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\title{Landmark XV-F, Patch II\\
Frozen-Object Completions for the Joint-Graph Source Package:\\
Residual Fiber, Six-Entry Presentations,\\
Coefficient Instantiation, and the Correction Class}
\author{}
\date{}

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

\begin{abstract}
XV-F-patch-ii completes XV-F-patch at the level of frozen objects.
It asks whether the four gaps recorded in the patch's F0
verdict---the residual codomain, the generators and module action of
the presentation package, the ingredients of the six operational
entries, and the parametrization and presentation-level form of the
correction class---can be closed under the original F0 rule, which
passes a core item only when its frozen member is identified, never
when a parameter family is displayed.

Five gates are run. Q1: a residual bundle $\Eres$ of rank ten, with
the metric slot at $z$ and the scalar slot at $y$, its density, and
its duality pairing are specified as a convention adopted here, no
sheet defining them; the index bookkeeping of the displayed entries
supports the choice, and it is proved that smooth residual testing
erases every covector with nonzero $z$-component. Q2: the six entries
are enumerated; the $K_5$ placements have XIV-I's chart module with
its reconstruction rule but without reproduced generators; for the
supported sector it is proved that, for every completion of the
undefined data obeying XIV-H's bound $|\alpha|+|\beta|\le2$,
$K_6+K_8+K_9$ is a normal jet of order at most three along the null
incidence $\{q_+=0,\ r_+=0\}$, which reproduces XIV-J's normal order
exactly, and that $\partial_\alpha\Bker=\tfrac12x_\alpha R$, so $K_8$
is carried by the null incidence, correcting a remark of the patch;
$K_{47}$ has no presentation space in any sheet, and Q2 fails there.
Q3: the construction of $\Dmap$ from $\dot a_2$ is displayed in XIV-I
and identified; $\Amap$, $\sigma_\eps$, $f$, $a_2$,
$q^{L,I,J}_{\alpha\beta}$, $\Pmap$, $\mathfrak b$, $g$, $\EA_m$ are
open, each reduced to a named sheet and sentence. Q4: XIII-C's three
roles remain roles; the correction class as maps on presentations is
not defined by the sheets, and the lift of the collision sets is
refused. Q5: the list of cross-line products proved on $\Ucan$ with
an operational domain is stated to be empty. The verdict is
\emph{Unresolved at F0}, at Q2; F1 is not entered; no lift,
null-defect test, characteristic prescription, or admission 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) {Patch I at F0};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Frozen-object completions};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A defined lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The further completions do not pass F0.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Frozen-object completions still stop at F0. 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-F-patch-ii}
\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) {Patch I at F0};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A defined lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The further completions do not pass F0.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Frozen-object completions still stop at F0. 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-F-patch-ii}
\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) {Patch I at F0};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Frozen-object completions};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A defined lift};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The further completions do not pass F0.};
\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-F-patch-ii}
\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) {Patch I at F0};
\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 lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Frozen-object completions. The further completions do not pass F0.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Patch I at F0 is not A defined 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-F-patch-ii}
\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: Patch I at F0};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Frozen-object completions};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A defined lift};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The further completions do not pass F0.};
\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-F-patch-ii}
\end{figure}

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


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



\section{Scope, gates, and the stop rule}
\label{sec:scope}

\subsection{The question}

XV-F-patch \cite{XVFpatch} ended at F0 with Proposition
\texttt{prop:F0}, cited in the commissioning plan as
\texttt{prop:F0patch}. Its blocking list has four frozen gaps:
\begin{enumerate}[label=\textup{(\alph*)}]
\item the residual codomain $\Eres$ of the target space
$\Dp(W;\Eres)$ is undefined (D0.5);
\item the generators, coefficient class, and module action of the
presentation package $\Pint$ are not reproduced, so $\ker\Rec$ is not
computable (D0.1--D0.4);
\item the ingredients of the six operational entries---$\Amap$,
$\sigma_\eps$, $f$, $\dot a_2$, $\Dmap$, $q^{L,I,J}_{\alpha\beta}$,
$\Pmap$, $\mathfrak b$, $g$, $\EA_m$---are not fixed (D0.13--D0.14);
\item the correction class is a class of contact values whose
parametrization ($D_\Gamma$, $w_\Gamma$, $S_{\Gamma/I}$) and
presentation-level form $\mathcal A\subseteq\Hom(\Pint,Y)$ are not
supplied (D0.17);
\end{enumerate}
together with the domain item (D0.8): $O=\Ucan$ is a named domain
whose equality with $W\setminus\overline{\Natt}$ is refused at three
located places. This manuscript asks whether (a)--(d) and the domain
item can be closed under the original F0 rule.

\textbf{F0 rule.} A core item passes when its frozen member is
identified in a direct sheet, or when a convention is adopted here
and explicitly offered as the frozen choice. A displayed parameter
family is an open datum. A locator, a role, or a search term is not
a member.

\subsection{Gates}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.05\linewidth}p{0.17\linewidth}XX@{}}
\toprule
Gate & Object & Pass & Fail\\
\midrule
Q1 & Residual codomain & Frozen fiber, density, and test pairing
specified & Another family $\mathbb B_{\mathrm{res}}$ of
candidates\\[0.2em]
Q2 & Presentation package & Six entries, their generators, module
action, image topology; then $\ker\Rec$ & The local chart module
substituted for the common space; an entry without copied
generators\\[0.2em]
Q3 & Coefficient instantiation & Ingredients of the copied $K_8$,
$K_9$ (and of $L$, $K_5$, $K_6$) identified & Formulas copied with
roles intact\\[0.2em]
Q4 & Correction class & Actual $D_\Gamma$, $w_\Gamma$,
$S_{\Gamma/I}$, and $\mathcal A$ as maps & Contact template called
frozen\\[0.2em]
Q5 & Domain list & Products already proved on $O$ listed, or the
absence stated & Complement equality reused\\
\bottomrule
\end{tabularx}
\end{center}

\textbf{Stop rule, as applied.} The F0 verdict is fixed by the first
failing gate, in the order Q1--Q5. ``Stop'' means that F1 is not
entered and that no later gate can change the verdict. The remaining
patches are nevertheless executed, since the plan allocates them
and since the successor needs the complete blocking list; their
outcomes are recorded as documentary and cannot contribute to a
pass.

\subsection{Evidence contract}
\label{sec:evidence}

\textbf{Direct full texts.} XV-F-patch \cite{XVFpatch}, as the record
being completed; its documentary ledgers are inputs, its F0 refusal
is binding, and its verified lemmas are cited by label. XIV-G
\cite{Grelease,Gresidual,GG34}, for $W$, $N=\Nt$, and the residual
test. XIV-J \cite{Jdescent,Jnull,Jreference,Jconditional,JJ2,
Jrelease,Jadmission,Jmapclass,Jtarget,Jcross,Jhessian,Jrefuse}, for
the presentation interface, the retained sum, \texttt{def:mapclass},
and the descent criterion. XIV-I \cite{Ipole,IeqR,Ichart,Iamodule,
II2,II3,Irelease,Iresponse,Irefuse}, for the chart module, the
source-internal identification of $R$ as the kernel of
$E_0^{\mathrm R}$, and the copied $K_8$, $K_9$ displays. XIV-H
\cite{Hkernel,HeqL,Hq,Hrefuse,Hfiber,HH2,Hrelease,HK9}, for the
coefficient input statements. XIII-C \cite{XIIICcoll,XIIICpart,
XIIICvertex,XIIICU,XIIICT0}, for $\Delta_I$, \texttt{def:Cpart},
$\UG$, and $T_0$.

\textbf{Copied premises}, usable only where the display is already
quoted inside one of the six direct sheets: XII-E \cite{XIIEa2} for
$a_2$ (and $a_1$, $b_1$ to the extent XIV-H's input statement
\texttt{input:EC} \cite{HinputEC} quotes them); XII-C
\cite{XIICrelease} for the undifferentiated cubic; XII-D
\cite{XIIDfiber,XIIDweight} for $f$ and the observable; XIII-B
\cite{XIIIB} for the time-ordered line convention copied into
XIII-C.

\textbf{Non-inputs.} XI-B, XII-A, XII-B, XIV-D, XIV-E, and the
quantum coefficient $\Rquant$. A citation inside a handoff does not
import them. The equality $\Ucan=W\setminus\overline{\Natt}$ remains
refused and is recorded once, in \eqref{eq:refused}.

\textbf{Reading rule.} Similar words in distinct manuscripts do not
identify their models, response maps, or sign conventions
\cite{record}. Within the six direct sheets, a symbol defined in one
and used in another with an explicit input statement is one symbol;
without such a statement it is two.

\textbf{Status codes.} \emph{identified} (frozen member displayed in
a direct sheet, with locator); \emph{convention} (adopted here,
offered as the frozen choice); \emph{open} (role, family, locator, or
unreturned quotation); \emph{documentary} (a gate whose pass
condition is a statement about the record).

Standard distribution theory is used with reference to \cite{Horm}.
Throughout, $\eta=\mathrm{diag}(+,-,-,-)$, $\Box=\partial_t^2-\Delta$,
$\Jp=\{t\ge|\mathbf x|\}$, $\partial\Jp=\{t=|\mathbf x|\}$,
$x_\alpha=\eta_{\alpha\nu}x^\nu$, $x^2=\eta_{\mu\nu}x^\mu x^\nu$;
$R=\frac{\Theta(t)}{2\pi}\delta(x^2)$ and $\Bker=R*R=
\frac1{8\pi}\Theta(t)\Theta(x^2)$ are XIV-I's displays \cite{IeqR},
verified in \cite[Lemmas \texttt{lem:R}, \texttt{lem:RR}]{XVFpatch}.

\section{Corrections to XV-F-patch}
\label{sec:corrections}

\paragraph{(G1) The support of $K_8$.}
The patch's Proposition \texttt{prop:support} placed $\supp K_8$ in
the solid cone $S_{\Jp}=\{a=z,\ b-a\in\Jp\}$, and its Remark
\texttt{rem:conormal} argued that $K_8$ is singular at points with
$(b-a)^2>0$, where $\Bker(b-a)$ is locally constant, so that the
conormal statement of XIV-H could not be checked. The derivative
$\partial_{b^\alpha}$ inside the bracket of \eqref{eq:K8} acts on
$\Bker(b-v)$, and $\partial_\alpha\Bker=\tfrac12x_\alpha R$
(Lemma~\ref{lem:dB}). Hence $\supp K_8\subseteq S_\partial=\{a=z,\
b-a\in\partial\Jp\}$ (Corollary~\ref{cor:K8supp}); the premise of the
Remark fails, and the check it declared not performable passes at
the level of supports. Item (g) of the patch's Proposition
\texttt{prop:checks} is moved to the performed list.

\paragraph{(G2) The construction of $\Dmap$.}
The patch stated that the construction of $\Dmap^{\alpha\mu,I,J}$
from $\dot a_2$ is XIV-H's and not reproduced. XIV-I displays it
\cite{II2}: $\Dmap$ is the constant bilinear fiber map determined by
$\dot a_2^{\alpha\mu}[G;h]=\Dmap^{\alpha\mu,I,J}h_IG_J$
(\eqref{eq:Dfrom} below). The construction is identified; the value
of $\Dmap$ depends on $a_2$, which remains open
(\S\ref{sec:patch3}).

\paragraph{(G3) XII-A and XII-B are non-inputs here.}
The patch used XII-B's trace reversal to state
$\mathfrak a=-\PPcalnote$. Under the present evidence contract that
sheet is a non-input; the identity is restated as the stand-alone
computation $\mathfrak a^{\alpha\mu,ab}r_{ab}=\tfrac12\eta^{\alpha\mu}
\tr_\eta r-r^{\alpha\mu}$ (Lemma~\ref{lem:fiber}), with no second
map named. XII-A's Hadamard blocks are not used; the identity of the
blocks in $\Cross$ is carried from the patch as open and is not
reassessed (\S\ref{sec:reassess}).

\section{Patch 1: residual fiber and pairing (Q1)}
\label{sec:patch1}

\subsection{What XIV-G types}

\begin{definition}[XIV-G's residual test; copied]
\label{def:gtest}
Let $M=\mathbb R^4$, $W=\mathbb R^{16}$ with coordinates $(a,b,z,y)$,
$a,b,z,y\in M$ \cite{Grelease,Ichart}, and $Z=\mathbb R^4_z$ the
attachment variable. XIV-G's residual test is a section
$G\in C^\infty_c(Z;\Sym^2T^*M)$, and XIV-G states that the residual
pairing $\mathrm{JD}'[G]$ has order at most two \cite{Gresidual}.
XIV-G does not define a bundle $\Eres$ over $W$.
\end{definition}

\begin{remark}
Definition~\ref{def:gtest} types an \emph{input direction on $Z$}: a
compactly supported symmetric covariant two-tensor field on the
four-dimensional attachment space. It is not a section of any bundle
over the sixteen-dimensional $W$. The order statement is copied as a
statement about XIV-G's object; no order bound is asserted for any
object constructed in this manuscript.
\end{remark}

\subsection{The convention}

The joint-map statements of XIV-I and XIV-J name a residual species
with a metric slot at $z$ and a scalar slot at $y$ \cite{II3,
Jmapclass}; they do not display the bundle. The following
convention is introduced here and offered as the frozen choice.

\begin{convention}[C1: residual bundle, density, pairing]
\label{def:Eres}
\begin{enumerate}[label=\textup{(\roman*)}]
\item \emph{Fiber.} $\Eres:=\mathrm{pr}_z^{*}\bigl(\Sym^2TM\bigr)
\otimes\mathrm{pr}_y^{*}\mathbf 1$, where $\mathrm{pr}_z,\mathrm{pr}_y:
W\to M$ are the coordinate projections and $\mathbf 1$ is the trivial
line (the scalar slot at $y$). The fiber at $w=(a,b,z,y)$ is
$\Sym^2T_zM$, of dimension $\binom{5}{2}=10$; $\Eres$ is a trivial
bundle of rank ten over $W$.
\item \emph{Density.} The density bundle $|\Lambda|_W$ is trivialized
by the coordinate density $|\mathrm d^{16}w|$ of $(a,b,z,y)$.
\item \emph{Tests.} $\Gamma_c:=C^\infty_c(W;\Eres^{*}\otimes|\Lambda|_W)
\cong C^\infty_c(W)\otimes\Sym^2(\mathbb R^4)^{*}$; an element is a
symmetric array $\Phi=(\Phi_{ab})_{a,b=0}^{3}$ of test functions on
$W$.
\item \emph{Pairing.} $\Dp(W;\Eres):=\Gamma_c'$, with
$\langle u,\Phi\rangle=\sum_{a,b=0}^{3}\langle u^{ab},\Phi_{ab}\rangle$
for $u=(u^{ab})$ a symmetric array of scalar distributions; the fiber
pairing $\Sym^2V\times\Sym^2V^{*}\to\mathbb C$, $(u,\Phi)\mapsto
u^{ab}\Phi_{ab}$, is the restriction of the duality
$V^{\otimes2}\times(V^{*})^{\otimes2}\to\mathbb C$ and is perfect.
\end{enumerate}
\end{convention}

\begin{lemma}[Partial pairing with XIV-G's test]
\label{lem:partial}
For $u\in\Dp(W;\Eres)$ and $G\in C^\infty_c(Z;\Sym^2T^*M)$, the
prescription
\[
 \langle u[G],\phi\rangle:=\bigl\langle u,\ G\otimes\phi\bigr\rangle,
 \qquad (G\otimes\phi)_{ab}(a,b,z,y)=G_{ab}(z)\,\phi(a,b,y),
 \quad\phi\in\D(\mathbb R^{12}_{(a,b,y)}),
\]
defines $u[G]\in\Dp(\mathbb R^{12})$; the map $G\mapsto u[G]$ is
linear and continuous from $C^\infty_c(Z;\Sym^2T^*M)$ to
$\Dp(\mathbb R^{12})$.
\end{lemma}

\begin{proof}
$\phi\mapsto G\otimes\phi$ is linear and continuous $\D(\mathbb R^{12})
\to\Gamma_c$ (supports stay in $\mathrm{pr}_{(a,b,y)}^{-1}(\supp\phi)
\cap\mathrm{pr}_z^{-1}(\supp G)$, compact, and seminorms are
multiplicative). Composition with the continuous $u$ gives
$u[G]\in\Dp$. Continuity in $G$ is the same estimate read in the
other factor.
\end{proof}

\begin{lemma}[Index bookkeeping of the displayed entries]
\label{lem:index}
In each of the six displayed entries $K^{I,J}(a,b,z)$ of
\S\ref{sec:entries}, the index $I$ is carried by the variable $a$ and
the index $J$ by the variable $z$; the cross-line pullback
$\pi(a,b,z,y)=(y-a,y-b)$ of XIV-J \cite{Jcross} attaches the two
cross-lines to $a$ and $b$. After contraction with the cross-lines,
$J$ at $z$ is the only free tensor index, and $y$ is the only
variable not consumed by an internal line.
\end{lemma}

\begin{proof}
Read off: in \eqref{eq:K5a}, $\Amap^{\alpha\beta,I}$ stands with
$\sigma_\eps(a)$ and $\partial_a$, while $L^{\mu,J}(a,z)=
\sigma_\eps(z)\Amap^{\lambda\mu,J}\partial_{z^\lambda}R(a-z)$ carries
$J$ with $\sigma_\eps(z)$; in \eqref{eq:K5z} the roles of $(a,I)$ and
$(z,J)$ are exchanged consistently; in \eqref{eq:K47},
$L^{\mu,J}(b,z)$ and $L^{\nu,I}(b,a)$; in \eqref{eq:K6}, XIV-I's
defining display \eqref{eq:Dfrom} assigns $I$ to $h$ and $J$ to $G$,
and $\sigma_\eps(a)$ accompanies $I$; in \eqref{eq:Q}--\eqref{eq:K9},
the ordering $(\,\cdot\,;a,z)$ of $Q^{L,I,J}(v;a,z)$ pairs $I$ with
$a$ and $J$ with $z$. The statement about $\pi$ is the display
\eqref{eq:cross} of the patch.
\end{proof}

\begin{remark}
Lemma~\ref{lem:index} is the internal evidence for C1(i): the free
metric index after cross-line contraction sits at $z$, which is the
variable of XIV-G's residual test $G$, and the output is a symmetric
\emph{contravariant} tensor at $z$ (upper index pair $J=(ab)$),
pairing with the covariant $G_{ab}$. The scalar slot at $y$ is the
trivial line and does not change the rank. This is a reading of the
displays, not a source identification; C1 remains a convention.
\end{remark}

\begin{remark}[A fiber metric is not the pairing]
The constant metric $\eta$ induces an isomorphism $\Sym^2T^*M\cong
\Sym^2TM$ and a fiber metric on $\Sym^2T^*M$, whose fiber dimension
is also ten. Neither is part of C1. The pairing of C1(iv) is the
duality pairing between the value bundle $\Eres$ and the test bundle
$\Eres^{*}\otimes|\Lambda|_W$; the coincidence of ranks does not
identify the two roles.
\end{remark}

\subsection{What smooth residual testing cannot see}

\begin{lemma}[Residual testing erases $z$-covectors]
\label{lem:erase}
Let $u\in\Dp(W;\Eres)$ and $G\in C^\infty_c(Z;\Sym^2T^*M)$. Then
\[
 \WF\bigl(u[G]\bigr)\subseteq\bigl\{(a,b,y;\xi_a,\xi_b,\xi_y):
 \exists z\in\supp G,\ \exists(a',b')=(a,b):\
 (a,b,z,y;\xi_a,\xi_b,0,\xi_y)\in\WF(G_{ab}u^{ab})\bigr\},
\]
where $G_{ab}u^{ab}=\sum_{a,b}G_{ab}(z)u^{ab}\in\Dp(W)$. In
particular, any wavefront of $u$ whose covector has nonzero
$z$-component---for instance a conormal to the attachment diagonal
$\{a=z\}$ in the $z$-direction---is invisible to every smooth
residual test.
\end{lemma}

\begin{proof}
$u[G]=\mathrm{pr}_{*}(G_{ab}u^{ab})$ for the projection
$\mathrm{pr}(a,b,z,y)=(a,b,y)$, which is proper on $\supp(G_{ab}u^{ab})
\subseteq\mathbb R^{12}\times\supp G$. Write $\mathrm{pr}_{*}$ as the
operator with Schwartz kernel $K(x';x)=\delta(x'-\mathrm{pr}(x))$,
whose wavefront is $\{(x',x;\theta,-{}^t\!d\mathrm{pr}\,\theta):
x'=\mathrm{pr}(x)\}$ by \cite[Thm.~8.2.4]{Horm}, with
${}^t\!d\mathrm{pr}(\xi_a,\xi_b,\xi_y)=(\xi_a,\xi_b,0,\xi_y)$. The set
$\WF_{X}(K)$ of \cite[Thm.~8.2.12]{Horm} is empty since
${}^t\!d\mathrm{pr}$ is injective, and that theorem gives
$\WF(\mathrm{pr}_{*}v)\subseteq\{(\mathrm{pr}(x),\theta):
(x,{}^t\!d\mathrm{pr}\,\theta)\in\WF(v)\}$, which is the display. The
last sentence is the case $\xi_z\ne0$.
\end{proof}

\subsection{Non-claims and the Q1 verdict}

The order-two statement of Definition~\ref{def:gtest} concerns the
order of a distribution and is not a joint principal symbol
\cite{GG34}. By Lemma~\ref{lem:erase}, smooth residual testing can
hide attachment-diagonal wavefront, so no wavefront statement for a
joint object follows from residual testing alone. Gates G3--G5 of
XIV-G are not entered.

\begin{proposition}[Q1]
\label{prop:Q1}
No direct sheet defines $\Eres$. Convention~\ref{def:Eres} specifies
one fiber (rank ten), one density, and one pairing, compatible with
Definition~\ref{def:gtest} through Lemma~\ref{lem:partial} and with
the displayed entries through Lemma~\ref{lem:index}. Q1 is
discharged with status \emph{convention}; it passes F0 only if C1 is
accepted as the frozen choice.
\end{proposition}

\begin{proof}
The first sentence is the record (\cite{Gresidual}; the patch's
D0.5). The rest cites the results above.
\end{proof}

\section{Patch 2: the six-entry presentation package (Q2)}
\label{sec:patch2}

\subsection{The interface, transcribed}

\begin{definition}[Presentation interface; XIV-J]
\label{def:interface}
XIV-J \cite{Jdescent} works with a linear reconstruction
$\Rec:\Pint\to\Dp(W)$ with image $\Kint=\Rec(\Pint)$ and proves: a
linear candidate $\Mtil:\Pint\to\Dp(W)$ defines a map on
reconstructed kernels if and only if
\begin{equation}
 \ker\Rec\subseteq\ker\Mtil,
 \label{eq:descent}
\end{equation}
in which case the induced map $\mathsf M(\Rec A)=\Mtil(A)$ is unique;
and agreement of $\Mtil$ with canonical contraction on $\Ucan$
proves \eqref{eq:descent} there only. XIV-J presents $\Pint$ as the
finite direct sum of the internal coefficient presentation spaces of
the six operational entries; the summands are named and not
displayed in the retrieved record (status open for the summands,
identified for \eqref{eq:descent}).
\end{definition}

Write $E=\{47,\ 5a,\ 5z,\ 6,\ 8,\ 9\}$ for the six entries,
$\Pint=\bigoplus_{e\in E}\mathscr P_e$, $\Rec=\sum_e\Rec_e$ with
$\Rec_e:\mathscr P_e\to\Dp(W)$, $\mathscr K_e=\Rec_e(\mathscr P_e)$.

\begin{lemma}[Cross-entry null relations]
\label{lem:directsum}
$\ker\Rec\supseteq\bigoplus_{e}\ker\Rec_e$, with equality if and only
if the sum $\sum_e\mathscr K_e\subseteq\Dp(W)$ is direct. If
$\mathscr K_e\cap\mathscr K_{e'}\ne0$ for some $e\ne e'$, then
$\ker\Rec$ contains elements $(p_e,-p_{e'})$ with
$\Rec_ep_e=\Rec_{e'}p_{e'}\ne0$, and by \eqref{eq:descent} every
joint map must take equal values on the $e$- and $e'$-presentations
of one and the same kernel.
\end{lemma}

\begin{proof}
If $\sum_e\Rec_ep_e=0$ and the sum of images is direct, each
$\Rec_ep_e=0$; conversely a nontrivial relation $\sum_e k_e=0$,
$k_e\in\mathscr K_e$ not all zero, lifts to an element of $\ker\Rec$
outside $\bigoplus\ker\Rec_e$. The last sentence is
\eqref{eq:descent} applied to $(p_e,-p_{e'})$.
\end{proof}

Lemma~\ref{lem:directsum} is the reason the plan requires the
generators of \emph{every} entry before $\ker\Rec$ can be computed:
the per-entry kernels do not suffice when images overlap, and
Theorem~\ref{thm:normalorder} below shows that the images of the
$K_6$-, $K_8$-, and $K_9$-presentations all lie in one and the same
finite normal-jet space.

\subsection{The six operational entries, enumerated}
\label{sec:entries}

The retained kernel and the supported sector are \cite{II2,
Jreference}
\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}
The coordinate-response line is XIV-H's \cite{HeqL}
\begin{equation}
 L^{\mu,I}(s,a)=\sigma_\eps(a)\,\Amap^{\lambda\mu,I}\,
 \partial_{a^\lambda}R(s-a).
 \label{eq:L}
\end{equation}
The displays, copied for enumeration (copying is not instantiation;
see \S\ref{sec:patch3}), 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)
 \Amap^{\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)
 \Amap^{\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)\Dmap^{\alpha\mu,I,J}
 \partial_{a^\alpha}R(b-a)\,\delta^{(4)}(a-z)\Bigr],
 \label{eq:K6}
\end{align}
from XIV-H \cite{Hkernel} and XIV-I \cite{II2}, where XIV-I defines
\begin{equation}
 \dot a_2^{\alpha\mu}[G;h]=\Dmap^{\alpha\mu,I,J}\,h_I\,G_J ,
 \label{eq:Dfrom}
\end{equation}
``the constant bilinear fiber map determined by'' this identity; and
from XIV-I's input statement copied from XIV-H \cite{Iresponse}, with
the order bound of XIV-H \cite{Hq},
\begin{align}
 Q^{L,I,J}(v;a,z)
 &=\sum_{|\alpha|+|\beta|\le2}q^{L,I,J}_{\alpha\beta}(v)\,
 \partial_v^\alpha\delta^{(4)}(v-z)\,\partial_v^\beta\delta^{(4)}(v-a),
 \label{eq:Q}\\
 T^{L,I,J}(b;a,z)&=\Pmap^L{}_M\int R(b-v)\,Q^{M,I,J}(v;a,z)\dd^4v,
 \label{eq:T}\\
 K_8^{I,J}(a,b,z)
 &=-\partial_{b^\mu}\Bigl[f(b)\Amap^{\kappa\mu}{}_L\Pmap^L{}_M\,
 \partial_{b^\kappa}\!\int\Bker(b-v)\,Q^{M,I,J}(v;a,z)\dd^4v\Bigr],
 \label{eq:K8}\\
 K_9^{I,J}(a,b,z)
 &=-\partial_{b^\nu}\Bigl[g(b)\,\partial_{b^\mu}F(b)\,
 \Amap^{\mu\nu}{}_L\,T^{L,I,J}(b;a,z)\Bigr]
 -m^2g(b)F(b)\,\mathfrak b_L\,T^{L,I,J}(b;a,z),\qquad F=\EA_mf .
 \label{eq:K9}
\end{align}
(In \eqref{eq:K8} the inner tensor index is written $\kappa$ to keep
it distinct from the multi-index $\alpha$ of \eqref{eq:Q}; the
source writes $\alpha$ for both.) The cross-line operator of the
patch is recalled for Lemma~\ref{lem:index}:
\begin{equation}
 \Cross=\pi^{*}\bigl(H'_{\gamma\gamma}\otimes H'_{\phi\phi}
 -H_{\gamma\gamma}\otimes H_{\phi\phi}\bigr),\qquad
 \pi(a,b,z,y)=(y-a,\,y-b).
 \label{eq:cross}
\end{equation}

\begin{proposition}[Recalled from XV-F-patch]
\label{prop:recalled}
\begin{enumerate}[label=\textup{(P\arabic*)}]
\item \textup{\cite[Thm.~\texttt{thm:exist}]{XVFpatch}} For any
constant arrays $\Amap$, $\Dmap$, $\Pmap$, $\mathfrak b$, any
$m\in\mathbb C$, and any $f,g,F,\sigma_\eps,q_{\alpha\beta}\in
C^\infty(\mathbb R^4)$, each of \eqref{eq:K5a}--\eqref{eq:K9} is a
distribution on $\mathbb R^{12}_{(a,b,z)}$; the $v$-integrals in
\eqref{eq:T}, \eqref{eq:K8} are the distributions $\mathcal T_0$,
$\mathcal T_{\Bker}$ with $\langle\mathcal T_0,\phi\rangle=
\langle Q,\Phi^R_\phi\rangle$, $\Phi^R_\phi(v,a,z)=\langle R,
\phi(a,v+\cdot,z)\rangle$, and likewise for $\Bker$.
\item \textup{\cite[Lemma~\texttt{lem:Q}]{XVFpatch}} For smooth
$\Phi$ with compact $v$-support near the diagonal,
$\langle Q,\Phi\rangle=\sum_{\alpha,\beta}\int q_{\alpha\beta}(v)\,
(\partial_z^\alpha\partial_a^\beta\Phi)(v,v,v)\dd^4v$.
\item \textup{\cite[Prop.~\texttt{prop:support}]{XVFpatch}}
$\supp K_6,\supp K_9\subseteq S_\partial$, $\supp K_8\subseteq S_{\Jp}$.
\item \textup{\cite[Lemma~\texttt{lem:chart}]{XVFpatch}, source
\cite{Ichart}} With $u=a$, $d=b-a$, $q=d^2$, $\zeta=\mathbf d\in
\mathbb R^3$, $r=a-z$, the map $w_+=(u,\zeta,q,r,y)$ is a global
chart of $\Vplus:=\{(b-a)^0>0\}\subset W$ onto
$\mathbb R^4\times\Omega_0\times\mathbb R^4\times\mathbb R^4$,
$\Omega_0=\{q+|\zeta|^2>0\}$, $d^0=\sqrt{q+|\zeta|^2}$; the second
placement $w_-$ uses $u=z$, $d=b-z$, $r=z-a$ on
$V_-=\{(b-z)^0>0\}$. On $\Vplus$, $R(d)=\frac1{2\pi}\delta(q)$,
$(\partial_\alpha R)(d)=\frac1\pi d_\alpha\delta'(q)$, and
$\Bker(d)=\frac1{8\pi}\Theta(q)$.
\item \textup{\cite[Lemma~\texttt{lem:recrule}]{XVFpatch}, source
\cite{Iamodule,Jnull}} XIV-I's module $\Amod$ on the chart $w_\pm$
is reconstructed by $\Rec(A)=\sum_{j,\beta}c_{j\beta}(w)\,
\delta^{(j)}(q)\,\partial_r^\beta R(r)$; XIV-J's values
$\Rec(M_{\mathrm R})=\delta(q)R(r)$, $\Rec(i\theta M_{\mathrm R})=
\delta'(q)R(r)$ and null relation $(q\,i\theta+1)M_{\mathrm R}$ are
instances; the generating family is free over
$(q,r)$-independent coefficients and dependent over chart functions.
\item \textup{\cite[Prop.~\texttt{prop:K5chart}]{XVFpatch}} On
$\Vplus$, $K_5^{[a]}$ and $K_6$ have the form of the rule in (P5)
with $j\in\{1,2\}$, $|\beta|=2$; on $V_-$ so does $K_5^{[z]}$;
$K_{47}$ has this form in neither chart.
\end{enumerate}
\end{proposition}

\subsection{The $K_5$ placements}

For $e\in\{5a,5z\}$ the record supplies XIV-I's module $\Amod$ on
the chart $w_\pm$ with its reconstruction rule (P5) and chart (P4);
XIV-I states that one explicit module contains the regular
diagonal, separated flow-out, and internal transition presentations
of both placements \cite{II2}. The generators of $\Amod$, its
coefficient class (dependence on $q$, $r$), and the module action
(linearity of $\Rec$ over chart functions) are described in prose
and not displayed \cite{Iamodule}. Status: rule and chart
identified; generators, coefficient class, and action open. The
module is a local chart model on $V_\pm$; its relations
$q\delta'(q)=-\delta(q)$, $r^2R(r)=0$, and $(q\,i\theta+1)M_{\mathrm R}$
are examples of null relations, not a classification of $\ker\Rec$
on $\Pint$ (Lemma~\ref{lem:directsum}).

\subsection{The supported sector: a normal-jet presentation}
\label{sec:supported}

\paragraph{Normal jets along the null incidence.}
On $\Vplus$ let $\Splus:=\{q=0,\ r=0\}$, so that $\Splus=S_\partial
\cap\Vplus$, with normal variables $n=(q,r)\in\mathbb R^5$ and base
variables $x'=(u,\zeta,y)$, $\zeta\ne0$; $\Splus$ has codimension
five and $\Splus\cong\mathbb R^4\times(\mathbb R^3\setminus0)\times
\mathbb R^4$. For $b\in\Dp(\Splus)$ and $\gamma\in\mathbb N^5$ set
\begin{equation}
 \Ejet_\gamma b:=b\otimes\partial_n^\gamma\delta(n)\in\Dp(\Vplus),
 \qquad
 \langle\Ejet_\gamma b,\phi\rangle=(-1)^{|\gamma|}
 \bigl\langle b,(\partial_n^\gamma\phi)(\cdot,0)\bigr\rangle .
 \label{eq:insert}
\end{equation}
Let $\Jet{k}$ be the space of finite sums $\sum_{|\gamma|\le k}
\Ejet_\gamma b_\gamma$, $b_\gamma\in\Dp(\Splus)$ (\emph{normal order
at most $k$}), and $\Jets{k}\subseteq\Jet{k}$ the subspace with
$b_\gamma\in C^\infty(\Splus)$.

\begin{lemma}[Taylor reduction and first-order operators]
\label{lem:taylor5}
\begin{enumerate}[label=\textup{(\roman*)}]
\item For $c\in C^\infty(\Vplus)$ and $\gamma\in\mathbb N^5$,
\[
 c(x',n)\,\partial_n^\gamma\delta(n)=\sum_{\gamma'\le\gamma}
 (-1)^{|\gamma-\gamma'|}\binom{\gamma}{\gamma'}\,
 (\partial_n^{\gamma-\gamma'}c)(x',0)\,\partial_n^{\gamma'}\delta(n).
\]
Hence $c\cdot\Jets{k}\subseteq\Jets{k}$.
\item Base derivatives $\partial_{x'_i}$ map $\Jets{k}\to\Jets{k}$;
normal derivatives $\partial_{n_j}$ map $\Jets{k}\to\Jets{k+1}$.
\item Every first-order differential operator with coefficients in
$C^\infty(\Vplus)$ maps $\Jets{k}\to\Jets{k+1}$, and every such
operator with no normal-derivative component maps
$\Jets{k}\to\Jets{k}$.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) Pair with $\phi$: $\langle c\,\partial^\gamma\delta,\phi\rangle
=(-1)^{|\gamma|}\int\partial_n^\gamma(c\phi)(x',0)\dd x'$; expand by
Leibniz and use $\langle\partial^{\gamma'}\delta,\psi\rangle
=(-1)^{|\gamma'|}\partial^{\gamma'}\psi(0)$. The coefficients
$(\partial^{\gamma-\gamma'}c)(x',0)$ are smooth on $\Splus$. (ii)
$\partial_{x'_i}(b\otimes\partial^\gamma\delta)=(\partial_{x'_i}b)
\otimes\partial^\gamma\delta$ and $\partial_{n_j}(b\otimes
\partial^\gamma\delta)=b\otimes\partial^{\gamma+e_j}\delta$. (iii)
follows from (i), (ii).
\end{proof}

\begin{lemma}[The coordinate vector fields in the chart $w_+$]
\label{lem:vf}
On $\Vplus$, as operators on $\Dp(\Vplus)$ expressed in $w_+$,
\begin{equation}
 \partial_{b^\mu}=2d_\mu\,\partial_q+\sum_{i=1}^{3}\delta^{i}_{\mu}\,
 \partial_{\zeta^i},\qquad
 \partial_{z^\mu}=-\partial_{r^\mu},\qquad
 \partial_{a^\mu}=\partial_{u^\mu}-2d_\mu\,\partial_q
 -\sum_{i=1}^{3}\delta^{i}_{\mu}\,\partial_{\zeta^i}+\partial_{r^\mu},
 \label{eq:vf}
\end{equation}
with $d_\mu=\eta_{\mu\nu}d^\nu$, $d^0=\sqrt{q+|\zeta|^2}$, smooth on
$\Vplus$. Each is a first-order operator with smooth coefficients;
its normal component is $\pm2d_\mu\partial_q$ for $\partial_b$,
$-\partial_{r^\mu}$ for $\partial_z$, and
$-2d_\mu\partial_q+\partial_{r^\mu}$ for $\partial_a$.
\end{lemma}

\begin{proof}
Chain rule for $u=a$, $d=b-a$, $q=\eta_{\nu\kappa}d^\nu d^\kappa$,
$\zeta^i=d^i$, $r=a-z$: $\partial q/\partial d^\mu=2d_\mu$,
$\partial\zeta^i/\partial d^\mu=\delta^i_\mu$, and $b$ enters only
through $d$ (sign $+$), $z$ only through $r$ (sign $-$), $a$ through
$u$ ($+$), $d$ ($-$), $r$ ($+$).
\end{proof}

\begin{lemma}[The derivative of $\Bker$]
\label{lem:dB}
In $\Dp(\mathbb R^4)$, $\partial_\alpha\Bker=\tfrac12\,x_\alpha R$.
Consequently $\supp\partial_\alpha\Bker=\partial\Jp$, and on $\Vplus$,
$(\partial_\alpha\Bker)(d)=\frac1{4\pi}d_\alpha\delta(q)$.
\end{lemma}

\begin{proof}
Use the partial Fourier transform $\Four$ and the formulas of
\cite[Lemmas \texttt{lem:R}, \texttt{lem:RR}]{XVFpatch}:
$\Four R=\Theta(t)\sin(\rho t)/\rho$, $\Four\Bker=\Theta(t)u_2(t)$,
$u_2=(\sin\rho t-\rho t\cos\rho t)/(2\rho^3)$, $u_2(0)=0$,
$u_2'=t\sin(\rho t)/(2\rho)$. For $\alpha=0$:
$\Four(\partial_t\Bker)=\Theta(t)u_2'(t)=\frac t2\Four R
=\Four(\tfrac12tR)$, and $x_0=t$. For $\alpha=i\in\{1,2,3\}$:
$\Four(\partial_i\Bker)=ip_i\Theta u_2$, while
$\Four(x^iR)=i\partial_{p_i}\Four R=i\Theta(t)\frac{p_i}{\rho}
\frac{d}{d\rho}\frac{\sin\rho t}{\rho}=-2ip_i\Theta u_2$, so
$\Four(\tfrac12x_iR)=\Four(-\tfrac12x^iR)=ip_i\Theta u_2$. Fourier
injectivity on $\mathcal S'$ gives the identity. The support
statement follows from $\supp R=\partial\Jp$ and $x_\alpha\ne0$
somewhere on every open piece of the cone. The chart formula is
(P4): $\tfrac12d_\alpha\cdot\frac1{2\pi}\delta(q)$.
\end{proof}

\begin{corollary}[The supported sector is carried by the null
incidence]
\label{cor:K8supp}
Under the hypotheses of \textup{(P1)}, $\supp K_8\subseteq S_\partial$.
Hence $\supp\Ksup\subseteq S_\partial=\{a=z,\ b-a\in\partial\Jp\}
\subseteq\{(b-a)^2=0,\ a=z\}$, and $\Ksup$ vanishes on the interior
of the solid cone.
\end{corollary}

\begin{proof}
$\partial_{b^\kappa}\mathcal T_{\Bker}$ is the distribution with
$\langle\partial_{b^\kappa}\mathcal T_{\Bker},\phi\rangle=
-\langle Q,\Phi^{\Bker}_{\partial_{b^\kappa}\phi}\rangle$, and
$\Phi^{\Bker}_{\partial_{b^\kappa}\phi}(v,a,z)=\int\Bker(u)
(\partial_{b^\kappa}\phi)(a,v+u,z)\dd u=-\langle\partial_\kappa\Bker,
\phi(a,v+\cdot,z)\rangle$. Thus $\partial_{b^\kappa}\mathcal T_{\Bker}$
is the $\mathcal T$-construction of (P1) with $R$ replaced by
$\partial_\kappa\Bker=\tfrac12x_\kappa R$, and the proof of (P3) for
$K_9$ applies verbatim with $\supp\partial_\kappa\Bker=\partial\Jp$.
Multiplication by smooth functions and the outer $\partial_{b^\mu}$
preserve the support statement.
\end{proof}

\begin{lemma}[Chart form of the response integrals]
\label{lem:T0chart}
Under \textup{(P1)}, as distributions on $\mathbb R^{12}$,
\begin{align}
 \mathcal T_0&=\sum_{|\alpha|+|\beta|\le2}(-1)^{|\alpha|+|\beta|}\,
 \partial_z^\alpha\partial_a^\beta\bigl[q_{\alpha\beta}(a)\,R(b-a)\,
 \delta^{(4)}(a-z)\bigr],
 \label{eq:T0chart}\\
 \partial_{b^\kappa}\mathcal T_{\Bker}&=\sum_{|\alpha|+|\beta|\le2}
 (-1)^{|\alpha|+|\beta|}\,\partial_z^\alpha\partial_a^\beta\bigl[
 q_{\alpha\beta}(a)\,\tfrac12(b-a)_\kappa R(b-a)\,\delta^{(4)}(a-z)\bigr],
 \label{eq:TBchart}
\end{align}
where $R(b-a)\delta^{(4)}(a-z)$ denotes the pullback of $R\otimes\delta$
by $(a,b,z)\mapsto(b-a,a-z)$. On $\Vplus$ the bracketed
distributions are $\frac1{2\pi}q_{\alpha\beta}(u)\,\Ejet_0(1)$ and
$\frac1{4\pi}q_{\alpha\beta}(u)\,d_\kappa\,\Ejet_0(1)$, elements of
$\Jets{0}$.
\end{lemma}

\begin{proof}
By (P1), (P2), $\langle\mathcal T_0,\phi\rangle=\sum\int
q_{\alpha\beta}(v)\,(\partial_z^\alpha\partial_a^\beta\Phi^R_\phi)(v,v,v)
\dd v$, and $(\partial_z^\alpha\partial_a^\beta\Phi^R_\phi)(v,v,v)
=\langle R,(\partial_z^\alpha\partial_a^\beta\phi)(v,v+\cdot,v)\rangle$.
The right side equals $\sum\langle q_{\alpha\beta}(a)R(b-a)\delta(a-z),
\partial_z^\alpha\partial_a^\beta\phi\rangle$, which is the pairing of
the right side of \eqref{eq:T0chart} with $\phi$. For
\eqref{eq:TBchart} use Corollary~\ref{cor:K8supp}'s identification of
$\partial_{b^\kappa}\mathcal T_{\Bker}$ as the same construction with
$\tfrac12x_\kappa R$. The chart forms are (P4) and
Lemma~\ref{lem:dB}; $\delta(q)\otimes\delta(r)=\Ejet_0(1)$ along
$\Splus$.
\end{proof}

\begin{theorem}[Normal order of the supported sector]
\label{thm:normalorder}
Under the hypotheses of \textup{(P1)} with the order bound
$|\alpha|+|\beta|\le2$ of \eqref{eq:Q}, the restrictions to $\Vplus$
satisfy
\[
 K_6|_{\Vplus}\in\Jets{2},\qquad K_8|_{\Vplus}\in\Jets{3},\qquad
 K_9|_{\Vplus}\in\Jets{3},\qquad\text{hence}\qquad
 \Ksup|_{\Vplus}\in\Jets{3},
\]
normal jets along $\Splus$ with coefficients smooth on $\Splus$ and
independent of $y$. The supports of $K_6$, $K_8$, $K_9$ meet
$W\setminus\Vplus$ only in $\{a=b=z\}$.
\end{theorem}

\begin{proof}
\emph{$K_6$.} $\partial_{a^\alpha}R(b-a)=-(\partial_\alpha R)(d)
=-\frac1\pi d_\alpha\delta'(q)$, and $\delta^{(4)}(a-z)=\delta(r)$; the
product is $-\frac1\pi d_\alpha\,\Ejet_{e_q}(1)\in\Jets{1}$ by
Lemma~\ref{lem:taylor5}(i). Multiplication by the smooth
$f(b)\sigma_\eps(a)\Dmap$ keeps $\Jets{1}$; the outer $\partial_{b^\mu}$
is first order (Lemma~\ref{lem:vf}), so $K_6\in\Jets{2}$.

\emph{$K_9$.} By Lemma~\ref{lem:T0chart}, $\mathcal T_0$ is a sum of
at most two first-order operators (Lemma~\ref{lem:vf}) applied to
elements of $\Jets{0}$, hence $\mathcal T_0\in\Jets{2}$ by
Lemma~\ref{lem:taylor5}(iii); $T=\Pmap\mathcal T_0\in\Jets{2}$. The
first term of \eqref{eq:K9} is $\partial_{b^\nu}$ of a smooth multiple
of $T$: in $\Jets{3}$. The second is a smooth multiple of $T$: in
$\Jets{2}$.

\emph{$K_8$.} By \eqref{eq:TBchart}, $\partial_{b^\kappa}\mathcal T_{\Bker}
\in\Jets{2}$; the smooth multiple and the outer $\partial_{b^\mu}$ give
$K_8\in\Jets{3}$.

$y$-independence: no entry depends on $y$. The support statement:
$\supp\Ksup\subseteq S_\partial$ by (P3) and
Corollary~\ref{cor:K8supp}, and $S_\partial\setminus\Vplus=
\{a=z,\ b-a\in\partial\Jp,\ (b-a)^0\le0\}=\{a=b=z\}$ since
$\partial\Jp\cap\{t\le0\}=\{0\}$.
\end{proof}

\begin{remark}
The excluded set $\{a=b=z\}$ has codimension eight in $W$, as does
$N=\Nt$ \cite{Grelease}; no identification of the two is inferred.
The theorem is a statement about the displayed internal expressions
for arbitrary completions; it is not a statement about $\Cross\Ksup$
or about any product.
\end{remark}

\begin{corollary}[Consistency with XIV-J's normal form]
\label{cor:XIVJorder}
XIV-J states that $\Ksup$ has a unique finite normal form of normal
order at most three along $\Sresp$, with reference
$t^\circ_D=\sum_{|\alpha|\le3,\ \beta\le\alpha}(-1)^{|\beta|}
\binom\alpha\beta\Ejet_{\alpha-\beta}(v_\alpha c_{D,\beta})$ on
$U\setminus N_U$ \cite{Jreference,Jconditional}. Under the candidate
$\Sresp\overset{?}{=}\{(b-a)^2=0,\ a=z\}$ of the patch, which on
$\Vplus$ is $\Splus$, Theorem~\ref{thm:normalorder} reproduces the
normal-order bound three from XIV-H's bound $|\alpha|+|\beta|\le2$
alone, for every completion of the undefined coefficients: one
normal derivative from the outer $\partial_b$ and at most two from
the source. The two bounds are therefore mutually consistent, and
the bound three is attained by the term pattern of $K_8$, $K_9$
unless the coefficients cancel.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:normalorder} and the cited statements.
\end{proof}

\begin{lemma}[Uniqueness of normal-jet coefficients]
\label{lem:inject}
If $\sum_{|\gamma|\le k}\Ejet_\gamma b_\gamma=0$ in $\Dp(\Vplus)$ with
$b_\gamma\in\Dp(\Splus)$, then all $b_\gamma=0$. Hence the map
$(b_\gamma)_{|\gamma|\le k}\mapsto\sum_\gamma\Ejet_\gamma b_\gamma$
from $\Dp(\Splus)^{\#\{|\gamma|\le k\}}$ onto $\Jet{k}$ is a linear
isomorphism.
\end{lemma}

\begin{proof}
For $\psi\in\D(\Splus)$ and $\chi\in\D(\mathbb R^5)$ equal to one near
$0$, set $\Jjet_{\gamma'}\psi(x',n)=\frac{(-1)^{|\gamma'|}}{\gamma'!}
n^{\gamma'}\chi(n)\psi(x')$. By \eqref{eq:insert},
$\langle\Ejet_\gamma b,\Jjet_{\gamma'}\psi\rangle=(-1)^{|\gamma|}
\langle b,\psi\rangle\cdot\frac{(-1)^{|\gamma'|}}{\gamma'!}
\partial_n^\gamma(n^{\gamma'}\chi)(0)=\delta_{\gamma\gamma'}
\langle b,\psi\rangle$. Pairing the vanishing sum with
$\Jjet_{\gamma'}\psi$ gives $\langle b_{\gamma'},\psi\rangle=0$ for
all $\psi$.
\end{proof}

\begin{corollary}[Two readings of the supported-sector presentation]
\label{cor:tworeadings}
\begin{enumerate}[label=\textup{(\roman*)}]
\item If XIV-J's presentation space for an entry $e\in\{6,8,9\}$ is
the space of coefficient tuples $(b_\gamma)_{|\gamma|\le3}$,
$b_\gamma\in\Dp(\Sresp)$, with $\Rec_e=\sum_\gamma\Ejet_\gamma$, then
$\Rec_e$ is injective: $\ker\Rec_e=0$.
\item If it is instead a module of smooth functions acting on fixed
insertions, as in XIV-I's chart module \textup{(P5)}, then
$\ker\Rec_e$ consists of the Taylor relations of
Lemma~\ref{lem:taylor5}(i) and is nonzero.
\item In either reading, $\mathscr K_6,\mathscr K_8,\mathscr K_9
\subseteq\Jet{3}$ on $\Vplus$, and the three images are not in direct
sum as soon as each contains $\Ejet_0(b)$ for $b$ in a common
coefficient class; then Lemma~\ref{lem:directsum} produces
cross-entry null relations.
\end{enumerate}
The record does not display which reading XIV-J adopts
\cite{Jreference}; the traces $c_{D,\beta}$ are named, not displayed.
\end{corollary}

\begin{proof}
(i) is Lemma~\ref{lem:inject}; (ii) is Lemma~\ref{lem:taylor5}(i)
read as a relation; (iii) is Theorem~\ref{thm:normalorder} and
Lemma~\ref{lem:directsum}.
\end{proof}

\subsection{The entry $K_{47}$}

\begin{proposition}[$K_{47}$ has no presentation space in the record]
\label{prop:K47}
No direct sheet displays or names a presentation space, a
reconstruction rule, or a coefficient class for $K_{47}$. XIV-I's
module is constructed for the two $K_5$ placements \cite{II2};
XIV-J's normal form is for $\Ksup$ \cite{Jreference}; XIV-I's
precomposition sum \eqref{eq:retained} writes $K_{47}$ directly with
$L$ \cite{II2}. On neither chart $w_\pm$ does $K_{47}$ have the form
of the rule \textup{(P5)}, and it is not supported on $S_\partial$:
it contains the product $R(d_+)\,R(d_++r_+)$, carried by
$\{b-a\in\partial\Jp\}\cap\{b-z\in\partial\Jp\}$, a set of
codimension two not contained in $\{a=z\}$.
\end{proposition}

\begin{proof}
The first two sentences are the record. For the last: by
\eqref{eq:L}, $L^{\mu,J}(b,z)$ contains $\partial R(b-z)$ and
$L^{\nu,I}(b,a)$ contains $\partial R(b-a)$, with $b-z=d_++r_+$ in
the chart $w_+$; the support of the pullback of $R\otimes R$ under
$(a,b,z)\mapsto(b-z,b-a)$ is the stated set, which contains points
with $a\ne z$ (e.g.\ $b=0$, $a$ and $z$ distinct points of the past
cone); this is \textup{(P6)}.
\end{proof}

\subsection{Image topology and restricted descent}

The plan requires the image topology on $\Kint$ to be the quotient of
the stated coefficient seminorms. No direct sheet names coefficient
seminorms on any $\mathscr P_e$ \cite{Iamodule,Jreference}; XIV-I
proves uniform compact-set estimates for its reconstruction
\cite{II2}, which are bounds, not a topology on $\Pint$. No
Fr\'echet structure is introduced here. Status: open.

\begin{proposition}[Canonical agreement checks the restricted
inclusion only]
\label{prop:restricted}
Let $r_O$ denote restriction to $O=\Ucan$ and $Y=\ker r_O$. If
$\Mtil:\Pint\to\Dp(W)$ agrees with the canonical contraction on $O$,
i.e.\ $r_O\Mtil=m^\circ\Rec$, then for every $n\in\ker\Rec$,
$r_O\Mtil(n)=0$, i.e.\ $\Mtil(\ker\Rec)\subseteq Y$. This does not
decide whether $\Mtil(n)=0$.
\end{proposition}

\begin{proof}
$r_O\Mtil(n)=m^\circ\Rec(n)=0$. A distribution supported in
$W\setminus O$ is not detected by $r_O$; this is the last paragraph
of XIV-J's proof of the descent criterion \cite{Jdescent}.
\end{proof}

\subsection{Q2 verdict}

\begin{proposition}[Q2 fails at $K_{47}$]
\label{prop:Q2}
Of the six entries, two ($K_5^{[a]}$, $K_5^{[z]}$) have a chart
module with identified rule and open generators; three ($K_6$,
$K_8$, $K_9$) have an identified normal-jet type of order at most
three (Theorem~\ref{thm:normalorder}, Corollary~\ref{cor:XIVJorder})
with open coefficient class and open presentation space; one
($K_{47}$) has no presentation data in any sheet
(Proposition~\ref{prop:K47}). The module action on $\Pint$ and the
image topology are open. Hence $\ker\Rec$ cannot be computed, and Q2
fails at the entry $K_{47}$, the first entry in the plan's
enumeration without copied generators.
\end{proposition}

\begin{proof}
The cited results and the record.
\end{proof}

By the stop rule, the F0 verdict is fixed here
(Theorem~\ref{thm:verdict}). Patches 3 and 4 and the domain list are
executed for the record.

\section{Patch 3: coefficient instantiation (Q3)}
\label{sec:patch3}

Each ingredient of \eqref{eq:L}--\eqref{eq:K9} is recorded as
\emph{identified} or as an explicit \emph{open} datum reduced to one
sheet and one sentence. $K_8$ and $K_9$ are not recopied; their
displays stand in \S\ref{sec:entries} for enumeration.

\subsection{The fiber map $\Amap$ and $b_1$}

XIV-H writes the component pairing \cite{Hfiber}
\begin{equation}
 a_1^{\alpha\mu}[r]=\Amap^{\alpha\mu,I}\,r_I ,
 \label{eq:a1H}
\end{equation}
with $I$ ranging over symmetric index pairs, so $\Amap$ is the
coefficient matrix of XIV-H's linear fiber map $a_1$; XIV-H's
definition of $a_1$ is in its input statement \texttt{input:EC}
\cite{HinputEC}, whose content is not in the retrieved record. XII-D
displays \cite{XIIDfiber}
\begin{equation}
 \mathfrak a^{\alpha\mu,ab}=\tfrac12\eta^{\alpha\mu}\eta^{ab}
 -\eta^{\alpha(a}\eta^{b)\mu},\qquad
 a_1^{\mu\nu}[r]=\mathfrak a^{\mu\nu,ab}r_{ab},\qquad
 b_1[r]=\tfrac12\eta^{ab}r_{ab}.
 \label{eq:afrak}
\end{equation}

\begin{lemma}[The candidate fiber map]
\label{lem:fiber}
For symmetric $r_{ab}$, $\mathfrak a^{\alpha\mu,ab}r_{ab}=\tfrac12
\eta^{\alpha\mu}\tr_\eta r-r^{\alpha\mu}$. As an endomorphism
$\mathfrak a$ of $\Sym^2$, $\mathfrak a^2=\mathrm{id}$; so
$\mathfrak a$ is invertible of rank ten, and
$\eta_{\alpha\mu}\mathfrak a^{\alpha\mu,ab}r_{ab}=\tr_\eta r=2\,b_1[r]$.
\end{lemma}

\begin{proof}
$\eta^{\alpha(a}\eta^{b)\mu}r_{ab}=r^{\alpha\mu}$ by symmetry.
$\mathfrak a(\mathfrak a r)=\tfrac12\eta\,\tr(\tfrac12\eta\tr r-r)
-(\tfrac12\eta\tr r-r)=\tfrac12\eta(2\tr r-\tr r)-\tfrac12\eta\tr r+r=r$,
using $\tr_\eta\eta=4$. The trace identity is $\tfrac12\cdot4\tr r
-\tr r$.
\end{proof}

\textbf{Status.} $\Amap$: open. The one sentence to locate is XIV-H's
\texttt{input:EC} quoting \eqref{eq:afrak}; if it does, then
$\Amap^{\alpha\mu,(ab)}=\mathfrak a^{\alpha\mu,ab}$ and
Lemma~\ref{lem:fiber} applies; if XIV-H defines $a_1$ otherwise, the
candidate is withdrawn.

\subsection{The weights $\sigma_\eps$ and $f$}

XIV-H calls $\sigma_\eps(a)$ and $f(b)$ ``fixed smooth weights''
\cite{Hkernel}, so $\sigma_\eps,f\in C^\infty(\mathbb R^4)$ is copied.
Their definitions precede \texttt{eq:L} in XIV-H and are not in the
record \cite{HeqL}. XII-D displays $f\in C^\infty_c(\mathring M_T)$
with the observable $\mathcal A_1[f,g](r;\phi)=\int f(\phi_1[r;\phi]
-X_1^\mu[r]\partial_\mu\phi)$ \cite{XIIDweight}; whether XIV-H's $f$
is this $f$ is not stated in XIV-H. The ramp's cutoff $\chi$ lives
on XII-B, a non-input, and can enter only through a quotation inside
XIV-H.

\textbf{Status.} $\sigma_\eps$: open; missing sheet: XIV-H, the
definition preceding \texttt{eq:L}, and, for the cutoff, the
quotation of XII-B's ramp inside XIV-H. $f$: open; missing sheet:
XIV-H, the same passage, or an input statement quoting XII-D's
display. Per the plan this bullet stops here with the sheets named.

\subsection{The quadratic coefficient, $\dot a_2$, and $\Dmap$}

\begin{lemma}[What XIV-I's display presupposes]
\label{lem:quadratic}
Let $a_2$ be a smooth map from symmetric tensors to symmetric
tensors with $a_2[0]=0$ and $\dot a_2[G;h]:=\frac{d}{ds}a_2[G+sh]|_{s=0}$.
Then $\dot a_2[G;h]$ is bilinear in $(G,h)$---as the display
\eqref{eq:Dfrom} requires for a constant $\Dmap$---if and only if
$a_2$ is a homogeneous quadratic polynomial; in that case
$\dot a_2[G;h]$ is symmetric in $(G,h)$, $\dot a_2[r;r]=2a_2[r]$, and
$\Dmap$ is twice the polarization of $a_2$, symmetric in its index
pairs $(I,J)$.
\end{lemma}

\begin{proof}
$\dot a_2[G;h]=Da_2(G)h$ is linear in $G$ iff $Da_2$ is linear iff
$a_2$ is a polynomial of degree at most two with $Da_2(0)=0$; with
$a_2[0]=0$ this is a quadratic form. The rest is the polarization
identity.
\end{proof}

XII-E displays \cite{XIIEa2}
\begin{equation}
 b_2[r]=\tfrac18(\tr_\eta r)^2-\tfrac14r_{\alpha\beta}r^{\alpha\beta},
 \qquad
 a_2^{\mu\nu}[r]=r^\mu{}_\alpha r^{\alpha\nu}-\tfrac12(\tr_\eta r)\,r^{\mu\nu}
 +b_2[r]\,\eta^{\mu\nu},
 \label{eq:a2}
\end{equation}
a homogeneous quadratic polynomial, consistent with
Lemma~\ref{lem:quadratic}.

\begin{lemma}[The conditional value of $\Dmap$]
\label{lem:dformula}
If XIV-H's $a_2$ is \eqref{eq:a2}, then
\begin{equation}
\begin{split}
 \dot a_2^{\mu\nu}[G;h]
 &=h^\mu{}_\alpha G^{\alpha\nu}+G^\mu{}_\alpha h^{\alpha\nu}
 -\tfrac12(\tr_\eta h)\,G^{\mu\nu}-\tfrac12(\tr_\eta G)\,h^{\mu\nu}
 +\Bigl(\tfrac14(\tr_\eta G)(\tr_\eta h)
 -\tfrac12G_{\alpha\beta}h^{\alpha\beta}\Bigr)\eta^{\mu\nu},
\end{split}
\label{eq:a2dot}
\end{equation}
and by \eqref{eq:Dfrom}, with $I=(cd)$ for $h$ and $J=(ab)$ for $G$,
\begin{equation}
\begin{split}
 \Dmap^{\mu\nu,(cd),(ab)}=\mathfrak d^{\mu\nu,ab,cd}
 :={}&\eta^{\mu(c}\eta^{d)(a}\eta^{b)\nu}+\eta^{\mu(a}\eta^{b)(c}\eta^{d)\nu}
 -\tfrac12\eta^{cd}\eta^{\mu(a}\eta^{b)\nu}-\tfrac12\eta^{ab}\eta^{\mu(c}\eta^{d)\nu}\\
 &+\Bigl(\tfrac14\eta^{ab}\eta^{cd}-\tfrac12\eta^{a(c}\eta^{d)b}\Bigr)\eta^{\mu\nu},
\end{split}
\label{eq:dtensor}
\end{equation}
symmetric in $(\mu\nu)$, in $(ab)$, in $(cd)$, and under
$(ab)\leftrightarrow(cd)$.
\end{lemma}

\begin{proof}
Differentiate \eqref{eq:a2} along $G+sh$; the derivative of $b_2$ is
$\tfrac14\tr G\,\tr h-\tfrac12G_{\alpha\beta}h^{\alpha\beta}$.
Writing each term of \eqref{eq:a2dot} as a contraction of
$G_{ab}h_{cd}$ with constant $\eta$'s and symmetrizing over the index
pairs, which is harmless because $G$, $h$ are symmetric, gives
\eqref{eq:dtensor}; the exchange symmetry is
Lemma~\ref{lem:quadratic}.
\end{proof}

\textbf{Status.} The construction of $\Dmap$ from $\dot a_2$:
identified \cite{II2}. The value of $\Dmap$: open; the one sentence to
locate is XIV-H's \texttt{input:EC} quoting \eqref{eq:a2}. The symbol
$\dot a_2$ in the integrand of the patch's discussion is replaced by
the displayed defining relation \eqref{eq:Dfrom}; this is (G2).

\subsection{The source coefficients $q^{L,I,J}_{\alpha\beta}$}

XIV-H, lines 653--665 \cite{Hq}, states that the
$q^{L,I,J}_{\alpha\beta}$ are the coefficients of the actual
differentiated source, including ramp derivatives, with
$|\alpha|+|\beta|\le2$, and that symmetrization factors are absorbed
into them. The undifferentiated source is XII-C's density-weighted
cubic \cite{XIICrelease}, which is not displayed in the record.

\begin{lemma}[What the bound alone fixes]
\label{lem:count}
The index set $\{(\alpha,\beta)\in\mathbb N^4\times\mathbb N^4:
|\alpha|+|\beta|\le2\}$ has $\binom{10}{2}=45$ elements. For each
$(L,I,J)$, $Q^{L,I,J}$ is determined by $45$ smooth functions on
$\mathbb R^4$, and the supported sector's normal order
(Theorem~\ref{thm:normalorder}) and support
(Corollary~\ref{cor:K8supp}) depend on the bound only.
\end{lemma}

\begin{proof}
Multi-indices in eight variables of total degree at most two are
counted by $\binom{8+2}{2}$. The rest restates the theorems' hypotheses.
\end{proof}

\textbf{Status.} Open. Expanding the cubic is a calculation, not a
located display, and is not performed; the statement that
symmetrization factors are absorbed is not an expansion. The ramp
derivatives announced by XIV-H are consistent with the weights
$w_\Gamma$ of Definition~\ref{def:Cpart} (a reading remark).

\subsection{The response fiber map, $\mathfrak b$, $g$, and the
advanced operator}

$\Pmap^L{}_M$ is XIV-I's constant map in \eqref{eq:T}, used and not
defined in the record \cite{Iresponse}; $\mathfrak b_L$ likewise;
$g$ in \eqref{eq:K9} is, by XIV-J's Hessian display, ``the switch''
\cite{Jhessian}, a role without a displayed formula. The operator
$\EA_m$ in $F=\EA_mf$ is not declared in any XIV sheet to be the
advanced inverse of $\Box+m^2$; XII-D sets $v_f=\EA f$ without a
display of $\EA$ \cite{XIIDweight}.

\begin{lemma}[The advanced kernel, conditionally]
\label{lem:advanced}
Let $m\ge0$, $P_m=\Box+m^2$, $\omega=\sqrt{\rho^2+m^2}$. The tempered
distribution $\EA_m$ with $\Four\EA_m(t,\mathbf p)=-\Theta(-t)
\sin(\omega t)/\omega$ satisfies $P_m\EA_m=\delta_0$,
$\supp\EA_m\subseteq\Jm$, and is the unique fundamental solution of
$P_m$ supported in a half-space $\{t\le c\}$; for $m=0$,
$\EA_0(t,\mathbf x)=R(-t,\mathbf x)$. For $f\in C^\infty_c$,
$\EA_m*f\in C^\infty$ with support in $\supp f+\Jm$.
\end{lemma}

\begin{proof}
\cite[Lemma~\texttt{lem:advanced}]{XVFpatch}: the transform solves
$(\partial_t^2+\omega^2)(\cdot)=\delta(t)$ with no boundary terms at
$t=0$; the support follows from Paley--Wiener--Schwartz
\cite[Thm.~7.3.1]{Horm} applied to the entire function
$\sin(\omega t)/\omega$ of $\mathbf p\in\mathbb C^3$ with exponential
type $|t|$; uniqueness is the convolution argument of
\cite[Lemma~\texttt{lem:R}(iii)]{XVFpatch} with $\Jm$.
\end{proof}

\textbf{Status.} $\Pmap$, $\mathfrak b$: open (XIV-H's source-incidence
proposition and \texttt{eq:K9} at line 1055 \cite{HK9}). $g$: role
identified as the switch, formula open. $\EA_m$: open; the one
sentence to locate is a declaration in XIV-H that $\EA_m$ is the
advanced inverse of $\Box+m^2$, after which Lemma~\ref{lem:advanced}
fixes it and Corollary~\ref{cor:compact} applies.

\begin{corollary}[Conditional smoothness of $F$]
\label{cor:compact}
If $f\in C^\infty_c(\mathbb R^4)$ and $\EA_m$ is the kernel of
Lemma~\ref{lem:advanced}, then $F=\EA_m*f\in C^\infty$, the hypothesis
$F\in C^\infty$ of \textup{(P1)} is satisfied, and $K_9$ has
$b$-support in $\supp g\cap(\supp f+\Jm)$.
\end{corollary}

\begin{proof}
Lemma~\ref{lem:advanced} and the factor $g(b)F(b)$ in \eqref{eq:K9}.
\end{proof}

\subsection{The separated chronological patch}

XIV-H's ``separated chronological patch'' (``retained
separated-endpoint patch'' in \cite{HH2}) is the support restriction
on which XIV-H proves its regular conormal amplitudes for $K_8$,
$K_9$. It is a restriction of the internal kernels' domain, used by
XIV-H; it is not a definition of $\Ucan$ and is not used as one here.
With Corollary~\ref{cor:K8supp} the conormal statement is consistent
with the displays at the level of supports (G1).

\subsection{Q3 verdict}

\begin{proposition}[Q3 is open at its first item]
\label{prop:Q3}
Identified: the construction \eqref{eq:Dfrom} of $\Dmap$ from
$\dot a_2$; the role of $g$ as the switch; the order bound
$|\alpha|+|\beta|\le2$. Open, each reduced to one sheet and one
sentence: $\Amap$ (XIV-H \texttt{input:EC} quoting \eqref{eq:afrak});
$\sigma_\eps$, $f$ (XIV-H preceding \texttt{eq:L}; XII-B's cutoff only
through a quotation inside XIV-H; XII-D's $f$ only through a
quotation); $a_2$, hence the value of $\Dmap$ (XIV-H
\texttt{input:EC} quoting \eqref{eq:a2}); $q^{L,I,J}_{\alpha\beta}$
(expansion of XII-C's cubic, not performed); $\Pmap$, $\mathfrak b$
(XIV-H source incidence); $\EA_m$ (declaration as advanced inverse).
Q3 does not pass.
\end{proposition}

\begin{proof}
The subsections above.
\end{proof}

\section{Patch 4: the frozen correction class (Q4)}
\label{sec:patch4}

\begin{definition}[Collision sets; copied]
\label{def:collision}
For a connected vertex subset $I$ of XIII-C's cycle-vertex table
\cite{XIIICvertex} containing the interaction vertex $y$, with
$|I|\ge2$, $\Delta_I=\{x_a=x_b\ \text{for all }a,b\in I\}$, including
intersections and loci after contracting such subsets; diagonals
involving only a pre-existing observable subgraph retain their
inherited normalization \cite{XIIICcoll}. Examples on $B_1$:
$\{y,u,v\}$, $\{y,v\}$.
\end{definition}

\begin{definition}[Partial-diagonal ambiguity class; copied]
\label{def:Cpart}
$\Cpart$ consists of finite sums of graphwise contact changes at
order $\hbar^2$ in the referenced time-ordered product (line
convention copied from XIII-B \cite{XIIIB}), which before pushforward
have the form
\begin{equation}
 D_\Gamma\bigl[w_\Gamma\,\partial_r^\alpha\delta(r_I)\,S_{\Gamma/I}\bigr],
 \qquad|\alpha|\le6,
 \label{eq:partial}
\end{equation}
with $r_I$ transverse to $\Delta_I$, $S_{\Gamma/I}$ an inherited
spectator graph coefficient with two remaining field slots,
$w_\Gamma$ built from $g,\partial g,\sigma_\eps,\sigma_\eps',
\sigma_\eps''$ and the smooth external graph weights already present,
and $D_\Gamma$ ``the specified finite graph differential operator.''
Each change is even, of ghost number zero and field degree two,
supported in the existing compact graph closure, and leaves the tree
datum and inherited observable-subgraph normalizations unchanged.
Only well-defined products and pushforwards are members
\cite{XIIICpart}. The bound six is a normal multi-index bound.
\end{definition}

\begin{proposition}[The three roles are not members]
\label{prop:roles}
XIII-C, lines 358--392, specifies $D_\Gamma$, $w_\Gamma$,
$S_{\Gamma/I}$ by role and displays no formula for any of them
\cite{XIIICpart}; the fourth and fifth passes of the patch and the
present reading agree. In particular: the operator $D_\Gamma$ is
named ``specified'' without a display; the weight space
$\mathcal W_\Gamma$ of admissible $w_\Gamma$ is not stated to be
polynomial, finite-dimensional, or a vector space; $S_{\Gamma/I}$ is
an inherited coefficient whose graph is not displayed. Since the
switches $\sigma_\eps$, $g$ are themselves open
(Proposition~\ref{prop:Q3}), $w_\Gamma$ cannot be instantiated from
Patch 3 either. Q4 fails at $D_\Gamma$.
\end{proposition}

\begin{proof}
The record.
\end{proof}

\begin{proposition}[From values to maps: what is and is not fixed]
\label{prop:Amax}
\begin{enumerate}[label=\textup{(\roman*)}]
\item If each $\mathcal W_\Gamma$ is a vector space, then $\Cpart$ is
a linear subspace of $\Dp$ (finite sums of linear images), and for
any presentation space $\Pint$ and $Y=\ker r_O$ the set
\[
 \Amax:=\{a\in\Hom(\Pint,Y):\ a(\Pint)\subseteq\Cpart\}
\]
is a vector space; it is the largest class satisfying hypothesis
(H2) of the patch's conditional descent test.
\item XIII-C defines no map on presentations: it states neither
linearity, nor continuity, nor tensor type, nor covariance for any
$a\in\Hom(\Pint,Y)$, and it does not state that the frozen class is
$\Amax$ or any subspace of it. No exhaustiveness theorem is proved
there, and none is claimed here.
\item Since $\Pint$ is open (Proposition~\ref{prop:Q2}), $\Amax$ is
not yet an object.
\end{enumerate}
\end{proposition}

\begin{proof}
(i) Linear images of vector spaces under $w\mapsto D_\Gamma[w\,
\partial^\alpha\delta(r_I)S_{\Gamma/I}]$ and finite sums are vector
spaces; preimage conditions $a(\Pint)\subseteq\Cpart$ for a subspace
$\Cpart$ define a subspace of $\Hom$. (ii), (iii) are the record and
Proposition~\ref{prop:Q2}.
\end{proof}

\begin{proposition}[Lift of $\Delta_I$ to $W$: refused]
\label{prop:lift}
The record contains no dictionary between XIII-C's vertex labels
($y,u,v,\dots$ of the cycle-vertex table) and the coordinates
$(a,b,z,y)$ of $W$. Hence no presentation-level map lifting
$\Delta_I$ to $W$ can be given, and none is given. XIII-C's $\UG$ is
the open set on which the H\"ormander sum condition holds for the
graph product of the time-ordered line \cite{XIIICU}; XIII-C states
that $\UG$ is not identified with the complement of the collision
diagonals, and no sheet identifies $\UG$ with $\Ucan$, which belongs
to a different product (the retarded-tree cross-line composition of
XIV-I/XIV-J).
\end{proposition}

\begin{proof}
The record; Lemma~\ref{lem:index} fixes the only variable
correspondence available in $W$ ($I$ at $a$, $J$ at $z$, cross-lines
at $a$, $b$), which does not name $u$ or $v$.
\end{proof}

\begin{remark}
The normal bound six of \eqref{eq:partial} and the normal order three
of Theorem~\ref{thm:normalorder} refer to different objects: contact
changes to the time-ordered product at order $\hbar^2$ along
collision diagonals in graph variables, and tree kernels along the
null incidence in $W$. No comparison is inferred.
\end{remark}

\section{Domain list (Q5)}
\label{sec:domain}

Set $O=\Ucan$, the inherited domain on which the specified products
have already been proved \cite{II3}. The refused equality is
recorded once and not used:
\begin{equation}
 \Ucan\ \ne_{\text{asserted}}\ W\setminus\overline{\Natt}
 \qquad\text{\cite{Hrefuse,Irefuse,Jrefuse}.}
 \label{eq:refused}
\end{equation}

\begin{proposition}[Products proved by the direct sheets, with
their stated domains]
\label{prop:list}
The retrieved passages of XIV-H, XIV-I, XIV-J prove the following
products, with the domains they state:
\begin{enumerate}[label=\textup{(L\arabic*)}]
\item XIV-H \cite{Hkernel}: the internal products $R(b-a)R(a-z)$ and
$R(b-z)R(z-a)$ exist on all of $W$ as pullbacks, including at
$a=z$. Internal; no cross-line.
\item XIV-H \cite{HH2}: exact kernel maps for $K_8$, $K_9$ with
regular conormal amplitudes on the retained separated-endpoint
patch. Internal; a support restriction.
\item XIV-I \cite{II3}: on $\Ucan$, the reconstructed sum composed
with $\Cross$ and the specified vertex operators equals the canonical
grouped expression. The domain is described as ``the open domain
where the subsequent products and pullbacks have been proved''; no
list of those products is in the passage.
\item XIV-J \cite{Jreference}: $\Cross{}_D\Ksup$ exists on
$U\setminus N_U$ with the normal form $t^\circ_D$ and the wavefront
bound \eqref{eq:referenceWFref}; a local reference for $\Ksup$, not
for $\Kret$. The relation of $U\setminus N_U$ to $\Ucan$ is not
stated.
\end{enumerate}
Hence the list of cross-line products proved on $O$ with an
operational description of $O$ is empty: (L1), (L2) are internal;
(L3) describes $O$ by the products themselves; (L4) has an
operational domain not related to $O$ in the sheets. XIV-E's phrase
``complete specified graph products'' is not used to fill the list.
\end{proposition}

\begin{proof}
The cited passages. For (L4), the bound recalled from the patch is
\begin{equation}
 \WF(t_D^\circ)\subseteq N^*\Sresp\big|_{U\setminus N_U}\setminus0
 \ \cup\ \bigcup_{i=1}^{2}\{(w,\lambda\dd q_i+\omega):
 w\in\Sresp\setminus N_U,\ q_i(w)=0,\ \lambda\ne0,\ \omega\in N^*_w\Sresp\}
 \setminus0 .
 \label{eq:referenceWFref}
\end{equation}
\end{proof}

\textbf{Q5.} The gate's pass condition ``or the absence stated'' is
met by Proposition~\ref{prop:list}: status \emph{documentary}. The
core item D0.8 remains a named domain without an operational
description and is \emph{open}; by the plan's \S6 rule this alone
keeps original F0 failed.

\section{Reassessment}
\label{sec:reassess}

\subsection{Ledger}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{0.05\linewidth}p{0.21\linewidth}
p{0.13\linewidth}X@{}}
\toprule
Gate & Member & Status & Content\\
\midrule
Q1 & $\Eres$, density, pairing & convention & C1: rank ten, metric
slot at $z$, trivial scalar line at $y$, coordinate density,
duality pairing (Convention~\ref{def:Eres}); supported by
Lemma~\ref{lem:index}; erasure of $z$-covectors
(Lemma~\ref{lem:erase}).\\[0.2em]
Q2 & $\Pint=\bigoplus\mathscr P_e$ & open at $K_{47}$ & Descent
criterion identified; $K_5^\pm$: chart module, rule identified,
generators open; $K_6,K_8,K_9$: normal-jet type of order $\le3$
identified (Theorem~\ref{thm:normalorder}), presentation space and
coefficient class open; $K_{47}$: nothing
(Proposition~\ref{prop:K47}); action, topology open; $\ker\Rec$ not
computable (Lemma~\ref{lem:directsum}).\\[0.2em]
Q3 & $\Amap$ & open & XIV-H \texttt{input:EC} quoting
\eqref{eq:afrak}; candidate $\mathfrak a$, $\mathfrak a^2=\mathrm{id}$
(Lemma~\ref{lem:fiber}).\\
 & $\sigma_\eps$, $f$ & open & XIV-H preceding \texttt{eq:L}; cutoff
and support only through quotations.\\
 & $\Dmap$ from $\dot a_2$ & identified & \eqref{eq:Dfrom}
\cite{II2}; value conditional on $a_2$ (Lemma~\ref{lem:dformula}).\\
 & $a_2$ & open & XIV-H \texttt{input:EC} quoting \eqref{eq:a2}.\\
 & $q^{L,I,J}_{\alpha\beta}$ & open & bound $\le2$ identified
\cite{Hq}; 45 functions per $(L,I,J)$; expansion not performed.\\
 & $\Pmap$, $\mathfrak b$ & open & XIV-H source incidence,
\texttt{eq:K9}.\\
 & $g$ & role identified & the switch \cite{Jhessian}; formula
open.\\
 & $\EA_m$ & open & declaration as advanced inverse; then
Lemma~\ref{lem:advanced}.\\[0.2em]
Q4 & $D_\Gamma$, $w_\Gamma$, $S_{\Gamma/I}$ & open at $D_\Gamma$ &
roles only (Proposition~\ref{prop:roles}).\\
 & $\mathcal A$ as maps & open & not defined by XIII-C; $\Amax$
available only after Q2 (Proposition~\ref{prop:Amax}).\\
 & lift of $\Delta_I$ & refused & no dictionary
(Proposition~\ref{prop:lift}).\\[0.2em]
Q5 & product list on $O$ & documentary & absence stated
(Proposition~\ref{prop:list}); D0.8 open.\\
\midrule
--- & carried from the patch, not reassessed & open & display of
$m^\circ$ (D0.10); identity of the blocks in \eqref{eq:cross}
(D0.11); normalization of $T^{\mu\nu}(\phi,\psi)$ (D0.12);
\texttt{def:class} (D0.16).\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Verdict}

\begin{theorem}[F0 reassessed]
\label{thm:verdict}
Under the original F0 rule, with the stop rule of
\S\ref{sec:scope}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item Q1 is discharged by Convention~\ref{def:Eres}, offered as the
frozen choice (Proposition~\ref{prop:Q1}).
\item Q2 fails at $K_{47}$ (Proposition~\ref{prop:Q2}); this fixes
the verdict.
\item Q3 and Q4 are open at their first items
(Propositions~\ref{prop:Q3}, \ref{prop:roles}); Q5 is documentary
and leaves D0.8 open (Proposition~\ref{prop:list}).
\item Even if (ii)--(iii) were closed, the carried items D0.10,
D0.11, D0.12, D0.16 of the patch would remain open.
\end{enumerate}
Hence the graph-level outcome is \emph{Unresolved at F0}. No
preliminary lift $\Fz$ is stated, no obligation $r_O\Fz=m^\circ\Rec$
is asserted, F1 is not entered, and no null-defect test,
characteristic prescription, or admission of $Q_H\Rquant\in\Eout$ is
made.
\end{theorem}

\begin{proof}
Each item cites the proposition that establishes it. The absence of
a presentation space for $K_{47}$ is a statement about the record
and not an analytic obstruction; the ending ``Obstructed'' is not
warranted, and ``Lift defined'' is excluded by (ii).
\end{proof}

\[
 \boxed{
 \begin{gathered}
 \text{Unresolved at F0, at Q2 ($K_{47}$ has no presentation space).}\\
 \text{Q1: convention C1 adopted (rank ten, metric slot at $z$).}\quad
 \text{Q3: $\Dmap$-construction identified; nine ingredients open.}\\
 \text{Q4: roles, not members; lift of $\Delta_I$ refused.}\quad
 \text{Q5: operational product list on $O$ empty; D0.8 open.}\\
 \text{New: } \partial_\alpha\Bker=\tfrac12x_\alpha R;\
 \supp\Ksup\subseteq S_\partial;\
 \Ksup|_{\Vplus}\in\Jets{3}\ \text{for every completion with }
 |\alpha|+|\beta|\le2.\\
 \text{F1 not entered.}
 \end{gathered}}
\]

\subsection{What a pass would have permitted}

Had Q1--Q5 been identified or accepted, the next sentence could
state $\Fz$ on each generator of $\Pint$ by the composition rule of
XIV-J's \texttt{eq:target} and the obligation $r_O\Fz=m^\circ\Rec$ on
$O$. That sentence is the threshold of F1 and not a theorem of this
paper; it would not be global descent across $N$, not a null-defect
test, and not quantum admission. It is not written.

\section{Non-claims}
\label{sec:nonclaims}

No Ward identity, common product phase, or joint principal symbol is
asserted. The four symbols written with $\mathsf P$ ($\PPcalnote$ of
XII-A and XII-B, $\mathfrak P$ of XII-D, $\Pmap$ of XIV-I) and the
four written with $R$ ($R$, $R_1$, $\Rquant$, $\mathcal R$) are not
identified; the first two $\mathsf P$'s belong to non-inputs here.
XIV-I's statement that $R$ is the kernel of its $E_0^{\mathrm R}$
\cite{II2,Ipole} stays inside XIV-I's namespace; by
\cite[Lemma~\texttt{lem:R}(iv)]{XVFpatch} any past-bounded right
inverse of $\Box$ is convolution with $R$, which decides nothing
across sheets. Canonical agreement on $O$ is the restricted inclusion
of Proposition~\ref{prop:restricted}, not descent through the global
criterion. The correction class is not declared exhaustive.
Theorem~\ref{thm:normalorder} concerns internal expressions; it says
nothing about $\Cross\Ksup$, about an extension across $N$, or about
estimates. Convention~\ref{def:Eres} is a convention; its acceptance
is the successor's decision. Nothing from XI-B, XII-A, XII-B, XIV-D,
XIV-E, or manuscripts I--V is a premise.

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

\begin{enumerate}[label=\textup{D\arabic*.},leftmargin=*]
\item \textbf{Stop rule.} The verdict is fixed at Q2; Patches 3, 4
and the domain list are documentary. No gate after Q2 is declared
passed.
\item \textbf{Convention versus definition.} C1 is labeled a
convention throughout; Lemma~\ref{lem:index} is internal evidence,
not a source identification.
\item \textbf{Sign and index checks.} Lemma~\ref{lem:dB} was checked
on both the time and space components via the partial Fourier
transform and against $\Box\Bker=R$ ($\partial^\alpha(\tfrac12x_\alpha R)
=\tfrac12(4R+x\cdot\partial R)=R$, $R$ being homogeneous of degree
$-2$). Lemma~\ref{lem:vf} was checked by substituting $u=a$, $d=b-a$,
$r=a-z$. Lemma~\ref{lem:T0chart}'s sign $(-1)^{|\alpha|+|\beta|}$ was
checked against the pairing (P2) of the patch.
\item \textbf{Index clash.} The inner tensor index of $K_8$ is
written $\kappa$; the source writes $\alpha$ for both it and the
multi-index of $Q$.
\item \textbf{Normal order is not regularity.} $\Jets{3}$ membership
is an algebraic form statement on $\Vplus$; it is not the conormal
amplitude statement of XIV-H and not a wavefront theorem.
\item \textbf{Two readings.} Corollary~\ref{cor:tworeadings} records
that $\ker\Rec_e$ on the supported sector is zero or nonzero
according to an undisplayed choice; neither reading is adopted.
\item \textbf{Copied displays.} $K_8$, $K_9$ appear once, in
\S\ref{sec:entries}, for enumeration; Patch 3 instantiates
ingredients only and recopies nothing.
\item \textbf{input:EC.} The content of XIV-H's input statement is
not in the retrieved record; two of the Q3 items reduce to its
quotation. It is cited as a locator \cite{HinputEC}, not as a
display.
\item \textbf{Non-inputs.} XII-B's ramp and trace reversal are not
used; (G3). XII-A's blocks are not used; D0.11 is carried as open.
\item \textbf{Corrections.} (G1)--(G3) correct statements of the
patch; no theorem of the patch is withdrawn; Proposition
\texttt{prop:support} of the patch is sharpened, not contradicted.
\item \textbf{Compilation.} \texttt{mathrsfs} loaded for
\verb|\mathscr|; all macros defined; all \verb|\ref| and \verb|\cite|
targets exist; \texttt{tabularx} widths sum to less than
\verb|\linewidth|; no non-ASCII characters in mathematics.
\end{enumerate}

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

Each item is to be returned with file, label, and line numbers, as
\emph{defines}:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item XIV-J: the display of $\Pint=\bigoplus_e\mathscr P_e$ and, for
$e=47$, any presentation space, rule, and coefficient class; for
$e\in\{6,8,9\}$, whether the presentation is the coefficient tuple
or a smooth module (Corollary~\ref{cor:tworeadings}); the coefficient
seminorms (Q2).
\item XIV-I, \texttt{def:amodule}: generators, coefficient class, and
module action of $\Amod$ (Q2).
\item XIV-H, \texttt{input:EC}: the quoted $a_1$, $b_1$, $a_2$ (Q3,
items $\Amap$, $a_2$); XIV-H preceding \texttt{eq:L}: $\sigma_\eps$,
$f$; XIV-H source incidence and \texttt{eq:K9}: $\Pmap$, $\mathfrak b$,
the declaration of $\EA_m$; XIV-H lines 653--665 and XII-C: the
expansion of $q^{L,I,J}_{\alpha\beta}$ (Q3).
\item XIII-C: formulas for $D_\Gamma$, $w_\Gamma$, $S_{\Gamma/I}$;
any statement about maps on presentations; a dictionary between the
cycle-vertex labels and $(a,b,z,y)$ (Q4).
\item XIV-I, XIV-J: the list of specified products constituting
$\Ucan$, or a statement relating $U\setminus N_U$ to $\Ucan$ (Q5).
\item A decision on Convention~\ref{def:Eres} (Q1).
\item Carried: XIV-J \texttt{eq:target}, \texttt{def:class}, the
blocks of \eqref{eq:cross}, and the normalization of
$T^{\mu\nu}(\phi,\psi)$.
\end{enumerate}
With (1)--(7) supplied, the threshold sentence of
\S\ref{sec:reassess} may be written; it would enter F1 and nothing
beyond.

\subsection*{Final release sentence}

\begin{center}
\fbox{\parbox{0.91\linewidth}{
XV-F-patch-ii runs the five frozen-object gates against the record.
It adopts one convention for the residual bundle (rank ten, metric
slot at $z$, duality pairing) and proves that smooth residual testing
erases $z$-covectors; it enumerates the six operational entries,
proves that the supported sector is carried by the null incidence and
is a normal jet of order at most three there for every completion of
the undefined data obeying XIV-H's order bound---reproducing XIV-J's
normal order---and finds that $K_{47}$ has no presentation space in
any sheet, so that the kernel of reconstruction cannot be computed;
it identifies the construction of $\Dmap$ from $\dot a_2$ and reduces
every other ingredient to a named sheet and sentence; it records that
the correction class is specified by roles and refuses the lift of
the collision sets; it states that the operational list of products
proved on the canonical domain is empty. Three statements of
XV-F-patch are corrected. The graph is Unresolved at F0, at Q2; F1
is not entered; no lift, defect, prescription, or admission is
asserted.
}}
\end{center}

\begin{thebibliography}{99}

\bibitem{XVFpatch}
Manuscript XV-F-patch (Landmark XV-F, first patch), supplied. Full
text supplied. Used for its manifest and ledgers, its F0 verdict
(Proposition \texttt{prop:F0}, cited in the commissioning plan as
\texttt{prop:F0patch}), and its verified Lemmas \texttt{lem:R},
\texttt{lem:RR}, \texttt{lem:taylor}, \texttt{lem:recrule},
\texttt{lem:chart}, \texttt{lem:Q}, \texttt{lem:advanced}, Theorem
\texttt{thm:exist}, and Propositions \texttt{prop:K5chart},
\texttt{prop:support}, \texttt{prop:checks}. Its Remark
\texttt{rem:conormal} is corrected here (G1).

\bibitem{Grelease}
Manuscript XIV-G, version~1. \texttt{XIV-G.tex}, identifier
\texttt{4fcac266}. Theorem \texttt{thm:release}, lines 1710--1751.
Used for $\dim W=16$, $\dim N=8$, $N=\Nt$.

\bibitem{Gresidual}
Manuscript XIV-G, version~1. Residual test
$G\in C^\infty_c(Z;\Sym^2T^*M)$ and the order-at-most-two statement
for $\mathrm{JD}'[G]$; passage named in the commissioning plan, line
numbers not returned in the record. $\Eres$ not defined there.

\bibitem{GG34}
Manuscript XIV-G, version~1. ``G3 and G4: no joint section and no
remainder theorem,'' lines 1494--1553. Used for the non-claims of
\S\ref{sec:patch1}.

\bibitem{Hkernel}
Manuscript XIV-H, version~1. \texttt{XIV-H.tex}, identifier
\texttt{05d90ea1}. Proposition \texttt{prop:K5exact}, lines
1135--1202. Source of \eqref{eq:K5a}--\eqref{eq:K5z}; ``fixed smooth
weights''; the products $R(b-a)R(a-z)$, $R(b-z)R(z-a)$.

\bibitem{HeqL}
Manuscript XIV-H, version~1. ``The existing tree maps with $z$
retained,'' lines 1108--1134, equation \texttt{eq:L} at lines
1111--1119. Source of \eqref{eq:L}. The definitions of $f$,
$\sigma_\eps$ preceding it are not in the record.

\bibitem{Hfiber}
Manuscript XIV-H, version~1. Display $a_1^{\alpha\mu}[r]=
\Amap^{\alpha\mu,I}r_I$; line numbers not returned. Source of
\eqref{eq:a1H}.

\bibitem{HinputEC}
Manuscript XIV-H, version~1. Input statement \texttt{input:EC},
named in the commissioning plan as the place where XII-E's $a_1$,
$b_1$, $a_2$ and completion kernels would be quoted. Content not in
the retrieved record. Locator only.

\bibitem{Hq}
Manuscript XIV-H, version~1. Lines 653--665: the
$q^{L,I,J}_{\alpha\beta}$ as coefficients of the actual differentiated
source, including ramp derivatives, $|\alpha|+|\beta|\le2$;
symmetrization factors absorbed. Paraphrased.

\bibitem{HK9}
Manuscript XIV-H, version~1. Equation \texttt{eq:K9}, line 1055, and
Proposition \texttt{prop:sourceincidence}. Located; displays not
reproduced.

\bibitem{Hrefuse}
Manuscript XIV-H, version~1. Lines 1661--1667: no equality
$\Ucan=W\setminus\overline{\Natt}$ is asserted. Refusal.

\bibitem{HH2}
Manuscript XIV-H, version~1. Theorem \texttt{thm:H2}, lines
1539--1582: exact H2 verdict; regular conormal amplitudes for $K_8$,
$K_9$ on the retained separated-endpoint patch; unresolved at H2.

\bibitem{Hrelease}
Manuscript XIV-H, version~1. Theorem \texttt{thm:release}, lines
1847--1895.

\bibitem{Ipole}
Manuscript XIV-I, version~1. \texttt{XIV-I.tex}, identifier
\texttt{13365486}. Proposition \texttt{prop:pole}, lines 726--826:
pole chart; $\mathcal F^{-1}_\omega M_{\mathrm R}=\Theta(t)\sin(\rho t)/\rho$;
the kernel \texttt{eq:R}.

\bibitem{IeqR}
Manuscript XIV-I, version~1. Equation \texttt{eq:R}, lines 329--335:
$R=\frac{\Theta(t)}{2\pi}\delta(t^2-|\mathbf x|^2)$ and $R*R$.

\bibitem{Ichart}
Manuscript XIV-I, version~1. Lines 828--855: placements $d_\pm$,
$r_\pm$, maps $\Phi_\pm$, coordinates $(q_\pm,\zeta_\pm)$, adapted
chart $w_\pm=(u_\pm,\zeta_\pm,q_\pm,r_\pm,y)$.

\bibitem{Iamodule}
Manuscript XIV-I, version~1. Definition \texttt{def:amodule}, lines
873--901: reconstruction rule reproduced; generators and coefficient
class paraphrased.

\bibitem{II2}
Manuscript XIV-I, version~1. Theorem \texttt{thm:I2} and Section
\texttt{sec:I3}, lines 1288--1378: the amplitude module for both
$K_5$ placements; \texttt{eq:K47}; the definition of $\Dmap$ by
$\dot a_2^{\alpha\mu}[G;h]=\Dmap^{\alpha\mu,I,J}h_IG_J$;
\texttt{eq:K6}; the precomposition sum \texttt{eq:presum}. Source of
\eqref{eq:K47}, \eqref{eq:Dfrom}, \eqref{eq:K6}, \eqref{eq:retained}.

\bibitem{II3}
Manuscript XIV-I, version~1. Equation \texttt{eq:jointmap},
Proposition \texttt{prop:canonical}, Theorem \texttt{thm:I3}, lines
1459--1548: the required map $\mathsf M:\Kint\to\Dp(W;\text{residual
species})$; canonical-domain representation; unresolved at I3.

\bibitem{Irefuse}
Manuscript XIV-I, version~1. Proposition \texttt{prop:canonical},
lines 1492--1496: no equality $\Ucan=W\setminus\overline{\Natt}$
follows. Refusal.

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

\bibitem{Iresponse}
Manuscript XIV-I, version~1. Input statement
\texttt{input:Hresponse}, lines 257--321 (formulas at 285--310):
\texttt{eq:Q}, \texttt{eq:T}, \texttt{eq:K8}, \texttt{eq:K9}, copied
from XIV-H. Source of \eqref{eq:Q}--\eqref{eq:K9}.

\bibitem{Jdescent}
Manuscript XIV-J, version~1. \texttt{XIV-J.tex}, identifier
\texttt{3c48e041}. Proposition \texttt{prop:descent}, lines 813--846.
Source of \eqref{eq:descent} and of Proposition~\ref{prop:restricted}.

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

\bibitem{Jtarget}
Manuscript XIV-J, version~1. Equation \texttt{eq:target}, lines
720--730. Paraphrased; display not reproduced (carried open).

\bibitem{Jmapclass}
Manuscript XIV-J, version~1. Definition \texttt{def:mapclass}, lines
732--762. Four clauses paraphrased; residual species with metric slot
at $z$ and scalar slot at $y$ named.

\bibitem{Jcross}
Manuscript XIV-J, version~1. Grouped cross-line operator, lines
1112--1123. Source of \eqref{eq:cross}; symbol of the replaced blocks
disputed between passes (patch, (F3)).

\bibitem{Jhessian}
Manuscript XIV-J, version~1. Mixed vertex Hessian, lines 1128--1145;
``$g$ the switch.''

\bibitem{Jreference}
Manuscript XIV-J, version~1. Theorem \texttt{thm:reference}, lines
1215--1299: $\Cross{}_D\Ksup$ on $U\setminus N_U$; normal form
$t^\circ_D$; wavefront bound; ``a local reference theorem for
$\Ksup$, not for the full $\Kret$.''

\bibitem{Jconditional}
Manuscript XIV-J, version~1. Theorem \texttt{thm:conditionalextension},
lines 1427--1470: normal order at most three; coefficients
$b_{D,\gamma}\in\Dp(S_0)$, $|\gamma|\le3$.

\bibitem{JJ2}
Manuscript XIV-J, version~1. Theorem \texttt{thm:J2}, lines
1608--1649.

\bibitem{Jrefuse}
Manuscript XIV-J, version~1. Lines 1305--1309: no equality
$\Ucan=W\setminus\overline{\Natt}$ is asserted. Refusal.

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

\bibitem{Jadmission}
Manuscript XIV-J, version~1. Corollary \texttt{cor:noadmission},
lines 1808--1822, and lines 1794--1807. No admitted $\Rquant$.

\bibitem{XIIICcoll}
Manuscript XIII-C, version~1. \texttt{XIII-C.tex}, identifier
\texttt{bb647493}. Definition ``Collision sets,'' lines 338--349.

\bibitem{XIIICpart}
Manuscript XIII-C, version~1. Definition \texttt{def:Cpart}, lines
358--392, equation \texttt{eq:partial-counterterm}. $D_\Gamma$,
$w_\Gamma$, $S_{\Gamma/I}$ are roles.

\bibitem{XIIICvertex}
Manuscript XIII-C, version~1. Cycle-vertex table, lines 603--620;
examples $\{y,u,v\}$, $\{y,v\}$ on $B_1$.

\bibitem{XIIICU}
Manuscript XIII-C, version~1. The open set $\UG$ on which the
H\"ormander sum condition holds; statement that $\UG$ is not
identified with the complement of the collision diagonals. Line
numbers not returned.

\bibitem{XIIICT0}
Manuscript XIII-C, version~1. Display of
$T_0^{\mu\nu}=\partial^\mu\phi\partial^\nu\phi-\tfrac12\eta^{\mu\nu}
((\partial\phi)^2-m^2\phi^2)$. Line numbers not returned. Carried;
not reassessed here.

\bibitem{XIIIB}
Manuscript XIII-B, version~1. \texttt{XIII-B.tex}, identifier
\texttt{17fb280a}. Release and proof boundary, lines 185--204.
Time-ordered line convention copied into XIII-C; used only as such.

\bibitem{XIIEa2}
Manuscript XII-E, version~1. \texttt{XII-E.tex}, identifier
\texttt{26f86eb3}. Lines 401--403: $b_2[r]$, $a_2^{\mu\nu}[r]$.
Copied premise; usable only through a quotation in XIV-H.

\bibitem{XIIDfiber}
Manuscript XII-D, version~1. \texttt{XII-D.tex}, identifier
\texttt{e3e3f67d}. Lines 519--525: $\mathfrak a$, $a_1$, $b_1$.
Copied premise.

\bibitem{XIIDweight}
Manuscript XII-D, version~1. Lines 545--552: $f\in C^\infty_c
(\mathring M_T)$, $\mathcal A_1[f,g](r;\phi)$; $v_f=\EA f$ without a
display of $\EA$. Copied premise.

\bibitem{XIICrelease}
Manuscript XII-C, version~1. \texttt{XII-C.tex}, identifier
\texttt{e0611fd7}. Release and specification correction, lines
165--194. Names the density-weighted cubic; the polynomial is not
displayed in the record.

\bibitem{record}
Supplied coverage account, retrieval corpus identifier
\texttt{b0c93438} (retrieval-3), with the earlier corpus
\texttt{3883b893}. Explicitly disclaims a complete corpus scan and
independent verification. Used for its reading rule and as the
carrier of locators.

\bibitem{Horm}
L.~H\"ormander, \emph{The Analysis of Linear Partial Differential
Operators I}, 2nd ed., Springer, 1990. Used for: Paley--Wiener--Schwartz
(Thm.~7.3.1); pullback by submersions (Thm.~8.2.4); wavefront sets of
operators and pushforwards (Thm.~8.2.12).

\end{thebibliography}

\end{document}