\documentclass[11pt,letterpaper]{article}

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
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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{\Recp}{\Rec^{\circ}}
\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}}
\newcommand{\Pjet}{\mathscr P_r^{\mathrm{jet}}}
\newcommand{\Fz}{F_0}
\newcommand{\Sresp}{S_{\mathrm{resp}}}
\newcommand{\Natt}{\mathcal N_{\mathrm{att}}}

\title{Landmark XV-D\\
Instantiating the Frozen Joint Graph:\\
Source Recovery, the Null Defect of Extension Lifts,\\
and the Boundary of the First Full Calculation}
\author{}
\date{}

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

\begin{abstract}
XV-D continues the supplied XV-C manuscript at its source-completeness
stop. Its task is to recover the frozen data for the six-branch joint
characteristic target, define the preliminary full lift, compute its
reconstruction-null defect, and test the inherited correction and
analytic requirements; or, if the data cannot be recovered, to report
precisely why the graph-level calculation cannot yet be performed.

The second alternative applies. A twenty-item provenance audit of the
supplied record (the full XV-A, XV-B, and XV-C texts and the retrieved
XIV-H, XIV-I, and XIV-J excerpts) shows that the presentation space,
the full reconstruction map, the grouped cross-line operators, the
canonical contraction, the frozen candidate and correction classes,
the coefficient distributions $v_\alpha$ and $c_{D,\beta}$, and the
output definitions $Q_H$, $\Eout$ are named but not defined in that
record. No preliminary lift for the actual graph is therefore
definable, and no graph-specific null defect, obstruction, or
estimate is asserted. The graph-level outcome is
\emph{Unresolved at D0}.

The new mathematical content is the one null-defect calculation that
the record does make source-complete. For the finite normal-jet
module with reconstruction to a punctured base, we define the
extension lifts $\Fz^T=\Rec_T$ associated with the distributional
extensions $T$ of the punctured coefficient, compute their
reconstruction-null defect exactly, prove that it is always supported
on the puncture, that it depends on the extension, that it induces an
injection of the quotient of the two reconstruction kernels, and that
the inherited identity $\Rec((q\,i\theta+1)M_{\mathrm R})=0$ never
contributes to it. An explicit nonzero defect is exhibited and
cancelled by a change of extension. A conditional reduction of
XV-C's coefficient estimates to extendability of the cross-line
traces is proved under an explicit smoothness hypothesis on
$v_\alpha$. None of these results is transferred to the inherited
graph; they fix what must be computed once the missing definitions
are supplied.
\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) {Incomplete source package};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Punctured-jet null defect};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A defined graph lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {No preliminary lift is defined. The defect is not a graph result.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The null defect is not transferred to the graph. 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-D}
\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) {Incomplete source package};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A defined graph lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {No preliminary lift is defined. The defect is not a graph result.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{The null defect is not transferred to the 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:split-XV-D}
\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) {Incomplete source package};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Punctured-jet null defect};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A defined graph lift};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {No preliminary lift is defined. The defect is not a graph result.};
\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-D}
\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) {Incomplete source package};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A defined graph lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Punctured-jet null defect. No preliminary lift is defined. The defect is not a graph result.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Incomplete source package is not A defined graph lift. 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-D}
\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: Incomplete source package};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Punctured-jet null defect};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A defined graph lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {No preliminary lift is defined. The defect is not a graph result.};
\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-D}
\end{figure}

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


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



\section{Purpose, scope, and outcomes}
\label{sec:scope}

\subsection{The question}

XV-C \cite{XVC} proves local reconstruction and extension tests and
stops at the gate it calls C0: the data defining the full joint
characteristic operation for the frozen six-branch graph are not in
the supplied record. XV-D asks the question that XV-C left open:

\begin{quote}
Given the exact frozen data for the six-branch graph, can one define
its full characteristic operation, prove that it descends under
reconstruction, and meet the inherited correction and analytic
requirements?
\end{quote}

Three endings are permitted for the graph-level problem.
\begin{enumerate}[label=\textup{(\alph*)}]
\item \textbf{Constructed:} an admissible, presentation-independent
full joint map with the required bounds.
\item \textbf{Obstructed in a specified exhaustive class:} a
graph-specific proof that no allowed map exists.
\item \textbf{Unresolved at a named gate:} the exact missing
definition, calculation, or estimate is identified.
\end{enumerate}
``Not constructed'' is not ``does not exist.'' None of the three
endings is assumed in advance.

\subsection{Gates and the stop rule}

We use five descriptive gates for this manuscript.
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.07\linewidth}X@{}}
\toprule
Gate & Content\\
\midrule
D0 & Source-complete data package: every operator, space, class,
and bound needed to specify the full operation and its allowed
corrections is defined in the record.\\
D1 & Definition of a preliminary full lift
$\Fz:\Pint\to\Dp(W;E_{\mathrm{res}})$ with $r_O\Fz=m^\circ\Rec$.\\
D2 & Evaluation of the null defect $d_0=\Fz|_{\ker\Rec}$ and of its
class relative to the inherited correction space.\\
D3 & Coefficient estimates for the actual $v_\alpha$, $c_{D,\beta}$,
and the characteristic compositions of $K_{47}$ and both $K_5$
placements.\\
D4 & Complete-graph microlocal and uniform estimates, output
operations, and admission.\\
\bottomrule
\end{tabularx}
\end{center}

\textbf{D0 stop rule.} If any definition needed to specify the full
operation or its allowed corrections is unavailable, this is stated
plainly; D1--D4 are then not entered for the actual graph, and no
graph-specific construction, obstruction, or null-defect value is
claimed. XV-D may then be a source audit together with any local
calculation that is source-complete.

\subsection{Evidence contract}

The direct inputs are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the supplied full texts of XV-A \cite{XVA}, XV-B \cite{XVB},
and XV-C \cite{XVC};
\item the retrieved XIV-H, XIV-I, and XIV-J excerpts
\cite{Hkernel,Hverdict,Ipole,Iamplitude,Ijoint,Irelease,
Jdescent,Jnull,Jreference,Jconditional,Jverdict,Jrelease,Jadmission};
\item the retrieved XIV-G release statement, used only for its
dimension bookkeeping \cite{Grelease};
\item the proposed XV-D plan, used as a statement of objectives and
acceptance criteria, not as a source of theorems.
\end{enumerate}
The coverage account \cite{record} supplies documentary context only.
A retrieved locator, a predecessor citation, or a sentence reporting
that a predecessor proved something is not treated as the missing
definition. Nothing from the relative classical branch (XIV-A, XIV-B)
or the XIII-D/XIII-E routes is imported. Distributional arguments are
local in Euclidean charts and extend componentwise to
finite-dimensional residual fibers.

\subsection{Numbering and admission}

The technical-continuation convention of XV-B and XV-C is retained:
XV-B, XV-C, XV-D study the unresolved construction before admission.
The admission success criterion remains
\begin{equation}
 Q_H\Rquant\in\Eout ,
 \label{eq:admission}
\end{equation}
a prerequisite for downstream quantum use, not for writing this
continuation. It is not asserted here.

\section{Source-completeness dossier (Gate D0)}
\label{sec:dossier}

\subsection{The data package}

\begin{definition}[Instantiation data, itemized]
\label{def:package}
A source-complete instantiation of the frozen joint target consists
of the following items, each given by an actual definition.
\begin{enumerate}[label=\textup{D0.\arabic*},leftmargin=*,itemsep=0pt]
\item the presentation space $\Pint$;
\item the reconstruction map $\Rec:\Pint\to\Kint$ on all of $\Pint$;
\item the carrier $\Kint$ and the topology required of the joint map;
\item a complete description, or complete method of handling, of
$\ker\Rec$;
\item the attachment domain $W$ and its coordinates;
\item the characteristic locus $N$ and its lift to $W$;
\item the response incidence $\Sresp$;
\item the canonical open domain $O=\Ucan$;
\item the collision strata and observable diagonals;
\item the canonical contraction $m^\circ$ and its proved domain;
\item the grouped cross-line operators $\Cross$;
\item the mixed vertex Hessian and vertex weights;
\item the tensor maps $\mathsf A^{\alpha\beta,I}$,
$\mathsf D^{\alpha\mu,I,J}$, the kernels $L^{\mu,J}$, $R$, and the
weights $f$, $\sigma_\eps$;
\item the explicit internal maps of $K_8$ and $K_9$;
\item the normalization of the pushed-forward coefficient;
\item the frozen candidate class of joint maps;
\item the frozen correction (ambiguity) class;
\item the coefficient distributions $v_\alpha$ and $c_{D,\beta}$;
\item the complete-graph target wavefront cone and uniform
estimates, including the operator $\mathcal L(\cdot)$ of the
XIV-J insertion wavefront lemma;
\item the output operations, $Q_H$, and $\Eout$.
\end{enumerate}
\end{definition}

\subsection{Provenance table}

Status codes: \textsf{S} = directly supplied in the record (formula or
definition reproduced); \textsf{T} = faithfully transcribed from a
named source excerpt, with the excerpt reproduced; \textsf{R} =
reported only (a locator, a name, or a sentence stating that a
predecessor defined or proved it); \textsf{U} = unavailable (no
definition in the record).

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.07\linewidth}
p{0.32\linewidth}p{0.08\linewidth}X@{}}
\toprule
Item & What the record contains & Status & What is missing\\
\midrule
D0.1 & The phrase ``finite retarded-pole module of Definition
def:amodule'' \cite{Iamplitude}; XV-B's abstract interface. &
\textsf{R} & The definition of the module and its elements.\\[0.3em]
D0.2 & Two values: $\Rec(M_{\mathrm R})=\delta(q)R(r)$,
$\Rec(i\theta M_{\mathrm R})=\delta'(q)R(r)$ \cite{Jnull}. &
\textsf{T} (two values) & $\Rec$ on all of $\Pint$; the chart in
which $q$, $\theta$, $r$ are defined.\\[0.3em]
D0.3 & ``$\Kint=\im\Rec$'' as an interface convention \cite{XVB}. &
\textsf{R} & The inherited carrier topology.\\[0.3em]
D0.4 & One null identity \eqref{eq:inheritednull}
\cite{Jnull}. & \textsf{T} (one relation) & Whether it generates
$\ker\Rec$; any other relation.\\[0.3em]
D0.5 & Name $W$; internal variables $(a,b,z,y)$ \cite{Hkernel};
``dimension sixteen'' \cite{Grelease}. & \textsf{R} & Defining
coordinates and the residual bundle $E_{\mathrm{res}}$.\\[0.3em]
D0.6 & Name $N=\Natt$; ``dimension eight'' \cite{Grelease};
``codimension eight'' \cite{record}. & \textsf{R} & Defining
equations; the lift to attachment variables.\\[0.3em]
D0.7 & Name $\Sresp$; ``codimension five'' \cite{record}; normal
coordinates $q_1,q_2$ appear in \eqref{eq:referenceWF}
\cite{Jreference}. & \textsf{R} & Defining equations.\\[0.3em]
D0.8 & Name $\Ucan$; the statement that
$\Ucan=W\setminus\overline{\Natt}$ does not follow \cite{Ijoint}. &
\textsf{R} & The definition of the microlocally admissible domain.\\[0.3em]
D0.9 & Names only \cite{record}. & \textsf{U} & Equations of the
collision strata and observable diagonals.\\[0.3em]
D0.10 & Name; ``canonical grouped expression eq:target''
\cite{Ijoint}. & \textsf{R} & The expression and its proved
domain.\\[0.3em]
D0.11 & Name $\Cross$; existence of a traceability lemma
lem:traces \cite{Jreference}. & \textsf{R} & The operators, their
wavefront sets, their normal traces.\\[0.3em]
D0.12 & Names; XIV-D equation labels \cite{record}. & \textsf{U} &
The Hessian and the weights.\\[0.3em]
D0.13 & $\mathsf D$ fixed by
$\dot a_2^{\alpha\mu}[G;h]=\mathsf D^{\alpha\mu,I,J}h_IG_J$
\cite{Iamplitude}; pole chart of $R$ \eqref{eq:pole}
\cite{Ipole}; names $\mathsf A$, $L$, $f$, $\sigma_\eps$. &
\textsf{T}/\textsf{R} & $\dot a_2$; cone formula eq:R; eq:L;
$\mathsf A$; $f$; $\sigma_\eps$.\\[0.3em]
D0.14 & ``Exact kernel maps eq:K8reduced and eq:K9''
\cite{Hverdict}. & \textsf{R} & The formulas.\\[0.3em]
D0.15 & ``Normalization eq:Rconvention'' \cite{Jadmission}. &
\textsf{R} & The convention.\\[0.3em]
D0.16 & ``The class fixed in Definition def:mapclass''
\cite{Jverdict}; ``Definition def:class'' \cite{Ijoint}. &
\textsf{R} & The definitions.\\[0.3em]
D0.17 & ``XIII-C's partial-diagonal ambiguity class''
\cite{record}. & \textsf{R} & Its definition and
parametrization.\\[0.3em]
D0.18 & Symbols in \eqref{eq:reference}; the proof sentence
``multiplication by $v_\alpha$ adds no wavefront''
\cite{Jreference}. & \textsf{R} & Formulas for $v_\alpha$,
$c_{D,\beta}$.\\[0.3em]
D0.19 & Supported-sector bound \eqref{eq:referenceWF}
\cite{Jreference}; ``$\WF(t_D)\subseteq\bigcup\mathcal L(b_{D,\gamma})$''
\cite{Jconditional}. & \textsf{T} (partial) & The full target cone;
the operator $\mathcal L$; the uniform estimates.\\[0.3em]
D0.20 & The line $Q_H\Rquant\in\Eout$ \cite{Jadmission}. &
\textsf{R} & $Q_H$, $\Eout$, the output operations.\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{The D0 verdict}

\begin{proposition}[D0 fails]
\label{prop:D0}
The supplied record does not contain definitions for items
D0.1, D0.2 (beyond two values), D0.4 (beyond one relation),
D0.5--D0.12, D0.14--D0.18, and D0.20 of
Definition~\ref{def:package}. In particular it does not define the
domain, codomain, or value of a preliminary full lift
$\Fz:\Pint\to\Dp(W;E_{\mathrm{res}})$, nor the correction space
$\mathcal A$ against which a null defect would be tested.
\end{proposition}

\begin{proof}
Each row of the provenance table lists the literal content of the
record for the item. For the items named, that content is a name,
a locator, a defining relation in terms of a further undefined
object ($\mathsf D$ via $\dot a_2$), or a sentence reporting a
predecessor statement. The full texts of XV-A, XV-B, and XV-C
themselves record that the source definitions of the cross-line
factors, the frozen map class, $Q_H$, and $\Eout$ were not
reproduced \cite{XVA,XVB,XVC}; the present audit confirms that the
retrieved excerpts do not add them.

A preliminary lift needs a domain ($\Pint$, D0.1), a codomain
($W$, $E_{\mathrm{res}}$, D0.5), and a defining operation on each
presentation (the products of the internal kernels with $\Cross$
and the vertex data, D0.10--D0.13). The correction class needs
D0.16--D0.17. Each is in status \textsf{R} or \textsf{U}. This is a
statement about the record, not about the existence of the
definitions elsewhere.
\end{proof}

\[
 \boxed{\text{D0: source-complete graph instantiation unavailable.}}
\]

Under the stop rule, Gates D1--D4 are not entered for the actual
graph. Sections~\ref{sec:branches} and \ref{sec:lift} record what
the freeze and the lift definition would require;
Sections~\ref{sec:nulljet} and \ref{sec:coefficients} contain the
calculations that are source-complete.

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

\subsection{Retained sum and operational entries}

The retained kernel and supported sector are
\begin{equation}
 \Kret=K_{47}+K_5+K_6+K_8+K_9,\qquad
 K_5=K_5^{[a]}+K_5^{[z]},\qquad
 \Ksup=K_6+K_8+K_9 .
 \label{eq:retained}
\end{equation}
$K_{47}$ groups the original branches $B_4+B_7$; the two $K_5$
placements belong to the original branch $B_5$. The six operational
entries are
\[
 K_{47},\quad K_5^{[a]},\quad K_5^{[z]},\quad K_6,\quad K_8,\quad K_9 .
\]

\subsection{Supplied internal expressions}

The supplied formulas are \cite{Hkernel,Iamplitude}
\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}\\
 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],
 \label{eq:K47}\\
 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{align}
Both $K_5$ placements are finite differential expressions in the
tensor-product pullbacks of $R(b-a)R(a-z)$ and $R(b-z)R(z-a)$, which
exist in the independent variables $(a,b,z)$ including at $a=z$
\cite{Hkernel,XVB}. The 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,\ |\omega-s\rho|<c\rho,\ 0<c<1,
 \label{eq:pole}
\end{equation}
with $(\lambda-i0)^{-1}=\PV\frac1\lambda+i\pi\delta(\lambda)$
\cite{Ipole}. The delta part is fixed by the retarded line and is
not adjusted.

\subsection{Freeze check}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.09\linewidth}
p{0.22\linewidth}p{0.20\linewidth}p{0.17\linewidth}X@{}}
\toprule
Entry & Internal expression & Cross-line factors & Domain of
definition & Status at $N$\\
\midrule
$K_{47}$ & \eqref{eq:K47}; $L$ not defined (D0.13) &
$\Cross$ undefined (D0.11) & Internal: retained patch
\cite{Iamplitude}; composed: $\Ucan$ only \cite{Ijoint} &
Not defined on tests meeting $N$.\\[0.3em]
$K_5^{[a]}$ & \eqref{eq:K5a}; pullback existence proved &
Undefined & Internal: all $(a,b,z)$; composed: $\Ucan$ &
Not defined; I3 stop \cite{Ijoint}.\\[0.3em]
$K_5^{[z]}$ & \eqref{eq:K5z}; pullback existence proved &
Undefined & Internal: all $(a,b,z)$; composed: $\Ucan$ &
Not defined; I3 stop.\\[0.3em]
$K_6$ & \eqref{eq:K6}; $\mathsf D$ via undefined $\dot a_2$ &
Normal traces $c_{D,\beta}$ named (D0.18) &
Composed: $U\setminus N_U$ \cite{Jreference} &
Conditional interface only \cite{Jconditional}.\\[0.3em]
$K_8$ & Reported, not reproduced (D0.14) & Normal traces named &
Composed: $U\setminus N_U$ & Conditional interface only.\\[0.3em]
$K_9$ & Reported, not reproduced (D0.14) & Normal traces named &
Composed: $U\setminus N_U$ & Conditional interface only.\\
\bottomrule
\end{tabularx}
\end{center}

This table is a freeze check. It records that the decomposition is
carried forward unchanged; it is not evidence that the full
operation exists. The propagator, retarded pole, regulator, carrier,
normalization, and ambiguity class are not altered in this
manuscript. No new model is introduced.

\section{What the preliminary lift would require}
\label{sec:lift}

\subsection{Three operations}

A preliminary lift $\Fz:\Pint\to\Dp(W;E_{\mathrm{res}})$ with
$r_O\Fz=m^\circ\Rec$ requires three distinct operations:
\begin{enumerate}[label=\textup{(\roman*)}]
\item reconstruction of the internal distributions
(defined in the named module, D0.1--D0.2);
\item canonical cross-line composition on $O=\Ucan$ (D0.8, D0.10,
D0.11);
\item the full operation on tests meeting $N$, including the
products of the internal distributions with the singular
cross-lines in the attachment variables.
\end{enumerate}
The internal product calculations for the $K_5$ placements settle
only (i). Operation (iii) is exactly the object left unresolved at
I3 and J2 \cite{Ijoint,Jverdict}.

\subsection{Row-by-row requirements}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.09\linewidth}
p{0.22\linewidth}p{0.14\linewidth}p{0.20\linewidth}X@{}}
\toprule
Entry & Factors to compose & Domain & Rule required & Unresolved
item\\
\midrule
$K_{47}$ & $L\cdot L$ (internal), then $\Cross$, vertex Hessian &
tests meeting $N$ & Composition theorem or explicit prescription
for $\Cross\cdot K_{47}$ at $N$ & D0.10--D0.13; no theorem in record.\\[0.3em]
$K_5^{[a]}$ & $R(b-a)R(a-z)$ pullback, $\mathsf A$, $L$, then
$\Cross$ & tests meeting $N$ & Same & D0.11, D0.13; opposite-covector
audit blocks the ordinary product theorem \cite{Ijoint}.\\[0.3em]
$K_5^{[z]}$ & $R(b-z)R(z-a)$ pullback, $\mathsf A$, $L$, then
$\Cross$ & tests meeting $N$ & Same & Same.\\[0.3em]
$K_6,K_8,K_9$ & finite normal-jet form, normal traces
$c_{D,\beta}$, weights $v_\alpha$ & $U\setminus N_U$ proved;
extension to $N_U$ required & Coefficient extension satisfying
XV-C's estimate and the frozen class & D0.16--D0.18.\\
\bottomrule
\end{tabularx}
\end{center}

\begin{proposition}[No evaluable preliminary lift in the record]
\label{prop:nolift}
No linear map $\Fz$ with domain $\Pint$ and codomain
$\Dp(W;E_{\mathrm{res}})$ is specified by the supplied record, and
no such map can be specified from it without supplying at least the
items D0.1, D0.2, D0.5, D0.10, D0.11, and D0.13.
\end{proposition}

\begin{proof}
A linear map is specified by its domain, codomain, and value on each
element of the domain. The domain $\Pint$ is not defined (D0.1), so
``for each element of the domain'' has no content. The codomain is
not defined (D0.5). Even granting a domain and codomain, the value on
a presentation of any of the six entries must be the composed
distribution of the entry with the cross-line operators and vertex
data (D0.10--D0.13), none of which is defined. Writing a formula
with the undefined symbols would be a plausible-looking operator,
not a definition.
\end{proof}

Consequently the equality $r_O\Fz=m^\circ\Rec$ has been proved
nowhere in this record and is not asserted here for any map. The
null defect of the actual graph is not an evaluable object in this
release. This proves the plan's requirement that the calculation be
reported blocked rather than filled.

\section{The null defect in the one source-complete module}
\label{sec:nulljet}

The only module in the record for which presentations,
reconstruction, canonical domain, and target are all defined is
XV-C's finite normal-jet module. XV-C computed its full
reconstruction kernel. XV-D now carries out, for this module, the
full D1--D2 programme: define the preliminary lifts, compute their
null defects, locate the defects, and test cancellation. Everything
below is for this module. It is not identified with $\Pint$.

\subsection{Chart, insertions, and two reconstructions}

Let $X\subset\mathbb R^d$ and $V_q\subset\mathbb R^m$ be open with
$0\in V_q$, let $Z\subset X$ be relatively closed, and put
\[
 X^\circ=X\setminus Z,\qquad
 U=X\times V_q,\qquad
 U^\circ=X^\circ\times V_q,\qquad
 S=X\times\{0\}.
\]
For $b\in\Dp(X)$ and $\phi\in\D(U)$ the insertion is
\begin{equation}
 \langle\Ejet_\gamma b,\phi\rangle
 =(-1)^{|\gamma|}\langle b,\partial_q^\gamma\phi(\cdot,0)\rangle .
 \label{eq:insert}
\end{equation}
The same formula with $X^\circ$, $U^\circ$ in place of $X$, $U$
defines $\Ejet^\circ_\gamma:\Dp(X^\circ)\to\Dp(U^\circ)$.

Fix $r\ge0$ and the presentation space
\[
 \Pjet=\bigoplus_{|\alpha|\le r}C^\infty(U),
 \qquad a=(a_\alpha)_{|\alpha|\le r}.
\]
For $a\in\Pjet$ and $|\gamma|\le r$ define the smooth coefficient
\begin{equation}
 \beta_\gamma(a)(x)
 =
 \sum_{\substack{\alpha\ge\gamma\\|\alpha|\le r}}
 (-1)^{|\alpha-\gamma|}\binom{\alpha}{\gamma}
 \bigl(\partial_q^{\alpha-\gamma}a_\alpha\bigr)(x,0)
 \in C^\infty(X).
 \label{eq:beta}
\end{equation}

\begin{definition}[Full and punctured reconstruction]
\label{def:tworec}
Fix $T^\circ\in\Dp(X^\circ)$. The \emph{punctured reconstruction} is
\[
 \Recp:\Pjet\longrightarrow\Dp(U^\circ),\qquad
 \Recp(a)=\sum_{|\alpha|\le r}a_\alpha|_{U^\circ}\,
 \Ejet^\circ_\alpha T^\circ .
\]
For an extension $T\in\Dp(X)$ of $T^\circ$ (that is,
$T|_{X^\circ}=T^\circ$), the \emph{full reconstruction} is
\[
 \Rec_T:\Pjet\longrightarrow\Dp(U),\qquad
 \Rec_T(a)=\sum_{|\alpha|\le r}a_\alpha\,\Ejet_\alpha T .
\]
Write $H=\ker\Recp$ and $H_T=\ker\Rec_T$.
\end{definition}

In the language of XV-B's interface \cite{XVB}, this module is the
instance
\[
 \Pint\leadsto\Pjet,\quad
 \Rec\leadsto\Recp,\quad
 W\leadsto U,\quad
 O\leadsto U^\circ,\quad
 m^\circ\leadsto\mathrm{id},\quad
 V=\Dp(U),\quad
 Y=\ker r=\{t:\supp t\subset Z\times V_q\},
\]
where $r:\Dp(U)\to\Dp(U^\circ)$ is restriction. The canonical
contraction is the identity because the punctured reconstruction
already is the composed object on the canonical domain. The role of
the characteristic set is played by $Z\times V_q$; the role of the
response incidence by $S$.

\subsection{Normal form, extraction, and support}

\begin{lemma}[Normal form of a reconstruction]
\label{lem:normalform}
For every $T\in\Dp(X)$ and $a\in\Pjet$,
\begin{equation}
 \Rec_T(a)=\sum_{|\gamma|\le r}\Ejet_\gamma\bigl(\beta_\gamma(a)\,T\bigr).
 \label{eq:normalform}
\end{equation}
The same identity holds for $\Recp$ with $T^\circ$,
$\Ejet^\circ_\gamma$, and $\beta_\gamma(a)|_{X^\circ}$.
\end{lemma}

\begin{proof}
For $\phi\in\D(U)$,
\[
 \langle a_\alpha\Ejet_\alpha T,\phi\rangle
 =(-1)^{|\alpha|}
 \bigl\langle T,\partial_q^\alpha(a_\alpha\phi)(\cdot,0)\bigr\rangle
 =(-1)^{|\alpha|}\sum_{\beta\le\alpha}\binom{\alpha}{\beta}
 \bigl\langle(\partial_q^\beta a_\alpha)(\cdot,0)\,T,
 \partial_q^{\alpha-\beta}\phi(\cdot,0)\bigr\rangle
\]
by the Leibniz rule. Since
$(-1)^{|\alpha|}=(-1)^{|\beta|}(-1)^{|\alpha-\beta|}$, each term is
$(-1)^{|\beta|}\binom{\alpha}{\beta}
\langle\Ejet_{\alpha-\beta}((\partial_q^\beta a_\alpha)(\cdot,0)T),\phi\rangle$.
Substituting $\gamma=\alpha-\beta$ and summing over $\alpha$ gives
\eqref{eq:normalform} with the coefficient \eqref{eq:beta}. The
punctured case is the same computation on $U^\circ$.
\end{proof}

\begin{lemma}[Extraction and support]
\label{lem:extraction}
Let $b_\gamma\in\Dp(X)$ for $|\gamma|\le r$ and
$t=\sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma$. Then
\begin{enumerate}[label=\textup{(\roman*)}]
\item $t=0$ if and only if $b_\gamma=0$ for all $|\gamma|\le r$;
\item $\supp t=\bigl(\bigcup_{|\gamma|\le r}\supp b_\gamma\bigr)\times\{0\}$;
\item $t|_{U^\circ}=\sum_{|\gamma|\le r}\Ejet^\circ_\gamma(b_\gamma|_{X^\circ})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Choose $\chi\in\D(V_q)$ equal to one near $0$ and set
$\Jjet_\gamma\psi(x,q)=\frac{(-1)^{|\gamma|}}{\gamma!}q^\gamma\chi(q)\psi(x)$
for $\psi\in\D(X)$. Since $\chi$ is constant near $0$,
$(-1)^{|\beta|}\partial_q^\beta\Jjet_\gamma\psi(x,0)=\delta_{\beta\gamma}\psi(x)$
for all $\beta$, so $\langle t,\Jjet_\gamma\psi\rangle=\langle b_\gamma,\psi\rangle$.
This proves (i) and the inclusion ``$\supseteq$'' in (ii): if $t$
vanishes near $(x_0,0)$, choose $\chi$ supported in the normal
neighborhood and $\psi$ supported near $x_0$ to see that every
$b_\gamma$ vanishes near $x_0$. The inclusion ``$\subseteq$'' in (ii)
is immediate from \eqref{eq:insert}: a test vanishing near
$(\bigcup\supp b_\gamma)\times\{0\}$ has normal jets vanishing on
$\bigcup\supp b_\gamma$. For (iii), a test $\phi\in\D(U^\circ)$ has
$\partial_q^\gamma\phi(\cdot,0)\in\D(X^\circ)$, so
$\langle\Ejet_\gamma b_\gamma,\phi\rangle
=(-1)^{|\gamma|}\langle b_\gamma|_{X^\circ},\partial_q^\gamma\phi(\cdot,0)\rangle$.
\end{proof}

\begin{theorem}[The two kernels]
\label{thm:kernels}
Let $T$ be an extension of $T^\circ$. Then
\begin{align}
 H&=\ker\Recp
 =\{a\in\Pjet:\ \beta_\gamma(a)|_{X^\circ}\,T^\circ=0
 \text{ in }\Dp(X^\circ)\text{ for all }|\gamma|\le r\},
 \label{eq:Hpunct}\\
 H_T&=\ker\Rec_T
 =\{a\in\Pjet:\ \beta_\gamma(a)\,T=0
 \text{ in }\Dp(X)\text{ for all }|\gamma|\le r\},
 \label{eq:Hfull}
\end{align}
and $H_T\subseteq H$. Moreover $r\circ\Rec_T=\Recp$, so every
$\Rec_T$ is a preliminary lift for the punctured reconstruction.
\end{theorem}

\begin{proof}
\eqref{eq:Hpunct} and \eqref{eq:Hfull} follow from
Lemma~\ref{lem:normalform} and Lemma~\ref{lem:extraction}(i) on
$X^\circ$ and on $X$ respectively. For the compatibility, apply
Lemma~\ref{lem:extraction}(iii) to \eqref{eq:normalform}:
$\Rec_T(a)|_{U^\circ}=\sum\Ejet^\circ_\gamma(\beta_\gamma(a)|_{X^\circ}T|_{X^\circ})
=\Recp(a)$. If $\Rec_T(a)=0$ then its restriction $\Recp(a)=0$,
giving $H_T\subseteq H$.
\end{proof}

\subsection{Extension lifts and their null defects}

\begin{definition}[Extension lift]
For an extension $T$ of $T^\circ$, the \emph{extension lift} is
$\Fz^T=\Rec_T:\Pjet\to\Dp(U)$. Its \emph{null defect} is
\[
 d_0^T=\Fz^T|_H:H\longrightarrow\Dp(U).
\]
\end{definition}

\begin{theorem}[Exact null defect of an extension lift]
\label{thm:defect}
Let $T$ extend $T^\circ$ and let $a\in H$. Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $d_0^T(a)=\sum_{|\gamma|\le r}\Ejet_\gamma\bigl(\beta_\gamma(a)T\bigr)$,
and every coefficient $\beta_\gamma(a)T$ is supported in $Z$;
\item $\supp d_0^T(a)\subseteq Z\times\{0\}$; in particular
$d_0^T(a)\in Y$ and $d_0^T(a)|_{U^\circ}=0$;
\item $\ker d_0^T=H_T$; hence $d_0^T$ induces an injective linear
map $H/H_T\to Y$, and $d_0^T=0$ if and only if $H_T=H$;
\item $d_0^T(a)=0$ on all of $U$ if and only if
$\beta_\gamma(a)T=0$ in $\Dp(X)$ for all $|\gamma|\le r$; otherwise
the defect is a nonzero distribution supported in $Z\times\{0\}$,
and there is no open $G\subset U\setminus(Z\times\{0\})$ with
$d_0^T(a)|_G\ne0$.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) is \eqref{eq:normalform}. For $a\in H$,
$(\beta_\gamma(a)T)|_{X^\circ}=\beta_\gamma(a)|_{X^\circ}T^\circ=0$
by \eqref{eq:Hpunct}, so $\supp(\beta_\gamma(a)T)\subseteq Z$.
(ii) follows from Lemma~\ref{lem:extraction}(ii) and the definition
of $Y$. (iii): $\ker d_0^T=H\cap\ker\Rec_T=H\cap H_T=H_T$ by
Theorem~\ref{thm:kernels}. (iv) is Lemma~\ref{lem:extraction}(i)--(ii).
\end{proof}

\begin{remark}[The plan's trichotomy in this module]
For every tested null presentation $a\in H$ the outcome is never
``defect on an open set away from the puncture.'' Either the defect
vanishes on all of $U$ (when $a\in H_T$) or it is a nonzero
distribution carried by $Z\times\{0\}$. The second case is the
genuine content of ``vanishes only on the canonical domain'': it is
produced not by the insertion structure but by tangential
annihilator relations $\beta T^\circ=0$ that fail for $T$.
\end{remark}

\subsection{An explicit nonzero defect}

\begin{proposition}[A computed null defect]
\label{prop:example}
Let $d=2$, $X=\mathbb R^2$ with coordinates $(x_1,x_2)$,
$Z=\{0\}$, $m=1$, $r=0$. Let $T^\circ\in\Dp(X^\circ)$ be the
restriction to $X^\circ$ of $\delta(x_1)$, that is,
$\langle T^\circ,\psi\rangle=\int\psi(0,x_2)\dd x_2$ for
$\psi\in\D(X^\circ)$. Consider the two extensions
\[
 T_1=\delta(x_1),\qquad
 T_2=\delta(x_1)+\partial_{x_1}\delta_0 ,
\]
and the presentation $a\in\Pjet=C^\infty(U)$, $a(x,q)=x_1$. Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $H=\{a\in C^\infty(U): a(0,x_2,0)=0\ \text{for all }x_2\}$,
and $a=x_1$ belongs to $H$;
\item $H_{T_1}=H$, so $d_0^{T_1}=0$;
\item $H_{T_2}=\{a\in H:\ (\partial_{x_1}a)(0,0,0)=0\}$, the quotient
$H/H_{T_2}$ is one-dimensional, and
\[
 d_0^{T_2}(x_1)=\Ejet_0\bigl(x_1\,\partial_{x_1}\delta_0\bigr)
 =-\Ejet_0\delta_0\ne0,
 \qquad \supp d_0^{T_2}(x_1)=\{(0,0,0)\}=Z\times\{0\}.
\]
\end{enumerate}
\end{proposition}

\begin{proof}
For $r=0$, $\beta_0(a)=a(\cdot,0)$. (i): $a(\cdot,0)T^\circ=0$ on
$X^\circ$ means $\int a(0,x_2,0)\psi(0,x_2)\dd x_2=0$ for all
$\psi\in\D(X^\circ)$, i.e. $a(0,x_2,0)=0$ for $x_2\ne0$, hence for
all $x_2$ by continuity. For $a=x_1$ this holds. (ii): the same
computation on $X$ gives $H_{T_1}=H$. (iii): for $a\in H$,
\[
 \langle a(\cdot,0)\partial_{x_1}\delta_0,\psi\rangle
 =-\partial_{x_1}\bigl(a(\cdot,0)\psi\bigr)(0)
 =-(\partial_{x_1}a)(0,0,0)\,\psi(0)-a(0,0,0)\,\partial_{x_1}\psi(0)
 =-(\partial_{x_1}a)(0,0,0)\,\psi(0),
\]
so $a(\cdot,0)T_2=-(\partial_{x_1}a)(0,0,0)\delta_0$. This vanishes
if and only if $(\partial_{x_1}a)(0,0,0)=0$, giving $H_{T_2}$; the
linear functional $a\mapsto(\partial_{x_1}a)(0,0,0)$ is onto
$\mathbb C$ on $H$ (take $a=x_1$), so $H/H_{T_2}$ is one-dimensional.
For $a=x_1$ the coefficient is $-\delta_0$, and
Lemma~\ref{lem:extraction} gives the nonvanishing and the support.
\end{proof}

\begin{corollary}[Extension dependence and cancellation by extension]
\label{cor:extension}
Let $T,T'$ be extensions of $T^\circ$ and $s=T'-T$, so
$\supp s\subseteq Z$. Then
\[
 \Fz^{T'}-\Fz^T=\Rec_s,\qquad
 \Rec_s(\Pjet)\subseteq V_{Z\times\{0\}}
 :=\{t\in\Dp(U):\supp t\subseteq Z\times\{0\}\},
\]
and $d_0^{T'}=d_0^T+\Rec_s|_H$. The defect of $\Fz^T$ is cancelled
by a change of extension if and only if there is an extension $T'$
with $H_{T'}=H$. In Proposition~\ref{prop:example}, $s=-\partial_{x_1}\delta_0$
cancels the defect of $\Fz^{T_2}$.
\end{corollary}

\begin{proof}
Linearity of $T\mapsto\Rec_T$ gives $\Fz^{T'}-\Fz^T=\Rec_s$. By
\eqref{eq:normalform}, $\Rec_s(a)=\sum\Ejet_\gamma(\beta_\gamma(a)s)$
with each coefficient supported in $Z$, so Lemma~\ref{lem:extraction}(ii)
gives the support statement. Cancellation means $d_0^{T'}=0$, which
by Theorem~\ref{thm:defect}(iii) is $H_{T'}=H$. In the example,
$T_2+s=T_1$ and $H_{T_1}=H$.
\end{proof}

\begin{proposition}[Relative obstruction in the unrestricted
puncture-supported class]
\label{prop:unrestricted}
Let $\mathcal A_Z=\Hom(\Pjet,V_{Z\times\{0\}})$. For every extension
$T$, the relative obstruction class
$\Ob_{\mathcal A_Z}(\Fz^T)=[d_0^T]\in\Hom(H,Y)/\{a|_H:a\in\mathcal A_Z\}$
vanishes. The cancelling correction constructed in the proof is
algebraic; it is not asserted to be continuous, to be a change of
extension, or to lie in any inherited class.
\end{proposition}

\begin{proof}
By Theorem~\ref{thm:defect}(ii), $d_0^T(H)\subseteq V_{Z\times\{0\}}$.
Choose an algebraic complement $\Pjet=H\oplus L$ with projection
$\pi_H$ and set $a=-d_0^T\pi_H\in\mathcal A_Z$. Then
$(\Fz^T+a)|_H=d_0^T-d_0^T=0$. This is the algebraic criterion of
XV-B \cite{XVB} instantiated with the actual spaces of this module.
\end{proof}

\begin{corollary}[No separated witness in this module]
If a closed $C\subseteq U\setminus U^\circ$ contains $Z\times\{0\}$,
then for every extension $T$ and every $a\in H$ there is no open
$G\subset U\setminus C$ with $\Fz^T(a)|_G\ne0$. The contact-support
obstruction test of XV-B therefore cannot produce an obstruction for
any extension lift of this module.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:defect}(ii).
\end{proof}

\subsection{The inherited null identity produces no defect}

The record supplies one null identity for the actual graph
\cite{Jnull}:
\begin{equation}
 \Rec(M_{\mathrm R})=\delta(q)R(r),\quad
 \Rec(i\theta M_{\mathrm R})=\delta'(q)R(r),\quad
 n:=(q\,i\theta+1)M_{\mathrm R},\quad \Rec(n)=0 .
 \label{eq:inheritednull}
\end{equation}
Its analogue in the jet module is the presentation
\begin{equation}
 a^{(j)}=\bigl(a^{(j)}_\alpha\bigr)_{|\alpha|\le r},\qquad
 a^{(j)}_0=1,\quad a^{(j)}_{e_j}=q_j,\quad
 a^{(j)}_\alpha=0\ \text{otherwise},\qquad r\ge1,\ 1\le j\le m,
 \label{eq:ajet}
\end{equation}
for which $\Rec_T(a^{(j)})=\Ejet_0T+q_j\Ejet_{e_j}T$.

\begin{proposition}[Universal relations carry no defect]
\label{prop:universal}
For every $T\in\Dp(X)$, $a^{(j)}\in H_T$. Consequently
$a^{(j)}\in H$ and $d_0^T(a^{(j)})=0$ for every extension $T$ of
every $T^\circ$. More generally, if $a\in\Pjet$ satisfies
$\beta_\gamma(a)=0$ in $C^\infty(X)$ for all $|\gamma|\le r$, then
$a\in H_T$ for every $T$ and $d_0^T(a)=0$.
\end{proposition}

\begin{proof}
From \eqref{eq:beta}, $\beta_0(a^{(j)})=a^{(j)}_0(x,0)-\partial_{q_j}a^{(j)}_{e_j}(x,0)=1-1=0$,
$\beta_{e_j}(a^{(j)})=a^{(j)}_{e_j}(x,0)=q_j|_{q=0}=0$, and all other
$\beta_\gamma(a^{(j)})$ vanish because the corresponding $a_\alpha$
vanish. Thus $\beta_\gamma(a^{(j)})T=0$ for every $T$ and
\eqref{eq:Hfull} gives $a^{(j)}\in H_T$. The general statement is the
same argument.
\end{proof}

\begin{remark}[What this does and does not say]
In the jet module the null relations split into two kinds:
\emph{universal} relations, with all $\beta_\gamma(a)=0$ as smooth
functions, which hold for every coefficient and never produce a
defect; and \emph{annihilator} relations, with some
$\beta_\gamma(a)\ne0$ but $\beta_\gamma(a)T^\circ=0$, which are the
only source of a nonzero defect (Proposition~\ref{prop:example}).
The supplied identity \eqref{eq:inheritednull} is of the first kind
in this module. For the actual graph this classification is not
available: $\Rec$ is defined on two elements only (D0.2), and the
record does not say whether $\ker\Rec$ contains annihilator
relations. The statement ``the inherited identity carries no
defect'' is proved here for \eqref{eq:ajet} in the jet module and is
not transferred to \eqref{eq:inheritednull} in $\Pint$.
\end{remark}

\subsection{The inherited identity for an arbitrary lift}

\begin{proposition}[Descent on the span of one null presentation]
\label{prop:onenull}
Let $\Pint$, $\Rec$, $O$, $m^\circ$, $Y$ be as in XV-B's linear
interface, let $n\in\ker\Rec$, and let $\Fz$ be any preliminary lift.
Then $\Fz(n)\in Y$, and for a correction class
$\mathcal A\subseteq\Hom(\Pint,Y)$ the component of the obstruction
class on $\operatorname{span}\{n\}$ is the class of $\Fz(n)$ in
$Y/\{a(n):a\in\mathcal A\}$. If every $a\in\mathcal A$ takes values
in $V_C$ for a closed $C\subseteq W\setminus O$, then
$\supp\Fz(n)\subseteq C$ is necessary for descent in
$\Fz+\mathcal A$. For $n=(q\,i\theta+1)M_{\mathrm R}$ the descent
equation is
\begin{equation}
 \Fz(q\,i\theta M_{\mathrm R})+\Fz(M_{\mathrm R})=0
 \quad\text{in }\Dp(W;E_{\mathrm{res}}).
 \label{eq:nulltest}
\end{equation}
\end{proposition}

\begin{proof}
$r_O\Fz(n)=m^\circ\Rec(n)=0$, so $\Fz(n)\in\ker r_O=Y$. A candidate
$\Fz+a$ annihilates $n$ iff $a(n)=-\Fz(n)$, which is the quotient
statement. If $a(n)\in V_C$ for all $a$, then $\Fz(n)=-a(n)\in V_C$
is necessary. The last display is linearity.
\end{proof}

For the actual graph, $\Fz$ is undefined (Proposition~\ref{prop:nolift}),
so \eqref{eq:nulltest} has no computed left-hand side in this
release.

\section{Coefficient and extension calculations}
\label{sec:coefficients}

\subsection{What the record says about $v_\alpha$ and $c_{D,\beta}$}

The supplied supported-sector reference is \cite{Jreference}
\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}
with wavefront bound
\begin{equation}
\begin{split}
 \WF(t_D^\circ)\subseteq\Gamma_{\mathrm{supp}}^\circ
 ={}&
 N^*\Sresp\big|_{U\setminus N_U}\setminus0\\
 &\cup
 \bigcup_{i=1}^2
 \left\{
 (w,\lambda\dd q_i+\omega):
 w\in\Sresp\setminus N_U,\ q_i(w)=0,\ \lambda\ne0,\
 \omega\in N^*_w\Sresp
 \right\}\setminus0 .
\end{split}
\label{eq:referenceWF}
\end{equation}
XV-C collected the unique coefficients
\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}
and proved that the unrestricted supported realization exists if and
only if each $b^\circ_{D,\gamma}$ satisfies the compact-uniform
estimate
\begin{equation}
 |\langle b_{D,\gamma}^\circ,\psi\rangle|\le C_{K,\gamma}\,p_{k_{K,\gamma}}(\psi),
 \qquad \psi\in\D(S_0\setminus Z),\ \supp\psi\subset K,
 \label{eq:jetbound}
\end{equation}
where $S_0$ is the base of the normal chart of $\Sresp$ and
$Z=\{x:(x,0)\in N_U\}$ \cite{XVC}.

The defining formulas of $v_\alpha$ and $c_{D,\beta}$ are not in the
record (D0.18). The supplied proof of the reference theorem contains
the sentence ``the normal traces have the tangential bounds in that
lemma; multiplication by $v_\alpha$ adds no wavefront''
\cite{Jreference}. We record the following \emph{reading}, not a
supplied definition: $c_{D,\beta}$ are normal traces of the
cross-line factors along $\Sresp$, and $v_\alpha$ are multipliers
whose multiplication adds no wavefront, as a smooth function would.
The next result is stated under an explicit hypothesis matching that
reading, so that it can be applied or discarded once the formulas are
supplied.

\subsection{A conditional reduction}

\begin{proposition}[Reduction of the coefficient estimates to the
traces]
\label{prop:reduction}
Assume
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item $v_\alpha\in C^\infty(S_0)$ for all $|\alpha|\le3$;
\item $c_{D,\beta}\in\Dp(S_0\setminus Z)$ for all $|\beta|\le3$.
\end{enumerate}
Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item If every $c_{D,\beta}$, $|\beta|\le3$, extends to a
distribution $\tilde c_{D,\beta}\in\Dp(S_0)$, then every
$b^\circ_{D,\gamma}$ satisfies \eqref{eq:jetbound}, the unrestricted
supported realization of normal order at most three exists, and one
realization is
\[
 t_D=\sum_{|\gamma|\le3}\Ejet_\gamma b_{D,\gamma},\qquad
 b_{D,\gamma}=\sum_{\substack{\beta\ge0\\|\gamma+\beta|\le3}}
 (-1)^{|\beta|}\binom{\gamma+\beta}{\gamma}
 v_{\gamma+\beta}\tilde c_{D,\beta}.
\]
\item Order by order, \eqref{eq:jetbound} for $|\gamma|=3$ involves
only $c_{D,0}$; for $|\gamma|=2$ only $c_{D,0}$ and $c_{D,e_j}$;
for $|\gamma|=1$ only $c_{D,\beta}$ with $|\beta|\le2$; for
$\gamma=0$ all $c_{D,\beta}$ with $|\beta|\le3$.
\item If for some $|\gamma|=3$ and some compact $K$ the estimate
\eqref{eq:jetbound} fails for $b^\circ_{D,\gamma}=v_\gamma c_{D,0}$,
then $c_{D,0}$ has no extension to $\Dp(S_0)$; the failure sequence
of XV-C for $b^\circ_{D,\gamma}$ is then a failure witness for the
trace $c_{D,0}$ itself.
\item Changing the extensions $\tilde c_{D,\beta}$ by $Z$-supported
distributions $s_\beta$ changes $b_{D,\gamma}$ by
$\sum(-1)^{|\beta|}\binom{\gamma+\beta}{\gamma}v_{\gamma+\beta}s_\beta$,
which is $Z$-supported. This family lies in XV-C's unrestricted
supported ambiguity but is not asserted to exhaust it.
\end{enumerate}
\end{proposition}

\begin{proof}
(i) Multiplication of a distribution on $S_0$ by a smooth function and
finite sums are defined and restrict compatibly to $S_0\setminus Z$,
so $b_{D,\gamma}|_{S_0\setminus Z}=b^\circ_{D,\gamma}$. Distributional
continuity of $b_{D,\gamma}$ on tests supported in $K$ gives
\eqref{eq:jetbound}; XV-C's supported realization theorem then gives
existence, and its realization formula gives the display.
(ii) is read off from \eqref{eq:collected}, since
$|\gamma+\beta|\le3$ forces $|\beta|\le3-|\gamma|$.
(iii) If $c_{D,0}$ extended, then $v_\gamma\tilde c_{D,0}$ would
extend $v_\gamma c_{D,0}$ and \eqref{eq:jetbound} would hold by (i).
(iv) Smooth multiples and finite sums of $Z$-supported distributions
are $Z$-supported; XV-C's ambiguity theorem identifies the
unrestricted ambiguity as all $Z$-supported coefficient changes, of
which these are some.
\end{proof}

\begin{remark}
The converse of (i)---that extendability of all $b^\circ_{D,\gamma}$
implies extendability of each $c_{D,\beta}$---is not claimed. It
would require rank conditions on the multipliers $v_\alpha$
(for instance, invertibility of the triangular system
\eqref{eq:collected} locally), which depend on the unsupplied
formulas. Nor does (i) yield a realization in the frozen class: it is
the unrestricted class, and compatibility with the frozen
prescription, continuity on $\Pint$, and the microlocal requirements
are separate obligations.
\end{remark}

\subsection{What is not computed}

No estimate \eqref{eq:jetbound} and no failure sequence is
established for an actual coefficient, because no actual coefficient
is defined. The characteristic compositions of $K_{47}$ and of both
$K_5$ placements are not of supported normal-jet form in the record
and are not addressed by Proposition~\ref{prop:reduction}. Gate D3
is not entered for the actual graph.

\section{Correction-class and contact analysis}
\label{sec:corrections}

The inherited correction class $\mathcal A$ is not defined
(D0.16--D0.17). The following statements are therefore all that can
be made.

\begin{enumerate}[label=\textup{(\roman*)}]
\item For the actual graph, neither $d_0$ nor the quotient
$\Hom(H,Y)/\{a|_H:a\in\mathcal A\}$ is evaluable
(Propositions~\ref{prop:D0} and \ref{prop:nolift}).
\item For the jet module, the relative obstruction vanishes in the
unrestricted puncture-supported class
(Proposition~\ref{prop:unrestricted}), and cancellation within the
narrower class of extension changes is equivalent to the existence of
an extension $T'$ with $H_{T'}=H$ (Corollary~\ref{cor:extension}).
Whether such an extension exists for a given $T^\circ$ is not decided
in general here; it is decided affirmatively in
Proposition~\ref{prop:example}.
\item No separated witness exists in the jet module for any
extension lift. This is a property of that module, not a statement
that the actual graph lacks separated defects.
\item A proposed cancellation for the actual graph must satisfy the
support, normalization, tensor, continuity, and microlocal conditions
of the frozen class, none of which is available to check.
\end{enumerate}

No obstruction is asserted. No cancellation is asserted for the
actual graph.

\section{Full-graph estimates and the admission boundary}
\label{sec:bounds}

The complete-graph estimates require the ambient variables and the
strata. The record supplies only bookkeeping:
$\dim W=16$ and $\dim N=\dim\Natt=8$ \cite{Grelease}; $\Sresp$ of
codimension five, $N_U$ of codimension eight, and a relative
codimension-three flag \cite{record}. These are reported numbers.
The defining equations of $W$, $N$, $\Sresp$, the collision strata,
and the observable diagonals are not in the record (D0.5--D0.9).

Accordingly no statement is made about:
a complete-graph wavefront bound across $N$; uniform transverse
estimates for a joint section; compatibility on intersections of
characteristic and collision strata; proper pushforward or
restriction estimates; or the admission \eqref{eq:admission}. The
local distributional-order bound of XV-C for a supported
normal-order-three realization is not a transverse estimate for the
graph. Gate D4 is not entered.

\section{Verdict and remaining gates}
\label{sec:verdict}

\begin{theorem}[XV-D local mathematical release]
\label{thm:release}
For the finite normal-jet module with punctured reconstruction
defined in Section~\ref{sec:nulljet}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item every extension $T$ of the punctured coefficient $T^\circ$
defines a preliminary lift $\Fz^T=\Rec_T$ with $r\Fz^T=\Recp$;
\item the two reconstruction kernels are \eqref{eq:Hpunct} and
\eqref{eq:Hfull}, with $H_T\subseteq H$;
\item the null defect is exactly
$d_0^T(a)=\sum_\gamma\Ejet_\gamma(\beta_\gamma(a)T)$, is supported
in $Z\times\{0\}$, and induces an injection $H/H_T\to Y$;
\item the defect can be nonzero (Proposition~\ref{prop:example}),
depends on the extension, and is cancelled by a change of extension
exactly when some extension $T'$ has $H_{T'}=H$;
\item the relative obstruction in the unrestricted
puncture-supported class vanishes, and no separated contact witness
exists;
\item universal relations, including the jet analogue
\eqref{eq:ajet} of the inherited identity
\eqref{eq:inheritednull}, carry no defect for any extension.
\end{enumerate}
In addition, under hypotheses \textup{(H1)}--\textup{(H2)},
extendability of the cross-line traces $c_{D,\beta}$ implies XV-C's
supported-sector estimates order by order
(Proposition~\ref{prop:reduction}).
\end{theorem}

\begin{proof}
(i)--(ii) are Theorem~\ref{thm:kernels}. (iii) is
Theorem~\ref{thm:defect}. (iv) is Proposition~\ref{prop:example} and
Corollary~\ref{cor:extension}. (v) is Proposition~\ref{prop:unrestricted}
and its corollary. (vi) is Proposition~\ref{prop:universal}. The
last sentence is Proposition~\ref{prop:reduction}.
\end{proof}

\subsection{Full-graph verdict}

\begin{equation}
 \boxed{
 \begin{gathered}
 \text{Unresolved at D0: source-complete instantiation unavailable.}\\
 \text{Gates D1--D4 not entered for the actual graph.}\\
 \text{The inherited full joint-map gate remains open.}
 \end{gathered}
 }
 \label{eq:verdict}
\end{equation}

This is justified by Propositions~\ref{prop:D0} and
\ref{prop:nolift}. It continues the J2 stop of XIV-J, the B1 stop of
XV-B, and the C0 stop of XV-C \cite{Jverdict,XVB,XVC}. It is not an
obstruction: no exhaustive class is defined and no incompatibility
is proved. It is not a construction: no map on $\Pint$ is defined.
The admission \eqref{eq:admission} is unasserted, and ``not
admitted'' is not ``nonexistent'' \cite{Jadmission}.

\subsection{Acceptance ledger}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.36\linewidth}X@{}}
\toprule
Acceptance item & XV-D verdict\\
\midrule
Actual frozen graph and every required operator defined &
No. Twenty-item audit; D0 fails (Proposition~\ref{prop:D0}).\\[0.3em]
All retained branches, including $K_{47}$ and both $K_5$ placements &
Tracked; freeze check passed; no characteristic evaluation.\\[0.3em]
Operation defined on tests meeting $N$ &
No (Proposition~\ref{prop:nolift}).\\[0.3em]
Descent under the full reconstruction kernel &
Not evaluable for the graph. Fully computed for the jet module
(Theorem~\ref{thm:defect}).\\[0.3em]
Corrections in the inherited permitted class &
Class undefined. Jet-module cancellation analysed in two
unrestricted classes.\\[0.3em]
Coefficient estimates for actual $v_\alpha$, $c_{D,\beta}$ &
Formulas unavailable; conditional reduction proved
(Proposition~\ref{prop:reduction}).\\[0.3em]
Complete-graph microlocal and uniform estimates &
Not entered; strata equations unavailable.\\[0.3em]
Obstruction in an exhaustive class &
Not asserted.\\[0.3em]
Quantum admission &
Unasserted.\\
\bottomrule
\end{tabularx}
\end{center}

\section{Debugging record}
\label{sec:debug}

\begin{enumerate}[label=\textup{D\arabic*.},leftmargin=*]
\item \textbf{Title.} The suggested title announced ``the First Full
Calculation.'' Since that calculation is blocked at D0, the title
names its boundary instead; no result is announced that the text does
not prove.
\item \textbf{Outcome selection.} The unresolved alternative is
taken at the gate named by the plan. No plausible-looking operator is
written with undefined symbols.
\item \textbf{Provenance discipline.} A sentence in a predecessor
proof (``multiplication by $v_\alpha$ adds no wavefront'') is
recorded as a reading and converted into an explicit hypothesis
(H1), not into a definition.
\item \textbf{Branch counting.} Six operational entries; $K_{47}$
groups two originals; both $K_5$ placements belong to $B_5$. The
supported sector is not the full sum.
\item \textbf{Model versus instance.} The jet module is explicitly
an instance of XV-B's interface with $m^\circ=\mathrm{id}$ and
$O=U^\circ$. It is not declared equal to $\Pint$; no result of
Section~\ref{sec:nulljet} is transferred to the graph.
\item \textbf{Two kernels.} The null defect is defined relative to
the \emph{punctured} kernel $H$, which is the reconstruction actually
available on the canonical domain; the full kernel $H_T$ depends on
the extension. Conflating them would make every defect vanish
trivially.
\item \textbf{Tangential annihilators.} The example uses
$x_1\delta(x_1)=0$ versus $x_1\partial_{x_1}\delta_0=-\delta_0$;
coefficient equations remain equations in $\Dp(X)$, as in XV-C.
\item \textbf{Algebraic corrections.} The cancelling correction in
Proposition~\ref{prop:unrestricted} is algebraic and unrestricted; it
is not offered as a graph prescription. The narrower class of
extension changes is analysed separately.
\item \textbf{Extendability versus admissibility.} Proposition~\ref{prop:reduction}
gives the unrestricted class only; membership in the frozen class is
not inferred.
\item \textbf{Dimensions.} The numbers 16, 8, and the codimensions
5, 8, 3 are reported bookkeeping and are not used in any proof.
\item \textbf{Prescription.} Pole, propagators, carrier, normalization,
and ambiguity class are unchanged. No regulator or finite part is
introduced.
\item \textbf{Obstruction language.} ``No separated witness in the
jet module'' is a theorem about that module, not a statement that the
graph has no separated defect.
\item \textbf{Compilation.} All macros used are defined in the
preamble; all \verb|\ref| and \verb|\cite| targets exist; tabularx
column widths sum to less than \verb|\linewidth|; no non-ASCII
characters occur in mathematics.
\end{enumerate}

\section{Successor requirement}
\label{sec:successor}

The next substantive step requires the items of
Definition~\ref{def:package} in status \textsf{S}. With them, the
calculation of Section~\ref{sec:nulljet} is to be repeated for the
actual module: define $\Fz$ on every presentation of the six entries
by the supplied composition rule; prove $r_O\Fz=m^\circ\Rec$ on the
supplied domain; compute $\ker\Rec$ and classify its relations into
universal and annihilator type; evaluate $d_0$ on each annihilator
relation and locate its support relative to the supplied contact
locus; and decide cancellation in the supplied class
$\mathcal A$. Proposition~\ref{prop:onenull} gives the form of the
test on the one supplied relation. Proposition~\ref{prop:reduction}
gives the first estimate to check for the supported sector once
$v_\alpha$ and $c_{D,\beta}$ are supplied. The characteristic
compositions of $K_{47}$ and both $K_5$ placements require a
composition theorem that is not in the record and is not supplied by
any result here.

\subsection*{Final release sentence}

\begin{center}
\fbox{\parbox{0.91\linewidth}{
XV-D audits the supplied record item by item and finds the frozen
graph data still source-incomplete at D0; it therefore defines no
preliminary lift for the actual graph and asserts no graph-specific
null defect, obstruction, estimate, or admission. For the one
source-complete module, the finite normal-jet module with punctured
reconstruction, it computes the null defect of every extension lift
exactly, proves it puncture-supported and extension-dependent,
exhibits a nonzero instance, and shows that the inherited null
identity carries no defect. The full joint characteristic map
remains unresolved before instantiation.
}}
\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.

\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. Used for the linear
interface, the relative-lifting criterion, the contact-support test,
and the unrestricted contact-valued criterion.

\bibitem{XVC}
Manuscript XV-C, supplied \texttt{XV-C-va.tex}.
\emph{The Full Joint Characteristic Target: Verified Reconstruction
Tests and the Instantiation Boundary}. Full text supplied. Used for
the finite normal-jet null criterion, the collected coefficients,
the exact extension criterion, and the supported realization
theorem.

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

\bibitem{Hverdict}
Manuscript XIV-H, version~1. Theorem \texttt{thm:H2}, lines
1539--1582; Theorem \texttt{thm:release}, lines 1847--1895. Direct
excerpts supplied. The maps \texttt{eq:K8reduced} and \texttt{eq:K9}
are named, not reproduced.

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

\bibitem{Iamplitude}
Manuscript XIV-I, version~1. Theorem \texttt{thm:I2} and adjacent
precomposition formulas, lines 1288--1378. Direct excerpt supplied;
Definition \texttt{def:amodule} is named, not reproduced.

\bibitem{Ijoint}
Manuscript XIV-I, version~1. Joint-map interface, Proposition
\texttt{prop:canonical}, Theorem \texttt{thm:I3}, lines 1459--1548.
Direct excerpt supplied; \texttt{eq:target} and
\texttt{def:class} are named, not reproduced.

\bibitem{Irelease}
Manuscript XIV-I, version~1. Theorem \texttt{thm:release}, lines
1729--1814. Direct excerpt supplied.

\bibitem{Jdescent}
Manuscript XIV-J, version~1. \texttt{XIV-J.tex}, identifier
\texttt{3c48e041}. Proposition \texttt{prop:descent}, lines
813--846. Direct excerpt supplied.

\bibitem{Jnull}
Manuscript XIV-J, version~1. Lemma \texttt{lem:null}, lines
853--883. Direct excerpt supplied.

\bibitem{Jreference}
Manuscript XIV-J, version~1. Theorem \texttt{thm:reference}, lines
1215--1299. Direct excerpt supplied; Lemma \texttt{lem:traces} and
the formulas of $v_\alpha$, $c_{D,\beta}$ are named, not reproduced.

\bibitem{Jconditional}
Manuscript XIV-J, version~1. Theorem
\texttt{thm:conditionalextension}, lines 1427--1470. Direct excerpt
supplied; Lemma \texttt{lem:insertWF} and the operator $\mathcal L$
are named, not reproduced.

\bibitem{Jverdict}
Manuscript XIV-J, version~1. Theorem \texttt{thm:J2}, lines
1608--1649. Direct excerpt supplied; Definition
\texttt{def:mapclass} is named, not reproduced.

\bibitem{Jrelease}
Manuscript XIV-J, version~1. Theorem \texttt{thm:release}, lines
1894--1939. Direct excerpt supplied.

\bibitem{Jadmission}
Manuscript XIV-J, version~1. Corollary \texttt{cor:noadmission},
lines 1808--1822, and admission-interface passage, lines
1794--1807. Direct excerpts supplied; \texttt{eq:Rconvention},
$Q_H$, and $\Eout$ are named, not reproduced.

\bibitem{Grelease}
Manuscript XIV-G, version~1. \texttt{XIV-G.tex}, identifier
\texttt{4fcac266}. Theorem \texttt{thm:release}, lines 1710--1751.
Excerpt supplied in the retrieval account. Used only for the stated
dimensions of $W$ and $N=\Natt$.

\bibitem{record}
Supplied coverage account, retrieval corpus identifier
\texttt{3883b893}, retrieval~3. Explicitly disclaims a complete
corpus scan and independent verification. Used only as documentary
context and for reported codimension bookkeeping.

\end{thebibliography}

\end{document}