\documentclass[11pt,letterpaper]{article}

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\newcommand{\D}{\mathcal D}
\newcommand{\Dp}{\mathcal D'}
\newcommand{\WF}{\operatorname{WF}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\Rec}{\mathsf{Rec}}
\newcommand{\Mmap}{\mathsf M}
\newcommand{\Pint}{\mathscr P_{\mathrm{int}}}
\newcommand{\Kint}{\mathscr K_{\mathrm{int}}}
\newcommand{\Kret}{K_{\mathrm{ret}}}
\newcommand{\Ksup}{K_{\mathrm{supp}}}
\newcommand{\Ucan}{U_{\mathrm{can}}}
\newcommand{\Cross}{\mathscr C_{\mathrm{cross}}}
\newcommand{\Rquant}{R_1^{(1)}}
\newcommand{\Eout}{\mathscr E_H}
\newcommand{\Ejet}{\mathsf E}
\newcommand{\Jjet}{\mathsf J}
\newcommand{\Ob}{\operatorname{Ob}}
\newcommand{\eps}{\varepsilon}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\PV}{\operatorname{PV}}

\title{Landmark XV-C\\
The Full Joint Characteristic Target:\\
Verified Reconstruction Tests and the Instantiation Boundary}
\author{}
\date{}

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

\begin{abstract}
XV-C addresses the proposed instantiation of XV-B's relative-lifting
framework for the frozen six-branch joint characteristic target.
The supplied data retain the grouped $K_{47}$ contribution, both
placements of $K_5$, and the supported sector $K_6+K_8+K_9$.
They do not supply the complete cross-line operators, reconstruction
space, correction class, or output definitions needed to construct
the full graph operation.

We therefore take the explicitly permitted unresolved outcome,
rather than inventing the missing graph data. The new mathematical
results are local tests needed by the proposed application.
We give a complete reconstruction-null criterion in a specified
finite normal-jet module, collect the supplied supported-sector
coefficients through normal order three, and prove an exact
distributional extension criterion expressed by test-function
estimates. This criterion yields a necessary and sufficient test for
unrestricted supported realization of the supplied normal form.
It also provides explicit failure witnesses and a support-sensitive
description of the remaining ambiguity.

These results do not evaluate the unsupplied cross-line coefficients
or establish membership in the frozen correction class. The full
joint map remains unresolved before graph-level instantiation.
No exhaustive obstruction, complete-graph wavefront theorem, or
admitted quantum coefficient is asserted.
\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) {Local reconstruction tests};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Instantiation boundary};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {The joint graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The graph stays unresolved. Local tests are not a graph obstruction.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Local tests stop at the instantiation gate. 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-XV-C}
\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) {Local reconstruction tests};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {The joint graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The graph stays unresolved. Local tests are not a graph obstruction.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Local tests stop at the instantiation gate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-XV-C}
\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) {Local reconstruction tests};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Instantiation boundary};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {The joint graph};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The graph stays unresolved. Local tests are not a graph obstruction.};
\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-XV-C}
\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) {Local reconstruction tests};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {The joint graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Instantiation boundary. The graph stays unresolved. Local tests are not a graph obstruction.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Local reconstruction tests is not The joint graph. 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-XV-C}
\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: Local reconstruction tests};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Instantiation boundary};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: The joint graph};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The graph stays unresolved. Local tests are not a graph obstruction.};
\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-XV-C}
\end{figure}

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


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



\section{Scope, conventions, and the permitted ending}
\label{sec:scope}

\subsection{What can be completed from the supplied material}

The intended target is a map
\begin{equation}
 \Mmap:\Kint\longrightarrow\Dp(W;E_{\mathrm{res}})
 \label{eq:fullmap}
\end{equation}
for the actual frozen joint graph. Its value on the retained sum must
use the prescribed grouped cross-lines and mixed vertex Hessian,
agree with canonical composition on its proved domain, descend under
reconstruction, and satisfy the required complete-graph analytic
bounds.

The supplied XV-C plan expressly requires a prerequisite stop if
the defining graph data cannot be recovered. That stop applies here.
A construction or a nonexistence theorem for the actual full graph
cannot be justified from operators and classes that have not been
specified.

Accordingly, this manuscript has two distinct conclusions:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the local mathematical tests stated below are proved
completely;
\item their application to the full frozen graph remains unresolved
at the source-completeness gate.
\end{enumerate}
The second conclusion is not a mathematical obstruction to existence.

\subsection{Numbering and downstream admission}

We retain the technical-continuation convention adopted in the
supplied XV-B manuscript \cite{XVB}.
Thus XV-B and XV-C may study the unresolved construction before
admission. The admission success criterion remains
\begin{equation}
 Q_H\Rquant\in\Eout.
 \label{eq:admission}
\end{equation}
It is a prerequisite for the intended downstream quantum use, not
a prerequisite for writing this technical continuation.

The supplied XV-A text uses the earlier convention under which
``XV-B remains closed'' \cite{XVA}. That naming convention is not
reimposed here. The retained mathematical boundary is that the
required quantum coefficient has not been admitted in the supplied
releases.

\subsection{Evidence contract}

The direct inputs are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the supplied full XV-A and XV-B texts;
\item the supplied XIV-H, XIV-I, and XIV-J excerpts;
\item the proposed XV-C plan, used as a statement of objectives,
not as a source of construction theorems.
\end{enumerate}

The broad retrieval account supplies documentary context only.
A passage having been retrieved from a manuscript does not supply
every definition in that manuscript or transfer its theorems to the
present problem.

No result from the relative classical branch or the unfinished
XIII-E route is imported. No claim about the current state of an
external manuscript registry is made.

All distributional arguments below are local in Euclidean charts.
They extend componentwise to finite-dimensional residual fibers.
Statements attributed to predecessors retain their supplied domains
and are not independently verified here.

\section{Frozen branch and placement ledger}
\label{sec:branches}

\subsection{The six original branches}

The retained kernel is
\begin{equation}
 \Kret=K_{47}+K_5+K_6+K_8+K_9,
 \qquad
 \Ksup=K_6+K_8+K_9.
 \label{eq:retained}
\end{equation}
The first sum contains six original branches $B_4,\ldots,B_9$:
$K_{47}$ groups $B_4+B_7$.
The $K_5$ branch has two retained placements.

Write
\[
 K_5=K_5^{[a]}+K_5^{[z]}.
\]
These superscripts label placements, not additional original
branches.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.16\linewidth}
p{0.19\linewidth}p{0.23\linewidth}X@{}}
\toprule
Original branch content & Retained block & Placement record &
Available information\\
\midrule
$B_4+B_7$ &
$K_{47}$ &
One grouped block &
Internal expression supplied; characteristic cross-line
composition not supplied.\\[0.35em]
$B_5$ &
$K_5^{[a]}+K_5^{[z]}$ &
Both placements retained &
Exact internal expressions supplied; neither full characteristic
cross-line operation supplied.\\[0.35em]
$B_6$ &
$K_6$ &
One block &
Internal expression supplied; included in the supported reference.\\[0.35em]
$B_8$ &
$K_8$ &
One block &
Exact internal map reported; explicit map not reproduced in the
supplied excerpts.\\[0.35em]
$B_9$ &
$K_9$ &
One block &
Exact internal map reported; explicit map not reproduced in the
supplied excerpts.\\
\bottomrule
\end{tabularx}
\end{center}

The operational list
\[
 K_{47},\ K_5^{[a]},\ K_5^{[z]},\ K_6,\ K_8,\ K_9
\]
has six entries, but it is not a list of six independently labelled
original branches: the first entry groups two originals and the
next two entries belong to the same original.

This ledger checks the supplied decomposition. It does not verify
unsupplied tensor coefficients in a proposed joint operation.

\subsection{Available internal expressions}

The two supplied $K_5$ placements are
\begin{align}
 K_5^{[a],I,J}(a,b,z)
 &=
 \partial_{b^\mu}\Bigl[
 f(b)\sigma_\eps(a)
 \partial_{a^\alpha}R(b-a)
 \mathsf A^{\alpha\beta,I}
 \partial_{a^\beta}L^{\mu,J}(a,z)
 \Bigr],
 \label{eq:K5a}\\
 K_5^{[z],I,J}(a,b,z)
 &=
 \partial_{b^\mu}\Bigl[
 f(b)\sigma_\eps(z)
 \partial_{z^\alpha}R(b-z)
 \mathsf A^{\alpha\beta,J}
 \partial_{z^\beta}L^{\mu,I}(z,a)
 \Bigr].
 \label{eq:K5z}
\end{align}
The supplied XIV-H proposition expresses them as finite differential
operations on the tensor-product pullbacks associated with
\[
 R(b-a)R(a-z),\qquad R(b-z)R(z-a)
\]
\cite{Hkernel}. XV-B independently checks the existence mechanism
for these internal products \cite{XVB}.

The other displayed source expressions are
\begin{equation}
\begin{split}
 K_{47}^{I,J}(a,b,z)
 =-\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]
\end{split}
\label{eq:K47}
\end{equation}
and
\begin{equation}
 K_6^{I,J}(a,b,z)
 =
 \partial_{b^\mu}\Bigl[
 f(b)\sigma_\eps(a)\mathsf D^{\alpha\mu,I,J}
 \partial_{a^\alpha}R(b-a)\delta^{(4)}(a-z)
 \Bigr].
 \label{eq:K6}
\end{equation}
Their complete inherited ingredients are not defined by these
formulas alone \cite{Iamplitude}.

In particular, the existence of internal products in independent
variables does not establish their product with additional singular
cross-lines in the attachment variables.

\subsection{The unchanged pole}

The supplied internal pole chart is
\begin{equation}
 M_{\mathrm R}
 =
 -\frac{s}{2\rho}(\omega-s\rho-i0)^{-1}
 +\frac{s}{2\rho(\omega+s\rho)},
 \quad
 \rho>0,\quad
 |\omega-s\rho|<c\rho,\quad 0<c<1,
 \label{eq:pole}
\end{equation}
with $s\in\{+1,-1\}$ and
\[
 (\lambda-i0)^{-1}
 =
 \PV\frac1\lambda+i\pi\delta(\lambda).
\]
It represents the existing retarded line \cite{Ipole}.
Its delta part is not adjusted in this manuscript.
No graph regulator, finite-part prescription, or restoring limit
is introduced.

\section{The source-completeness gate}
\label{sec:freeze}

\begin{definition}[Instantiation data]
A source-complete instantiation requires:
\begin{enumerate}[label=\textup{F\arabic*.},leftmargin=*]
\item
the actual presentation space $\Pint$, reconstruction map $\Rec$,
carrier $\Kint=\im\Rec$, and the topology required of the map;

\item
the attachment domain $W$, characteristic loci, canonical open
domain $O=\Ucan$, and the relevant incidence equations;

\item
the canonical contraction $m^\circ$, the grouped cross-line
operators, mixed vertex Hessian, tensor coefficients, weights, and
normalization;

\item
the permitted candidate and correction classes, with their contact,
partial-diagonal, continuity, tensor, and microlocal restrictions;

\item
the complete-graph target bounds and the definitions of the output
operations, $Q_H$, and $\Eout$.
\end{enumerate}
\end{definition}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.12\linewidth}
p{0.36\linewidth}X@{}}
\toprule
Item & What is supplied & What remains unspecified\\
\midrule
F1 &
Abstract reconstruction interfaces and one explicit null identity &
Actual full presentation module, its complete relations, and
inherited topology.\\[0.35em]
F2 &
Names of the domains and loci; internal variables $(a,b,z,y)$ &
Defining equations and complete microlocally admissible domain.\\[0.35em]
F3 &
Selected internal kernels and the existence of a canonical-domain
representation &
Explicit grouped cross-lines, full mixed vertex contraction,
and complete normalization data.\\[0.35em]
F4 &
An abstract relative-lifting theorem and reported inherited
ambiguity restrictions &
The actual exhaustive correction class and its parametrization.\\[0.35em]
F5 &
A partial supported-sector wavefront bound and an unasserted
admission line &
Full graph target cone, uniform estimates, output definitions,
$Q_H$, and $\Eout$.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{proposition}[Prerequisite verdict]
\label{prop:prerequisite}
The supplied material does not satisfy the instantiation-data
requirement above. In particular, it does not define an evaluable
preliminary full characteristic lift for the actual graph.
\end{proposition}

\begin{proof}
The supplied full XV-B text explicitly records that the source
definitions of the cross-line factors, frozen map class, $Q_H$, and
$\Eout$ were not reproduced. Its reconstruction interface is
introduced as an abstract interface, separate from the unsupplied
physical definitions.

The additional XIV-H--XIV-J excerpts reproduce internal formulas,
open-gate statements, a supported reference, and a conditional
extension interface. They do not reproduce the missing grouped
cross-line operators or the definition of the frozen map class.
The supported coefficients are displayed using $v_\alpha$ and
$c_{D,\beta}$ without explicit formulas for those ingredients.

Consequently F1--F5 are not all supplied. This is a statement about
the provided text, not a claim that the missing definitions do not
exist elsewhere.
\end{proof}

The stop in Proposition~\ref{prop:prerequisite} is documentary:
\[
 \boxed{\text{C0: source-complete graph instantiation unavailable.}}
\]
It neither proves failure of an analytic construction nor changes the
inherited analytic stop at the full joint map.

The remaining sections prove local tests that identify exactly what
must be checked once the missing definitions are supplied.

\section{Complete null tests in a finite normal-jet module}
\label{sec:relations}

\subsection{A specified local module}

Let
\[
 U=X\times V_q,\qquad
 S=X\times\{0\},
\]
where $X\subset\mathbb R^d$ and $V_q\subset\mathbb R^m$ are open,
and $0\in V_q$.

For a multi-index $\gamma\in\mathbb N^m$ and
$b\in\Dp(X)$, define
\begin{equation}
 \langle\Ejet_\gamma b,\phi\rangle
 =
 (-1)^{|\gamma|}
 \langle b,\partial_q^\gamma\phi(x,0)\rangle,
 \qquad \phi\in\D(U).
 \label{eq:insert}
\end{equation}

Fix $T\in\Dp(X)$ and an integer $r\ge0$. Define the specific
presentation space
\[
 \mathscr P_r^{\mathrm{jet}}(T)
 =
 \bigoplus_{|\alpha|\le r}C^\infty(U)
\]
and its reconstruction
\begin{equation}
 \Rec_T(a)
 =
 \sum_{|\alpha|\le r}a_\alpha\,\Ejet_\alpha T.
 \label{eq:jetrec}
\end{equation}
Multiplication here is by smooth functions and is defined.

This is a local test module. It is not declared to equal the
inherited $\Pint$.

\begin{lemma}[Coefficient extraction]
\label{lem:extraction}
For distributions $b_\gamma\in\Dp(X)$,
\[
 \sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma=0
 \quad\Longleftrightarrow\quad
 b_\gamma=0\ \text{for every }|\gamma|\le r.
\]
\end{lemma}

\begin{proof}
Choose $\chi\in\D(V_q)$ equal to one near zero and define
\[
 \Jjet_\gamma\psi(x,q)
 =
 \frac{(-1)^{|\gamma|}}{\gamma!}
 q^\gamma\chi(q)\psi(x),
 \qquad \psi\in\D(X).
\]
For all multi-indices $\beta$,
\[
 (-1)^{|\beta|}
 \partial_q^\beta\Jjet_\gamma\psi(x,0)
 =
 \delta_{\beta\gamma}\psi(x).
\]
The fact that $\chi$ is constant near zero makes all its positive
order derivatives at zero vanish.

Testing the finite insertion sum against $\Jjet_\gamma\psi$
therefore gives $\langle b_\gamma,\psi\rangle$.
This proves the forward implication; the reverse is immediate.
\end{proof}

\begin{theorem}[Complete null criterion for the specified module]
\label{thm:jetnull}
For $a\in\mathscr P_r^{\mathrm{jet}}(T)$, set
\begin{equation}
 B_\gamma(a)
 =
 \sum_{\substack{\alpha\ge\gamma\\|\alpha|\le r}}
 (-1)^{|\alpha-\gamma|}
 \binom{\alpha}{\gamma}
 \bigl(\partial_q^{\alpha-\gamma}a_\alpha\bigr)(x,0)\,T.
 \label{eq:B}
\end{equation}
Then
\begin{equation}
 \Rec_T(a)=
 \sum_{|\gamma|\le r}\Ejet_\gamma B_\gamma(a),
 \label{eq:jetnormal}
\end{equation}
and
\begin{equation}
 \ker\Rec_T
 =
 \left\{
 a:B_\gamma(a)=0
 \text{ in }\Dp(X)\text{ for all }|\gamma|\le r
 \right\}.
 \label{eq:completekernel}
\end{equation}
Thus \eqref{eq:completekernel} is a complete null-relation test for
this module, not merely a list of necessary identities.
\end{theorem}

\begin{proof}
For one summand and any test $\phi$,
\[
 \langle a_\alpha\Ejet_\alpha T,\phi\rangle
 =
 (-1)^{|\alpha|}
 \left\langle
 T,\partial_q^\alpha(a_\alpha\phi)(x,0)
 \right\rangle.
\]
The Leibniz rule gives
\[
 \partial_q^\alpha(a_\alpha\phi)
 =
 \sum_{\beta\le\alpha}
 \binom{\alpha}{\beta}
 (\partial_q^\beta a_\alpha)
 (\partial_q^{\alpha-\beta}\phi).
\]
Using \eqref{eq:insert}, this becomes
\[
 a_\alpha\Ejet_\alpha T
 =
 \sum_{\beta\le\alpha}
 (-1)^{|\beta|}
 \binom{\alpha}{\beta}
 \Ejet_{\alpha-\beta}
 \left(
   (\partial_q^\beta a_\alpha)(x,0)\,T
 \right).
\]
Collecting the coefficient of $\Ejet_\gamma$ yields
\eqref{eq:B} and \eqref{eq:jetnormal}.
Lemma~\ref{lem:extraction} then proves
\eqref{eq:completekernel}.
\end{proof}

\begin{remark}[Tangential annihilators are retained]
The equations in \eqref{eq:completekernel} are distributional
equations. They cannot be replaced by equations saying that the
smooth scalar coefficients vanish pointwise.
A nonzero distribution $T$ can be annihilated by a nonzero smooth
function. For example, $x\delta(x)=0$.
The criterion therefore retains, rather than suppresses,
tangential annihilator relations.
\end{remark}

\subsection{The inherited elementary null identity}

Let $e_j$ denote a coordinate multi-index. Directly from
\eqref{eq:insert},
\begin{equation}
 q_j\Ejet_\alpha T
 =
 \begin{cases}
 -\alpha_j\Ejet_{\alpha-e_j}T,&\alpha_j>0,\\
 0,&\alpha_j=0.
 \end{cases}
 \label{eq:qidentity}
\end{equation}
For $\alpha=e_j$ this gives
\[
 q_j\Ejet_{e_j}T+\Ejet_0T=0.
\]

The supplied XIV-J chart uses the corresponding one-variable
reconstruction convention
\[
 \Rec(M_{\mathrm R})=\delta(q)R(r),\qquad
 \Rec(i\theta M_{\mathrm R})=\delta'(q)R(r).
\]
Consequently its reported null presentation is
\begin{equation}
 n=(q\,i\theta+1)M_{\mathrm R},
 \qquad \Rec(n)=0.
 \label{eq:inheritednull}
\end{equation}
Theorem~\ref{thm:jetnull} includes this identity in its local model
\cite{Jnull}.

For a candidate $F$ defined on the two supplied summands, descent
requires the concrete equality
\begin{equation}
 F(q\,i\theta M_{\mathrm R})+F(M_{\mathrm R})=0
 \quad\text{on the full domain}.
 \label{eq:actualnulltest}
\end{equation}
Canonical agreement establishes only its restriction to the proved
canonical domain.

\subsection{What has not been proved about the full kernel}

Theorem~\ref{thm:jetnull} does not prove that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the full inherited presentation space is a finite normal-jet
module;
\item all its null relations are generated by
\eqref{eq:inheritednull} or by \eqref{eq:qidentity};
\item both complete $K_5$ placements, $K_{47}$, and every residual
term are covered by this local module;
\item the actual joint operation acts continuously on this module.
\end{enumerate}

The full descent test remains
\[
 F|_{\ker\Rec}=0.
\]
A verified local null test cannot replace a complete description or
other complete treatment of the actual reconstruction kernel.

\section{The supplied supported coefficients through order three}
\label{sec:coefficients}

\subsection{Exact collection of the source formula}

For a vertex component $D$, the supplied supported reference is
\begin{equation}
 t_D^\circ
 =
 \sum_{\substack{|\alpha|\le3\\\beta\le\alpha}}
 (-1)^{|\beta|}\binom{\alpha}{\beta}
 \Ejet_{\alpha-\beta}
       \bigl(v_\alpha c_{D,\beta}\bigr)
 \quad\text{on }U\setminus N_U.
 \label{eq:reference}
\end{equation}
It is a predecessor input for $\Ksup$ only \cite{Jreference}.

The products in \eqref{eq:reference} are understood on their proved
source domain. This manuscript does not define them at the omitted
characteristic points.

\begin{proposition}[Collected coefficients]
\label{prop:collected}
The unique coefficients in the supplied normal chart are
\begin{equation}
 b_{D,\gamma}^\circ
 =
 \sum_{\substack{\beta\ge0\\|\gamma+\beta|\le3}}
 (-1)^{|\beta|}
 \binom{\gamma+\beta}{\gamma}
 v_{\gamma+\beta}c_{D,\beta}.
 \label{eq:collected}
\end{equation}
In particular:
\begin{align}
 |\gamma|=3:\qquad
 b_{D,\gamma}^\circ
 &=
 v_\gamma c_{D,0},
 \label{eq:degree3}\\
 |\gamma|=2:\qquad
 b_{D,\gamma}^\circ
 &=
 v_\gamma c_{D,0}
 -
 \sum_{j=1}^m
 (\gamma_j+1)v_{\gamma+e_j}c_{D,e_j},
 \label{eq:degree2}\\
 |\gamma|=1:\qquad
 b_{D,\gamma}^\circ
 &=
 v_\gamma c_{D,0}
 -
 \sum_{j=1}^m
 (\gamma_j+1)v_{\gamma+e_j}c_{D,e_j}
 \notag\\
 &\quad+
 \sum_{|\beta|=2}
 \binom{\gamma+\beta}{\gamma}
 v_{\gamma+\beta}c_{D,\beta},
 \label{eq:degree1}\\
 b_{D,0}^\circ
 &=
 \sum_{|\alpha|\le3}
 (-1)^{|\alpha|}v_\alpha c_{D,\alpha}.
 \label{eq:degree0}
\end{align}
\end{proposition}

\begin{proof}
In \eqref{eq:reference}, put $\gamma=\alpha-\beta$.
Then $\alpha=\gamma+\beta$ and
\[
 \binom{\alpha}{\beta}
 =
 \binom{\gamma+\beta}{\gamma}.
\]
This gives \eqref{eq:collected}.
The displayed degree formulas are the finite sums according to
$|\beta|\le3-|\gamma|$.

Uniqueness is Lemma~\ref{lem:extraction}, applied locally on the
source domain.
\end{proof}

The formulas collect the actual supplied reference expression.
They are not a numerical or analytic evaluation of $v_\alpha$ or
$c_{D,\beta}$, whose defining formulas are absent.

\begin{corollary}[No cancellation between distinct insertion orders]
\label{cor:nocancel}
In a realization supported on the response incidence and preserving
normal order at most three, lower insertion orders cannot replace a
missing coefficient at a different insertion order.
\end{corollary}

\begin{proof}
Every coefficient is independently extracted by the corresponding
$\Jjet_\gamma$ test. Thus a finite sum of changes at other insertion
orders has zero extracted coefficient at order $\gamma$.
\end{proof}

This does not rule out cancellations inside a collected coefficient
in \eqref{eq:collected}. Such cancellations would have to be proved
using its actual ingredients.

\section{An exact local distributional extension test}
\label{sec:extension}

This section supplies the estimate that must be checked for the
coefficients in Section~\ref{sec:coefficients}.

For an open $X\subset\mathbb R^d$ and
$\phi\in\D(X)$, extend $\phi$ by zero outside $X$ and set
\[
 p_k(\phi)
 =
 \max_{|\alpha|\le k}
 \|\partial^\alpha\phi\|_{L^\infty(\mathbb R^d)}.
\]
For a compact $K\subset X$, tests with support in $K$ are understood
to have the usual test-function topology.

\begin{theorem}[Exact extension criterion]
\label{thm:extension}
Let $Z\subset X$ be relatively closed and
$b^\circ\in\Dp(X\setminus Z)$.
The following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item There exists $b\in\Dp(X)$ with
\[
 b|_{X\setminus Z}=b^\circ.
\]
\item For every compact $K\subset X$, there are an integer
$k_K\ge0$ and a constant $C_K<\infty$ such that
\begin{equation}
 |\langle b^\circ,\phi\rangle|
 \le C_K p_{k_K}(\phi)
 \label{eq:extensionbound}
\end{equation}
for every $\phi\in\D(X\setminus Z)$ with $\supp\phi\subset K$.
\end{enumerate}
When extensions exist, their differences are exactly the
distributions in $\Dp(X)$ supported in $Z$.
\end{theorem}

\begin{proof}
Suppose (i) holds. Distributional continuity on tests supported in
a fixed compact $K$ gives a finite-order estimate
\[
 |\langle b,\phi\rangle|\le C_Kp_{k_K}(\phi).
\]
On the tests in (ii), the pairing with $b$ is the pairing with
$b^\circ$. This proves necessity.

Assume (ii). Choose a relatively compact open set $V\subset X$ and
a compact $K\subset X$ containing $\overline V$.
On the linear subspace $\D(V\setminus Z)\subset\D(V)$,
the functional defined by $b^\circ$ is bounded by
$C_Kp_{k_K}$.

The real or complex Hahn--Banach theorem extends this functional to
a linear functional $B_V$ on $\D(V)$ satisfying the same bound.
Since $p_{k_K}$ is continuous on every fixed-compact test space,
$B_V$ is a distribution on $V$. It agrees with $b^\circ$ on
$V\setminus Z$.

Take a locally finite cover $\{V_i\}$ of $X$ by such relatively
compact open sets and a smooth partition of unity $\{\chi_i\}$
subordinate to it. Choose the extensions $B_{V_i}$ above and define
\[
 \langle b,\phi\rangle
 =
 \sum_i\langle B_{V_i},\chi_i\phi\rangle.
\]
For each compactly supported test, only finitely many terms
contribute. On each fixed compact test space this is a finite sum
of continuous functionals, hence a distribution.

If $\phi$ is supported in $X\setminus Z$, every term is the pairing
of $b^\circ$ with $\chi_i\phi$. The partition of unity gives
\[
 \langle b,\phi\rangle=\langle b^\circ,\phi\rangle.
\]
This proves sufficiency.

Finally, two extensions have zero difference after restriction to
$X\setminus Z$, which is equivalent to support in $Z$.
Conversely, adding a distribution supported in $Z$ changes no such
restriction.
\end{proof}

\begin{remark}[What Hahn--Banach does not select]
The proof constructs unrestricted distributional extensions.
It does not select a normalized characteristic prescription, ensure
a prescribed wavefront cone, or prove membership in the inherited
contact class. Its arbitrary extension choices are not adopted as
a prescription for the frozen joint graph.
\end{remark}

\begin{proposition}[Explicit failure witnesses]
\label{prop:failure}
If \eqref{eq:extensionbound} fails for a compact $K$, then there
exist tests
\[
 \phi_n\in\D(X\setminus Z),\qquad
 \supp\phi_n\subset K,
\]
such that
\begin{equation}
 p_n(\phi_n)\le1,\qquad
 |\langle b^\circ,\phi_n\rangle|>n.
 \label{eq:failuresequence}
\end{equation}
Any such sequence excludes a distributional extension of
$b^\circ$ to $X$.
\end{proposition}

\begin{proof}
Failure of the estimate means that for each $n$ the bound with
order $n$ and constant $n$ fails.
Choose a nonzero test $\psi_n$ with
\[
 |\langle b^\circ,\psi_n\rangle|>n\,p_n(\psi_n).
\]
The seminorm $p_n(\psi_n)$ is positive because it includes the
zeroth derivative. Dividing by it gives \eqref{eq:failuresequence}.

If an extension existed, on $K$ it would satisfy
$|\langle b,\phi\rangle|\le Cp_k(\phi)$ for some $C,k$.
For $n\ge k$, the sequence obeys $p_k(\phi_n)\le p_n(\phi_n)\le1$,
whereas its pairings exceed $n$. This contradicts the estimate for
all sufficiently large $n$.
\end{proof}

This is an exact failure test. No sequence of the form
\eqref{eq:failuresequence} is asserted for an inherited coefficient
in this release.

\section{The supported-sector criterion in evaluable form}
\label{sec:supported}

\subsection{Local geometric hypotheses}

Continue in a product normal chart
\[
 U=X\times V_q,\qquad S=X\times\{0\}.
\]
Let $A\subset U$ be relatively closed and set
\[
 Z=\{x\in X:(x,0)\in A\}.
\]
Suppose $t^\circ\in\Dp(U\setminus A)$ is supported on
$S\setminus A$ and has the local finite normal form
\begin{equation}
 t^\circ=
 \sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma^\circ,
 \qquad
 b_\gamma^\circ\in\Dp(X\setminus Z).
 \label{eq:puncturedform}
\end{equation}
The equality is understood in product neighborhoods contained in
$U\setminus A$ that meet $S$; away from $S$, the distribution is zero.

A supported realization of normal order at most $r$ means a
distribution supported in $S$ that annihilates tests whose normal
jets through order $r$ vanish. This is the support-explicit
convention of XV-B.

\begin{theorem}[Exact supported realization test]
\label{thm:supported}
Under these hypotheses, the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item There exists a supported realization $t\in\Dp(U)$ of normal
order at most $r$ with
\[
 t|_{U\setminus A}=t^\circ.
\]
\item For every compact $K\subset X$ and every $|\gamma|\le r$,
there are constants $C_{K,\gamma}$ and integers $k_{K,\gamma}$
such that
\begin{equation}
 |\langle b_\gamma^\circ,\psi\rangle|
 \le C_{K,\gamma}p_{k_{K,\gamma}}(\psi)
 \label{eq:jetbound}
\end{equation}
whenever $\psi\in\D(X\setminus Z)$ and $\supp\psi\subset K$.
\end{enumerate}

When these conditions hold, every such realization has a unique
representation
\begin{equation}
 t=\sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma
 \label{eq:realization}
\end{equation}
with $b_\gamma$ extending $b_\gamma^\circ$.
The unrestricted supported ambiguity is
\begin{equation}
 \left\{
 \sum_{|\gamma|\le r}\Ejet_\gamma d_\gamma:
 d_\gamma\in\Dp(X),\ \supp d_\gamma\subset Z
 \right\}.
 \label{eq:ambiguity}
\end{equation}
\end{theorem}

\begin{proof}
A distribution of the stated supported normal order has a unique
finite insertion representation. For completeness, define its
coefficients by
\[
 \langle b_\gamma,\psi\rangle
 =
 \langle t,\Jjet_\gamma\psi\rangle.
\]
For a test $\phi$, the difference between $\phi$ and the finite
sum of its lifted signed normal jets has zero normal jets through
order $r$. The defining annihilation condition gives
\eqref{eq:realization}. Lemma~\ref{lem:extraction} proves uniqueness.

If (i) holds, near each point of $S$ outside $A$ choose a product
neighborhood disjoint from $A$. The realization and the prescribed
reference agree there. Extracting coefficients shows that each
$b_\gamma$ restricts to $b_\gamma^\circ$.
Theorem~\ref{thm:extension} gives \eqref{eq:jetbound}.

Conversely, (ii) and Theorem~\ref{thm:extension} extend each
$b_\gamma^\circ$ to a distribution on $X$.
The finite insertion sum is defined on $U$, supported in $S$,
and has normal order at most $r$.
On each product neighborhood outside $A$ meeting $S$, its coefficients
agree with those in \eqref{eq:puncturedform}; hence the distributions
agree by coefficient uniqueness. Off $S$, both vanish.
These local equalities prove the required restriction.

The extension differences of the coefficients are exactly the
distributions supported in $Z$, by
Theorem~\ref{thm:extension}. Uniqueness of the insertion representation
gives \eqref{eq:ambiguity}.
\end{proof}

\begin{corollary}[The exact missing unrestricted extension estimate]
\label{cor:sourceestimate}
For the supplied supported reference, take
\[
 r=3,\qquad S=S_{\mathrm{resp}},\qquad A=N_U.
\]
Its unrestricted supported normal-order-preserving extension exists
locally if and only if the collected distributions
\eqref{eq:degree3}--\eqref{eq:degree0} satisfy
\eqref{eq:jetbound} on the tangential characteristic complement.
\end{corollary}

\begin{proof}
The source provides the supported normal form on its stated domain.
Proposition~\ref{prop:collected} gives its unique coefficients.
Apply Theorem~\ref{thm:supported}.
\end{proof}

Corollary~\ref{cor:sourceestimate} replaces the bare assertion
``the coefficients extend'' by a precise test-function estimate.
It does not prove that estimate for the actual coefficients.

In particular, the top-order requirement is a bound for
\[
 v_\gamma c_{D,0}\qquad(|\gamma|=3),
\]
and the order-two requirement is a bound for the full difference
in \eqref{eq:degree2}, not merely a bound for each internal line
before the cross-line traces are taken.

\subsection{A finite-order bound for the resulting realization}

\begin{proposition}[Local distributional order]
\label{prop:order}
Suppose, on a relatively compact tangent neighborhood, coefficient
extensions can be chosen with a common estimate
\[
 |\langle b_\gamma,\psi\rangle|\le C_\gamma p_k(\psi),
 \qquad |\gamma|\le r.
\]
Then the corresponding realization satisfies a local estimate of
total derivative order at most $k+r$.
\end{proposition}

\begin{proof}
For a test $\phi$ in a fixed product neighborhood, put
\[
 \psi_\gamma(x)=\partial_q^\gamma\phi(x,0).
\]
Tangential derivatives of $\psi_\gamma$ through order $k$ are
bounded by derivatives of $\phi$ through total order
$k+|\gamma|\le k+r$.
Thus
\[
 \begin{split}
 |\langle t,\phi\rangle|
 &\le
 \sum_{|\gamma|\le r}
 C_\gamma p_k(\psi_\gamma)\\
 &\le
 \left(\sum_{|\gamma|\le r}C_\gamma\right)
 p_{k+r}(\phi),
 \end{split}
\]
with the last seminorm taken in the product coordinates.
\end{proof}

For the supplied sector, the normal-order contribution is at most
three. The estimate is ordinary local distributional continuity.
It is not a target transverse scaling degree, a complete-graph
wavefront estimate, or a normalized admission theorem.

\subsection{Supported ambiguity versus permitted contacts}

\begin{proposition}[Support of a finite normal-jet distribution]
\label{prop:support}
For
\[
 t=\sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma
\]
one has
\[
 \supp t
 =
 \left\{(x,0):x\in
        \bigcup_{|\gamma|\le r}\supp b_\gamma\right\}.
\]
Consequently, if $C\subset U$ is relatively closed, a supported
normal-order-preserving change
$\Delta t=\sum\Ejet_\gamma d_\gamma$ is supported in $C$ if and only
if every $d_\gamma$ is supported in
\[
 Z_C=\{x:(x,0)\in C\}.
\]
\end{proposition}

\begin{proof}
The inclusion from right to left as an upper bound on the support
follows from the insertion definition.

For the reverse inclusion, suppose $t$ vanishes in a product
neighborhood of $(x,0)$. Choose the cutoff in $\Jjet_\gamma$ inside
that normal neighborhood. Testing there extracts every coefficient,
so all $b_\gamma$ vanish near $x$.
This proves the support equality and then the stated equivalence.
\end{proof}

Thus identifying \eqref{eq:ambiguity} with a contact class requires
both a support comparison and all further frozen tensor,
normalization, continuity, and microlocal restrictions.
The support comparison alone is not enough.

\section{Readiness of the full relative-lifting calculation}
\label{sec:lifting}

Once the actual reconstruction and canonical contraction are
specified, write
\[
 H=\ker\Rec,\qquad
 V=\Dp(W;E_{\mathrm{res}}),\qquad
 r_O:V\longrightarrow\Dp(O;E_{\mathrm{res}}),
\]
and
\[
 Y=\ker r_O
   =\{t\in V:\supp t\subset W\setminus O\}.
\]
No equality between $W\setminus O$ and the characteristic locus is
assumed.

A preliminary lift must first be defined:
\[
 F_0:\Pint\longrightarrow V,\qquad
 r_OF_0=m^\circ\Rec.
\]
Only then is its null defect
\[
 d_0=F_0|_H
\]
an evaluable object.

If the actual permitted candidate family is proved to be
$F_0+\mathcal A$, with
$\mathcal A\subset\Hom(\Pint,Y)$ a specified vector space, XV-B gives
the exact descent criterion
\begin{equation}
 [d_0]=0
 \quad\text{in}\quad
 \Hom(H,Y)/\{a|_H:a\in\mathcal A\}.
 \label{eq:relative}
\end{equation}
It uses membership in the actual correction image, not merely its
closure \cite{XVB}.

\begin{proposition}[The obstruction has not been evaluated]
\label{prop:noevaluation}
The supplied data do not provide an evaluation of $d_0$ or of
\eqref{eq:relative} for the full frozen graph.
\end{proposition}

\begin{proof}
Proposition~\ref{prop:prerequisite} identifies the missing full
cross-line operation, full presentation space, and actual correction
class. No characteristic preliminary lift $F_0$ for these data is
defined in this release.

Without such a lift, $F_0|_H$ has no computed value. Without the
actual correction class, its restriction image and relative quotient
cannot be evaluated for the inherited instance.
The local equations in Sections~\ref{sec:relations} and
\ref{sec:supported} do not supply either missing object.
\end{proof}

\subsection{The contact-support witness that would suffice}

If all permitted corrections have support in a closed contact locus
$C$, a computed null presentation $A\in H$ and an open set
$G\subset W\setminus C$ satisfying
\[
 F_0(A)|_G\ne0
\]
would exclude descent in that correction family. Indeed, for every
permitted $a$,
\[
 (F_0+a)(A)|_G=F_0(A)|_G\ne0.
\]
This is the contact-support test of XV-B.

No such separated defect is computed here. Nor is the actual
correction class proved to be exhausted by a contact-valued affine
family. Therefore no obstruction outcome is asserted.

\subsection{Why individual failure would not suffice}

Failure of one selected extension, one correction, or one ordinary
product criterion would not exclude all permitted constructions.
Conversely, arbitrary Hahn--Banach extensions or algebraic
corrections would not establish a construction in the frozen class.

The full decision requires either:
\begin{enumerate}[label=\textup{(\roman*)}]
\item an actual admissible full map, with its defining operations
and all required bounds proved; or
\item an incompatibility theorem in a class proved to exhaust the
permitted maps.
\end{enumerate}
Neither is supplied by the present local tests.

\section{Graph-level bounds and the admission boundary}
\label{sec:bounds}

The predecessor data distinguish three operations:
\begin{enumerate}[label=\textup{(\roman*)}]
\item reconstruction of defined internal distributions;
\item canonical cross-line composition on its proved open domain;
\item the full characteristic graph operation on tests meeting $N$.
\end{enumerate}
The XIV-I internal amplitude result supplies the first operation in
its named module, not the third \cite{Iamplitude,Ijoint}.

The supported reference provides a wavefront bound only for
$\Ksup$ on $U\setminus N_U$ \cite{Jreference}.
The estimates in this paper concern unrestricted coefficient
extendability and finite local distributional order.
They do not prove:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a complete-graph wavefront bound across the characteristic
locus;
\item uniform transverse estimates for the actual joint section;
\item compatibility on intersections of characteristic and
collision strata;
\item compatibility with the frozen contact and partial-diagonal
ambiguity;
\item proper output pushforward or restriction estimates;
\item the normalized output admission \eqref{eq:admission}.
\end{enumerate}

A response-normal insertion order is not a transverse degree of the
quantum graph. Likewise, internal frequency homogeneity is not a
transverse estimate for the later composition. The supplied null
presentation is an additional reason presentation degree must not
be treated as an intrinsic reconstructed filtration without a
descent proof.

The ambient characteristic locus, its attachment-variable lift, the
response incidence, and collision loci are not identified with one
another. Their equations and intersections must be supplied before
stratum-specific estimates can be formulated.

No interacting differential is applied to an unconstructed
coefficient, and no quantum Ward defect is formed.

\section{Acceptance ledger and debugging}
\label{sec:debug}

\subsection{Acceptance ledger}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.33\linewidth}X@{}}
\toprule
Required acceptance item & XV-C verdict\\
\midrule
Complete frozen graph, including cross-lines &
Not supplied. The source-completeness gate fails.\\[0.35em]

Six original branches and both $K_5$ placements &
Tracked explicitly in the source decomposition.
No characteristic evaluation of all blocks is constructed.\\[0.35em]

Operation on tests meeting $N$ &
Not constructed for the full graph.\\[0.35em]

Actual supported coefficients shown to extend &
Exact collected formulas and an equivalent estimate test are proved.
The estimates for the unsupplied ingredients are not verified.\\[0.35em]

Complete reconstruction independence &
A complete criterion is proved for a specified finite normal-jet
module. The actual full reconstruction kernel is not determined.\\[0.35em]

Inherited permitted correction class &
Not supplied or replaced by an unrestricted class.\\[0.35em]

Actual null defect cancelled or genuinely obstructed &
Neither. No full preliminary lift or exhaustive class is available.\\[0.35em]

Complete-graph wavefront and uniform estimates &
Not proved.\\[0.35em]

Quantum admission &
Unasserted.\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Repairs made in the final formulation}

\begin{enumerate}[label=\textup{D\arabic*.},leftmargin=*]
\item
\textbf{Outcome selection.}
The manuscript takes the plan's unresolved alternative.
It does not turn a proposed title or target into a proved construction.

\item
\textbf{Numbering.}
XV-B denotes the supplied technical-continuation manuscript.
Older downstream exit language is retained only as an admission
boundary.

\item
\textbf{Evidence.}
A broad retrieval inventory is not treated as the complete text of
every retrieved manuscript.

\item
\textbf{Branch counting.}
$K_{47}$ groups two original branches; the two $K_5$ placements
belong to one original branch.
The supported sector is not the full retained sum.

\item
\textbf{Local versus full relations.}
The normal-jet null criterion is complete for its explicitly defined
module only. No complete generating claim about the inherited
$\ker\Rec$ is made.

\item
\textbf{Tangential annihilators.}
Coefficient equations remain equations in $\Dp(X)$.
Smooth multiplication of a fixed nonzero distribution is not
assumed injective.

\item
\textbf{Extendability versus admissibility.}
The test-function estimate is equivalent to unrestricted
distributional extension. It is not equivalent to membership in the
frozen graph prescription class.

\item
\textbf{Simultaneous maps.}
Pointwise extension of every displayed coefficient does not itself
construct a continuous linear operation on the entire inherited
presentation space.

\item
\textbf{Support.}
The supported coefficient theorem explicitly requires support on the
response incidence. Additional characteristic terms outside that
incidence are not silently discarded.

\item
\textbf{Domain.}
The canonical open domain is not identified with an entire
characteristic complement.

\item
\textbf{Estimates.}
Ordinary finite distributional order, response-normal order, internal
frequency degree, and target transverse degree remain distinct.

\item
\textbf{Prescription.}
The retarded pole, propagators, carrier, ambiguity class, and
normalization are not changed. Hahn--Banach choices are not selected
as a graph regulator or finite part.

\item
\textbf{Obstruction.}
A separated-contact witness is stated as a conditional test.
No witness or exhaustive no-go class for the actual graph is invented.

\item
\textbf{Admission.}
The coefficient is not asserted to be nonexistent.
It is not constructed or admitted in this release.
\end{enumerate}

\section{Exact release and successor requirement}
\label{sec:release}

\begin{theorem}[XV-C local mathematical release]
\label{thm:localrelease}
For the local objects explicitly defined in this manuscript:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the reconstruction-null space of
$\mathscr P_r^{\mathrm{jet}}(T)$ is exactly the simultaneous kernel
of the collected coefficient maps \eqref{eq:B};

\item the supplied supported-sector normal form has the unique
collected coefficients \eqref{eq:degree3}--\eqref{eq:degree0};

\item a distribution on a closed-set complement extends
distributionally exactly when it satisfies the compact-uniform
finite-order test estimate \eqref{eq:extensionbound};

\item a supported normal-order-preserving realization of the supplied
local normal form exists in the unrestricted distribution class
exactly when its collected coefficients satisfy
\eqref{eq:jetbound};

\item its unrestricted supported ambiguity is
\eqref{eq:ambiguity}, with contact support characterized by
Proposition~\ref{prop:support};

\item a sequence \eqref{eq:failuresequence} for an actual coefficient
would exclude its unrestricted distributional extension.
\end{enumerate}
\end{theorem}

\begin{proof}
Assertion (i) is Theorem~\ref{thm:jetnull}.
Assertion (ii) is Proposition~\ref{prop:collected}.
Assertion (iii) is Theorem~\ref{thm:extension}.
Assertion (iv) is Theorem~\ref{thm:supported} and
Corollary~\ref{cor:sourceestimate}.
Assertion (v) is the ambiguity part of
Theorem~\ref{thm:supported}, together with
Proposition~\ref{prop:support}.
Assertion (vi) is Proposition~\ref{prop:failure}.
\end{proof}

\subsection{Full-graph verdict}

For the inherited physical instance, the outcome is:
\begin{equation}
 \boxed{
 \begin{gathered}
 \text{Unresolved at C0: source-complete instantiation unavailable.}\\
 \text{The inherited full joint-map gate remains open.}
 \end{gathered}
 }
 \label{eq:verdict}
\end{equation}

This conclusion is justified by
Proposition~\ref{prop:prerequisite} and
Proposition~\ref{prop:noevaluation}.
The supplied XIV-J release already leaves the full map unresolved,
and XV-B supplies interfaces rather than an instantiation
\cite{Jverdict,Jrelease,XVB}.

The local release theorem does not supply a value for the full sum
on a test meeting the characteristic locus.
It does not establish the admissibility of an extension, evaluate a
full null defect, or prove complete-graph bounds.
Therefore it neither closes the full-map gate nor establishes
\eqref{eq:admission}.

No full joint section or controlled remainder is constructed.
The subsequent transverse, contact, output, and admission
obligations remain undischarged for the required coefficient.
The no-admission statement is not a universal nonexistence theorem
\cite{Jadmission}.

\subsection{What the next input must contain}

The next substantive calculation requires an explicit package:
\begin{enumerate}[label=\textup{(\roman*)}]
\item
the complete frozen cross-line and vertex contraction, including all
six retained operational blocks;

\item
the actual reconstruction space and a complete method of handling
its null relations;

\item
the incidence equations and the actual coefficient distributions
$v_\alpha$, $c_{D,\beta}$;

\item
the exhaustive permitted construction and correction class;

\item
the full graph microlocal target and the output definitions.
\end{enumerate}

Once these are supplied, the supported-sector extension subtask can
be attacked by proving \eqref{eq:jetbound} for
\eqref{eq:degree3}--\eqref{eq:degree0}, or by producing an actual
failure sequence. A preliminary full lift can then be tested on
\eqref{eq:actualnulltest} and on the remaining reconstruction
relations.

Successful supported extension would still leave the characteristic
compositions of $K_{47}$ and both $K_5$ placements to be joined into
one coherent map. A construction of that full map would still leave
the subsequent output and admission obligations.

\subsection*{Final release sentence}

\begin{center}
\fbox{\parbox{0.91\linewidth}{
XV-C proves the local reconstruction and coefficient-extension tests
stated in this manuscript and records the complete retained branch
ledger. It does not replace the missing frozen graph data.
The full joint characteristic map remains unresolved before
instantiation; no exhaustive obstruction or admitted quantum
coefficient is asserted.
}}
\end{center}

\begin{thebibliography}{99}

\bibitem{XVA}
Manuscript XV-A, supplied \texttt{XV-A-va.tex}.
\emph{A Model-and-Domain Audit of the Ramp, Graviton, and
Joint-Extension Tracks}.
Full text supplied in the present record.
Its earlier downstream numbering convention is distinguished from
the technical-continuation convention used here.

\bibitem{XVB}
Manuscript XV-B, supplied \texttt{XV-B-va.tex}.
\emph{Relative Lifting and Reconstruction Descent for the Full Joint
Characteristic Target}.
Full text supplied in the present record.
Used for the abstract relative-lifting theorem, supported normal-jet
interface, and explicit application boundary.

\bibitem{Hkernel}
Manuscript XIV-H, version~1.
\texttt{XIV-H.tex}, identifier \texttt{05d90ea1}.
Proposition \texttt{prop:K5exact},
``Exact two-placement kernel formula,'' lines 1135--1202.
Direct excerpt supplied.

\bibitem{Hverdict}
Manuscript XIV-H, version~1.
Theorem \texttt{thm:H2},
``Exact H2 verdict,'' lines 1539--1582.
Direct excerpt supplied.
Historical internal-amplitude stop subsequently addressed by XIV-I.

\bibitem{Ipole}
Manuscript XIV-I, version~1.
\texttt{XIV-I.tex}, identifier \texttt{13365486}.
Proposition \texttt{prop:pole},
``Exact pole decomposition and chart matching,''
lines 726--826.
Direct excerpt supplied.

\bibitem{Iamplitude}
Manuscript XIV-I, version~1.
Theorem \texttt{thm:I2},
``The internal amplitude gate is Constructed,''
and adjacent precomposition formulas, lines 1288--1378.
Direct excerpt supplied.
The adjacent precomposition proposition is reproduced partially.

\bibitem{Ijoint}
Manuscript XIV-I, version~1.
Joint-map interface, Proposition \texttt{prop:canonical}, and
Theorem \texttt{thm:I3}, lines 1459--1548.
Direct excerpt supplied.

\bibitem{Jdescent}
Manuscript XIV-J, version~1.
\texttt{XIV-J.tex}, identifier \texttt{3c48e041}.
Proposition \texttt{prop:descent},
``Descent criterion for a coefficient-level construction,''
lines 813--846.
Direct excerpt supplied.

\bibitem{Jnull}
Manuscript XIV-J, version~1.
Lemma \texttt{lem:null},
``A normal identity invisible to polynomial-degree bookkeeping,''
lines 853--883.
Direct excerpt supplied.

\bibitem{Jreference}
Manuscript XIV-J, version~1.
Theorem \texttt{thm:reference},
``Supported-sector reference distribution,''
lines 1215--1299.
Direct excerpt supplied.
Used only on its stated off-characteristic domain.

\bibitem{Jconditional}
Manuscript XIV-J, version~1.
Theorem \texttt{thm:conditionalextension},
``Conditional supported realization,''
lines 1427--1470.
Direct excerpt supplied.
The local formulation in this manuscript retains an explicit
response-support condition.

\bibitem{Jverdict}
Manuscript XIV-J, version~1.
Theorem \texttt{thm:J2},
``Exact J2 verdict,'' lines 1608--1649.
Direct excerpt supplied.

\bibitem{Jrelease}
Manuscript XIV-J, version~1.
Theorem \texttt{thm:release},
``XIV-J mathematical release,'' lines 1894--1939.
Direct excerpt supplied.
The predecessor exit language uses the earlier downstream
numbering convention.

\bibitem{Jadmission}
Manuscript XIV-J, version~1.
Corollary \texttt{cor:noadmission},
``No quantum admission in this release,''
lines 1808--1822.
Direct excerpt supplied.

\end{thebibliography}

\end{document}