
\begin{equation}
 |\Rec A(\Psi)|
 \le C_Lp_{L,4}(A)\|\Psi\|_4,
 \qquad \supp\Psi\subset L.
 \label{eq:internal-order}
\end{equation}
Consequently reconstruction is strongly continuous,
\(\ker\Rec\) is closed, and the quotient is Hausdorff.
\end{lemma}

\begin{proof}
The retarded kernel \(\Rker\) is a locally finite positive
measure. Indeed its integral on a compact future cone segment
has the radial form
\[
 \frac1{4\pi}\int_0^{T}t\dd t\int_{S^2}(\cdots)\dd\Omega.
\]
Thus the tensor-product seeds are locally order zero.
The normal-delta seeds are likewise locally finite measures.
Smooth changes of coordinates preserve order-zero estimates
on compact subsets, with their density factors included.

Every reconstructed generator is a derivative of order at
most four of one of these measures, multiplied by a smooth
coefficient. Transferring derivatives to the product of the
coefficient and test proves \eqref{eq:internal-order}.
The seminorm may be taken as the maximum of coefficient
derivatives through order four on a slightly enlarged
fixed compact chart set.

A bounded test set has common support and bounded derivatives,
so the estimate proves strong continuity. Closedness of the
kernel and Hausdorffness of the quotient follow.
\end{proof}

\subsection{Exact retarded multiplier}

Write \(\kappa=(\omega,\xi)\), \(\rho=|\xi|\), and
\(p(\kappa)=\omega^2-\rho^2\).

\begin{lemma}[Retarded multiplier and pole identity]
\label{lem:pole}
The Fourier transform of the fixed kernel is
\[
 M_{\mathrm R}(\omega,\xi)
 =
 -\lim_{\delta\downarrow0}
        [(\omega-i\delta)^2-\rho^2]^{-1}.
\]
For \(\rho>0\),
\begin{equation}
 M_{\mathrm R}
 =-\frac1{2\rho}
 \left[(\omega-\rho-i0)^{-1}
                 -(\omega+\rho-i0)^{-1}\right].
 \label{eq:partial-fractions}
\end{equation}
Near a nonzero characteristic sheet
\(\lambda_s=\omega-s\rho=0\), with
\(|\lambda_s|<c\rho\), \(0<c<1\),
\begin{equation}
 M_{\mathrm R}
 =-\frac{s}{2\rho}(\lambda_s-i0)^{-1}
       +\frac{s}{2\rho(\omega+s\rho)}.
 \label{eq:pole}
\end{equation}
The full multiplier is homogeneous of degree \(-2\).
\end{lemma}

\begin{proof}
The spatial Fourier transform of \(\Rker\) is
\(\Theta(t)\sin(\rho t)/\rho\).
Temporal damping gives
\[
 \int_0^\infty e^{-(\delta+i\omega)t}
                 \frac{\sin(\rho t)}{\rho}\dd t
 =-\bigl[(\omega-i\delta)^2-\rho^2\bigr]^{-1}.
\]
The damped existing kernels converge in tempered
distributions. Partial fractions give the two identities.

The physical kernel has homogeneity degree \(-2\) in
dimension four, so its Fourier transform has degree \(-2\).
Alternatively positive dilation in the damped identity
gives the same distributional conclusion.
\end{proof}

The temporal inverse transform of the complete
\eqref{eq:partial-fractions} is
\(\Theta(t)\sin(\rho t)/\rho\).
A restriction of \eqref{eq:pole} to a pole chart is an
identity on that chart; after a frequency cutoff, its inverse
transform is the corresponding microlocalized kernel, not
the whole uncut kernel.

\subsection{A global compact-frequency bound}

Let \(B_-(\lambda)=(\lambda-i0)^{-1}\).

\begin{lemma}[Frequency control through the origin]
\label{lem:origin}
For every \(R>0\), there is \(C_R\) such that
\begin{equation}
 |\pair{M_{\mathrm R}}{\psi}|
 \le C_R\|\psi\|_1,
 \qquad \supp\psi\subset[-R,R]\times\{|\xi|\le R\}.
 \label{eq:MR-order}
\end{equation}
This includes \(\kappa=0\).
Every frequency derivative of order \(k\) has a corresponding
bound using \(\|\psi\|_{k+1}\).
\end{lemma}

\begin{proof}
On fixed compact intervals, the principal-value subtraction
and the delta term give
\[
 |\pair{B_-}{h}|\le C_R\|h\|_1.
\]
The same bound holds uniformly for shifts by
\(\pm\rho\), \(0\le\rho\le R\), after enlarging the
fixed interval.

The global distributional interpretation of
\eqref{eq:partial-fractions} is therefore
\[
 -\frac12\int_{\R^3}\frac{\dd^3\xi}{|\xi|}
 \left[
 \pair{B_-}{\psi(\,\cdot+|\xi|,\xi)}
 -
 \pair{B_-}{\psi(\,\cdot-|\xi|,\xi)}
 \right].
\]
The shifted tests have a common compact temporal support.
The integrand is bounded by \(C_R\|\psi\|_1/|\xi|\).
Its spatial integral is finite:
\[
 \int_{|\xi|\le R}\frac{\dd^3\xi}{|\xi|}=2\pi R^2.
\]
The expression is the Fourier transform of the retarded
kernel, as follows by the temporal transform of its
spatial sine representation. It thus defines the same
distribution at the origin, rather than an extra extension
chosen there. This proves \eqref{eq:MR-order}.
For derivatives, transfer the derivatives to the test.
\end{proof}

Define five-frequency generators
\[
 A_{j\beta}(\theta,\kappa)
 =(i\theta)^j(i\kappa)^\beta M_{\mathrm R}(\kappa),
 \qquad d=j+|\beta|.
\]

\begin{theorem}[All-region internal-frequency estimates]
\label{thm:frequency}
Fix \(j+|\beta|\le4\), a frequency multiindex \(\nu\), and
a compact set \(B\subset\R^5\). There is \(C_{B,j,\beta,\nu}\)
such that
\begin{equation}
 \left|
 \left\langle
 L^{-(d-2-|\nu|)}
       (\partial^\nu A_{j\beta})(L\,\cdot),\psi
 \right\rangle
 \right|
 \le C_{B,j,\beta,\nu}\|\psi\|_{1+|\nu|},
 \qquad L>0,\quad \supp\psi\subset B.
 \label{eq:frequency-bound}
\end{equation}
In particular, the estimate covers every compact normalized
annulus
\[
 B\subset\{1/2\le|(\theta,\kappa)|\le2\},
\]
including its part where \(\kappa=0\) and \(\theta\ne0\).
It also gives a fixed finite-order bound on low-frequency
compact sets.

For finite smooth coefficient arrays, coefficient derivatives
through order \(r\) multiply the right-hand side by their
order-\(r\) compact-set seminorm. The degree-at-most-three
part satisfies
\[
 \bigl|\pair{L^{-2}A_{[\le3]}(w;L\,\cdot)}{\psi}\bigr|
 \le C_B L^{-1}p_{H,0}(A)\|\psi\|_1,\qquad L\ge1,
\]
uniformly for \(w\) in a fixed compact parameter set \(H\).
\end{theorem}

\begin{proof}
Lemma~\ref{lem:origin}, applied after integration in the
compact \(\theta\)-variable, bounds \(A_{j\beta}\) by a
first-derivative test seminorm. The polynomial factors are
bounded with their needed derivatives on \(B\).
Transferring \(\nu\) derivatives gives the order
\(1+|\nu|\) bound.

Homogeneity is exact:
\[
 A_{j\beta}(L\,\cdot)=L^{d-2}A_{j\beta},\qquad
 (\partial^\nu A_{j\beta})(L\,\cdot)
       =L^{d-2-|\nu|}\partial^\nu A_{j\beta}.
\]
This proves \eqref{eq:frequency-bound}.

For clarity, a compact normalized annulus has a finite
cover consisting of a neighborhood of \(\kappa=0\), the
two nonzero-sheet neighborhoods, and a noncharacteristic
remainder. For example, first separate
\(|\kappa|<2a\) with a fixed \(a>0\). On its complement,
use sheet neighborhoods
\(|\omega-s\rho|<c\rho\); outside smaller such neighborhoods
the quantity \(|p(\kappa)|/|\kappa|^2\) has a positive
lower bound. The pole formula controls the sheet pieces,
the smooth multiplier \(-1/p\) controls the remainder,
and Lemma~\ref{lem:origin} controls the first region.
Thus no nonzero-sheet formula is substituted for the
origin region. The global estimate already controls the
whole cover with one finite test order.

Smooth parameter differentiation acts only on coefficients.
Finally, degree \(d\le3\) scales as \(L^{d-2}\), and
division by \(L^2\) costs at most \(L^{-1}\).
The finite sum gives the last bound.
\end{proof}

Separate inverse Fourier transforms reconstruct
\(\delta^{(j)}(q)\partial_r^\beta R(r)\), followed by
smooth coefficient multiplication. These presentation
estimates are not transverse scaling estimates of the
crossed graph.

%=====================================================================
\section{Intrinsic regularization and local canonical convergence}
\label{sec:regularization}
%=====================================================================

\subsection{Definition and exact descent}

Fix \(U\Subset W\). Choose
\(\chi\in C_c^\infty(\R^8)\) equal to one near
\(\mathfrak f(\overline U)\), and
\(\rho\in C_c^\infty(\R^8)\) with \(\int\rho=1\).
Put
\[
 \rho_\delta(x)=\delta^{-8}\rho(x/\delta),\qquad
 C_\delta=\rho_\delta*(\chi C).
\]
Define
\begin{equation}
 \Fmap_\delta(K)=i\,\Vertex(K,C_\delta)|_U.
 \label{eq:F}
\end{equation}
The smoothing parameter \(\delta\) is not the frozen
switch width \(\eps\).

\begin{theorem}[Regularized map and reconstruction descent]
\label{thm:regularization}
For each \(\delta>0\), \eqref{eq:F} is a continuous linear map
from the quotient space \(\Kint\) to strong distributions.
Its lift satisfies
\[
 \widetilde{\Fmap}_\delta=\Fmap_\delta\Rec,\qquad
 \widetilde{\Fmap}_\delta(\ker\Rec)=0.
\]
It is defined on every compact joint test, including tests
meeting \(N\). Its value on \(\Kret\) uses the complete
branch and vertex inventory.
\end{theorem}

\begin{proof}
The array \(C_\delta\) is smooth. The test formula
\eqref{eq:vertex-test} uses smooth multiplication, finite
component contractions, and output differentiation.
These operations, for fixed smooth coefficients, are
continuous on strong distributions.

Composing with Lemma~\ref{lem:reconstruction} gives a
continuous map on presentations. It depends only on
their actual reconstructed distribution and therefore
descends continuously for the quotient topology.
A null reconstruction pairs to zero with every smooth
test in \eqref{eq:vertex-test}. This proves all null
identities simultaneously.

Linearity in \(K\), together with \eqref{eq:Kret}, retains
every term. Appendix~\ref{app:inventory} specifies the
finite vertex expansion without a leading-term truncation.
\end{proof}

Here ``intrinsic'' means representation-independent on
\(\Kint\). It does not claim independence of the flat
relative-variable coordinates or of the mollifier.

\subsection{The approximation lemma}

\begin{lemma}[Constant-cone mollification]
\label{lem:mollification}
Let \(u\in\mathcal E'(\R^d)\) and let
\(A\subset\R^d\setminus0\) be a closed angular cone with
\(\WF(u)\subset\R^d\times A\). Then
\[
 \rho_\delta*u\longrightarrow u
\]
in the normal topology of
\(\Dist_{\R^d\times A}(\R^d)\).
\end{lemma}

\begin{proof}
Strong convergence follows by transposing convolution:
\[
 \pair{\rho_\delta*u-u}{\varphi}
 =\pair u{\check\rho_\delta*\varphi-\varphi}.
\]
The expression converges in smooth seminorms uniformly
on bounded test sets.

Compact support gives polynomial Fourier growth, and the
wavefront condition gives rapid decay on every closed
cone separated from \(A\). Since
\[
 \widehat{\rho_\delta*u-u}
 =[\widehat\rho(\delta\,\cdot)-1]\widehat u,
\]
a frequency ball--tail split gives weighted Fourier
convergence on such cones.

For localized Fourier seminorms, convolve this identity
with \((2\pi)^{-d}\widehat\chi\).
In a slightly enlarged tested cone, use rapid decrease
of \(\widehat u\). In its complement, angular separation
forces the Schwartz argument of \(\widehat\chi\) to
control the total frequency, dominating the polynomial
bound of \(\widehat u\). This gives uniformly small
weighted tails. Strong convergence controls the bounded
frequency region. All normal seminorms therefore converge.
\end{proof}

\begin{lemma}[Localized relative smoothing before a fixed product]
\label{lem:local-product}
Let \(f:\Omega\subset\R^n\to\R^d\) be a linear submersion,
\(u\in\Dist(\Omega)\), \(v\in\Dist(\R^d)\), and \(D\) a
constant-coefficient differential operator.
Let \(L\Subset\Omega\), and let \(\chi=1\) near \(f(L)\).
If on a neighborhood of \(L\),
\begin{equation}
 \WF(u)\cap\bigl(-f^*\WF(Dv)\bigr)=\varnothing,
 \label{eq:product-condition}
\end{equation}
then
\[
 u\,f^*\bigl(D[\rho_\delta*(\chi v)]\bigr)
 \longrightarrow u\,f^*(Dv)
\]
strongly in distributions locally near \(L\).
A finite list of component pairings can be treated
simultaneously.
\end{lemma}

\begin{proof}
At a fixed point of \(L\), compactness of normalized
directions and closedness of wavefront sets give small
base neighborhoods and closed angular cones containing
the relevant wavefronts, with a strict nonopposition
margin after pullback by \(df^T\).

Differentiate before smoothing:
\[
 D[\rho_\delta*(\chi v)]
 =\rho_\delta*D(\chi v).
\]
Choose a relative cutoff \(\sigma\) equal to one on a
smaller neighborhood of the image and supported within
the chosen relative neighborhood. Compact support of
\(\rho\) ensures that, for sufficiently small \(\delta\),
the smoothed factor on that smaller neighborhood equals
\(\rho_\delta*[\sigma D(\chi v)]\).

Lemma~\ref{lem:mollification} applies to this compact
distribution in its constant angular cone. Submersion
pullback continuity and fixed-factor product continuity
then give the local convergence.
A finite partition covers \(L\).

For a finite list of pairings, make the neighborhood
small enough for each pairing and apply the argument
with its own cones. There is no need to take a union
of incompatible component cones.
\end{proof}

\subsection{Application to every vertex term}

The crossed operators are exactly \(\id\) and the four
\(\partial_{2,\mu}\); raised derivatives are fixed linear
combinations. They do not enlarge wavefront sets.
The map \(\mathfrak f\) is a linear submersion.

For a fixed \(K\), define \(\mathcal O_K\) as the open set
where, componentwise for every vertex pairing,
\begin{equation}
 \WF(K^{I,J})
 \cap\bigl(-\mathfrak f^*\WF(DC_I^{\mu\nu})\bigr)
 =\varnothing.
 \label{eq:OP}
\end{equation}

\begin{corollary}[Complete ordinary-product convergence]
\label{cor:canonical}
On compact subsets of \(\mathcal O_K\),
\[
 \Fmap_\delta(K)\longrightarrow i\,\Vertex(K,C)
\]
strongly in distributions, independently of the normalized
compact mollifier and of a cutoff equal to one near the
relative image.

The limit agrees with the inherited canonical value
where that value is the ordinary contraction satisfying
\eqref{eq:OP}.
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:local-product} to every pairing
in Appendix~\ref{app:inventory}.
Smooth weights, finite sums, and output derivatives
preserve convergence. Each admissible mollifier has
the same ordinary product as its limit.
\end{proof}

This does not identify \(\mathcal O_K\) with an entire
characteristic complement. A generalized canonical
composition needs its own approximation theorem.

\subsection{Exact cutoff locality}

\begin{proposition}[Eventual cutoff independence]
\label{prop:cutoff}
Let \(\chi_1,\chi_2\) both equal one on a neighborhood of
\(\mathfrak f(L)\), where \(L\Subset U\).
For a fixed compact mollifier, the two regularized vertex
maps have identical values on tests supported in \(L\)
for every sufficiently small \(\delta\).
\end{proposition}

\begin{proof}
The distribution \((\chi_1-\chi_2)C\) vanishes on a
relative neighborhood of the compact \(\mathfrak f(L)\).
Its distance from a smaller compact neighborhood is positive.
For sufficiently small \(\delta\), convolution with
\(\rho_\delta\), and all its derivatives, therefore vanish
on that smaller neighborhood. The crossed coefficients
and the test action \eqref{eq:vertex-test} coincide.
\end{proof}

This is valid even for tests meeting \(N\). It proves locality
in the cutoff, not independence of the mollifier at \(N\).

%=====================================================================
\section{An actual estimate for the complete retained family}
\label{sec:moderate}
%=====================================================================

\subsection{Crossed-distribution order}

The compact array \(\chi C\) has finite distributional order.
Fix any integer \(q\ge0\) and constant \(B\) such that,
componentwise,
\begin{equation}
 |\pair{\chi C_I^{\mu\nu}}{\psi}|
 \le B\|\psi\|_q
 \label{eq:cross-order}
\end{equation}
on a fixed compact neighborhood of its support.
A compact distribution always admits such a choice.
No universal minimal \(q\) is claimed.

\begin{lemma}[Smooth crossed-coefficient bounds]
\label{lem:smooth-bounds}
For every integer \(r\ge0\),
\begin{equation}
 \max_{|\alpha|\le r}
 \sup_x|\partial^\alpha C_\delta(x)|
 \le B_r\delta^{-8-q-r},
 \qquad 0<\delta\le1.
 \label{eq:Cdelta}
\end{equation}
The constants depend on the fixed compact array and finitely
many derivatives of \(\rho\).
\end{lemma}

\begin{proof}
Use
\[
 \partial^\alpha C_\delta(x)
 =
 \pair{\chi C}{\partial_x^\alpha\rho_\delta(x-\,\cdot)}.
\]
If a compact-support cutoff is needed for the pairing,
take it equal to one near \(\supp(\chi C)\).
Each derivative in the \(q\)-order distribution estimate
costs at most \(\delta^{-1}\), and the kernel contributes
\(\delta^{-8-|\alpha|}\). The cutoff derivatives contribute
only lower powers, absorbed for \(\delta\le1\).
\end{proof}

\subsection{Complete-contraction estimate on characteristic tests}

\begin{theorem}[Polynomial smoothing growth of the complete contraction]
\label{thm:moderate}
For every compact \(L\Subset U\), there are \(C_L>0\) and
a continuous coefficient seminorm \(p_{L,4}\) such that
\begin{equation}
 \boxed{
 |\Fmap_\delta(\Rec A)(\Phi)|
 \le C_Lp_{L,4}(A)\delta^{-q-13}\|\Phi\|_5,
 \qquad \supp\Phi\subset L,\quad 0<\delta\le1.
 }
 \label{eq:moderate}
\end{equation}
Here \(q\) is any fixed order in \eqref{eq:cross-order}.

The estimate applies to arbitrary joint tests meeting \(N\),
to the complete finite vertex operation, and in particular
to the named retained sum \(\Kret\).
It proves a finite order-five, polynomial-growth bound.
It does not prove a bound independent of \(\delta\).
\end{theorem}

\begin{proof}
Apply the order-four internal estimate
\eqref{eq:internal-order} to the smooth test appearing
in \eqref{eq:vertex-test}.

Four derivatives of
\(gZ(C_\delta)\partial_y\Phi\) require at most five
derivatives of \(C_\delta\) and at most five derivatives
of \(\Phi\). The mass contribution needs at most four
derivatives of \(C_\delta\), and is bounded by the same
right-hand side. The fixed linear map \(\mathfrak f\),
metric coefficients, smooth \(g\), and finite component
sums contribute fixed constants.

Lemma~\ref{lem:smooth-bounds}, with \(r=5\), gives
\(\delta^{-8-q-5}=\delta^{-q-13}\).
All internal blocks are included in the reconstruction
estimate, so this is an estimate of the complete
regularized expression. No cancellation or branch
nonvanishing assumption is used.
\end{proof}

For the fixed array \(A_{\mathrm{ret}}\) representing
\(\Kret\), \eqref{eq:moderate} is a nonconditional estimate
\[
 |\Fmap_\delta(\Kret)(\Phi)|
 \le C_{L,\mathrm{ret}}\delta^{-q-13}\|\Phi\|_5.
\]
It establishes moderate growth of the actual retained family.
Its power is deliberately coarse. Removing that loss or
proving summable differences remains a substantive
characteristic problem.

\begin{corollary}[A two-parameter bound]
\label{cor:two-parameter}
Choose coordinates \((\tau,\nu)\) with
\(N=\{\nu=0\}\), \(\nu\in\R^8\), and fixed compact test
supports. Set
\[
 \Phi_\lambda(\tau,\nu)
 =\lambda^{-8}\Phi(\tau,\nu/\lambda).
\]
Whenever the scaled supports remain in the chart,
\begin{equation}
 |\Fmap_\delta(\Rec A)(\Phi_\lambda)|
 \le Cp_{L,4}(A)
       \delta^{-q-13}\lambda^{-13}\|\Phi\|_5.
 \label{eq:two-parameter}
\end{equation}
\end{corollary}

\begin{proof}
The fixed coordinate transformation and its density
conversion have bounded derivatives on the compact chart.
The order-five seminorm of the scaled test is bounded
by \(C\lambda^{-8-5}\|\Phi\|_5\).
Apply \eqref{eq:moderate}.
\end{proof}

This estimate is uniform after the displayed powers are
included. It is neither a smoothing-uniform transverse
bound nor an intrinsic transverse degree of the limit.

\subsection{Estimate ledger}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.23\linewidth}p{.30\linewidth}Y@{}}
\toprule
Estimate & Complete expression controlled & Status\\
\midrule
Finite test order with polynomial smoothing growth &
All internal blocks and all vertex terms &
Proved: order five and loss \(\delta^{-q-13}\).\\[0.3em]
Normalized two-parameter growth &
Tests scaled toward \(N\), including the normal origin &
Proved: \eqref{eq:two-parameter}; still has smoothing loss.\\[0.3em]
Smoothing-uniform finite-order bound &
The named complete sum near \(N\) &
Not proved. Internal order alone does not remove
\(\delta^{-q-13}\).\\[0.3em]
Fourier decay off a selected output cone &
Localized complete crossed sum &
Conditional criterion below; no characteristic
cone estimate is verified.\\[0.3em]
Summable complete-sum differences &
Dyadic regularizations of \(\Kret\) &
Not proved. This would establish a fixed-kernel limit.\\[0.3em]
Uniform input continuity for a limit map &
Every reconstructed input &
Stronger than a limit for \(\Kret\); remains conditional.\\
\bottomrule
\end{tabularx}
\end{center}

The proved row is not relabeled as the missing
smoothing-uniform estimate. A future improvement must preserve
the complete tensor and branch sum before asserting a
cancellation or a smaller growth exponent.

%=====================================================================
\section{Reconstruction obstruction and conditional limit tools}
\label{sec:limits}
%=====================================================================

\subsection{Descent and the degree-four diagnostic}

\begin{proposition}[Descent criterion]
\label{prop:descent}
A linear map \(\widetilde{\Mmap}:\Pres\to\Dist(U)\)
defines a unique map on \(\Kint\) exactly when
\[
 \widetilde{\Mmap}(\ker\Rec)=0.
\]
Continuity of the lifted map gives continuity of the
descended map for the quotient topology.
\end{proposition}

\begin{proof}
Two presentations have the same reconstruction exactly
when their difference belongs to \(\ker\Rec\).
Annihilating this kernel makes the proposed image
independent of the representative. The quotient universal
property proves the continuity statement.
\end{proof}

In one \(K_5\) block and one tensor component, choose a
compact coefficient cutoff \(b\), and define
\begin{equation}
 c_{4,0}=bq,\qquad c_{3,0}=4b,\qquad
 c_{j,\beta}=0\quad\text{otherwise}.
 \label{eq:null-array}
\end{equation}

\begin{proposition}[Failure of highest-degree reconstruction]
\label{prop:degree-four}
The array \eqref{eq:null-array} belongs to the defined
degree-at-most-four space and has zero reconstruction:
\[
 b\bigl(q\delta^{(4)}(q)+4\delta^{(3)}(q)\bigr)R(r)=0.
\]
For a suitable \(b\), its degree-four part reconstructs
to the nonzero distribution
\[
 -4b\delta^{(3)}(q)R(r).
\]
Thus \(A\mapsto\Rec(A_{[4]})\) does not descend.
\end{proposition}

\begin{proof}
The identity \(q\delta^{(j)}=-j\delta^{(j-1)}\) follows
by applying the derivative distribution to \(q\varphi(q)\).
Take \(j=4\), tensor with \(R(r)\), and multiply by \(b\).

Choose \(b=1\) near a product region meeting the support
of \(R\). A test with a nonzero third \(q\)-jet and
nonzero retarded-kernel pairing detects the degree-four
part. Proposition~\ref{prop:descent} then excludes the rule.
\end{proof}

For \(F_d=\Rec(P_{5,\pm,d})\), the quotient class
\[
 [\Rec(A_{[d]})]\in F_d/F_{d-1}
\]
is nevertheless well-defined: the discarded terms are
already in \(F_{d-1}\). This algebraic associated-graded
statement selects no representative or joint symbol.

A continuous section of a specified quotient \(q:B\to S\)
requires a complemented kernel. Indeed a section gives
the projection \(I-sq\); conversely a continuous projection
\(P\) onto \(\ker q\) gives \(s(qb)=(I-P)b\).
No such actual joint quotient and section are constructed here.

\subsection{A limit for one kernel is weaker than a map theorem}

\begin{theorem}[Fixed-kernel distributional criterion]
\label{thm:fixed-limit}
Fix \(K_*\in\Kint\).
Suppose every test pairing of \(t_\delta=\Fmap_\delta(K_*)\)
has a limit, and for every compact \(L\Subset U\) there are
\(C_L,m_L\) such that
\[
 |t_\delta(\Phi)|\le C_L\|\Phi\|_{m_L},
 \qquad \supp\Phi\subset L,
\]
uniformly in \(\delta\).
Then \(t_\delta\) converges strongly to a distribution \(t\).

This conclusion concerns \(K_*\) only, not a continuous
map on all of \(\Kint\).
\end{theorem}

\begin{proof}
The limit inherits the local finite-order estimates, so is
a distribution. On a bounded test set supported in \(L\),
the tests are totally bounded in the \(C^{m_L}\) norm.
Use a finite net, the uniform estimate, and convergence
on its finitely many tests to obtain uniform convergence
on the bounded set.
\end{proof}

\begin{theorem}[Summable fixed-kernel test]
\label{thm:dyadic-limit}
Fix \(K_*\) and put \(t_n=\Fmap_{2^{-n}}(K_*)\).
Suppose on every compact \(L\Subset U\) there are an order
\(m_L\), summable numbers \(a_{n,L}\), and numbers
\(b_{n,L}\to0\) such that
\[
 |(t_{n+1}-t_n)(\Phi)|\le a_{n,L}\|\Phi\|_{m_L},
\]
and
\[
 \sup_{2^{-n-1}\le\delta\le2^{-n}}
 |(\Fmap_\delta(K_*)-t_n)(\Phi)|
 \le b_{n,L}\|\Phi\|_{m_L}.
\]
Then the full family has a strong distributional limit,
with remainder bounded by
\[
 \left(\sum_{j\ge n}a_{j,L}+b_{n,L}\right)\|\Phi\|_{m_L}
\]
on that smoothing shell.
\end{theorem}

\begin{proof}
The first estimate makes the telescoping dyadic sequence
Cauchy in a fixed finite-order test seminorm. Its limit is
a distribution because the initial term and the sum of
increments satisfy a common local bound.
The second estimate controls every value between dyadic
parameters. The displayed tail estimate follows by
telescoping and implies strong convergence.
\end{proof}

For the named \(K_*=\Kret\), this theorem is a concrete
research criterion. No summability is supplied by the
polynomial bound of Theorem~\ref{thm:moderate}.

\subsection{A continuous characteristic-map criterion}

\begin{assumption}[Uniform map-limit hypotheses]
\label{ass:map-limit}
Fix a closed cone \(\Gamma\subset T^*U\setminus0\).
For the complete family suppose:
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every pairing \(\Fmap_\delta(\Rec A)(\Phi)\) converges.
\item For each compact \(L\), there are \(C_L,m_L\) and a
continuous input seminorm \(p_L\) such that
\[
 |\Fmap_\delta(\Rec A)(\Phi)|
 \le C_Lp_L(A)\|\Phi\|_{m_L}
\]
uniformly in \(\delta\).
\item Every Fourier seminorm admissible for \(\Gamma\)
has a bound
\[
 q_{r,\chi,V}(\Fmap_\delta(\Rec A))\le Cp(A)
\]
with a continuous input seminorm, uniformly in \(\delta\).
\end{enumerate}
\end{assumption}

\begin{theorem}[Conditional continuous map]
\label{thm:map-limit}
Under Assumption~\ref{ass:map-limit}, the limits define a
continuous linear map
\[
 \Mmap:\Kint\to\Dist_\Gamma(U;V_{\mathrm{out}}),
\]
and each fixed-input family converges in the normal topology.
The map has exact reconstruction descent and agrees with
the ordinary canonical contraction on its verified domain.
\end{theorem}

\begin{proof}
The local finite-order bounds make the pointwise limits
distributions and imply strong convergence by the
finite-net argument of Theorem~\ref{thm:fixed-limit}.
The Fourier bounds pass to the limit and give the
wavefront inclusion.

The same inequalities bound every strong and Fourier
seminorm of the lifted limit map by continuous input
seminorms. It is therefore continuous.
Every finite regularization annihilates \(\ker\Rec\),
so the limit does too; Proposition~\ref{prop:descent}
gives the descended map.

For convergence of a degree-\(r\) Fourier seminorm, the
uniform degree-\(r+1\) bound makes the tail of the difference
at most \(C_A/(1+R)\) above radius \(R\).
On the frequency ball the tests
\(\chi e^{-i\,\cdot\,\xi}\) form a bounded test set,
and strong convergence gives uniform convergence there.
This proves normal convergence.
Canonical agreement follows from
Corollary~\ref{cor:canonical}.
\end{proof}

The hypotheses are not verified for the full frozen graph.
In particular, a fixed-kernel cancellation need not persist
for arbitrary coefficients in \(\Pres\).

\subsection{Transverse tests and supported references}

In coordinates \((\tau,\nu)\) for \(N\), transverse tests
must include the normal origin. A term
\(a(\tau)\partial_\nu^\alpha\delta^{(8)}(\nu)\) pairs with
\(\Phi_\lambda\) at scale \(\lambda^{-8-|\alpha|}\), but
vanishes on every normal annular test avoiding zero.

Any smoothing-uniform transverse estimate passes to an
existing distributional limit by fixing \(\lambda\) and
taking the limit in its test pairing. The estimate
\eqref{eq:two-parameter} is not of this form, because
its smoothing loss remains.

For the narrower supported sector, \(\Sresp\) has five
normal coordinates and
\[
 \Ksup=\sum_{|\alpha|\le3}
 v_\alpha(s)\partial_n^\alpha\delta^{(5)}(n).
\]
On \(\Sresp\setminus N\), at most one crossed cone is
singular: simultaneous vanishing of \(q_1,q_2\), together
with \(q_R=0\), forces aligned future null segments.
A crossed conormal there has a nonzero \(y\)-component,
whereas \(N^*\Sresp\) has zero \(y\)-component. The
ordinary product and trace criteria hold.

Its component reference is consequently
\[
 \sum_{\alpha,\ \beta\le\alpha}
 (-1)^{|\beta|}\binom{\alpha}{\beta}
 \Ejet_{\alpha-\beta}
 \left(v_\alpha\,i^*(\partial_n^\beta C_D)\right),
\]
where
\[
 \pair{\Ejet_\gamma b}{\Phi}
 =(-1)^{|\gamma|}
       \pair b{\partial_n^\gamma\Phi(s,0)}.
\]
This is only the \(K_6+K_8+K_9\) reference. Extending it
within a fixed supported normal-order class is equivalent
to extending its unique tangential coefficient distributions.
Polynomial normal-jet tests extract those coefficients,
and their insertion gives the converse. Neither their
extension nor a full-sum characteristic composition follows
from the finite jet count \cite{XIVJ}.

%=====================================================================
\section{Auxiliary choices, local limits, and normalization}
\label{sec:choices}
%=====================================================================

\subsection{Cutoff, mollifier, and chart issues are different}

Proposition~\ref{prop:cutoff} gives eventual exact equality
for different cutoffs that equal one near the relevant
relative image. It also gives exact compatibility on
compact overlaps for the same flat relative coordinates
and the same mollifier.

Different presentations of the same internal distribution
give identical regularized values for every \(\delta\).
Thus changing internal adapted charts through valid
reconstruction identities does not change the family.

A change of the coordinates in which the crossed distribution
is mollified is a different auxiliary construction.
General coordinate invariance of that smoothing procedure
is not claimed.

\begin{proposition}[Comparison of two auxiliary limits]
\label{prop:comparison}
Fix \(K\), and suppose two local regularizations, possibly
with different mollifiers, have distributional limits
\(t^\rho,t^{\widetilde\rho}\) on \(U\).
Then
\[
 (t^\rho-t^{\widetilde\rho})|_{\mathcal O_K\cap U}=0,
\]
so
\[
 \supp(t^\rho-t^{\widetilde\rho})
 \subset U\setminus\mathcal O_K.
\]
If a common reference on a larger open set \(V\subset U\)
has separately been proved for both choices, the support
is contained in \(U\setminus V\).

No automatic conclusion that the difference is supported
only in \(N\), or is a permitted collision contact, follows.
\end{proposition}

\begin{proof}
Both limits agree with the ordinary product on
\(\mathcal O_K\) by Corollary~\ref{cor:canonical}.
Equality of restrictions is exactly the stated support
property. The same argument applies to any separately
proved common reference domain.
\end{proof}

The exact finite-parameter comparison is
\[
 \Fmap_\delta^\rho(K)-\Fmap_\delta^{\widetilde\rho}(K)
 =i\,\Vertex\left(
 K,(\rho_\delta-\widetilde\rho_\delta)*(\chi C)\right).
\]
It has the same coarse polynomial estimate as
\eqref{eq:moderate}. That estimate does not make the
comparison tend to zero.

\begin{proposition}[A sufficient independence test]
\label{prop:independence}
If both limits exist and, on each compact \(L\Subset U\),
\[
 |(\Fmap_\delta^\rho(K)-\Fmap_\delta^{\widetilde\rho}(K))(\Phi)|
 \le a_L(\delta)\|\Phi\|_{m_L},
 \qquad a_L(\delta)\to0,
\]
then the limits are equal.
If compatible local limits agree on all overlaps, they
glue uniquely to a distribution on the covered domain.
\end{proposition}

\begin{proof}
Take the limit in the comparison estimate.
Overlap agreement gives distributional gluing by applying
a partition of unity to tests; independence of the partition
follows from the agreement.
\end{proof}

Therefore a future limit is first a \emph{local limit for
specified auxiliary choices}. It becomes an independent,
compatible, normalized construction only after the
corresponding comparison and normalization conditions
have been proved.

%=====================================================================
\section{Output geometry and contact ambiguity}
\label{sec:output}
%=====================================================================

Let
\[
 \pi(a,b,z,y)=(z,y).
\]
For a compactly supported existing distribution,
\begin{equation}
 \WF(\pi_*t)
 \subset
 \{(z,y;p,q):\exists a,b,\
 (a,b,z,y;0,0,p,q)\in\WF(t)\}.
 \label{eq:pushforward}
\end{equation}
This operation does not define a missing joint product.

Set
\[
 \mathfrak Z_W
 =\{(w;\xi_a,\xi_b,\xi_z,\xi_y):
             \xi_a+\xi_b+\xi_z+\xi_y=0\}\setminus0,
\]
\[
 Z_2=\{(z,p;y,q):p+q=0\}\setminus0.
\]
If \(\WF(t)\subset\mathfrak Z_W\), then
\(\WF(\pi_*t)\subset Z_2\): the eliminated covectors
in \eqref{eq:pushforward} are zero. Smooth weights and
differentiation preserve this intermediate inclusion.

Write \(\pi_\#\Gamma\) for the corresponding cotangent
projection relation, not for distributional pushforward.

\begin{theorem}[Characteristic-conormal projection]
\label{thm:projection}
At \(w\in N\), with direction \(\ell_{\mathbf n}\), the
conormals surviving \(\pi\) are exactly
\[
 (w;0,0,-c\ell_{\mathbf n}^{\flat},
                  c\ell_{\mathbf n}^{\flat}),\qquad c\ne0.
\]
Consequently, for
\[
 \mathcal H=\{(z,y):(y-z)^2=0,\ y^0>z^0\},
\]
\[
 \pi_\#(N^*N\setminus0)
 =(N^*\mathcal H\setminus0)|_{\pi(N)}\subset Z_2,
 \qquad N^*N\setminus0\subset\mathfrak Z_W.
\]
The equality is only over \(\pi(N)\).
\end{theorem}

\begin{proof}
Simultaneous translation is tangent to \(N\), giving
total covector balance. For a surviving covector write
\(\xi_a=\xi_b=0,\xi_z=p,\xi_y=q\).
Variation of \(a\) gives \(p+q=0\), variation of
\(\alpha\) gives \(q(\ell_{\mathbf n})=0\), and
variation of \(\mathbf n\) gives
\(q(0,\delta\mathbf n)=0\) for
\(\delta\mathbf n\perp\mathbf n\).
Thus \(q=c(1,-\mathbf n)=c\ell_{\mathbf n}^{\flat}\)
and \(p=-q\).

Conversely these covectors annihilate every tangent
variation. They are precisely the residual null-cone
conormals because \(y-z=\alpha\ell_{\mathbf n}\).
Locally the permitted \(\beta\)-values are open when a
retained point exists, so \(\pi(N)\) is locally an open
part of \(\mathcal H\).
\end{proof}

The inherited contact input states that permitted
interaction collision contacts vanish on this retained
separated patch \cite{XIVJ}. The following statements
use exactly that input.

\begin{proposition}[Contact correction criterion]
\label{prop:contacts}
Corrections in that frozen collision-contact class do not
change any value on tests supported in the retained patch.
They cannot repair a convergence failure or a nonzero
reconstruction-null defect detected there.
\end{proposition}

\begin{proof}
Their restrictions to the patch are zero.
\end{proof}

\begin{proposition}[Balanced characteristic ambiguity]
\label{prop:ambiguity}
Let \(\iota:N\hookrightarrow W\), and choose a nonzero
nonnegative smooth compact density \(a\) on a small
portion of \(N\). Then
\[
 A_N=\iota_*a
\]
is nonzero, supported in \(N\), and has wavefront in
\(N^*N\setminus0\). Moreover
\[
 \pi_*A_N\ne0,\qquad \WF(\pi_*A_N)\subset Z_2.
\]
It is not a permitted collision contact on this patch.

Thus off-\(N\) agreement and a balanced residual cone
do not alone select an extension if its intermediate
allowance contains this characteristic conormal.
\end{proposition}

\begin{proof}
In local coordinates \(A_N\) is a smooth compact
tangential density times an eight-normal-variable delta.
It is nonzero and conormal.
A residual nonnegative test equal to one on the projected
support gives the positive integral of \(a\), proving
nonzero pushforward. The wavefront bound follows from
Theorem~\ref{thm:projection} and \eqref{eq:pushforward}.
It does not vanish on the patch, unlike every allowed
collision contact.
\end{proof}

No such term is added to the actual graph or to \(\Part\).
Additional normalization, tensor identities, or growth
conditions could exclude it. The proposition only shows
that agreement and an output cone are not those conditions.

%=====================================================================
\section{Conclusions and remaining admission obligations}
\label{sec:conclusion}
%=====================================================================

\begin{theorem}[Mathematical release]
\label{thm:release}
For the specified finite internal module and crossed array:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The complete regularized retained contraction is a
continuous map on actual reconstructed kernels with exact
descent.
\item The fixed retarded-frequency generators have compact-set
finite-order estimates covering all frequency regions.
\item The family converges to ordinary canonical products
under the stated localized wavefront hypotheses.
\item The complete family has the nonconditional estimate
\eqref{eq:moderate} on arbitrary characteristic tests.
\item Degree-four highest-degree reconstruction fails descent,
while its associated-graded class is well-defined.
\item Fixed-kernel and continuous-map limits follow under their
separately stated additional estimates.
\item Cutoff locality, auxiliary-limit comparison, characteristic
projection, and ambiguity have the precise domains proved above.
\end{enumerate}
No characteristic limit or quantum admission for the full
frozen graph is concluded.
\end{theorem}

\begin{proof}
Use Theorems~\ref{thm:regularization},
\ref{thm:frequency}, and \ref{thm:moderate},
Corollary~\ref{cor:canonical},
Proposition~\ref{prop:degree-four},
Theorems~\ref{thm:fixed-limit}--\ref{thm:map-limit},
Propositions~\ref{prop:cutoff},
\ref{prop:comparison}, \ref{prop:independence},
Theorem~\ref{thm:projection}, and
Proposition~\ref{prop:ambiguity}.
\end{proof}

The retained estimate is an analytic advance at the level
of moderate growth. It is not the missing smoothing-uniform
estimate or a proof that the family converges.

The prospective admission statement remains
\[
 Q_H\Rquant\in\Ealg,\qquad
 \Ealg=(\Fpol[[\hbar,\kappa]],\star_H)/
                        \Nsol[[\hbar,\kappa]],
\]
with normalization
\[
 R_1(\Vcal;\Ocal)
 =\frac{i}{\hbar}
 \left(\mathsf T_2^{\mathrm{ren}}(\Vcal,\Ocal)
                    -\Vcal\star_H\Ocal\right).
\]
To obtain it one still needs a full characteristic
construction, its joint section and complete remainder,
appropriate uniform transverse estimates, contact gluing,
the actual global output theorem, and normalization.

The historical I3/J2 full-map problem therefore remains
unresolved. No nonexistence theorem for every admissible
composition is proved. XV-B's downstream admission exit
is not opened by these partial results.

The VII interface and reflection-triangle constructions,
the V--VII-B linear-current bridge, and the separate
prepared-source programmes are not premises replacing
these graph obligations.

\[
 \boxed{
 \begin{gathered}
 \widetilde{\Fmap}_\delta(\ker\Rec)=0,\\
 |\Fmap_\delta(\Kret)(\Phi)|
       \le C_L\delta^{-q-13}\|\Phi\|_5,\\
 \text{moderate growth does not establish a characteristic limit},\\
 Q_H\Rquant\in\Ealg:\ \text{not established}.
 \end{gathered}}
\]

%=====================================================================
\appendix
\section{Complete branch--vertex inventory}
\label{app:inventory}
%=====================================================================

\subsection{Finite vertex index set}

Let \(\mathcal I,\mathcal J\) be the fixed incoming and
residual tensor-component sets, and let
\(\mathcal S=\{0,1,2,3\}\).
The finite expansion of \(\Fmap_\delta\) is the union
of three types:
\[
 \mathfrak D_{\mathrm{kin}}
 =\{(I,J,\mu,\nu):I\in\mathcal I,J\in\mathcal J,
                              \mu,\nu\in\mathcal S\},
\]
\[
 \mathfrak D_{\mathrm{tr}}
 =\{(I,J,\mu,\lambda,\nu):
    I\in\mathcal I,J\in\mathcal J,
                         \mu,\lambda,\nu\in\mathcal S\},
\]
\[
 \mathfrak D_{\mathrm{mass}}
 =\{(I,J,\mu,\nu):I\in\mathcal I,J\in\mathcal J,
                              \mu,\nu\in\mathcal S\}.
\]
Terms with zero metric coefficient simply contribute zero.

For a given internal row \(K_b\), the operations are:

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.14\linewidth}p{.23\linewidth}p{.17\linewidth}Y@{}}
\toprule
Type & Crossed entry and operator \(D\) &
Output \(O\) & Coefficient inside \(O\)\\
\midrule
Kinetic &
\(\partial_{2,\mu}C_I^{\mu\nu}\) &
\(\partial_{y^\nu}\) &
\(i\,g(y)K_b^{I,J}\).\\[0.3em]
Trace kinetic &
\(\partial_2^\nu C_I^{\mu\lambda}\) &
\(\partial_{y^\nu}\) &
\(-\frac{i}{2}g(y)\eta_{\mu\lambda}K_b^{I,J}\).\\[0.3em]
Mass &
\(C_I^{\mu\nu}\), \(D=\id\) &
\(\id\) &
\(-\frac{i m^2}{2}g(y)\eta_{\mu\nu}K_b^{I,J}\).\\
\bottomrule
\end{tabularx}
\end{center}

In every row, pull back the crossed entry by \(\mathfrak f\),
multiply by the displayed coefficient, and apply \(O\).
The fibre contraction is the basis--dual-basis contraction
of the repeated \(I\), leaving \(J\).
Summing this explicitly defined finite index set gives
\(i\Vertex(K_b,C_\delta)\).

The output derivative is applied after multiplication.
When expanded, it differentiates the switch, internal
kernel, and both relative-variable dependences of the
pulled-back crossed entry. In contrast, \(D\) differentiates
only the crossed scalar slot before pullback.

\subsection{Internal placement inventory}

Split \(K_9=K_{9,\mathrm{der}}+K_{9,\mathrm{mass}}\) using
its two displayed lines. This produces seven bookkeeping
rows, not seven new physical branches.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.13\linewidth}p{.32\linewidth}p{.12\linewidth}Y@{}}
\toprule
Internal row & Placement or seed & Internal derivative bound &
Crossed/output operations\\
\midrule
\(K_{47}\) &
\(R(b-a)R(b-z)\);
exact coefficient \eqref{eq:K47} &
\(4\) &
All three vertex types above.\\[0.3em]
\(K_{5,+}\) &
\(R(b-a)R(a-z)\);
coefficient \eqref{eq:K5plus} &
\(4\) &
All three vertex types above.\\[0.3em]
\(K_{5,-}\) &
\(R(b-z)R(z-a)\);
coefficient \eqref{eq:K5minus} &
\(4\) &
All three vertex types above.\\[0.3em]
\(K_6\) &
Local \(a=z\), outer \(q_R=0\);
coefficient \eqref{eq:K6} &
\(2\) &
All three vertex types above.\\[0.3em]
\(K_8\) &
Actual source incidence, \(B=R*R\),
and coordinate divergence; \eqref{eq:K8} &
\(3\) &
All three vertex types above.\\[0.3em]
\(K_{9,\mathrm{der}}\) &
Source response \(T\), derivative scalar root;
first line of \eqref{eq:K9} &
\(3\) &
All three vertex types above.\\[0.3em]
\(K_{9,\mathrm{mass}}\) &
Same \(T\), mass scalar root;
second line of \eqref{eq:K9} &
\(2\) &
All three vertex types above, including the
additional crossed-vertex mass type.\\
\bottomrule
\end{tabularx}
\end{center}

Thus another reader obtains the complete finite contraction
by taking the product of the seven internal bookkeeping rows
with the three explicitly indexed vertex types. There is no
unspecified ``complete expansion.''

\subsection{Source-coefficient reproducibility}

The analytic theorem is stated for the fixed smooth arrays
\(q_{\alpha\beta}^{L,I,J}\), rather than pretending that their
numerical components were independently expanded here.

For the designated physical source, these arrays may be
recovered algorithmically from its local differential
expression
\[
 (\JD'[G]h)^L(v)
 =\sum_{|\alpha|+|\beta|\le2}
 q_{\alpha\beta}^{L,I,J}(v)
               \partial^\alpha G_J(v)\partial^\beta h_I(v).
\]
At a fixed point \(v\), choose polynomial jet tests
\[
 G_J(x)=e_J\frac{(x-v)^\alpha}{\alpha!},\qquad
 h_I(x)=e_I\frac{(x-v)^\beta}{\beta!},
\]
multiplied by a cutoff equal to one near \(v\).
Evaluation of \(\JD'[G]h\) at \(v\) isolates the stated
coefficient. This preserves derivatives of the frozen
switch in the actual Euler source.

The designated source is inherited as
\[
 \JD=\EL_\gamma(\sig\ell_3),\qquad
 \ell_3=-2[\rho^3]
       \bigl(\sqrt{-\det(\eta+\rho\gamma)}
                         R[\eta+\rho\gamma]\bigr),
\]
with its fixed curvature and unintegrated-density
representative. A numerical evaluation of the physical
coefficients requires that fixed source data; the theorems
do not invent or modify it.

%=====================================================================
\section{Evidence and verification record}
\label{app:evidence}
%=====================================================================

The complete supplied \texttt{XIV-H.tex},
\texttt{XIV-I.tex}, and \texttt{XIV-J.tex} are the designated
source texts. No later registry state, byte identity,
cryptographic digest, or external source archive is certified.

Their distinct reported stops are:
\begin{itemize}
\item XIV-H: exact kernels and regular charts, but its
uniform internal transition requirement remained unresolved
at H2.
\item XIV-I: the named finite internal module was constructed
at I2; the crossed characteristic operation remained
unresolved at I3.
\item XIV-J: the supported \(K_6+K_8+K_9\) reference was
constructed off \(N\); the complete joint map remained
unresolved at J2.
\end{itemize}
These are source statements with their own domains,
not automatic premise transfers or equivalent gate definitions.

This revision independently proves the displayed
regularization, frequency, growth, convergence-criterion,
and geometry statements. It does not verify all
predecessor graph identities, nor does it collapse the
various later manuscript stopping labels.

The source is intended for pdfLaTeX, with a second run
for cross-references. No compiler was executed in the
production of this text. Accordingly, successful compilation,
pagination, proof-assistant verification, independent
symbolic computation, referee acceptance, and journal
acceptance are not claimed.

%=====================================================================
\begin{thebibliography}{99}

\bibitem{XIVH}
N.~Cheung,
\emph{Landmark XIV-H: Internal kernel relations on the
operator route}.
Complete supplied manuscript source \texttt{XIV-H.tex}.
Relevant labels:
\texttt{prop:sourceincidence},
\texttt{prop:K8exact},
\texttt{prop:K9},
\texttt{prop:K5exact},
\texttt{prop:branching},
\texttt{thm:H2}.
Copied predecessor inputs retain their stated scope.

\bibitem{XIVI}
N.~Cheung,
\emph{Landmark XIV-I: Uniform internal retarded amplitudes
at the diagonal--flow-out transition}.
Complete supplied manuscript source \texttt{XIV-I.tex}.
Relevant labels:
\texttt{def:amodule},
\texttt{prop:pole},
\texttt{thm:dyadic},
\texttt{thm:reconstruction},
\texttt{thm:I2},
\texttt{prop:precomposition},
\texttt{prop:canonical},
\texttt{thm:I3}.

\bibitem{XIVJ}
N.~Cheung,
\emph{Landmark XIV-J: The joint characteristic map on the
unchanged carrier}.
Complete supplied manuscript source \texttt{XIV-J.tex}.
Relevant labels:
\texttt{def:mapclass},
\texttt{prop:descent},
\texttt{lem:null},
\texttt{prop:normalform},
\texttt{thm:reference},
\texttt{thm:conditionalextension},
\texttt{thm:J2},
\texttt{cor:noadmission}.

\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, restrictions, products,
and wavefront calculus.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
``Continuity of the fundamental operations on distributions
having a specified wave front set
(with a counter example by Semyon Alesker),''
\emph{Studia Mathematica} \textbf{232} (2016), 201--226.
arXiv:1409.7662.
doi:10.4064/sm8316-3-2016.
Proposition~5.1 and Theorems~6.1, 6.3:
normal-topology pullback, product, and proper-support
pushforward results.

\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.
Microlocal extension hypotheses; not a theorem supplying
the unresolved full characteristic limit in this paper.

\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.
Distribution-extension and scaling framework; no
normalization of the present full graph is inferred.

\end{thebibliography}

\end{document}