\documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlength{\parskip}{0.35em} \setlength{\parindent}{1.2em} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\D}{\mathcal D} \newcommand{\Dp}{\mathcal D'} \newcommand{\Sp}{\mathcal S'} \newcommand{\WF}{\operatorname{WF}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\im}{\operatorname{im}} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Recp}{\Rec^{\circ}} \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{\Cpart}{\mathscr C_{\mathrm{part}}} \newcommand{\Rquant}{R_1^{(1)}} \newcommand{\Eout}{\mathscr E_H} \newcommand{\Ejet}{\mathsf E} \newcommand{\Jjet}{\mathsf J} \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}}} \newcommand{\SZ}{\mathcal S_Z} \newcommand{\Vjet}{\mathcal V^{\mathrm{jet}}} \newcommand{\PPcal}{\mathcal P} \newcommand{\PPfrak}{\mathfrak P} \newcommand{\Pmap}{\mathsf P} \newcommand{\Bker}{\mathcal B} \newcommand{\Amap}{\mathsf A} \newcommand{\Amod}{\mathcal A^{m}(V)} \newcommand{\Jp}{J^{+}} \newcommand{\Jm}{J^{-}} \newcommand{\Four}{\mathcal F_{\mathbf x}} \newcommand{\Tren}{\mathsf T_2^{\mathrm{ren}}} \newcommand{\EA}{E^{\mathrm A}} \newcommand{\Tpol}{T^{\mathrm{pol}}} \newcommand{\Tsym}{T^{\mathrm{sym}}} \newcommand{\Tst}{\textsf{T}$^{\ast}$} \title{Landmark XV-F\\ Source Recovery from the Registered Full Texts\\ for the Frozen Joint Graph,\\ the Reconstruction-Fixed Ambiguity of Descending Extensions,\\ and the F0 Stop} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} XV-F continues XV-E at its gate E0. Its first task is to resolve the source gap that XV-E documented: either recover, from the supplied record, the core definitions needed to state the preliminary lift of the frozen six-branch joint target and its correction class, or report exactly which items remain unavailable. Five retrieval passes are recorded. The second reproduces XIV-H's coordinate-response line, the $K_8$, $K_9$ kernels, XIII-C's collision sets and partial-diagonal class, and XII-A's Hadamard matrix; the third returns locators only; the fourth reaches the registered full texts and reproduces XIV-I's reconstruction rule, XIV-I's retarded kernel $R$ and its square, XIV-J's grouped cross-line operator, mixed vertex Hessian, and output convention; the fifth yields exactly one identification---XIV-I's adapted chart $w_\pm=(u_\pm,\zeta_\pm,q_\pm,r_\pm,y)$ in terms of $(a,b,z,y)$---three located refusals of the equality $\Ucan=W\setminus\overline{\Natt}$, and four candidate displays from the XII sheet (the fiber tensor $\mathfrak a$, the observation weight $f$ as a compactly supported test function, XII-E's quadratic coefficient $a_2$, and the scalar stress tensor $T_0$). Every reproduced display is checked. New checks: in the adapted charts both $K_5$ placements and $K_6$ have exactly the form of the reconstruction rule with explicit coefficients; the fiber map $\mathfrak a$ is minus the trace reversal; the derivative $\dot a_2$ is recomputed; the polarization of $T_0$ matches the displayed Hessian for one normalization and is off by a factor two for the other; the advanced kernel $\EA_m$ is constructed and shown unique for all $m\ge0$, conditionally on a declaration the XIV texts do not contain. Three corrections to the circulated draft are recorded: the argument of the reconstruction coefficients is the chart point; with chart-function coefficients the generating family is not free, and XIV-J's null relation is an instance of this dependence; and no manuscript contains the symbol $\widehat H$ returned by the fourth pass. The tensor $\Amap$, the symbols $\sigma_\eps$, $\dot a_2$, $\mathsf D$, the ingredients of $K_8$, $K_9$, the normalization of the polarized stress, the displays of the target contraction, joint-map class and canonical domain, the residual bundle, the presentation-level correction class, the lift of the collision sets, and all candidate identifications remain unsupplied. The graph-level outcome is \emph{Unresolved at F0}. The mathematics on the finite normal-jet module with punctured reconstruction is unchanged: the disjoint classification of null relations; jet-detection as ``some extension carries a defect''; the affine space of descending extensions over the reconstruction-fixed freedom, infinite-dimensional in XV-D's example; and the theorem that a normal-order bound alone never makes the puncture-supported ambiguity finite-dimensional. Two phrasing errata to XV-E are recorded. Nothing is transferred to the inherited graph. \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) {Adapted chart, identified}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Canonical-domain sentences, refused}; \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 graph stays unresolved at F0.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One chart identification does not pass 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} \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) {Adapted chart, identified}; \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 graph stays unresolved at F0.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One chart identification does not pass 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} \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) {Adapted chart, identified}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Canonical-domain sentences, refused}; \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 graph stays unresolved at 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} \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) {Adapted chart, identified}; \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) {Canonical-domain sentences, refused. The graph stays unresolved at F0.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Adapted chart, identified 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} \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: Adapted chart, identified}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Canonical-domain sentences, refused}; \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 graph stays unresolved at 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} \end{figure} Figure~\ref{fig:split-XV-F} records one view of the same separation. Figure~\ref{fig:chain-XV-F} records one view of the same separation. Figure~\ref{fig:pair-XV-F} records one view of the same separation. Figure~\ref{fig:gate-XV-F} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XV-F} records that separation. The drawing adds no hypothesis. \section{Purpose, scope, and outcomes} \label{sec:scope} \subsection{The question and the permitted endings} XV-E \cite{XVE} performed a source recovery pass and found the data defining the full joint characteristic operation incomplete at its gate E0. XV-F asks: \begin{quote} Does the recoverable source material define the frozen six-branch operation well enough to construct its preliminary lift and test descent under the actual reconstruction map? \end{quote} Exactly one of three endings is permitted for the graph-level problem: \textbf{Lift defined and descended}; \textbf{Obstructed in the frozen class}, with the class proved exhaustive; or \textbf{Unresolved at a named gate}. Source incompleteness is not an analytic obstruction. Descent would not be quantum admission. \subsection{Gates and stop rule} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.07\linewidth}X@{}} \toprule Gate & Content\\ \midrule F0 & Lift-ready source package: the \emph{core} items of XV-D's Definition D0.1--D0.20 \cite{XVD} (Definition~\ref{def:core}) are reproduced in the record and mutually consistent.\\ F1 & Definition of $\Fz:\Pint\to\Dp(W;E_{\mathrm{res}})$ on the actual presentation space with $r_O\Fz=m^\circ\Rec$ proved on the source-defined domain $O$.\\ F2 & Determination of $\ker\Rec$; evaluation of $d_0=\Fz|_{\ker\Rec}$; cancellation or exhaustive noncancellation in the actual class $\mathcal A$.\\ F3 & Supported-sector estimates for the actual $v_\alpha$, $c_{D,\beta}$; characteristic compositions of $K_{47}$ and both $K_5$ placements.\\ F4 & Complete-graph estimates, output operations, admission.\\ \bottomrule \end{tabularx} \end{center} \textbf{F0 stop rule.} If a core item remains unavailable, this is said plainly; F1--F4 are not entered for the actual graph; no graph formula with placeholders is written and called a lift. The split into core and downstream items is descriptive and does not redefine XV-D's D0 or XV-E's E0. \subsection{Evidence contract} \label{sec:evidence} Direct inputs: the full texts of XV-A through XV-E \cite{XVA,XVB,XVC,XVD,XVE} and of XII-B \cite{XIIB}; the XIV-H, XIV-I, XIV-J excerpts of the first retrieval \cite{Hkernel,Hverdict,Ipole,Iamplitude,Ijoint,Irelease,Jdescent, Jnull,Jreference,Jconditional,Jverdict,Jrelease,Jadmission}; the XIV-C and XIV-G release statements \cite{Crelease,Grelease}; the XII-A and XII-D passages transcribed in XV-E \cite{XIIA,XIID}; the passages of the second retrieval \cite{HeqL,Ipackage,Iresponse, XIIAhad,XIIICcoll,XIIICpart,XIIDsource}; the one new passage of the third retrieval \cite{XIIDrelease}; the displays and prose descriptions of the fourth retrieval \cite{Iamodule,IeqR,Jcross, Jhessian,Jmapclass,Jtarget,Jrconv,HK9}; and the displays, refusals, and prose descriptions of the fifth retrieval \cite{Ichart,Hrefuse,Irefuse,Jrefuse,Hfiber,Hq,XIIDfiber, XIIDweight,XIIEa2,Jstress,XIIICvertex,XIICsource}. Locators cited without content \cite{XIVArel,Ione} and the X-B summary \cite{XBsum} are non-inputs. The coverage account \cite{record} is documentary context. Its reading rule---similar words in distinct manuscripts do not identify their models, response maps, or sign conventions---is applied throughout, including to the four symbols $\PPcal$ (XII-A), $\PPcal$ (XII-B), $\PPfrak$ (XII-D), $\Pmap$ (XIV-I) of \S\ref{sec:threeP}, to the four symbols $R$, $R_1$, $\Rquant$, $\mathcal R$ of \S\ref{sec:fourR}, and to the letters $a_1$, $X$, $f$, $g$, $E^{\mathrm A}$, $T$ shared between the XII and XIV sheets. Nothing from XIV-A, XIV-B, XIII-D, XIII-E, X-B, or manuscripts I--V is imported as a premise. \textbf{Displays, paraphrases, locators, refusals.} A reproduced display with file, label, and line numbers is a definition for the purposes of the manifest (status \textsf{T}). A prose description of a located definition, with locators but without its display, is a \emph{paraphrase} (status \textsf{P}); it fixes what the definition is about and does not pass F0. A locator, a search term, or a sentence stating where a definition should be found is not a definition (status \textsf{R}). A located sentence stating that an equality is \emph{not} asserted is a \emph{refusal}; it is recorded as such and closes the corresponding search. Standard distribution theory (pullback by submersions, tensor products, convolution under support conditions, point-supported distributions, Paley--Wiener--Schwartz) is used with reference to \cite{Horm}. \subsection{Numbering and admission} The technical-continuation convention of XV-B--XV-E is retained. The admission criterion \begin{equation} Q_H\Rquant\in\Eout \label{eq:admission} \end{equation} is a later project gate and is not asserted here. One companion rendering of the record writes the output space with a different macro; the source line \cite{Jadmission} is \eqref{eq:admission} with $\Eout$, and no second output space is introduced. \section{Errata to XV-E and corrections to the circulated draft} \label{sec:errata} \paragraph{(E1) Disjoint classification.} XV-E's Definition \texttt{def:kinds} calls $a\in H$ universal if $\beta(a)=0$, a flat-annihilator relation if each $\beta_\gamma(a)$ is flat on $Z$, and jet-detected otherwise. As worded, every universal relation is also flat-annihilator. The disjoint convention is Definition~\ref{def:kinds}; every theorem of XV-E holds under either reading, since each statement about the first two classes is a vanishing statement. \paragraph{(E2) Jet-detection is necessary, not sufficient.} The sentence in XV-E preceding its Lemma \texttt{lem:cutoff}, ``jet-detected relations produce a defect,'' overstates. What XV-E's Theorem \texttt{thm:jetdep}(ii) proves is that a nonzero defect is produced \emph{only} by jet-detected relations. Theorem~\ref{thm:jetchar} settles the exact relationship. \paragraph{(E3) Multiplier statement retained.} XV-E's Proposition \texttt{prop:erratum} is retained unchanged: for $v\in C^\infty(S_0)$, nonextendability of $vc$ implies nonextendability of $c$; a failure sequence for $vc$ transfers to a test family for $c$ that excludes extension but need not remain normalized in every increasing seminorm. \label{sec:corrections} \paragraph{(F1) The argument of the reconstruction coefficients.} The circulated draft glossed the symbol $w$ in XIV-I's reconstruction rule \eqref{eq:Recrule} as ``the remaining variables.'' The fifth pass \cite{Ichart} shows that $w_\pm$ is XIV-I's name for the full adapted chart point $(u_\pm,\zeta_\pm,q_\pm,r_\pm,y)$. The coefficients $c_{j\beta}(w)$ are therefore smooth functions on the chart, including of $q_\pm$ and $r_\pm$, unless XIV-I's unreproduced coefficient class restricts them. \paragraph{(F2) The free-module claim.} The circulated draft's Lemma \texttt{lem:recrule}(iii) asserted that $\Kint$ is free on the family $\{\delta^{(j)}(q)\partial_r^\beta R(r)\}$ and that $\ker\Rec$ is the coefficient kernel. This is correct only for coefficients independent of $(q,r)$. Over chart functions the family is dependent---$q\,\delta^{(j)}(q)=-j\,\delta^{(j-1)}(q)$ and $r^2R(r)=0$---and XIV-J's null relation $(q\,i\theta+1)M_{\mathrm R}$ is the instance $j=1$ of the first identity. Lemma~\ref{lem:recrule} states both cases; manifest items D0.3 and D0.4 are revised. \paragraph{(F3) The symbol $\widehat H$.} The fourth pass returned XIV-J's grouped cross-line operator with hatted replacement blocks $\widehat H_{\gamma\gamma}$, $\widehat H_{\phi\phi}$. The fifth pass reports that no manuscript in the corpus contains a symbol $\widehat H$ (the only hatted symbols in XIV-I are Fourier transforms and $\widehat s_0$, which XIV-I excludes from the present construction). The content of \eqref{eq:cross}---pullback by $\pi$ of a difference of two tensor products of blocks---is retained; the replaced blocks are written $H'_{\gamma\gamma}$, $H'_{\phi\phi}$ as this document's placeholder, and D0.11 carries the annotated status \Tst\ (display reproduced in content; one symbol disputed between passes). \section{Source manifest (Gate F0)} \label{sec:manifest} \subsection{Core and downstream items} \begin{definition}[Core and downstream] \label{def:core} The \emph{core} items of \cite{XVD}, Definition D0.1--D0.20, are those required to state a preliminary lift and its correction class: D0.1 ($\Pint$), D0.2 ($\Rec$ on $\Pint$), D0.3 ($\Kint$, topology), D0.4 ($\ker\Rec$), D0.5 ($W$, $E_{\mathrm{res}}$), D0.8 ($O=\Ucan$), D0.10 ($m^\circ$), D0.11 ($\Cross$), D0.12 (vertex Hessian, weights), D0.13 (tensor maps, $L$, $R$, $f$, $\sigma_\eps$), D0.14 ($K_8$, $K_9$), D0.16 (candidate class), D0.17 (correction class). The \emph{downstream} items are D0.6, D0.7, D0.9 (loci and strata), D0.15 (normalization), D0.18 ($v_\alpha$, $c_{D,\beta}$), D0.19 (target cone, estimates), D0.20 ($Q_H$, $\Eout$, output operations). \end{definition} Status codes: \textsf{S} supplied; \textsf{T} transcribed from a reproduced display or excerpt; \Tst\ transcribed in content with a symbol disputed between passes; \textsf{P} paraphrased (located definition described in prose, display not reproduced); \textsf{C} candidate (formula for a differently named object, identification not supplied); \textsf{R} reported only (name, locator, or sentence); \textsf{U} unavailable. \subsection{Manifest after five passes} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.055\linewidth}p{0.07\linewidth} p{0.09\linewidth}Xp{0.18\linewidth}@{}} \toprule Item & Group & Status & Record content & Change in XV-F\\ \midrule D0.1 & core & \textsf{T}/\textsf{P} & Finite retarded-pole module $\Amod$ \cite{Iamodule}; reconstruction rule \eqref{eq:Recrule} reproduced; adapted chart \eqref{eq:chart} reproduced \cite{Ichart}; generators and coefficient class paraphrased. & Pass 4: rule; pass 5: chart.\\[0.2em] D0.2 & core & \textsf{T}/\textsf{P} & Rule \eqref{eq:Recrule}; agrees with the two XIV-J values (Lemma~\ref{lem:recrule}(i)); linearity of $\Rec$ over chart functions not reproduced. & Pass 4: rule.\\[0.2em] D0.3 & core & \textsf{P} & $\Kint=\im\Rec$ is the span of the family $\{\delta^{(j)}(q_\pm)\partial_r^\beta R(r_\pm)\}$ over the unreproduced coefficient class; free iff that class is $(q,r)$-independent (Lemma~\ref{lem:recrule}); topology not named. & Pass 5: revised (F2).\\[0.2em] D0.4 & core & \textsf{T} (1)/\textsf{P} & Relation $(q\,i\theta+1)M_{\mathrm R}$ \cite{Jnull}, an instance of $q\delta'(q)=-\delta(q)$; further chart identities (Lemma~\ref{lem:recrule}(iv)) are null if $\Rec$ is $C^\infty$-linear; the general kernel depends on the coefficient class. & Pass 5: revised (F2).\\[0.2em] D0.5 & core & \textsf{R}/\textsf{P} & $W$ carries coordinates $(a,b,z,y)\in\mathbb R^{16}$ \cite{Grelease,Ichart}; $E_{\mathrm{res}}$ not defined. & Pass 5: coordinates confirmed.\\[0.2em] D0.6 & downstr. & \textsf{R} & Name; $\dim N=8$; codimension-four stratum named \cite{Crelease}. & None.\\[0.2em] D0.7 & downstr. & \textsf{R}/\textsf{C} & Name; codimension five; $q_1,q_2$ in \eqref{eq:referenceWF}. Candidate \eqref{eq:Srespcand}. & Pass 2: candidate; pass 4: support refined.\\[0.2em] D0.8 & core & \textsf{P} & ``The inherited domain on which the specified products are already proved''; the equality $\Ucan=W\setminus\overline{\Natt}$ is refused at three located places \cite{Hrefuse,Irefuse,Jrefuse}; XIII-C's $\mathcal U_\Gamma$ (H\"ormander sum condition) is not identified with the complement of the collision diagonals. & Pass 5: refusals located.\\[0.2em] D0.9 & downstr. & \textsf{T}/\textsf{R} & Collision sets $\Delta_I$ in XIII-C's graph variables \cite{XIIICcoll}; connected subsets of the cycle-vertex table \cite{XIIICvertex} (e.g.\ $\{y,u,v\}$, $\{y,v\}$ on $B_1$); lift to $W$ not supplied. & Pass 5: table located.\\[0.2em] D0.10 & core & \textsf{P} & \texttt{eq:target}: ``the formal six-branch contraction against the grouped Hadamard difference'' \cite{Jtarget}. & Pass 4.\\[0.2em] D0.11 & core & \Tst/\textsf{C} & $\Cross$ as the submersion pullback \eqref{eq:cross} \cite{Jcross}; symbol of the replaced blocks disputed (F3); identification with XII-A's admissible replacement \cite{XIIAhad} is a candidate. & Pass 4: display; pass 5: (F3).\\[0.2em] D0.12 & core & \textsf{T}/\textsf{P}/\textsf{C} & Mixed Hessian \eqref{eq:hess} \cite{Jhessian}; $T_0$ displayed \eqref{eq:T0} \cite{Jstress}; normalization of the polarized $T^{\mu\nu}(\phi,\psi)$ two-valued (Lemma~\ref{lem:polar}); remaining weights not reproduced. & Pass 4: Hessian; pass 5: $T_0$.\\[0.2em] D0.13 & core & \textsf{T}/\textsf{C} & \texttt{eq:L} \eqref{eq:L} \cite{HeqL}; \texttt{eq:R} \eqref{eq:R}--\eqref{eq:RR} \cite{IeqR}, verified; $\Amap$ defined as the matrix of XIV-H's $a_1$ \eqref{eq:a1H} \cite{Hfiber}, $a_1$ not reproduced; candidates $\mathfrak a$ \eqref{eq:a1}, $f\in C^\infty_c$, $\dot a_2$ \eqref{eq:a2dot}, $\sigma_\eps$ (\S\ref{sec:candidates}). & Pass 2: $L$; pass 4: $R$; pass 5: role of $\Amap$, three candidate displays.\\[0.2em] D0.14 & core & \textsf{T}/\textsf{P}/\textsf{C} & $K_8$, $K_9$ \eqref{eq:K8}--\eqref{eq:K9} \cite{Iresponse}; $\Bker$ explicit; $q^{L,I,J}_{\alpha\beta}$ paraphrased with $|\alpha|+|\beta|\le2$ \cite{Hq}; $\EA_m$ candidate \eqref{eq:EA}; $\Pmap$, $\mathfrak b$, $g$ not reproduced; existence proved (Theorem~\ref{thm:exist}). & Pass 5: order bound; advanced candidate.\\[0.2em] D0.15 & downstr. & \textsf{T} & \texttt{eq:Rconvention} \eqref{eq:Rconv} \cite{Jrconv}. & Pass 4.\\[0.2em] D0.16 & core & \textsf{P} & \texttt{def:mapclass}: four clauses paraphrased \cite{Jmapclass}; \texttt{def:class} not reproduced. & Pass 4.\\[0.2em] D0.17 & core & \textsf{T}/\textsf{R} & $\Cpart$ defined \cite{XIIICpart} (Definition~\ref{def:Cpart}); $D_\Gamma$, $w_\Gamma$, $S_{\Gamma/I}$ are roles, not formulas (confirmed by pass 5); relation to $\Hom(\Pint,Y)$ not supplied. & Pass 2.\\[0.2em] D0.18 & downstr. & \textsf{R} & Symbols in \eqref{eq:reference}. & None.\\[0.2em] D0.19 & downstr. & \textsf{T} (partial) & Bound \eqref{eq:referenceWF}; $\mathcal L$ named. & None.\\[0.2em] D0.20 & downstr. & \textsf{R}/\textsf{T} & $Q_H$, $\Eout$ named \cite{Jadmission}; output convention \eqref{eq:Rconv}. & Pass 4.\\ \bottomrule \end{tabularx} \end{center} \subsection{The third pass: locators only} \label{sec:thirdpass} The third retrieval pass returned the excerpts already audited, one new passage, and a locator guide. The new passage is XII-D's release boundary \cite{XIIDrelease}: it lists what is constructed on the XII sheet and states that no carrier enlargement, quantum retarded product, state, loop, interacting differential, or $\eps\to0$ limit is opened. It names none of $L$, $R$, $\Amap$, $\sigma_\eps$, $\Cross$, $\Ucan$, $m^\circ$ and is a non-input. The locator guide changed no status; it was used as the search plan for the fourth and fifth passes. XIV-A's relative master equation \cite{XIVArel} and manuscript I's context-transport passage \cite{Ione} are not imported. The XII-D local-source passage \cite{XIIDsource} identifies none of the required symbols and is not used. The X-B retrieval summary \cite{XBsum} concerns the scalar-only one-contraction normalization of a different track; its retarded-product line has the same $\frac{i}{\hbar}(\cdot-\cdot\star\cdot)$ form as \eqref{eq:Rconv}, and by the reading rule this is not an identification; it is a non-input. \subsection{The fourth pass: displays from the registered full texts} \label{sec:fourthpass} \paragraph{Presentation module and reconstruction (D0.1--D0.3).} XIV-I, \texttt{def:amodule}, lines 873--901 \cite{Iamodule}: the presentation module is the finite retarded-pole module $\Amod$ on the adapted chart $w=w_\pm$ of \eqref{eq:chart} below, reconstructed by \begin{equation} \Rec(A)=\sum_{j,\beta}c_{j\beta}(w)\,\delta^{(j)}(q)\, \partial_r^{\beta}R(r),\qquad A\in\Amod, \label{eq:Recrule} \end{equation} a finite sum, $q=q_\pm$, $r=r_\pm$, with coefficients $c_{j\beta}$ smooth on the chart and determined by $A$. The generators of $\Amod$ and the coefficient class (in particular whether $c_{j\beta}$ may depend on $q$, on $r$, or on both) are described in prose and not reproduced (status \textsf{P}). \paragraph{Retarded kernel (D0.13).} XIV-I, \texttt{eq:R}, lines 329--335 \cite{IeqR}: \begin{equation} R(t,\mathbf x)=\frac{\Theta(t)}{2\pi}\,\delta\bigl(t^2-|\mathbf x|^2\bigr), \label{eq:R} \end{equation} \begin{equation} \Bker:=R*R=\frac{1}{8\pi}\,\Theta(t)\,\Theta\bigl(t^2-|\mathbf x|^2\bigr). \label{eq:RR} \end{equation} \paragraph{Grouped cross-line operator (D0.11).} XIV-J, lines 1112--1123 \cite{Jcross}: with $\pi:W\to\mathbb R^4\times\mathbb R^4$, $\pi(a,b,z,y)=(y-a,\,y-b)$, \begin{equation} \Cross=\pi^{*}\bigl(H'_{\gamma\gamma}\otimes H'_{\phi\phi} -H_{\gamma\gamma}\otimes H_{\phi\phi}\bigr), \label{eq:cross} \end{equation} the pullback by the submersion $\pi$ of the difference of two tensor products of blocks written as distributions in the difference variables; $H'$ is this document's placeholder for XIV-J's replaced blocks (F3). Whether these are XII-A's frozen blocks \eqref{eq:Hblocks} and an admissible replacement in XII-A's sense is a candidate. \paragraph{Mixed vertex Hessian (D0.12).} XIV-J, lines 1128--1145 \cite{Jhessian}: \begin{equation} \mathcal V''_{\gamma\phi}[h,\psi] =-\int g(y)\,h_{\mu\nu}(y)\,T^{\mu\nu}(\phi,\psi)(y)\dd^4y , \label{eq:hess} \end{equation} bilinear in the metric direction $h$ and the scalar direction $\psi$ at the scalar background $\phi$, with $g$ the switch. XIV-J calls $T^{\mu\nu}(\phi,\psi)$ the inherited polarized tensor and does not expand it. \paragraph{Output convention (D0.15).} XIV-J, \texttt{eq:Rconvention}, lines 631--636 \cite{Jrconv}: \begin{equation} R_1(\mathcal V;\mathcal O) =\frac{i}{\hbar}\bigl(\Tren(\mathcal V,\mathcal O) -\mathcal V\star_H\mathcal O\bigr). \label{eq:Rconv} \end{equation} This is a map on functionals; it is not the kernel $R$ of \eqref{eq:R} (\S\ref{sec:fourR}). \paragraph{Paraphrased items (status \textsf{P}).} XIV-J, \texttt{eq:target}, lines 720--730 \cite{Jtarget}: $m^\circ$ is ``the formal six-branch contraction against the grouped Hadamard difference''; its display is not reproduced. XIV-J, \texttt{def:mapclass}, lines 732--762 \cite{Jmapclass}: the joint-map class consists of linear maps on reconstructed kernels that (a) contract with the cross-lines, (b) retain both $K_5$ placements, (c) agree with the canonical expression on the proved domain, and (d) meet $N$; the display and the meaning of (d) are not reproduced. XIV-H's own \texttt{eq:K9} at line 1055 \cite{HK9} is located and not reproduced. \subsection{The fifth pass: one identification, three refusals, four candidate displays} \label{sec:fifthpass} \paragraph{The adapted chart (D0.1, D0.5).} XIV-I, lines 828--855 \cite{Ichart}: the two placements are \begin{equation} \begin{gathered} d_+=b-a,\quad r_+=a-z,\qquad d_-=b-z,\quad r_-=z-a,\\ \Phi_+(a,b,z,y)=(a,d_+,r_+,y),\qquad \Phi_-(a,b,z,y)=(z,d_-,r_-,y), \end{gathered} \label{eq:chart} \end{equation} diffeomorphisms onto their images; near the null cone $q_\pm=d_\pm^2$ and $\zeta_\pm\in\mathbb R^3$ are ordinary coordinates; with $u_+=a$, $u_-=z$, the adapted chart is $w_\pm=(u_\pm,\zeta_\pm,q_\pm,r_\pm,y)$, of dimension sixteen, and $\Amod$ is defined on it. The particular choice of $\zeta_\pm$ is not reproduced. This relates the chart domain $V$ to $W$; it does not identify $V$ with $W$ and does not lift $\Delta_I$ to $W$. \paragraph{Three refusals (D0.8).} XIV-H, lines 1661--1667 \cite{Hrefuse}; XIV-I, Proposition \texttt{prop:canonical}, lines 1492--1496 \cite{Irefuse}; XIV-J, lines 1305--1309 \cite{Jrefuse}: each states that no equality $\Ucan=W\setminus\overline{\Natt}$ is asserted. $\Ucan$ remains the inherited domain on which the specified products have already been proved. XIII-C's $\mathcal U_\Gamma$ is the open set on which the H\"ormander sum condition holds, and XIII-C states that this does not identify $\mathcal U_\Gamma$ with the complement of the collision diagonals. \paragraph{Fiber map (D0.13).} XIV-H writes \cite{Hfiber} \begin{equation} a_1^{\alpha\mu}[r]=\Amap^{\alpha\mu,I}\,r_I , \label{eq:a1H} \end{equation} so that $\Amap$ is the coefficient matrix of XIV-H's linear fiber map $a_1$; $a_1$ itself is not reproduced. XII-D, lines 519--525 \cite{XIIDfiber}, and XII-A, lines 974--976 \cite{XIIA}, display \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:a1} \end{equation} The candidate is $\Amap^{\alpha\mu,I}=\mathfrak a^{\alpha\mu,ab}$ with $I$ the symmetric pair $(ab)$; no sentence in XIV-H states that its $a_1$ is XII-D's. \paragraph{Observation weight (D0.13).} XII-D, lines 545--552 \cite{XIIDweight}, takes an arbitrary $f\in C^\infty_c(\mathring M_T)$ on the open slab and sets \[ \mathcal A_1[f,g](r;\phi) =\int f\bigl(\phi_1[r;\phi]-X_1^\mu[r]\,\partial_\mu\phi\bigr)\dd^4x . \] On the XII sheet $f$ is a compactly supported test weight, not a missing formula. Whether XIV-H's $f$ in \eqref{eq:K5a}--\eqref{eq:K9} is this $f$ is a candidate (\S\ref{sec:candidates}). \paragraph{Quadratic coefficient (D0.13, $K_6$).} XII-E, lines 401--403 \cite{XIIEa2}: \begin{equation} b_2[r]=\tfrac18(\tr_\eta r)^2-\tfrac14 r_{\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} XIV uses $\dot a_2$ in the definition of $\mathsf D$ in \eqref{eq:K6} and does not record its expansion; Lemma~\ref{lem:a2} computes the derivative of \eqref{eq:a2}. \paragraph{Source coefficients (D0.14).} XIV-H, lines 653--665 \cite{Hq}: the $q^{L,I,J}_{\alpha\beta}$ are the coefficients of the actual differentiated source, including ramp derivatives, with $|\alpha|+|\beta|\le2$. The undifferentiated source is XII-C's density-weighted cubic \cite{XIICsource}, not a displayed polynomial. Expanding it is a calculation, not a located display; it is not performed here. \paragraph{Advanced factor (D0.14).} The record reports a normalization line $\Box E_0^{\mathrm R}= \mathrm{id}$ in XIV-I (locator not returned) alongside \eqref{eq:R}. XII-D sets $v_f=\EA f$ and does not display $\EA$. No XIV text declares $\EA_m$ to be the advanced inverse of $\Box+m^2$. Lemma~\ref{lem:advanced} constructs that inverse and proves its uniqueness; the identification remains a candidate. \paragraph{Polarized stress (D0.12).} XIII-C and XIV-J display \cite{Jstress} \begin{equation} T_0^{\mu\nu}(\phi)=\partial^\mu\phi\,\partial^\nu\phi -\tfrac12\eta^{\mu\nu}\bigl((\partial\phi)^2-m^2\phi^2\bigr), \label{eq:T0} \end{equation} and the record reports the vertex as $-\tfrac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}$ (locator not returned). XIV-J does not display the normalization of $T^{\mu\nu}(\phi,\psi)$ in \eqref{eq:hess}; Lemma~\ref{lem:polar} shows the factor is two-valued. \paragraph{Admissible replacement (D0.11).} XII-A, Definition \texttt{def:admissible}, lines 518--528 \cite{XIIAhad}, defines an admissible $H'$ by four clauses: the same shell equations in both slots, the same graded antisymmetric part $i\Delta$, a smooth difference $H'-H$, and vanishing unless $|A|+|B|$ is even and the ghost numbers sum to zero. No manuscript equates a XIV object with this $H'$ (F3). \paragraph{Collision sets (D0.9, D0.17).} XIII-C, lines 603--620 \cite{XIIICvertex}: the $\Delta_I$ are the connected collision subsets of the cycle-vertex table, for instance $\{y,u,v\}$ and $\{y,v\}$ on $B_1$. $D_\Gamma$, $w_\Gamma$, $S_{\Gamma/I}$ remain roles. \subsection{Symbols kept distinct} \label{sec:threeP} Four maps are written with the letter P: XII-A's $\PPcal$ in $H_{\gamma\gamma}=-\PPcal h_0$ \cite{XIIAhad}, with $\PPcal^2= \mathrm{id}$ stated there; XII-B's $\PPcal$, defined explicitly as trace reversal $\PPcal r=r-\tfrac12\eta\tr_\eta r$ \cite{XIIB}; XII-D's $\PPfrak$ in $G^{\mathrm R}_{\mathrm{gr}}=-\PPfrak E_0^{\mathrm R}$ \cite{XIID}; and XIV-I's constant map $\Pmap^L{}_M$ in \eqref{eq:T} \cite{Iresponse}. XII-B's trace reversal satisfies $\PPcal^2=\mathrm{id}$ in four dimensions: $\tr_\eta(\PPcal r)=\tr_\eta r-2\tr_\eta r=-\tr_\eta r$, so $\PPcal\PPcal r=\PPcal r+\tfrac12\eta\tr_\eta r=r$. This is consistent with XII-A's statement and identifies nothing. Lemma~\ref{lem:fiber} adds that XII-D's $a_1$ equals $-\PPcal$ on symmetric tensors, a relation among XII objects only. \label{sec:fourR} Four symbols are written with the letter R: the kernel $R$ of \eqref{eq:R}; the output map $R_1$ of \eqref{eq:Rconv}; the admission coefficient $\Rquant$ of \eqref{eq:admission}, whose relation to $R_1$ is not reproduced; and XII-B's $\mathcal R=\tfrac12K$ \cite{XIIB}. None is identified with another. Likewise XII-D's $a_1$, $X_1$, $f$, $\EA$ and XIV-H's $a_1$, $X$, $f$, $\EA_m$ are distinct symbols until a sentence identifies them. \subsection{Verification of the reproduced displays} \label{sec:verify} Throughout, $\eta=\mathrm{diag}(+,-,-,-)$, $\Box=\eta^{\lambda\kappa}\partial_\lambda\partial_\kappa =\partial_t^2-\Delta$ on $\mathbb R^4$, $\Jp=\{(t,\mathbf x):t\ge|\mathbf x|\}$, $\Jm=-\Jp$, $\partial\Jp=\{t=|\mathbf x|\}$, and $\Four$ the partial Fourier transform in $\mathbf x$, $\Four\phi(t,\mathbf p)=\int e^{-i\mathbf p\cdot\mathbf x}\phi(t,\mathbf x)\dd^3\mathbf x$, extended to $\Sp(\mathbb R^4)$ as an isomorphism; $\rho=|\mathbf p|$. \begin{lemma}[The kernel $R$] \label{lem:R} \begin{enumerate}[label=\textup{(\roman*)}] \item \eqref{eq:R} defines the positive measure $\langle R,\phi\rangle=\frac{1}{4\pi}\int_0^\infty t \int_{S^2}\phi(t,t\omega)\dd\omega\dd t$, equal to $\Theta(t)\,\delta(t-|\mathbf x|)/(4\pi|\mathbf x|)$; it is tempered, with $\supp R=\partial\Jp$. \item $\Four R(t,\mathbf p)=\Theta(t)\sin(\rho t)/\rho$. \item $\Box R=\delta_0$. If $R'\in\Dp(\mathbb R^4)$ satisfies $\Box R'=\delta_0$ and $\supp R'\subseteq\{t\ge c\}$ for some $c\in\mathbb R$, then $R'=R$. In particular $R$ is the unique fundamental solution supported in $\Jp$. \item If $E:\D(\mathbb R^4)\to\Dp(\mathbb R^4)$ is linear with $\Box E\phi=\phi$ and $\supp E\phi\subseteq\{t\ge c_\phi\}$ for every $\phi$, then $E\phi=R*\phi$. \end{enumerate} In particular \eqref{eq:R} agrees with XIV-I's pole chart \cite{Ipole}, which characterizes $R$ by the transform in \textup{(ii)}. \end{lemma} \begin{proof} (i) In polar coordinates $\mathbf x=r\omega$, for $t>0$ the composite $\delta(t^2-r^2)$ as a distribution in $r>0$ equals $\delta(r-t)/(2t)$, so $\frac1{2\pi}\int\Theta(t)\delta(t^2-r^2) \phi\,r^2\dd r\dd\omega\dd t=\frac1{4\pi}\int_0^\infty t\int_{S^2} \phi(t,t\omega)\dd\omega\dd t$; the same computation with $\delta(t-r)/(4\pi r)$ gives the same value. The mass of $R$ in the ball of radius $\varrho$ is at most $\varrho^2/2$, so $R$ is tempered. The support is the cone $t=r$. (ii) For $\psi\in\mathcal S(\mathbb R^4_{(t,\mathbf p)})$, by (i) and Fubini, $\langle R,\Four\psi\rangle=\int_0^\infty\!\!\int\psi(t,\mathbf p) \Bigl[\frac{t}{4\pi}\int_{S^2}e^{-it\mathbf p\cdot\omega}\dd\omega\Bigr] \dd^3\mathbf p\dd t$, and $\int_{S^2}e^{-it\mathbf p\cdot\omega}\dd\omega=4\pi\sin(\rho t)/(\rho t)$. (iii) Let $u(t)=\sin(\rho t)/\rho$; $u(0)=0$, $u'(0)=1$, $u''+\rho^2u=0$. Then $(\partial_t^2+\rho^2)(\Theta u) =\Theta(u''+\rho^2u)+u(0)\delta'+u'(0)\delta=\delta(t)$ for every $\mathbf p$, including $\rho=0$ where $u=t$. Since $\Four(\Box R)=(\partial_t^2+\rho^2)\Four R$, $\Box R=\delta_0$. Uniqueness: $v=R-R'$ satisfies $\Box v=0$ and $\supp v\subseteq\{t\ge c\}$. The addition map is proper on $\Jp\times\{t\ge c\}$: if $x\in\Jp$, $y^0\ge c$ and $x+y$ lies in a compact set, then $x^0\le C-c$, $|\mathbf x|\le x^0$, so $x$ and hence $y$ are bounded. Thus $R*v$ is defined and $v=\delta_0*v=(\Box R)*v=\Box(R*v)=R*(\Box v)=0$ \cite[\S4.2]{Horm}. (iv) $E\phi-R*\phi$ solves $\Box(\cdot)=0$ with support in a half-space $\{t\ge c\}$; apply the argument of (iii). \end{proof} \begin{lemma}[The display for $R*R$] \label{lem:RR} $R*R$ is well defined and equals $\Bker=\frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2) =\frac1{8\pi}\Theta(t-|\mathbf x|)$ almost everywhere. Hence \eqref{eq:RR} holds, $\Bker\in L^\infty(\mathbb R^4)$, $\supp\Bker=\Jp$, $\Box\Bker=R$, and $\Bker$ is constant on the interior of $\Jp$. On $\{t>0\}$, $\partial_\alpha\Bker =\frac{1}{4\pi}x_\alpha\,\delta(x^2)$, where $x_\alpha=\eta_{\alpha\nu}x^\nu$ and $x^2=t^2-|\mathbf x|^2$. \end{lemma} \begin{proof} $R*R$ exists since both factors are supported in $\Jp$, and $\supp(R*R)\subseteq\Jp+\Jp=\Jp$, $\Box(R*R)=(\Box R)*R=R$. For $\Bker$: $\Four\Bker(t,\mathbf p)=\frac{\Theta(t)}{8\pi} \int_{|\mathbf x|0\}$, where $x\mapsto x^2$ has nonvanishing differential away from $x=0$. \end{proof} \begin{lemma}[Taylor reduction of coefficients at a delta derivative] \label{lem:taylor} Let $c$ be smooth in $(q,\xi)$ near $q=0$ and $j\ge0$. Then, as distributions in $q$ depending smoothly on $\xi$, \[ c(q,\xi)\,\delta^{(j)}(q) =\sum_{i=0}^{j}(-1)^{j-i}\binom{j}{i}\, (\partial_q^{\,j-i}c)(0,\xi)\,\delta^{(i)}(q). \] In particular $q\,\delta^{(j)}(q)=-j\,\delta^{(j-1)}(q)$ and $q\,\delta(q)=0$. \end{lemma} \begin{proof} $\langle c\delta^{(j)},\phi\rangle=(-1)^j\partial_q^j(c\phi)(0) =\sum_i\binom ji(-1)^{j-i}\partial_q^{j-i}c(0)\,(-1)^i\partial_q^i\phi(0)$ by Leibniz, and $(-1)^i\partial_q^i\phi(0)=\langle\delta^{(i)},\phi\rangle$. \end{proof} \begin{lemma}[The reconstruction rule and its coefficient class] \label{lem:recrule} \begin{enumerate}[label=\textup{(\roman*)}] \item The two XIV-J values $\Rec(M_{\mathrm R})=\delta(q)R(r)$, $\Rec(i\theta M_{\mathrm R})=\delta'(q)R(r)$ \cite{Jnull} are instances of \eqref{eq:Recrule} with $c_{00}=1$, resp.\ $c_{10}=1$, all other coefficients zero. \item If $\Rec$ is linear over multiplication by the chart function $q$ (a property not reproduced), then \eqref{eq:Recrule} annihilates the null relation $(q\,i\theta+1)M_{\mathrm R}$ of \cite{Jnull}: $q\,\delta'(q)R(r)+\delta(q)R(r)=0$ by Lemma~\ref{lem:taylor}. \item Suppose the coefficients $c_{j\beta}$ are independent of $(q,r)$. Then the family $\{\delta^{(j)}(q)\partial_r^\beta R(r)\}$ is linearly independent over that class, $\Kint$ is free on it, and $\Rec(A)=0$ iff all $c_{j\beta}(A)=0$. In particular $\{\partial_r^\beta R\}_\beta$ is linearly independent over $\mathbb C$. \item Suppose the coefficient class contains the chart functions $q$ or $r^2$. Then the family is linearly dependent over that class: \[ q\,\delta^{(j)}(q)\,\partial_r^\beta R(r)=-j\,\delta^{(j-1)}(q)\,\partial_r^\beta R(r), \qquad r^2R(r)=0,\qquad r^2\partial_\lambda R(r)=-2r_\lambda R(r). \] If $\Rec$ is linear over the coefficient class, each of these is a null relation of $\Amod$; XIV-J's relation is the first with $j=1$, $\beta=0$. In this case $\ker\Rec$ strictly contains the coefficient kernel, and $\Kint$ is not free on the family. \end{enumerate} \end{lemma} \begin{proof} (i) is read off; (ii) is Lemma~\ref{lem:taylor}. (iii) First, $\{\partial^\beta_rR\}_\beta$ is independent over $\mathbb C$: if $\sum_\beta a_\beta\partial^\beta R=0$, apply the Fourier transform on $\mathbb R^4$ ($R$ is tempered). From $\Box R=\delta_0$, $\widehat R$ coincides on the open set $\Omega=\{p_0^2\ne|\mathbf p|^2\}$ with a smooth nowhere-vanishing function. The relation gives $P(p)\widehat R=0$ with $P(p)=\sum_\beta a_\beta(\pm ip)^\beta$, so $P=0$ on $\Omega$, hence $P\equiv0$ and all $a_\beta=0$. Now pair a vanishing sum $\sum c_{j\beta}(\xi)\delta^{(j)}(q)\partial^\beta R(r)$ with $\psi(\xi)\chi(q)\theta(r)$, where $\xi$ denotes the chart variables other than $(q,r)$: $\sum_{j,\beta}\langle c_{j\beta},\psi\rangle (-1)^j\chi^{(j)}(0)\langle\partial^\beta R,\theta\rangle=0$. Choose $\chi$ with $\chi^{(j)}(0)=\delta_{jj_0}$ for the finitely many $j$ occurring; then $\sum_\beta\langle c_{j_0\beta},\psi\rangle \partial^\beta R=0$, so $\langle c_{j_0\beta},\psi\rangle=0$ for all $\beta,\psi$. (iv) The first identity is Lemma~\ref{lem:taylor}. $r^2R(r)=\frac{1}{2\pi}\Theta(r^0)\,r^2\delta(r^2)=0$ by $q\delta(q)=0$ pulled back along $r\mapsto r^2$ on $r\ne0$, and $R$ has no mass at $r=0$ by Lemma~\ref{lem:R}(i). Differentiating $r^2R=0$ gives $2r_\lambda R+r^2\partial_\lambda R=0$. The remaining assertions are restatements. \end{proof} \begin{lemma}[The adapted chart] \label{lem:chart} \begin{enumerate}[label=\textup{(\roman*)}] \item $\Phi_\pm$ in \eqref{eq:chart} are linear automorphisms of $\mathbb R^{16}$. \item On $\{d\in\mathbb R^4:d^0>0\}$ the map $\kappa(d)=(d^2,\mathbf d)=(q,\zeta)$ is a diffeomorphism onto $\Omega_0=\{(q,\zeta)\in\mathbb R\times\mathbb R^3:q+|\zeta|^2>0\}$, with inverse $d^0=\sqrt{q+|\zeta|^2}$, $\mathbf d=\zeta$. The set $\partial\Jp\setminus\{0\}$ is contained in $\{d^0>0\}$ and is $\{q=0\}$ there. \item Hence, with $\zeta_\pm=\mathbf d_\pm$, $w_\pm=(u_\pm,\zeta_\pm,q_\pm,r_\pm,y)$ is a global chart of $V_\pm:=\{d_\pm^0>0\}\subset\mathbb R^{16}$ onto $\mathbb R^4\times\Omega_0\times\mathbb R^4\times\mathbb R^4$, of dimension $4+3+1+4+4=16$, consistent with \cite{Ichart} and with $\dim W=16$ \cite{Grelease}. \item On $V_\pm$, \[ R(d_\pm)=\tfrac{1}{2\pi}\,\delta(q_\pm),\qquad (\partial_\alpha R)(d_\pm)=\tfrac1\pi\,d_{\pm\alpha}\,\delta'(q_\pm),\qquad (\partial_\mu\partial_\alpha R)(d_\pm)=\tfrac1\pi\bigl[\eta_{\mu\alpha}\delta'(q_\pm) +2d_{\pm\mu}d_{\pm\alpha}\delta''(q_\pm)\bigr], \] where $d_{\pm\alpha}=\eta_{\alpha\nu}d_\pm^\nu$ is smooth on the chart, and $\Bker(d_\pm)=\frac{1}{8\pi}\Theta(q_\pm)$. \end{enumerate} \end{lemma} \begin{proof} (i) is immediate. (ii) $q=(d^0)^2-|\mathbf d|^2$ and $d^0>0$ give $d^0=\sqrt{q+|\zeta|^2}$, smooth on $\Omega_0$; bijectivity is clear. On $\partial\Jp\setminus\{0\}$, $d^0=|\mathbf d|>0$ and $q=0$. (iii) follows from (i), (ii). (iv) On $V_\pm$, $\Theta(d^0)=1$, and $\delta(d^2)$ is the pullback of $\delta(q)$ under the diffeomorphism $\kappa$; the derivatives follow from the chain rule $\partial_{d^\alpha}=2d_\alpha\partial_q+\delta_\alpha^i \partial_{\zeta^i}$, since $\delta(q)$ is independent of $\zeta$. The statement for $\Bker$ is Lemma~\ref{lem:RR}. \end{proof} \begin{proposition}[The $K_5$ placements and $K_6$ have the form of the reconstruction rule] \label{prop:K5chart} Let $\Amap$, $\mathsf D$ be constant arrays and $f,\sigma_\eps\in C^\infty(\mathbb R^4)$ (any completion). On $V_+=\{(b-a)^0>0\}$, in the chart $w_+$ with $b=a+d_+$, $z=a-r_+$, \begin{align*} K_5^{[a],I,J} &=\sum_{j=1}^{2}c^{[a]}_{j,\beta\lambda}(w_+)\,\delta^{(j)}(q_+)\, (\partial_\beta\partial_\lambda R)(r_+),\\ c^{[a]}_{1,\beta\lambda} &=\tfrac1\pi\,\sigma_\eps(a)\sigma_\eps(z)\,\Amap^{\alpha\beta,I}\Amap^{\lambda\mu,J} \bigl[(\partial_\mu f)(b)\,d_{+\alpha}+f(b)\,\eta_{\mu\alpha}\bigr],\\ c^{[a]}_{2,\beta\lambda} &=\tfrac2\pi\,\sigma_\eps(a)\sigma_\eps(z)\,\Amap^{\alpha\beta,I}\Amap^{\lambda\mu,J} f(b)\,d_{+\alpha}d_{+\mu}, \end{align*} and \[ K_6^{I,J}=-\sum_{j=1}^{2}e_j(w_+)\,\delta^{(j)}(q_+)\, \eta^{\beta\lambda}(\partial_\beta\partial_\lambda R)(r_+),\quad e_1=\tfrac1\pi\sigma_\eps(a)\mathsf D^{\alpha\mu,I,J} \bigl[(\partial_\mu f)(b)d_{+\alpha}+f(b)\eta_{\mu\alpha}\bigr],\quad e_2=\tfrac2\pi\sigma_\eps(a)\mathsf D^{\alpha\mu,I,J}f(b)d_{+\alpha}d_{+\mu}. \] On $V_-=\{(b-z)^0>0\}$, $K_5^{[z],I,J}$ has the same form in the chart $w_-$ with $(a,I)\leftrightarrow(z,J)$, $d_-$, $r_-$. All coefficients are smooth on the chart; they are of the form $c_{j\beta}(w_\pm)$ in \eqref{eq:Recrule} with $j\in\{1,2\}$ and $|\beta|=2$. The supports of $K_5^{[a]}$, $K_6$ (resp.\ $K_5^{[z]}$) meet the complement of $V_+$ (resp.\ $V_-$) only on $\{b=a\}$ (resp.\ $\{b=z\}$). \end{proposition} \begin{proof} From \eqref{eq:L}, $L^{\mu,J}(a,z)=\sigma_\eps(z)\Amap^{\lambda\mu,J} \partial_{z^\lambda}R(a-z)=-\sigma_\eps(z)\Amap^{\lambda\mu,J} (\partial_\lambda R)(r_+)$, so $\partial_{a^\beta}L^{\mu,J}(a,z) =-\sigma_\eps(z)\Amap^{\lambda\mu,J}(\partial_\beta\partial_\lambda R)(r_+)$; and $\partial_{a^\alpha}R(b-a)=-(\partial_\alpha R)(d_+)$. The bracket in \eqref{eq:K5a} is therefore $f(b)\sigma_\eps(a)\sigma_\eps(z)\Amap^{\alpha\beta,I}\Amap^{\lambda\mu,J} (\partial_\alpha R)(d_+)(\partial_\beta\partial_\lambda R)(r_+)$. With $a,z,y$ fixed, $\partial_{b^\mu}=\partial_{d_+^\mu}$ acts on $f(a+d_+)$ and on $(\partial_\alpha R)(d_+)$; insert Lemma~\ref{lem:chart}(iv). For $K_6$, write $\delta^{(4)}(a-z)=\delta(r_+)=\Box_rR(r_+) =\eta^{\beta\lambda}\partial_\beta\partial_\lambda R(r_+)$ (Lemma~\ref{lem:R}(iii)) and proceed identically; the overall sign is that of $\partial_{a^\alpha}R(b-a)=-(\partial_\alpha R)(d_+)$. $K_5^{[z]}$ follows by the symmetry of \eqref{eq:K5z} with \eqref{eq:K5a}. The support statement: each kernel contains the factor $(\partial_\alpha R)(d_\pm)$ or its $b$-derivative, supported in $\{d_\pm\in\partial\Jp\}$, and $\partial\Jp\setminus\{0\}\subset \{d^0>0\}$ by Lemma~\ref{lem:chart}(ii). \end{proof} \begin{remark} Proposition~\ref{prop:K5chart} is the first mutual-consistency check between D0.1--D0.2 and the six entries: the internal expressions of the two $K_5$ placements and of $K_6$ are, on the respective charts, elements of the span described by \eqref{eq:Recrule}, with the placement $\pm$ selecting the chart---consistent with XIV-J's clause (b) ``retain both $K_5$ placements.'' The coefficients depend on $q_\pm$ (through $d_\pm^0$ and $f(b)$) and on $r_\pm$ (through $\sigma_\eps(z)$); the $q$-dependence can be removed by Lemma~\ref{lem:taylor} at the cost of lower-order terms, the $r$-dependence cannot in general. Whether XIV-I's coefficient class admits this dependence is part of the unreproduced definition; by (F1) it is the natural reading. By the same mechanism $K_9$ and $K_8$ are sums of terms $\partial^\gamma R(d_+)$ or $\partial^{\gamma}\Bker(d_+)$ with $|\gamma|\ge1$ times $\partial^{\gamma'}\delta(r_+)=\partial^{\gamma'}\Box_rR(r_+)$ and smooth coefficients, hence also of the form \eqref{eq:Recrule} on $V_+$ once \eqref{eq:Q} is expanded; this is not carried out, since the $q_{\alpha\beta}$ are not reproduced. $K_{47}$ contains the factors $R(d_+)$ and $R(d_++r_+)$ and is not of this form in either chart. \end{remark} \begin{lemma}[The advanced kernel] \label{lem:advanced} Let $m\ge0$, $P_m=\Box+m^2$, $\omega=\sqrt{\rho^2+m^2}$. The tempered distribution $\EA_m$ with \begin{equation} \Four\EA_m(t,\mathbf p)=-\Theta(-t)\,\frac{\sin(\omega t)}{\omega} \label{eq:EA} \end{equation} satisfies $P_m\EA_m=\delta_0$ and $\supp\EA_m\subseteq\Jm$, and it is the unique fundamental solution of $P_m$ supported in a half-space $\{t\le c\}$. For $m=0$, $\EA_0(t,\mathbf x)=R(-t,\mathbf x)=\frac{\Theta(-t)}{2\pi} \delta(t^2-|\mathbf x|^2)$. For $f\in C^\infty_c(\mathbb R^4)$, $\EA_m*f\in C^\infty(\mathbb R^4)$ with support in $\supp f+\Jm$. \end{lemma} \begin{proof} With $u(t)=\sin(\omega t)/\omega$, $u(0)=0$, $u'(0)=1$: $\partial_t(-\Theta(-t)u)=\delta(t)u(0)-\Theta(-t)u'=-\Theta(-t)u'$, $\partial_t^2(-\Theta(-t)u)=\delta(t)u'(0)-\Theta(-t)u''$, so $(\partial_t^2+\omega^2)(-\Theta(-t)u)=\delta(t)-\Theta(-t)(u''+\omega^2u) =\delta(t)$; hence $P_m\EA_m=\delta_0$. Support: fix $t<0$. The function $\mathbf p\mapsto\sin(\omega t)/\omega$ is even in $\omega$, hence an entire function of $\omega^2=\mathbf p\cdot\mathbf p+m^2$, hence entire on $\mathbb C^3$, and $|\sin(\omega t)/\omega| =|t|\,|\sin(\omega t)/(\omega t)|\le|t|e^{|t||\operatorname{Im}\omega|}$ since $\sin z/z=\int_0^1\cos(sz)\dd s$. For $\mathbf p=\mathbf a+i\mathbf b$, $\omega^2=|\mathbf a|^2-|\mathbf b|^2 +m^2+2i\mathbf a\cdot\mathbf b$ and $(\operatorname{Im}\omega)^2 =\frac12(|\omega^2|-\operatorname{Re}\omega^2)\le|\mathbf b|^2$, because $|\omega^2|^2\le(|\mathbf a|^2+|\mathbf b|^2+m^2)^2$ (the difference of the squares is $4|\mathbf b|^2m^2\ge0$). By Paley--Wiener--Schwartz \cite[Thm.~7.3.1]{Horm}, the inverse transform in $\mathbf p$ is supported in $\{|\mathbf x|\le|t|\}$. If $\phi\in\D(\mathbb R^4)$ vanishes near $\Jm$, then for each $t$, $\phi(t,\cdot)$ vanishes near the ball of radius $|t|$, so $\langle\EA_m,\phi\rangle=0$. Uniqueness: as in Lemma~\ref{lem:R}(iii) with $\Jm$ and $\{t\le c\}$. For $m=0$, $\Four[R(-t,\mathbf x)]=\Theta(-t)\sin(-\rho t)/\rho$, which is \eqref{eq:EA}. The last sentence: convolution of a distribution with a test function is smooth, and $\supp(\EA_m*f)\subseteq\supp\EA_m+\supp f$. \end{proof} \begin{lemma}[The fiber map is minus the trace reversal] \label{lem:fiber} For symmetric $r_{ab}$, with $\mathfrak a$ as in \eqref{eq:a1}, $\mathfrak a^{\alpha\mu,ab}r_{ab}=\tfrac12\eta^{\alpha\mu}\tr_\eta r -r^{\alpha\mu}=-(\PPcal r)^{\alpha\mu}$, where $\PPcal$ is XII-B's trace reversal. $\mathfrak a^{\alpha\mu,ab}$ is symmetric in $(\alpha\mu)$ and in $(ab)$, and $b_1[r]=\tfrac12\tr_\eta r$. \end{lemma} \begin{proof} $\eta^{\alpha(a}\eta^{b)\mu}r_{ab}=\tfrac12(r^{\alpha\mu}+r^{\mu\alpha}) =r^{\alpha\mu}$. The symmetries are read off from \eqref{eq:a1}. \end{proof} \begin{lemma}[The derivative of XII-E's quadratic coefficient] \label{lem:a2} With $a_2$, $b_2$ as in \eqref{eq:a2}, the directional derivative at $G$ in the direction $h$ is \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}\\ &\quad+\Bigl(\tfrac14(\tr_\eta G)(\tr_\eta h) -\tfrac12 G_{\alpha\beta}h^{\alpha\beta}\Bigr)\eta^{\mu\nu} =:\mathfrak d^{\mu\nu,ab,cd}\,G_{ab}\,h_{cd}, \end{split} \label{eq:a2dot} \end{equation} bilinear and symmetric in $(G,h)$, with $\dot a_2[r;r]=2a_2[r]$. \end{lemma} \begin{proof} Differentiate each term of \eqref{eq:a2} at $s=0$ along $G+sh$; $\frac{d}{ds}b_2[G+sh]|_0=\tfrac14(\tr G)(\tr h)-\tfrac12G_{\alpha\beta}h^{\alpha\beta}$. Symmetry and the Euler identity follow from homogeneity of degree two. \end{proof} \begin{lemma}[Polarization of the scalar stress and the factor in the Hessian] \label{lem:polar} Let $T_0$ be as in \eqref{eq:T0} and define \begin{equation} \Tpol^{\mu\nu}(\phi,\psi) =\tfrac12\bigl(\partial^\mu\phi\,\partial^\nu\psi+\partial^\mu\psi\,\partial^\nu\phi\bigr) -\tfrac12\eta^{\mu\nu}\bigl(\partial\phi\cdot\partial\psi-m^2\phi\psi\bigr), \qquad \Tsym:=2\,\Tpol . \label{eq:Tpol} \end{equation} $\Tpol$ is the unique symmetric bilinear form with $\Tpol(\phi,\phi)=T_0(\phi)$, and $\frac{d}{ds}T_0(\phi+s\psi)|_{s=0} =2\Tpol(\phi,\psi)=\Tsym(\phi,\psi)$. For the vertex $\mathcal V[\gamma,\phi]=-\tfrac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}(\phi)$, \[ \frac{\partial^2}{\partial s\,\partial t}\mathcal V[\gamma+th,\phi+s\psi]\Big|_{0} =-\int g\,h_{\mu\nu}\,\Tpol^{\mu\nu}(\phi,\psi) =-\tfrac12\int g\,h_{\mu\nu}\,\Tsym^{\mu\nu}(\phi,\psi). \] Hence \eqref{eq:hess} is exactly the mixed Hessian of this vertex if XIV-J's $T^{\mu\nu}(\phi,\psi)$ is $\Tpol$, and is twice it if $T^{\mu\nu}(\phi,\psi)$ is $\Tsym$. The record does not display which. \end{lemma} \begin{proof} $T_0(\phi+s\psi)=T_0(\phi)+2s\Tpol(\phi,\psi)+s^2T_0(\psi)$ by bilinearity; the $t$-derivative of $\mathcal V$ is linear in $\gamma$. Uniqueness of the symmetric polarization of a quadratic form is standard. \end{proof} \begin{lemma}[The cross-line pullback] \label{lem:cross} Let $D\in\Dp(\mathbb R^4_u\times\mathbb R^4_v)$ be any distribution (in particular the difference in \eqref{eq:cross}) and $\pi(a,b,z,y)=(y-a,y-b)$ on $\mathbb R^{16}$. \begin{enumerate}[label=\textup{(\roman*)}] \item $\pi$ is a surjective linear map, hence a submersion; $\pi^*D\in\Dp(\mathbb R^{16})$ exists, depends continuously on $D$, and $\supp\pi^*D=\pi^{-1}(\supp D)$. \item $\pi^*D$ is independent of $z$ and invariant under simultaneous translation of $(a,b,z,y)$. \item $\WF(\pi^*D)\subseteq\{(a,b,z,y;-\xi,-\eta,0,\xi+\eta): (y-a,y-b;\xi,\eta)\in\WF(D)\}$. In particular every covector of $\Cross$ has zero $z$-component and $y$-component equal to minus the sum of its $a$- and $b$-components. \item If $D=D_1\otimes H'_2+H_1\otimes D_2$ with $D_1=H'_1-H_1$, $D_2=H'_2-H_2$ smooth (the smooth-difference clause of XII-A's admissibility \cite{XIIAhad}, \emph{if} XIV-J's blocks satisfy it), then $\WF(D)\subseteq\{\xi=0\}\cup\{\eta=0\}$ and $\WF(\pi^*D)\subseteq\{(0,-\eta,0,\eta)\}\cup\{(-\xi,0,0,\xi)\}$. \end{enumerate} \end{lemma} \begin{proof} (i) The Jacobian of $\pi$ has rank $8$. Existence, continuity, and the support statement are \cite[Thm.~8.2.4]{Horm}. (ii) $\pi$ is invariant under $z\mapsto z+c$ and under $(a,b,z,y)\mapsto(a+c,b+c,z+c,y+c)$. (iii) is the wavefront bound of \cite[Thm.~8.2.4]{Horm} with ${}^t\!d\pi(\xi,\eta)=(-\xi,-\eta,0,\xi+\eta)$. (iv) For $D_1$ smooth, $\WF(D_1\otimes H'_2)\subseteq\{(u,v;0,\eta)\}$ \cite[Thm.~8.2.9]{Horm}; similarly for the second term; insert into (iii). \end{proof} \subsection{The coordinate-response line and the candidate reduction} \label{sec:candidates} XIV-H's \texttt{eq:L} \cite{HeqL}: \begin{equation} L^{\mu,I}(s,a)=\sigma_\eps(a)\,\Amap^{\lambda\mu,I}\, \partial_{a^\lambda}R(s-a), \qquad X^\mu[h](s)=\int L^{\mu,I}(s,a)h_I(a)\dd^4a . \label{eq:L} \end{equation} XV-E transcribed from XII-A and XII-D \cite{XIIA,XIID} \begin{equation} \Lambda^{\mu,ab}(x,z) =\sigma_\eps(z)\,\mathfrak a^{\alpha\mu,ab}\, \partial^z_\alpha E_0^{\mathrm R}(x-z), \qquad G^{\mathrm R}_{\mathrm{gr}}=-\PPfrak E_0^{\mathrm R}, \label{eq:XIIA} \end{equation} with $\mathfrak a$ as in \eqref{eq:a1}. \begin{proposition}[Reduction of the candidate identifications] \label{prop:reduction} Let $I\leftrightarrow(ab)$ be a bijection of index sets. If $\Amap^{\lambda\mu,I}=\mathfrak a^{\lambda\mu,ab}$, $R=E_0^{\mathrm R}$ as distributions on $\mathbb R^4$, and the two symbols $\sigma_\eps$ denote the same function, then $L^{\mu,I}(s,a)=\Lambda^{\mu,ab}(s,a)$ for all $(s,a)$. None of the three equalities is supplied by the record. After the fifth pass each, and each further candidate, reduces to one sentence to be located: \begin{enumerate}[label=\textup{(\alph*)}] \item $R=E_0^{\mathrm R}$ holds if and only if $E_0^{\mathrm R}$ is a fundamental solution of $\partial_t^2-\Delta$ with support in a past-bounded half-space (Lemma~\ref{lem:R}(iii)--(iv)); the reported normalization $\Box E_0^{\mathrm R}=\mathrm{id}$ supplies the first half, the support sentence is missing. \item $\Amap=\mathfrak a$ holds if and only if XIV-H's $a_1$ in \eqref{eq:a1H} is XII-D's $a_1$ in \eqref{eq:a1}; then $a_1=-\PPcal$ (Lemma~\ref{lem:fiber}) and $X^\mu[h]$ in \eqref{eq:L} has the form of XII-D's $X_1^\mu$. \item $\sigma_\eps=\sigma_\eps$ requires XIV-H's definition of $\sigma_\eps(a)$ as a function on $\mathbb R^4$; XII-B displays the XII sheet's ramp $\sigma_\eps(t)=\chi(t/\eps)$ \cite{XIIB}. \item $f$: XIV-H's $f$ is a compactly supported test weight if and only if it is XII-D's $f\in C^\infty_c(\mathring M_T)$ \cite{XIIDweight}; XIV-H's definition is not reproduced. \item $\dot a_2$, $\mathsf D$: XIV-H's $\dot a_2$ is \eqref{eq:a2dot} if and only if XIV-H's $a_2$ is XII-E's \eqref{eq:a2}; the construction of $\mathsf D^{\alpha\mu,I,J}$ from $\dot a_2$ is XIV-H's and not reproduced, so even then $\mathsf D\overset{?}{=}\mathfrak d$ is a second step. \item $\EA_m$ is the kernel of Lemma~\ref{lem:advanced} if and only if XIV declares it the advanced inverse of $\Box+m^2$; no such declaration is in the record. \item $T^{\mu\nu}(\phi,\psi)$ in \eqref{eq:hess} is $\Tpol$ or $\Tsym$ of \eqref{eq:Tpol}; the choice fixes a factor two (Lemma~\ref{lem:polar}). \end{enumerate} \end{proposition} \begin{proof} Substituting the three equalities into \eqref{eq:L} gives $\sigma_\eps(a)\mathfrak a^{\lambda\mu,ab}\partial_{a^\lambda} E_0^{\mathrm R}(s-a)$, which is \eqref{eq:XIIA} with $(x,z)=(s,a)$. Items (a)--(g) cite the lemmas that reduce each to a single sentence, and the manifest records that the sentence is absent. The reading rule of \cite{record} forbids identification by notation. \end{proof} \begin{remark} Proposition~\ref{prop:reduction} is one-directional. The index placement $\Amap^{\alpha\mu}{}_L$ in \eqref{eq:K8}--\eqref{eq:K9} versus $\Amap^{\alpha\beta,I}$ in \eqref{eq:K5a} is XIV-I's; no lowering convention is supplied, and none is introduced. \end{remark} \subsection{The $K_8$ and $K_9$ kernels} \label{sec:K8K9} XIV-I's input statement \cite{Iresponse}, copied from XIV-H, is as follows. The differentiated source kernel is \begin{equation} 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} \end{equation} supported on $v=a=z$, with coefficients taken from the density-weighted source and not reproduced; the order bound is XIV-H's \cite{Hq}. The response kernel is \begin{equation} 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} \end{equation} With $\Bker=R*R$ as in \eqref{eq:RR} and $F=\EA_mf$, \begin{align} K_8^{I,J}(a,b,z) &=-\partial_{b^\mu}\Bigl[f(b)\Amap^{\alpha\mu}{}_L\Pmap^L{}_M\, \partial_{b^\alpha}\!\int\Bker(b-v)\,Q^{M,I,J}(v;a,z)\dd^4v\Bigr], \label{eq:K8}\\ K_9^{I,J}(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). \label{eq:K9} \end{align} The source states that on the separated chronological patch both kernels have regular conormal representations of upper order $3/2$ on \begin{equation} \Lambda_{\mathrm{resp}} =N^*\{(b-a)^2=0,\ a=z\}\setminus0\subset T^*W, \label{eq:Lresp} \end{equation} with zero $y$ covector. The symbols $\Pmap$, $\mathfrak b$, $g$ are not defined in the record; $q_{\alpha\beta}$ is paraphrased; $\EA_m$ is a candidate. XII-B's weighted Euler identity \cite{XIIB} shows that any density-weighted cubic source carries terms in $\partial\sigma_\eps$, consistent with XIV-H's ``including ramp derivatives'' and with the weights $w_\Gamma$ of Definition~\ref{def:Cpart}. This is a reading remark, not a datum. \subsection{Existence and supports of the internal expressions} \label{sec:supportcheck} \begin{lemma}[The source kernel exists and is carried by $v=a=z$] \label{lem:Q} For smooth $q^{L,I,J}_{\alpha\beta}$, the distribution \eqref{eq:Q} is defined on $\mathbb R^{12}_{(v,a,z)}$ by pullback under the diffeomorphism $\kappa(v,a,z)=(v,v-z,v-a)$ and $\supp Q\subseteq\{v=a=z\}$. For $\Phi\in C^\infty(\mathbb R^{12})$ whose restriction to a neighborhood of the diagonal $\{v=a=z\}$ has compact support in $v$ there, \begin{equation} \langle Q,\Phi\rangle=\sum_{\alpha,\beta}(-1)^{|\alpha|+|\beta|} \int q_{\alpha\beta}(v)\,\bigl[\partial_\xi^\alpha\partial_\eta^\beta \Phi(v,v-\eta,v-\xi)\bigr]_{\xi=\eta=0}\dd^4v \label{eq:Qpair} \end{equation} is finite and depends only on the $2$-jet of $\Phi$ along the diagonal. \end{lemma} \begin{proof} Each term is $q_{\alpha\beta}(v)$ times the pullback of $1\otimes\partial^\alpha\delta\otimes\partial^\beta\delta$ by $\kappa$, whose support is $\kappa^{-1}(\{\xi=\eta=0\})=\{v=a=z\}$; \eqref{eq:Qpair} is the definition of that pullback paired with $\Phi\circ\kappa^{-1}$, $\kappa^{-1}(v,\xi,\eta)=(v,v-\eta,v-\xi)$, Jacobian one; the jet order is $|\alpha|+|\beta|\le2$. \end{proof} \begin{theorem}[The six internal expressions exist for every completion of the undefined data] \label{thm:exist} Let $R$, $\Bker$ be as in \eqref{eq:R}--\eqref{eq:RR}. Let the arrays $\Amap^{\alpha\beta,I}$, $\Amap^{\alpha\mu}{}_L$, $\mathsf D^{\alpha\mu,I,J}$, $\Pmap^L{}_M$, $\mathfrak b_L$ be any constants, $m\in\mathbb C$, and $f,g,F,\sigma_\eps, q^{L,I,J}_{\alpha\beta}\in C^\infty(\mathbb R^4)$. Then \eqref{eq:L} defines $L\in\Dp(\mathbb R^8)$ with $X^\mu[h]\in C^\infty(\mathbb R^4)$ for $h\in\D$, and each of \eqref{eq:K5a}, \eqref{eq:K5z}, \eqref{eq:K47}, \eqref{eq:K6}, \eqref{eq:K8}, \eqref{eq:K9} defines a distribution on $\mathbb R^{12}_{(a,b,z)}$. No microlocal hypothesis and no restriction to a patch is needed for existence. \end{theorem} \begin{proof} \emph{$L$.} $(s,a)\mapsto s-a$ is a surjective linear map; the pullback of $\partial_\lambda R$ exists \cite[Thm.~8.2.4]{Horm}; multiply by the smooth $\sigma_\eps(a)\Amap$. $X^\mu[h]= \Amap^{\lambda\mu,I}(\partial_\lambda R)*(\sigma_\eps h_I)$ is a distribution convolved with a test function, hence smooth. \emph{$K_5^{[a]}$, $K_5^{[z]}$.} The bracket of \eqref{eq:K5a} is a smooth function of $(a,b,z)$ times constants times $(\partial_\alpha R)(b-a)\cdot(\partial_\beta\partial_\lambda R)(a-z)$, defined as $\Phi^*[(\partial_\alpha R)\otimes(\partial_\beta\partial_\lambda R)]$ with $\Phi(a,b,z)=(b-a,a-z)$, a surjective linear map $\mathbb R^{12}\to\mathbb R^8$ (this is XIV-H's pullback existence \cite{Hkernel}). Apply $\partial_{b^\mu}$. For \eqref{eq:K5z} use $\Phi'(a,b,z)=(b-z,z-a)$. \emph{$K_{47}$.} Each product in \eqref{eq:K47} is a smooth function times a derivative tensor of $R\otimes R$ pulled back by $\Psi(a,b,z)=(b-z,b-a)$, surjective linear. \emph{$K_6$.} $\Phi^*[(\partial_\alpha R)\otimes\delta]$ times a smooth function, then $\partial_{b^\mu}$. \emph{$K_8$, $K_9$.} For $\phi\in\D(\mathbb R^{12}_{(a,b,z)})$ set \[ \Phi^R_\phi(v,a,z)=\bigl\langle R,\ \phi(a,v+\cdot,z)\bigr\rangle, \qquad \Phi^{\Bker}_\phi(v,a,z)=\int\Bker(u)\,\phi(a,v+u,z)\dd^4u . \] Both are $C^\infty$ ($R$ paired with a smooth family of test functions with supports in a fixed compact; $\Bker\in L^\infty$ with compact $u$-support of the integrand), and their derivatives in $(a,z)$ vanish when $a\notin\mathrm{pr}_a(\supp\phi)$, so along the diagonal they are compactly supported in $v$. By Lemma~\ref{lem:Q}, $\langle\mathcal T_0,\phi\rangle:=\langle Q,\Phi^R_\phi\rangle$ and $\langle\mathcal T_{\Bker},\phi\rangle:=\langle Q,\Phi^{\Bker}_\phi\rangle$ are finite, linear, and bounded by $C\,p_N(\phi)$ for $\phi$ supported in a fixed compact; they define distributions on $\mathbb R^{12}$ agreeing with the displayed integrals when $Q$ is a test function. Then $T=\Pmap\mathcal T_0$, $K_8=-\partial_{b^\mu}[f\Amap\Pmap\partial_{b^\alpha}\mathcal T_{\Bker}]$, and $K_9=-\partial_{b^\nu}[g\,\partial_\mu F\,\Amap\,T]-m^2gF\mathfrak bT$ are distributions on $\mathbb R^{12}$. \end{proof} \begin{corollary}[Conditional consequences of the candidate readings of $f$ and $\EA_m$] \label{cor:compact} If $f\in C^\infty_c(\mathbb R^4)$ (candidate (d)), then $K_{47}$, $K_5^{[a]}$, $K_5^{[z]}$, $K_6$, $K_8$ have supports in $\{b\in\supp f\}$. If moreover $\EA_m$ is the kernel of Lemma~\ref{lem:advanced} (candidate (f)), then $F=\EA_m*f$ is smooth with $\supp F\subseteq\supp f+\Jm$, so the hypothesis $F\in C^\infty$ of Theorem~\ref{thm:exist} is satisfied, and $K_9$ has $b$-support in $\supp g\cap(\supp f+\Jm)$. \end{corollary} \begin{proof} Each listed kernel is $\partial_b$ of a bracket containing the factor $f(b)$; the rest is Lemma~\ref{lem:advanced}. \end{proof} \begin{proposition}[Supports of the supported sector] \label{prop:support} Under the hypotheses of Theorem~\ref{thm:exist}, \[ \supp K_6,\ \supp K_9\subseteq S_{\partial} :=\{(a,b,z):a=z,\ b-a\in\partial\Jp\},\qquad \supp K_8\subseteq S_{\Jp}:=\{(a,b,z):a=z,\ b-a\in\Jp\}, \] so $\supp\Ksup\subseteq S_{\Jp}\subseteq\{a=z\}\cap\{b^0\ge a^0\}$, and $S_\partial\subseteq\{(b-a)^2=0,\ a=z\}$. \end{proposition} \begin{proof} $K_6$ is a smooth multiple of $\partial_{b}$ of $\Phi^*[(\partial_\alpha R)\otimes\delta]$, supported in $\Phi^{-1}(\partial\Jp\times\{0\})=S_\partial$. For $K_9$: by \eqref{eq:Qpair}, $\langle\mathcal T_0,\phi\rangle$ is a finite sum of integrals over $v$ of derivatives $\partial_a^{\beta'} \partial_z^{\alpha'}\Phi^R_\phi(v,v,v) =\langle R(u),(\partial_a^{\beta'}\partial_z^{\alpha'}\phi)(v,v+u,v)\rangle$, which involve derivatives of $\phi$ only at points $(v,v+u,v)$ with $u\in\supp R=\partial\Jp$, i.e.\ on $S_\partial$. If $\phi$ vanishes on a neighborhood of $S_\partial$, all its derivatives vanish there and $\langle\mathcal T_0,\phi\rangle=0$. Multiplication by smooth functions and differentiation in $b$ preserve the statement. The same argument with $\Bker$, $\supp\Bker=\Jp$, gives $\supp\mathcal T_{\Bker}\subseteq S_{\Jp}$ and hence $\supp K_8\subseteq S_{\Jp}$. \end{proof} \begin{remark}[A candidate for the response incidence] The set $\{(b-a)^2=0,\ a=z\}$ in \eqref{eq:Lresp} has codimension $1+4=5$ in $W$, the reported codimension of $\Sresp$ \cite{record,Jreference}; it contains the supports of $K_6$ and $K_9$ and the singular locus of $\Bker(b-a)$; in the chart $w_+$ it is $\{q_+=0,\ r_+=0\}$. By the reading rule none of this identifies the two. We record \begin{equation} \Sresp\ \overset{?}{=}\ \{(b-a)^2=0,\ a=z\}\subset W \label{eq:Srespcand} \end{equation} as a candidate (status \textsf{C} for D0.7). Promotion requires the defining equation of $\Sresp$ in XIV-J. \end{remark} \begin{remark}[A check that cannot be performed] \label{rem:conormal} With the factor $\delta(a-z)$ and its derivatives present in \eqref{eq:Q}, $K_8$ is singular along all of $S_{\Jp}$, including points with $(b-a)^2>0$, where $\Bker(b-a)$ is locally constant. A distribution conormal to $S=\{(b-a)^2=0,a=z\}$ in the strict sense is smooth off $S$. The reported ``regular conormal representation of upper order $3/2$ on $\Lambda_{\mathrm{resp}}$ on the separated chronological patch'' \cite{Iresponse} is therefore compatible with \eqref{eq:K8} only under a reading of ``separated chronological patch'' or of ``regular conormal representation'' that the record does not reproduce. No inconsistency is asserted and none is excluded; the item is in the successor requirement. \end{remark} \subsection{The recovered correction class and collision sets} \label{sec:class} \begin{definition}[Collision sets; transcribed] \label{def:collision} For a connected vertex subset $I$ containing the interaction vertex $y$ with $|I|\ge2$, \[ \Delta_I=\{x_a=x_b\ \text{for every }a,b\in I\}, \] including intersections and loci after contracting such subsets. Diagonals involving only a pre-existing observable subgraph retain their inherited normalization. The admissible $I$ are the connected collision subsets of XIII-C's cycle-vertex table \cite{XIIICvertex}. \end{definition} \begin{definition}[Partial-diagonal ambiguity class; transcribed] \label{def:Cpart} $\Cpart$ consists of finite sums of graphwise contact changes at order $\hbar^2$ in the referenced time-ordered product, 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. The bound six is a conservative frozen bound. $\mathscr C_{\mathrm{tot}}\subseteq\Cpart$ denotes the subclass supported on the total graph diagonal. \end{definition} \begin{proposition}[What the recovered class does and does not supply] \label{prop:classstatus} \begin{enumerate}[label=\textup{(\roman*)}] \item $\Cpart$ is a class of coefficient \emph{values} (distributions), each supported in the closed set $C_{\mathrm{coll}}:=\overline{\bigcup_I\Delta_I}\cap \mathcal K_{\mathrm{gr}}$, where $\mathcal K_{\mathrm{gr}}$ is the compact graph closure named in Definition~\ref{def:Cpart}. \item XV-B's correction class is a vector space $\mathcal A\subseteq\Hom(\Pint,Y)$ of \emph{maps}. The record does not state which presentation-level maps are permitted; in particular it does not state that $\mathcal A=\{a\in\Hom(\Pint,Y):a(\Pint)\subseteq\Cpart\}$, nor any continuity, tensor, or covariance restriction on $a$. \item The collision sets are defined in XIII-C's graph variables; their lift to $W$ with coordinates $(a,b,z,y)$, and the inclusion $C_{\mathrm{coll}}\subseteq W\setminus\Ucan$ needed for $\Cpart\subseteq Y=\ker r_O$, are not supplied; the three refusals of \S\ref{sec:fifthpass} show the inclusion cannot be read off from $\Natt$. \item The parametrization of $\Cpart$ ($D_\Gamma$, the admissible $w_\Gamma$, $S_{\Gamma/I}$) consists of roles specified in XIII-C, not formulas. Exhaustiveness of any affine family built from $\Cpart$ cannot be checked. \end{enumerate} Hence D0.17 is \textsf{T} for the definition and \textsf{R} for the parametrization, and D0.17 does not pass to status \textsf{S}. \end{proposition} \begin{proof} (i) is the support clause of Definition~\ref{def:Cpart}. (ii)--(iv) are statements about the record: the reproduced passages \cite{XIIICcoll,XIIICpart,XIIICvertex} contain the two definitions above, the vertex table, and nothing further; the fifth pass confirms that $D_\Gamma$, $w_\Gamma$, $S_{\Gamma/I}$ are roles; $\Ucan$ is in status \textsf{P} (D0.8). \end{proof} \subsection{The Hadamard matrix and the blocks of the cross-line} \label{sec:hadamard} XII-A's frozen Hadamard data \cite{XIIAhad} are $h_\mu=-W_\mu$ with \[ W_\mu(x,y)=\int_{\mathbb R^3} \frac{e^{-i\omega_\mu(\mathbf p)(x^0-y^0)+i\mathbf p\cdot(\mathbf x-\mathbf y)}} {(2\pi)^3\,2\omega_\mu(\mathbf p)}\dd^3\mathbf p, \qquad \omega_\mu(\mathbf p)=\sqrt{|\mathbf p|^2+\mu^2}, \] $h_\mu-h_\mu^{\mathsf T}=iE_\mu$, blocks \begin{equation} H_{\gamma\gamma}=-\PPcal h_0,\quad H_{\phi\phi}=-h_m,\quad H_{\gamma b}=D_yH_{\gamma\gamma},\quad H_{b\gamma}=D_xH_{\gamma\gamma},\quad H_{bb}=D_xD_yH_{\gamma\gamma},\quad H_{c\bar c}=-ih_0,\quad H_{\bar cc}=ih_0, \label{eq:Hblocks} \end{equation} all other blocks zero; for the frozen translation-invariant data $H_{bb}=0$. An admissible replacement $H'$ is defined by the four clauses of \S\ref{sec:fifthpass}. The fourth pass shows that $\Cross$ is built from blocks named $H_{\gamma\gamma}$, $H_{\phi\phi}$ and replaced blocks \eqref{eq:cross}; the fifth pass shows that the replaced blocks are not written $\widehat H$ in any manuscript and that no manuscript equates them with XII-A's $H'$. Whether XIV-J's replaced blocks are translation invariant (as the difference-variable pullback requires) is also not stated. The Hadamard data are \textsf{T} for XII-A's own objects and \textsf{C} as ingredients of \eqref{eq:cross}. \subsection{Consistency checks} \begin{proposition}[Checks performed and not performable] \label{prop:checks} The following mutual-consistency checks among reproduced items pass: \textup{(a)} the reconstruction rule against XIV-J's two values (Lemma~\ref{lem:recrule}(i)) and, under $q$-linearity, against its null relation (Lemma~\ref{lem:recrule}(ii)); \textup{(b)} \eqref{eq:R} against the pole chart (Lemma~\ref{lem:R}); \textup{(c)} the display \eqref{eq:RR} (Lemma~\ref{lem:RR}); \textup{(d)} the adapted chart against $\dim W=16$ and the pullback domain of \eqref{eq:cross} (Lemmas~\ref{lem:chart}, \ref{lem:cross}); \textup{(e)} the supports of $K_6$, $K_9$ and the singular locus of $\Bker$ against the base of $\Lambda_{\mathrm{resp}}$ (Proposition~\ref{prop:support}); \textup{(f)} XII-B's trace reversal against XII-A's $\PPcal^2=\mathrm{id}$ and XII-D's $a_1=-\PPcal$ (Lemma~\ref{lem:fiber}); \textup{(g)} the form of both $K_5$ placements and of $K_6$ against the reconstruction rule in the adapted charts (Proposition~\ref{prop:K5chart}); \textup{(h)} the displayed Hessian against the polarization of $T_0$, which passes for exactly one of the two normalizations (Lemma~\ref{lem:polar}). The following cannot be performed from the record: \textup{(i)} the conormal statement for $K_8$ against its singular support (Remark~\ref{rem:conormal}); \textup{(j)} XIV-J's blocks against XII-A's \eqref{eq:Hblocks}; \textup{(k)} the candidate identifications (a)--(g) of Proposition~\ref{prop:reduction}; \textup{(l)} freeness of $\Kint$, which depends on the unreproduced coefficient class (Lemma~\ref{lem:recrule}(iii)--(iv)). \end{proposition} \begin{proof} Each item cites the result that establishes it or the passage whose absence prevents it. \end{proof} \subsection{F0 verdict} \begin{proposition}[F0 fails] \label{prop:F0} The core package is incomplete. Not reproduced in the record: the generators and coefficient class of $\Amod$ and the module action (D0.1--D0.4); the topology of $\Kint$ (D0.3); $E_{\mathrm{res}}$ (D0.5); a set equation for $O=\Ucan$, refused at three places (D0.8); the display of $m^\circ$ (D0.10); the identity of the blocks in $\Cross$ (D0.11); the normalization of $T^{\mu\nu}(\phi,\psi)$ and the remaining weights (D0.12); $\Amap$ beyond its role, $\sigma_\eps$, $f$, $\dot a_2$, $\mathsf D$ (D0.13); the $q^{L,I,J}_{\alpha\beta}$ beyond their order bound, $\Pmap$, $\mathfrak b$, $g$, $\EA_m$ (D0.14); the display of the candidate class (D0.16); and the parametrization and presentation-level form of the correction class (D0.17). Hence no preliminary lift $\Fz:\Pint\to\Dp(W;E_{\mathrm{res}})$ with $r_O\Fz=m^\circ\Rec$ can be stated, and no space $\mathcal A\subseteq\Hom(\Pint,Y)$ is available against which a null defect could be tested. \end{proposition} \begin{proof} A linear map requires a domain, a codomain, and a value on each element of the domain. The domain is known through its chart and its reconstruction rule but not its generators or coefficient class (D0.1, \textsf{T}/\textsf{P}); the codomain is undefined (D0.5). The value on a presentation of any of the six entries is its composition with $\Cross$, the vertex Hessian, and weights according to $m^\circ$ (D0.10, \textsf{P}); the internal kernels exist for every completion of the undefined data (Theorem~\ref{thm:exist}) but $\Amap$, $f$, $\sigma_\eps$, $\mathsf D$ and the ingredients of $K_8$, $K_9$ are not fixed, so no particular distribution is singled out. Proposition~\ref{prop:classstatus} shows that $\Cpart$ does not yield $\mathcal A$. Each cited item is in status \textsf{P}, \textsf{R}, \textsf{U}, \textsf{C}, or \Tst\ in the manifest. \end{proof} \[ \boxed{\text{F0: core package incomplete; one identification, three refusals, four candidate displays; no lift for the actual graph is definable.}} \] Gates F1--F4 are not entered for the actual graph. \section{Frozen ledger} \label{sec:branches} 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} with six operational entries $K_{47}$, $K_5^{[a]}$, $K_5^{[z]}$, $K_6$, $K_8$, $K_9$; $K_{47}$ groups $B_4+B_7$ and both $K_5$ placements belong to $B_5$. The supplied internal expressions 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) \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)\mathsf D^{\alpha\mu,I,J} \partial_{a^\alpha}R(b-a)\delta^{(4)}(a-z)\Bigr], \label{eq:K6} \end{align} together with \eqref{eq:K8}--\eqref{eq:K9}, the line $L$ of \eqref{eq:L}, $R$, $\Bker$ of \eqref{eq:R}--\eqref{eq:RR}, and the chart \eqref{eq:chart}. 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)},\qquad \rho>0,\ |\omega-s\rho|0$ with the closed $\varrho$-neighborhood $K'$ of $K$ contained in $X$, and $|\langle T,\psi\rangle|\le C_Tp_k(\psi)$ for all $\psi\in\D(X)$ with $\supp\psi\subset K'$. If $g\in\Ann(T^\circ)$ has vanishing $k$-jet at every point of $Z$, then $\langle gT,\phi\rangle=0$ for every $\phi\in\D(X)$ with $\supp\phi\subset K$. \item \textup{\cite{XVE}, Theorem \texttt{thm:point}} If $Z$ is finite and $T^\circ$ has an extension, it has an extension $T$ with $\Ann(T)=\Ann(T^\circ)$. \item \textup{\cite{XVD}, Proposition \texttt{prop:example}} Let $d=2$, $X=\mathbb R^2$, $Z=\{0\}$, $m=1$, $r=0$, $T^\circ=\delta(x_1)|_{X^\circ}$, $T_1=\delta(x_1)$, $T_2=\delta(x_1)+\partial_1\delta_0$, $a(x,q)=x_1$. Then $\Ann(T^\circ)=\{g:g(0,x_2)=0\ \forall x_2\}$, $a\in H$, $d_0^{T_1}(a)=0$, $d_0^{T_2}(a)=-\Ejet_0\delta_0\ne0$. \end{enumerate} \end{proposition} \subsection{Disjoint classification and the exact role of jet-detection} \begin{definition}[Three disjoint kinds of null relation] \label{def:kinds} Let $a\in H$. Call $a$ \emph{universal} if $\beta_\gamma(a)=0$ for all $|\gamma|\le r$; a \emph{flat-annihilator relation} if $a$ is not universal and every $\beta_\gamma(a)$ is flat on $Z$; \emph{jet-detected} if some $\beta_\gamma(a)$ is not flat at some point of $Z$. The three classes are disjoint and exhaust $H$. \end{definition} \begin{lemma}[Multiplying a delta derivative by a function] \label{lem:deltamult} Let $z_0\in X$, $g\in C^\infty(X)$, and let $\mu$ be of minimal length among the multi-indices with $\partial^\mu g(z_0)\ne0$. Then $g\,\partial^\mu\delta_{z_0}=(-1)^{|\mu|}\partial^\mu g(z_0)\,\delta_{z_0}$. \end{lemma} \begin{proof} For $\psi\in\D(X)$, $\langle g\partial^\mu\delta_{z_0},\psi\rangle =(-1)^{|\mu|}\partial^\mu(g\psi)(z_0) =(-1)^{|\mu|}\sum_{\nu\le\mu}\binom{\mu}{\nu}\partial^\nu g(z_0) \partial^{\mu-\nu}\psi(z_0)$. For $\nu\le\mu$, $\nu\ne\mu$, $|\nu|<|\mu|$ and $\partial^\nu g(z_0)=0$ by minimality. \end{proof} \begin{theorem}[Jet-detection is exactly ``some extension carries a defect''] \label{thm:jetchar} Suppose $T^\circ$ has an extension. For $a\in H$ the following are equivalent: \textup{(i)} $a$ is jet-detected; \textup{(ii)} there is an extension $T$ with $d_0^T(a)\ne0$. For a fixed extension a jet-detected relation may have zero defect: in \textup{(R5)}, $a=x_1$ is jet-detected, $d_0^{T_1}(a)=0$, $d_0^{T_2}(a)\ne0$. If $Z$ is finite, every jet-detected $a$ admits extensions $T'$, $T''$ with $d_0^{T'}(a)=0$ and $d_0^{T''}(a)\ne0$. \end{theorem} \begin{proof} (ii)$\Rightarrow$(i). If $a$ is not jet-detected, every $\beta_\gamma(a)$ is flat on $Z$ and lies in $\Ann(T^\circ)$ by (R2). Given $T$ and a compact $K$, choose $\varrho$, $K'$ as in (R3) and let $k$ be an order of $T$ on $K'$; (R3) gives $\langle\beta_\gamma(a)T,\phi\rangle=0$ for $\supp\phi\subset K$. As $K$ is arbitrary, $\beta_\gamma(a)T=0$ for all $\gamma$, so $d_0^T(a)=0$ by (R1). (i)$\Rightarrow$(ii). Let $\tilde T$ extend $T^\circ$; choose $\gamma$ and $z_0\in Z$ with $g=\beta_\gamma(a)$ not flat at $z_0$, and $\mu$ minimal with $\partial^\mu g(z_0)\ne0$. For $c\in\mathbb C$ set $T_c=\tilde T+c\,\partial^\mu\delta_{z_0}$, an extension of $T^\circ$ since $\supp\partial^\mu\delta_{z_0}\subset Z$. By Lemma~\ref{lem:deltamult}, $gT_c=g\tilde T+c(-1)^{|\mu|}\partial^\mu g(z_0)\delta_{z_0}$, an affine function of $c$ with nonzero direction; it vanishes for at most one $c$. For any other $c$, $\beta_\gamma(a)T_c\ne0$, hence $d_0^{T_c}(a)\ne0$ by (R1). The example is (R5); the last sentence is (R4) together with the construction of $T_c$. \end{proof} \subsection{Descending extensions form an affine space} \begin{definition} \label{def:SZ} $\SZ(T^\circ)=\{s\in\Dp(X):\supp s\subseteq Z,\ \langle s,h\rangle=0\ \forall h\in\Ann(T^\circ)\cap\D(X)\}$. \end{definition} \begin{lemma}[Null defect of an extension change] \label{lem:recs} Let $s\in\Dp(X)$ with $\supp s\subseteq Z$. Then $\Rec_s|_H=0$ iff $s\in\SZ(T^\circ)$. \end{lemma} \begin{proof} By (R1), $\Rec_s(a)=0$ iff $\beta_\gamma(a)s=0$ for all $\gamma$. As $a$ ranges over $H$, $\beta(a)$ ranges over all of $\Ann(T^\circ)^{M_r}$ by (R2). So $\Rec_s|_H=0$ iff $gs=0$ for all $g\in\Ann(T^\circ)$, i.e. iff $\langle s,g\phi\rangle=0$ for all $g\in\Ann(T^\circ)$, $\phi\in\D(X)$. The functions $g\phi$ are exactly the elements of $\Ann(T^\circ)\cap\D(X)$: each $g\phi$ lies there since $\Ann(T^\circ)$ is an ideal, and each $h\in\Ann(T^\circ)\cap\D(X)$ equals $h\phi$ with $\phi\in\D(X)$, $\phi=1$ on $\supp h$. \end{proof} \begin{theorem}[Descending extensions] \label{thm:affine} Let $T$ extend $T^\circ$ with $\Ann(T)=\Ann(T^\circ)$. An extension $T'$ satisfies $\Ann(T')=\Ann(T^\circ)$ iff $T'-T\in\SZ(T^\circ)$. Hence the descending extensions form the affine space $T+\SZ(T^\circ)$, the descending extension lifts are $\Fz^T+\{\Rec_s:s\in\SZ(T^\circ)\}$, and for $\mathcal A_{\mathrm{ext}}=\{\Rec_s:\supp s\subseteq Z\}$ the space $\mathcal A_{\mathrm{ext},0}=\{a\in\mathcal A_{\mathrm{ext}}:a|_H=0\}$ of XV-B's Theorem \texttt{thm:lifting}(iii) is $\{\Rec_s:s\in\SZ(T^\circ)\}$. \end{theorem} \begin{proof} $T'$ extends $T^\circ$ iff $s=T'-T$ is supported in $Z$. By (R2), $\Ann(T')=\Ann(T^\circ)$ iff $\Rec_{T'}|_H=0$; since $\Rec_{T'}=\Rec_T+\Rec_s$ and $\Rec_T|_H=0$, this is $\Rec_s|_H=0$, i.e. $s\in\SZ(T^\circ)$ by Lemma~\ref{lem:recs}. \end{proof} \begin{proposition}[The remaining freedom in XV-D's example] \label{prop:exampleSZ} In \textup{(R5)}, $\SZ(T^\circ)=\operatorname{span}\{\partial_2^{\,j}\delta_0:j\ge0\}$, which is infinite-dimensional, and the descending extensions are exactly $\delta(x_1)+\sum_{j\le k}c_j\partial_2^{\,j}\delta_0$, $k\ge0$, $c_j\in\mathbb C$. \end{proposition} \begin{proof} An $s$ supported at $0$ has finite order $k$ on a compact neighborhood of $0$, so by Theorem~\ref{thm:dimension}(i) below $s=\sum_{|\mu|\le k}c_\mu(-1)^{|\mu|}\partial^\mu\delta_0$ and $\langle s,h\rangle=\sum_\mu c_\mu\partial^\mu h(0)$. By (R5), $h\in\Ann(T^\circ)\cap\D(X)$ iff $h$ is compactly supported with $h(0,x_2)=0$ for all $x_2$. For such $h$, $\partial^\mu h(0)=0$ whenever $\mu_1=0$. For $\mu_1\ge1$, $h_\mu=x^\mu\chi$ ($\chi\in\D(X)$, $\chi=1$ near $0$) lies in $\Ann(T^\circ)\cap\D(X)$ and $\partial^\nu h_\mu(0)=\mu!\,\delta_{\mu\nu}$ for all $\nu$. Testing on all $h_\mu$ with $\mu_1\ge1$ gives $c_\mu=0$ for those $\mu$; conversely if these vanish, $\langle s,h\rangle=0$ for all admissible $h$. The $\partial_2^{\,j}\delta_0$ are independent (test on $x_2^{\,j}\chi$). Since $T_1$ descends by (R5), Theorem~\ref{thm:affine} gives the last assertion. \end{proof} \subsection{A normal-order bound does not make the ambiguity finite} Let $N_k=\binom{k+d}{d}=\#\{\mu\in\mathbb N^d:|\mu|\le k\}$ and \begin{equation} \Vjet_Z=\Bigl\{\sum_{|\gamma|\le r}\Ejet_\gamma d_\gamma:\ d_\gamma\in\Dp(X),\ \supp d_\gamma\subseteq Z\Bigr\}, \label{eq:Vjet} \end{equation} XV-C's ambiguity space \texttt{eq:ambiguity} \cite{XVC} for $S=X\times\{0\}$, $A=Z\times V_q$. \begin{theorem}[Structure and dimension of the puncture-supported ambiguity] \label{thm:dimension} \begin{enumerate}[label=\textup{(\roman*)}] \item Let $z_0\in X$ and $s\in\Dp(X)$ with $\supp s\subseteq\{z_0\}$ satisfy $|\langle s,\psi\rangle|\le Cp_k(\psi)$ for all $\psi\in\D(X)$ supported in a compact neighborhood $K'_0$ of $z_0$. Then there are unique $c_\mu\in\mathbb C$, $|\mu|\le k$, with $s=\sum_{|\mu|\le k}c_\mu(-1)^{|\mu|}\partial^\mu\delta_{z_0}$, i.e. $\langle s,\psi\rangle=\sum_{|\mu|\le k}c_\mu\partial^\mu\psi(z_0)$. \item The $\Ejet_\gamma\partial^\mu\delta_{z_0}$, $|\gamma|\le r$, $\mu\in\mathbb N^d$, are linearly independent in $\Dp(U)$. \item If $Z\ne\emptyset$, $\Vjet_Z$ is infinite-dimensional. \item If $Z=\{z_0\}$, the subspace $\Vjet_{z_0,k}$ of elements all of whose coefficients have order at most $k$ near $z_0$ has basis $\{\Ejet_\gamma\partial^\mu\delta_{z_0}:|\gamma|\le r,|\mu|\le k\}$ and dimension $M_rN_k$. \item $s\mapsto\Rec_s$ is injective on $\{s:\supp s\subseteq Z\}$; hence $\mathcal A_{\mathrm{ext}}$ is infinite-dimensional whenever $Z\ne\emptyset$, and for $Z=\{z_0\}$ its order-$\le k$ subspace has dimension $N_k$. \end{enumerate} \end{theorem} \begin{proof} (i) Let $Z_0=\{z_0\}$ and $T^\circ_0=0\in\Dp(X\setminus Z_0)$; $s$ extends $T^\circ_0$ and $\Ann(T^\circ_0)=C^\infty(X)$. Choose a closed ball $K$ centered at $z_0$ and $\varrho>0$ with closed $\varrho$-neighborhood $K'\subset K'_0$. Let $\psi\in\D(X)$ have vanishing $k$-jet at $z_0$ and $\chi\in\D(\operatorname{int}K)$ with $\chi=1$ near $z_0$. Then $\langle s,(1-\chi)\psi\rangle=0$ since $(1-\chi)\psi$ vanishes near $z_0$, and (R3) applied to $T=s$, $g=\psi$, $\phi=\chi$ gives $\langle s,\chi\psi\rangle=\langle\psi s,\chi\rangle=0$. So $\langle s,\psi\rangle=0$: the functional factors through the $k$-jet map $j^k_{z_0}:\D(X)\to\mathbb C^{N_k}$, which is surjective (use $(x-z_0)^\mu\chi$). A linear form on $\mathbb C^{N_k}$ is $j\mapsto\sum c_\mu j_\mu$; uniqueness follows from surjectivity. (This is the classical structure theorem \cite[Thm.~2.3.4]{Horm}, rederived from (R3) to keep the order bookkeeping explicit.) (ii) If $\sum c_{\gamma\mu}\Ejet_\gamma\partial^\mu\delta_{z_0}=0$ (finite sum), pair with $\Jjet_{\gamma'}\psi$ using \eqref{eq:extract}: $\sum_\mu c_{\gamma'\mu}(-1)^{|\mu|}\partial^\mu\psi(z_0)=0$ for all $\psi$; with $\psi=(x-z_0)^\nu\chi$ this is $c_{\gamma'\nu}(-1)^{|\nu|}\nu!=0$. (iii) Pick $z_0\in Z$; the $\Ejet_0\partial^\mu\delta_{z_0}$ lie in $\Vjet_Z$ and are independent by (ii). (iv) By (i) the displayed family spans; by (ii) it is independent; its cardinality is $M_rN_k$. (v) For the presentation $e_0$ with $(e_0)_0=1$ and all other components zero, $\Rec_s(e_0)=\Ejet_0s$, which vanishes iff $s=0$ by (R1). The dimension statements follow from (i)--(iii). \end{proof} \begin{corollary}[What finiteness of the ambiguity requires] \label{cor:finite} In the jet module, the unrestricted supported ambiguity of normal order at most $r$ is finite-dimensional only if $Z=\emptyset$. A normal-order bound alone never makes it finite. It becomes finite-dimensional, of dimension $M_r\binom{k+d}{d}$ per puncture point, exactly when the puncture is finite \emph{and} a tangential order bound $k$ is imposed. The same holds for $\mathcal A_{\mathrm{ext}}$ with $M_r$ replaced by $1$, and for $\SZ(T^\circ)$ in Proposition~\ref{prop:exampleSZ}, which has dimension $k+1$ under the bound $k$ and is infinite-dimensional without it. \end{corollary} \begin{proof} Theorem~\ref{thm:dimension}(iii)--(v) and Proposition~\ref{prop:exampleSZ}. \end{proof} \subsection{The same dichotomy for the recovered class} \begin{proposition}[Finiteness of $\Cpart$ is a question about the weights] \label{prop:Cpartdim} For each pair $(\Gamma,I)$ in Definition~\ref{def:Cpart} let $\mathcal W_\Gamma$ be the vector space of admissible weights $w_\Gamma$, $c_I$ the codimension of $\Delta_I$, and suppose $D_\Gamma$, $S_{\Gamma/I}$ fixed. If finitely many $(\Gamma,I)$ occur and each $\mathcal W_\Gamma$ is finite-dimensional, then $\Cpart$ is finite-dimensional with \[ \dim\Cpart\ \le\ \sum_{(\Gamma,I)}\binom{6+c_I}{c_I}\dim\mathcal W_\Gamma . \] If some $\mathcal W_\Gamma$ is infinite-dimensional, the bound $|\alpha|\le6$ gives no finite-dimensionality by itself; whether $\Cpart$ is finite-dimensional then depends on the kernel of $w\mapsto D_\Gamma[w\,\partial_r^\alpha\delta(r_I)S_{\Gamma/I}]$, which is not determined by the record. \end{proposition} \begin{proof} The map $(w,\alpha)\mapsto D_\Gamma[w\,\partial_r^\alpha\delta(r_I) S_{\Gamma/I}]$ is linear in $w$ for fixed $\alpha$, and $\Cpart$ is the span of its images over the finitely many $(\Gamma,I)$ and the $\binom{6+c_I}{c_I}$ multi-indices $|\alpha|\le6$ in $c_I$ transverse variables. A finite sum of finite-dimensional images is finite-dimensional with the stated bound. The second sentence is the observation that the normal-order bound restricts only $\alpha$; compare Corollary~\ref{cor:finite}. \end{proof} \begin{remark} The fifth pass confirms that XIII-C's ``built from'' $g,\partial g,\sigma_\eps,\sigma_\eps',\sigma_\eps''$ does not say whether the weight space is polynomial or arbitrary smooth. The record's caution that ``a bound on normal derivative order does not by itself make the ambiguity finite-dimensional'' \cite{record} is thus a theorem in the jet module (Corollary~\ref{cor:finite}) and an open reading question for $\Cpart$. \end{remark} \section{Gates F1--F4 for the actual graph} \label{sec:rest} F1: no $\Fz$ is definable (Proposition~\ref{prop:F0}); the composition rule $m^\circ$ is paraphrased only. F2: $\ker\Rec$ depends on the coefficient class and module action (Lemma~\ref{lem:recrule}); one relation is known (D0.4); $d_0$ is not an evaluable object; the form of the test is Proposition~\ref{prop:conditional}. F3: no actual $v_\alpha$, $c_{D,\beta}$ (D0.18); XV-E's Proposition \texttt{prop:erratum} is the first estimate to check; the characteristic compositions of $K_{47}$ and both $K_5$ placements require a composition theorem not in the record; the covector pattern of $\Cross$ (Lemma~\ref{lem:cross}) and the chart forms of Proposition~\ref{prop:K5chart} are the only microlocal data in the record. F4: the equations of $W$, $N$, $\Sresp$ (candidate only), and the lift of the collision strata are not in the record; no complete-graph bound, output operation beyond \eqref{eq:Rconv}, or admission \eqref{eq:admission} is addressed. \section{Verdict and release} \label{sec:verdict} \begin{theorem}[XV-F local mathematical release] \label{thm:release} \begin{enumerate}[label=\textup{(\roman*)}] \item \textup{(Source.)} The fourth and fifth passes reproduce the reconstruction rule \eqref{eq:Recrule}, the adapted chart \eqref{eq:chart}, the kernel $R$ and its square \eqref{eq:R}--\eqref{eq:RR}, the cross-line definition \eqref{eq:cross} (one symbol disputed), the mixed vertex Hessian \eqref{eq:hess}, the scalar stress \eqref{eq:T0}, the output convention \eqref{eq:Rconv}, the role of $\Amap$ \eqref{eq:a1H}, and three refusals of $\Ucan=W\setminus\overline{\Natt}$; they paraphrase $m^\circ$, the joint-map class, the generators and coefficient class of $\Amod$, and the order bound on $q_{\alpha\beta}$; and they return candidate displays \eqref{eq:a1}, \eqref{eq:a2}, $f\in C^\infty_c$. Verified: $R$ is the unique past-bounded fundamental solution of $\partial_t^2-\Delta$ (Lemma~\ref{lem:R}); the $R*R$ display (Lemma~\ref{lem:RR}); the chart is a global chart of $\{d_\pm^0>0\}$ of dimension sixteen (Lemma~\ref{lem:chart}); both $K_5$ placements and $K_6$ have the form of the reconstruction rule in the charts (Proposition~\ref{prop:K5chart}); the reconstruction family is free over $(q,r)$-independent coefficients and dependent over chart functions, XIV-J's null relation being an instance (Lemma~\ref{lem:recrule}); $\mathfrak a=-\PPcal$ (Lemma~\ref{lem:fiber}); $\dot a_2$ (Lemma~\ref{lem:a2}); the polarization factor is two-valued (Lemma~\ref{lem:polar}); the advanced kernel exists uniquely for all $m\ge0$ (Lemma~\ref{lem:advanced}); all six internal expressions exist for every completion (Theorem~\ref{thm:exist}), with supports as in Proposition~\ref{prop:support}; the candidates reduce to seven located sentences (Proposition~\ref{prop:reduction}); $\Cpart$ is a contact-supported class of values whose presentation-level form and parametrization are not supplied (Proposition~\ref{prop:classstatus}). One check cannot be performed (Remark~\ref{rem:conormal}). \item \textup{(Conditional.)} Under hypotheses (H1)--(H3), any lift must send each null relation into $\Cpart$, and a defect on an open set off the collision closure or a separating functional is an obstruction (Proposition~\ref{prop:conditional}). \item \textup{(Jet module.)} Null relations are, disjointly, universal, flat-annihilator, or jet-detected; for extendable $T^\circ$, jet-detected means that some extension carries a nonzero defect (Theorem~\ref{thm:jetchar}); descending extensions form the affine space $T+\SZ(T^\circ)$, infinite-dimensional in XV-D's example (Theorem~\ref{thm:affine}, Proposition~\ref{prop:exampleSZ}); puncture-supported distributions of finite order are finite delta sums, and the puncture-supported ambiguity of bounded normal order is infinite-dimensional unless the puncture is empty, with dimension $M_r\binom{k+d}{d}$ per point under a tangential bound $k$ (Theorem~\ref{thm:dimension}, Corollary~\ref{cor:finite}); the same dichotomy governs $\Cpart$ through its weight space (Proposition~\ref{prop:Cpartdim}). \end{enumerate} In addition, Section~\ref{sec:errata} records errata (E1)--(E2) to XV-E and corrections (F1)--(F3) to the circulated draft. \end{theorem} \begin{proof} Each item cites the result that proves it. \end{proof} \subsection{Full-graph verdict} \begin{equation} \boxed{ \begin{gathered} \text{Unresolved at F0: core source package incomplete.}\\ \text{Pass 5: one identification (adapted chart), three refusals ($\Ucan$), four candidate displays ($\mathfrak a$, $f$, $a_2$, $T_0$).}\\ \text{Still unreproduced: generators/coefficient class of }\Amod;\ E_{\mathrm{res}};\ m^\circ;\ \text{blocks of }\Cross;\ T^{\mu\nu}\text{ normalization};\\ \Amap,\ \sigma_\eps,\ f,\ \dot a_2,\ \mathsf D;\ q_{\alpha\beta},\ \Pmap,\ \mathfrak b,\ g,\ \EA_m;\ \text{candidate class};\ \text{presentation-level }\mathcal A;\ \text{lift of }\Delta_I;\ \text{seven candidates.}\\ \text{Gates F1--F4 not entered for the actual graph.} \end{gathered} } \label{eq:verdict} \end{equation} This is justified by Propositions~\ref{prop:reduction}, \ref{prop:classstatus}, \ref{prop:checks}, and \ref{prop:F0}. It continues the J2, B1, C0, D0, and E0 stops \cite{Jverdict,XVB,XVC,XVD,XVE}. It is neither an obstruction nor a construction. The admission \eqref{eq:admission} is unasserted; ``not admitted'' is not ``nonexistent'' \cite{Jadmission}. \subsection{Acceptance ledger} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.40\linewidth}X@{}} \toprule Acceptance item & XV-F verdict\\ \midrule Actual $\Pint$, $\Rec$, $\Kint$, $O$ defined & Chart and reconstruction rule: yes; generators, coefficient class, module action, topology, $O$: no ($O$ refused as an equation). F0 fails.\\[0.2em] All six operational entries with frozen coefficients and signs & All six formulas reproduced and shown to exist for every completion (Theorem~\ref{thm:exist}); three of them of reconstruction form in the charts (Proposition~\ref{prop:K5chart}); $\Amap$, $f$, $\sigma_\eps$, $\mathsf D$, $q_{\alpha\beta}$, $\Pmap$, $\mathfrak b$, $g$, $\EA_m$ not fixed.\\[0.2em] Lift defined on claimed domain & No (Proposition~\ref{prop:F0}).\\[0.2em] Canonical agreement proved & Not applicable.\\[0.2em] Full reconstruction kernel handled & Graph: depends on the coefficient class (Lemma~\ref{lem:recrule}); not evaluable. Jet module: complete (Theorem~\ref{thm:jetchar}).\\[0.2em] Defect computed and cancellation in the actual class & Graph: no; class definition recovered, form of the test fixed (Proposition~\ref{prop:conditional}). Jet module: descending extensions classified (Theorem~\ref{thm:affine}).\\[0.2em] Estimates for actual coefficients and full graph & Not entered.\\[0.2em] Admission after output specification & Convention \eqref{eq:Rconv} recorded; admission unasserted.\\ \bottomrule \end{tabularx} \end{center} \section{Debugging record} \label{sec:debug} \begin{enumerate}[label=\textup{D\arabic*.},leftmargin=*] \item \textbf{Displays, paraphrases, locators, refusals.} Each pass is graded line by line; only \textsf{T} and \textsf{S} can pass F0; a refusal closes a search and does not supply a definition. \item \textbf{Verification of displays.} Every reproduced display that admits an internal check was checked (Proposition~\ref{prop:checks}); the checks that cannot be performed are listed, not suppressed. \item \textbf{Corrections to the draft.} (F1)--(F3) are recorded in \S\ref{sec:errata}; the free-module statement is replaced by the two-case Lemma~\ref{lem:recrule}; the dependent instance is XIV-J's own null relation, which is the strongest internal evidence that the coefficient class contains chart functions. \item \textbf{Existence is not regularity.} Theorem~\ref{thm:exist} asserts existence as distributions for arbitrary completions; it asserts nothing about conormal regularity, compositions with $\Cross$ or the vertex data, or estimates. \item \textbf{Module form is not membership.} Proposition~\ref{prop:K5chart} shows that three internal expressions have the \emph{form} of \eqref{eq:Recrule}; whether they are values of $\Rec$ on elements of $\Amod$ requires the generators. \item \textbf{Candidates versus definitions.} \eqref{eq:a1H} defines $\Amap$ through XIV-H's $a_1$ and does not define $a_1$; \eqref{eq:a1}, \eqref{eq:a2}, $f\in C^\infty_c$, \eqref{eq:T0} are XII-sheet displays; no XIV sentence identifies them. The reduction in Proposition~\ref{prop:reduction} is one-directional. \item \textbf{Two normalizations.} Lemma~\ref{lem:polar} shows that \eqref{eq:hess} is consistent with the reported vertex for exactly one of the two natural polarizations; the factor two is an open item, not an error attributed to any text. \item \textbf{Advanced kernel.} Lemma~\ref{lem:advanced} is self-contained (Paley--Wiener--Schwartz); its identification with XIV's $\EA_m$ is conditional on a declaration not in the record. \item \textbf{Four P's and four R's.} $\PPcal$ (XII-A), $\PPcal$ (XII-B, explicit), $\PPfrak$ (XII-D), $\Pmap$ (XIV-I); $R$ (kernel), $R_1$ (output map), $\Rquant$ (admission), $\mathcal R$ (XII-B). None identified. Lemma~\ref{lem:fiber} relates XII-D's $a_1$ to XII-B's $\PPcal$ only. \item \textbf{Copied inputs.} The $K_8$, $K_9$ formulas are XIV-I input statements copied from XIV-H; XIV-H's own \texttt{eq:K9} is located and not reproduced. \item \textbf{Values versus maps.} XIII-C's class is a class of coefficient changes; XV-B's $\mathcal A$ is a class of maps on presentations. The passage from one to the other is hypothesis (H2). \item \textbf{Variables.} XIII-C's $\Delta_I$ live in XIII-C's graph variables (vertex table located); XIV-I's chart lives on $\{d_\pm^0>0\}\subset\mathbb R^{16}$; the lift of $\Delta_I$ to $W$ is not reproduced. \item \textbf{$q$-linearity.} Lemma~\ref{lem:recrule}(ii), (iv) are conditional on linearity of $\Rec$ over chart functions; the record displays the relation as null and does not display the module action. \item \textbf{Cross-line blocks.} \eqref{eq:cross} is \Tst; the blocks are \textsf{C}; Lemma~\ref{lem:cross}(iv) is conditional on the smooth-difference clause and on translation invariance of the replaced blocks. \item \textbf{XII-B, XII-E, X-B.} XII-B is used for its own objects only; XII-E only for the display \eqref{eq:a2}; the X-B summary is a non-input. \item \textbf{Disjointness and necessity.} Errata (E1)--(E2) to XV-E are phrasing corrections; no theorem of XV-E is withdrawn. \item \textbf{$T^\circ=0$ instance.} In Theorem~\ref{thm:dimension}(i), XV-E's Lemma \texttt{lem:flat} is applied to $T=s$ as an extension of the zero distribution; all hypotheses are checked. \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{Prescription.} Pole, propagators, carrier, normalization, and ambiguity class unchanged; recovered displays are transcribed, not modified. \item \textbf{Compilation.} \texttt{mathrsfs} loaded for \verb|\mathscr|; all macros defined; all \verb|\ref| and \verb|\cite| targets exist; \texttt{tabularx} column widths sum to less than \verb|\linewidth|; \verb|\allowdisplaybreaks| set; no non-ASCII characters in mathematics. \end{enumerate} \section{Successor requirement} \label{sec:successor} \subsection{Remaining items} The next step requires the following, each with file, label, and line numbers, recorded as \emph{defines} rather than \emph{cites}: \begin{enumerate}[label=\textup{(\arabic*)}] \item XIV-I, \texttt{def:amodule}: the generators of $\Amod$, the coefficient class (in particular its dependence on $q_\pm$, $r_\pm$), and the module action (linearity over chart functions); with these, $\ker\Rec$ follows from Lemma~\ref{lem:recrule}. \item XIV-G/XIV-H: $E_{\mathrm{res}}$ as an object (D0.5). \item XIV-I: the inherited list of ``specified products already proved'' that constitutes $\Ucan$ (D0.8); the equality with $W\setminus\overline{\Natt}$ is refused and is not to be searched again. \item XIV-J, \texttt{eq:target}: the display of $m^\circ$ (D0.10). \item XIV-J, \texttt{def:mapclass} and \texttt{def:class}: the displays, including the meaning of ``meet $N$'' (D0.16). \item XIV-J, lines 1112--1145: XIV-J's own symbol and definition for the replaced blocks in \eqref{eq:cross}; the normalization of $T^{\mu\nu}(\phi,\psi)$ in \eqref{eq:hess} ($\Tpol$ or $\Tsym$) and the display of the vertex (D0.11, D0.12). \item XIV-H, preceding \texttt{eq:L}: the definitions of $a_1$, $\sigma_\eps(a)$, $f$, $a_2$ and the construction of $\mathsf D$ from $\dot a_2$ (D0.13); XIV-H, lines 653--665 and \texttt{eq:K9} (line 1055): the expansion of $q^{L,I,J}_{\alpha\beta}$ from XII-C's density-weighted cubic, $\Pmap$, $\mathfrak b$, $g$, the declaration of $\EA_m$, and the meaning of ``separated chronological patch'' and ``regular conormal representation'' (D0.14, Remark~\ref{rem:conormal}). \item XIII-C: the formulas for $D_\Gamma$, the admissible $w_\Gamma$, $S_{\Gamma/I}$, and the statement of which presentation-level maps the frozen class permits (D0.17); the lift of $\Delta_I$ to $W$ and its position relative to $\Ucan$. \item XII-A or XI-B: the support sentence for $E_0^{\mathrm R}$; with it candidate (a) is decided by Lemma~\ref{lem:R}(iii)--(iv). Candidates (b)--(g) are decided by items (6)--(7). \end{enumerate} \subsection{Programme once the items are supplied} With (1)--(9) in status \textsf{S}: write $\Fz$ on each generator of $\Amod$ by the composition rule of \texttt{eq:target}; prove $r_O\Fz=m^\circ\Rec$ on $\Ucan$; determine $\ker\Rec$ from the coefficient class and module action; evaluate $d_0$ on each null relation, beginning with XIV-J's and the chart identities of Lemma~\ref{lem:recrule}(iv); and decide cancellation by Proposition~\ref{prop:conditional}. That step would state the preliminary lift and enter F1. It would not by itself enter F2--F4, and it would not admit the quantum coefficient \eqref{eq:admission}. If the actual target has a finite normal-jet structure along a finite puncture, Theorems~\ref{thm:jetchar}--\ref{thm:dimension} apply; whether it does is not known from the record. The question of XV-E's Remark \texttt{rem:open} remains open. The characteristic compositions of $K_{47}$ and both $K_5$ placements require a composition theorem not in the record and not supplied here. \subsection*{Final release sentence} \begin{center} \fbox{\parbox{0.91\linewidth}{ XV-F performs four further source recovery passes; it recovers and verifies XIV-I's reconstruction rule, adapted chart, retarded kernel and kernel square, XIV-J's cross-line definition, mixed vertex Hessian and output convention, XIV-H's coordinate-response line and the role of its fiber tensor, the $K_8$ and $K_9$ kernels with their order bound, XIII-C's collision sets, vertex table and partial-diagonal class, and XII-A's Hadamard matrix; locates three refusals of the canonical-domain equality; verifies four candidate displays from the XII sheet without identifying them; proves that all six internal expressions exist for every completion of the still-undefined data and that three of them take the reconstruction form in the adapted charts; corrects the circulated draft on the coefficient argument, the free-module claim, and a disputed symbol; reduces the candidate identifications to seven located sentences; and finds the core package still incomplete. It therefore defines no lift for the actual graph and asserts no graph-specific defect, obstruction, estimate, or admission, while fixing the form the descent test will take in the recovered class. For the punctured normal-jet module it characterizes jet-detected relations, proves that descending extensions form an affine space over the reconstruction-fixed freedom, and proves that a normal-order bound alone never makes the supported ambiguity finite-dimensional. 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. Used for Proposition \texttt{prop:transfer} (the transfer rule: same notation is not an identification). \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, Proposition \texttt{prop:descent}, Theorem \texttt{thm:lifting}, Theorem \texttt{thm:contact}, and Proposition \texttt{prop:allcontacts}. \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 ambiguity space \texttt{eq:ambiguity}. \bibitem{XVD} Manuscript XV-D, supplied as \texttt{XV-D.tex}. \emph{Instantiating the Frozen Joint Graph: Source Recovery, the Null Defect of Extension Lifts, and the Boundary of the First Full Calculation}. Full text supplied. Used for Definition D0.1--D0.20, Lemmas \texttt{lem:normalform}, \texttt{lem:extraction}, Theorems \texttt{thm:kernels}, \texttt{thm:defect}, Propositions \texttt{prop:example} and \texttt{prop:onenull}. \bibitem{XVE} Manuscript XV-E, supplied \texttt{XV-E-va.tex}. \emph{Source Recovery for the Frozen Joint Graph, Descending Extensions in the Punctured Normal-Jet Module, and the Boundary of the Full Lift}. Full text supplied. Used for the E0 manifest, the transcription of the XII-A/XII-D candidate formulas, Lemma \texttt{lem:surj}, Theorem \texttt{thm:ann}, Definition \texttt{def:kinds} (corrected here), Lemma \texttt{lem:flat}, Theorems \texttt{thm:jetdep}, \texttt{thm:point}, \texttt{thm:relative}, Remark \texttt{rem:open}, Proposition \texttt{prop:erratum}. \bibitem{XIIB} Manuscript XII-B, supplied \texttt{XII-B.tex}. \emph{Admission and one retarded response of the ramped cubic Einstein source, at fixed regulator width}. Full text supplied. Used only for its own objects: the ramp $\sigma_\eps(t)=\chi(t/\eps)$ with explicit $\chi$, trace reversal $\PPcal r=r-\tfrac12\eta\tr_\eta r$, $\mathcal R=\tfrac12K$, the conventions $\Box=\partial_t^2-\Delta$, $P_m=\Box+m^2$, $G^{\mathrm R}_{\mathrm{gr}}=-\PPcal E_0^{\mathrm R}$, its Lemma \texttt{lem:weighted}, and its own B0 stop. It does not display $E_0^{\mathrm R}$ and defines no response. \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{HeqL} Manuscript XIV-H, version~1. ``The existing tree maps with $z$ retained,'' lines 1108--1134, equation \texttt{eq:L} at lines 1111--1119. Direct excerpt supplied in the second retrieval; confirmed in the fourth. Source of \eqref{eq:L}. \bibitem{Hfiber} Manuscript XIV-H, version~1. Display $a_1^{\alpha\mu}[r]=\Amap^{\alpha\mu,I}r_I$, returned in the fifth retrieval; line numbers not returned. Source of \eqref{eq:a1H}; $a_1$ not defined there. \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, with $|\alpha|+|\beta|\le2$. Paraphrased in the fifth retrieval. \bibitem{Hrefuse} Manuscript XIV-H, version~1. Lines 1661--1667: no equality $\Ucan=W\setminus\overline{\Natt}$ is asserted. Located in the fifth retrieval (refusal). \bibitem{HK9} Manuscript XIV-H, version~1. Equation \texttt{eq:K9}, line 1055. Located in the fourth retrieval; display not reproduced. \bibitem{Hverdict} Manuscript XIV-H, version~1. Theorem \texttt{thm:H2}, lines 1539--1582; Theorem \texttt{thm:release}, lines 1847--1895. Direct excerpts supplied. \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{IeqR} Manuscript XIV-I, version~1. Equation \texttt{eq:R}, lines 329--335. Display reproduced in the fourth retrieval. Source of \eqref{eq:R}--\eqref{eq:RR}. \bibitem{Ichart} Manuscript XIV-I, version~1. Lines 828--855: the two placements $d_\pm$, $r_\pm$, the maps $\Phi_\pm$, the coordinates $(q_\pm,\zeta_\pm)$ near the null cone, and the adapted chart $w_\pm=(u_\pm,\zeta_\pm,q_\pm,r_\pm,y)$ of dimension sixteen. Reproduced in the fifth retrieval. Source of \eqref{eq:chart}. \bibitem{Iamodule} Manuscript XIV-I, version~1. Definition \texttt{def:amodule}, lines 873--901. Reconstruction rule reproduced in the fourth retrieval; generators and coefficient class paraphrased. Source of \eqref{eq:Recrule}. \bibitem{Iamplitude} Manuscript XIV-I, version~1. Theorem \texttt{thm:I2} and adjacent precomposition formulas, lines 1288--1378. Direct excerpt supplied. \bibitem{Ijoint} Manuscript XIV-I, version~1. Proposition \texttt{prop:canonical}, Theorem \texttt{thm:I3}, lines 1459--1548. Direct excerpt supplied. \bibitem{Irefuse} Manuscript XIV-I, version~1. Proposition \texttt{prop:canonical}, lines 1492--1496: no equality $\Ucan=W\setminus\overline{\Natt}$ is asserted. Located in the fifth retrieval (refusal). \bibitem{Irelease} Manuscript XIV-I, version~1. Theorem \texttt{thm:release}, lines 1729--1814. Direct excerpt supplied. \bibitem{Ipackage} Manuscript XIV-I, version~1. Input statement \texttt{input:Hpackage}, lines 224--245. Direct excerpt supplied in the second retrieval. \bibitem{Iresponse} Manuscript XIV-I, version~1. Input statement \texttt{input:Hresponse}, lines 257--321 (formulas at 285--310), equations \texttt{eq:Q}, \texttt{eq:T}, \texttt{eq:K8}, \texttt{eq:K9}. Direct excerpt supplied in the second retrieval; confirmed in the fourth. Source of \eqref{eq:Q}--\eqref{eq:Lresp}. \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{Jrconv} Manuscript XIV-J, version~1. Equation \texttt{eq:Rconvention}, lines 631--636. Display reproduced in the fourth retrieval. Source of \eqref{eq:Rconv}. \bibitem{Jtarget} Manuscript XIV-J, version~1. Equation \texttt{eq:target}, lines 720--730. Paraphrased in the fourth retrieval; display not reproduced. \bibitem{Jmapclass} Manuscript XIV-J, version~1. Definition \texttt{def:mapclass}, lines 732--762. Four clauses paraphrased in the fourth retrieval; display not reproduced. \bibitem{Jcross} Manuscript XIV-J, version~1. Grouped cross-line operator, lines 1112--1123. Definition reproduced in content in the fourth retrieval; the symbol for the replaced blocks returned by that pass ($\widehat H$) is not found by the fifth (F3). Source of \eqref{eq:cross}. \bibitem{Jhessian} Manuscript XIV-J, version~1. Mixed vertex Hessian, lines 1128--1145. Display reproduced in the fourth retrieval; $T^{\mu\nu}(\phi,\psi)$ named ``the inherited polarized tensor,'' not expanded. Source of \eqref{eq:hess}. \bibitem{Jstress} Manuscripts XIII-C and XIV-J, version~1. Display of $T_0^{\mu\nu}=\partial^\mu\phi\partial^\nu\phi-\tfrac12\eta^{\mu\nu} ((\partial\phi)^2-m^2\phi^2)$ and the vertex $-\tfrac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}$, returned in the fifth retrieval; line numbers not returned. Source of \eqref{eq:T0}. \bibitem{Jrefuse} Manuscript XIV-J, version~1. Lines 1305--1309: no equality $\Ucan=W\setminus\overline{\Natt}$ is asserted. Located in the fifth retrieval (refusal). \bibitem{Jreference} Manuscript XIV-J, version~1. Theorem \texttt{thm:reference}, lines 1215--1299. Direct excerpt supplied; $v_\alpha$, $c_{D,\beta}$, and the equation of $\Sresp$ not defined there. \bibitem{Jconditional} Manuscript XIV-J, version~1. Theorem \texttt{thm:conditionalextension}, lines 1427--1470. Direct excerpt supplied. \bibitem{Jverdict} Manuscript XIV-J, version~1. Theorem \texttt{thm:J2}, lines 1608--1649. Direct excerpt supplied. \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. \bibitem{Crelease} Manuscript XIV-C, version~1. \texttt{XIV-C.tex}, identifier \texttt{97aa7215}. Theorem \texttt{thm:release}, lines 1307--1340. Used only for the named codimension-four stratum. \bibitem{Grelease} Manuscript XIV-G, version~1. \texttt{XIV-G.tex}, identifier \texttt{4fcac266}. Theorem \texttt{thm:release}, lines 1710--1751. Used only for the dimensions of $W$ and $N$. \bibitem{XIIA} Manuscript XII-A, version~2. \texttt{XII-A-v2.tex}, identifier \texttt{acb05c0f}. ``Named composites and their formal sector,'' lines 969--1059; the tensor $\mathfrak a$ and the composite $\Lambda$ at lines 974--980. Transcribed in XV-E; confirmed in the fourth and fifth retrievals. Source of \eqref{eq:XIIA} and (with \cite{XIIDfiber}) of \eqref{eq:a1}. \bibitem{XIIAhad} Manuscript XII-A, version~2. ``The frozen Hadamard matrix,'' lines 442--492, equations \texttt{eq:H}, \texttt{eq:Hfrozen}; Definition \texttt{def:admissible}, lines 518--528 (four clauses). Direct excerpt supplied in the second retrieval; clauses confirmed in the fifth. Source of \eqref{eq:Hblocks}. \bibitem{XIID} Manuscript XII-D, version~1. \texttt{XII-D.tex}, identifier \texttt{e3e3f67d}. ``The first root and its transpose,'' lines 571--611. Transcribed in XV-E. \bibitem{XIIDfiber} Manuscript XII-D, version~1. Lines 519--525: $\mathfrak a$, $a_1^{\mu\nu}[r]=\mathfrak a^{\mu\nu,ab}r_{ab}$, $b_1[r]=\tfrac12\eta^{ab}r_{ab}$. Reproduced in the fifth retrieval. Source of \eqref{eq:a1}. \bibitem{XIIDweight} Manuscript XII-D, version~1. Lines 545--552: arbitrary $f\in C^\infty_c(\mathring M_T)$ and the observable $\mathcal A_1[f,g](r;\phi)$. Reproduced in the fifth retrieval. \bibitem{XIIDsource} Manuscript XII-D, version~1. ``The actual local source kernel,'' lines 841--860. Direct excerpt supplied in the second retrieval. Not used. \bibitem{XIIDrelease} Manuscript XII-D, version~1. Front matter, ``Release and proof boundary,'' lines 171--195. Direct excerpt supplied in the third retrieval. Non-input (\S\ref{sec:thirdpass}). \bibitem{XIIEa2} Manuscript XII-E, version~1. Lines 401--403: $b_2[r]$ and $a_2^{\mu\nu}[r]$. Reproduced in the fifth retrieval. Source of \eqref{eq:a2}. Used only for this display. \bibitem{XIICsource} Manuscript XII-C. The density-weighted cubic source, named as the undifferentiated source behind \cite{Hq}; locator not returned; display not reproduced. \bibitem{XIIICcoll} Manuscript XIII-C, version~1. \texttt{XIII-C.tex}, identifier \texttt{bb647493}. Definition ``Collision sets,'' lines 338--349. Direct excerpt supplied in the second retrieval. Source of Definition~\ref{def:collision}. \bibitem{XIIICpart} Manuscript XIII-C, version~1. Definition \texttt{def:Cpart}, lines 358--392, equation \texttt{eq:partial-counterterm}. Direct excerpt supplied in the second retrieval; confirmed in the fourth and fifth, which record that $D_\Gamma$, $w_\Gamma$, $S_{\Gamma/I}$ are roles, not formulas. Source of Definition~\ref{def:Cpart}. \bibitem{XIIICvertex} Manuscript XIII-C, version~1. Cycle-vertex table, lines 603--620; examples $\{y,u,v\}$, $\{y,v\}$ on $B_1$. Located in the fifth retrieval. \bibitem{XIVArel} Manuscript XIV-A, version~1. \texttt{XIV-A.tex}, identifier \texttt{c10ee4de}. Theorem \texttt{thm:relative}, lines 887--915. Relative classical branch; not imported. \bibitem{Ione} Manuscript I, version~1. \texttt{I.tex}, identifier \texttt{87fdaa5e}. ``Context transport and the Frauchiger--Renner example,'' lines 2189--2268. Not imported. \bibitem{XBsum} Retrieval summary of Manuscript X-B, version~3 (\texttt{X-B-v3.tex}, identifier \texttt{103c6c32}), lines 202--236, 846--859, 920--958, 982--990, with an accompanying plan. Scalar-only one-contraction normalization of a different track. Non-input. \bibitem{record} Supplied coverage account, retrieval corpus identifier \texttt{3883b893}, retrievals~3--5. Explicitly disclaims a complete corpus scan and independent verification. Used only as documentary context, for reported codimension bookkeeping, for its reading rule and cautions, and as the carrier of the file, label, and line locators of the reproduced displays and refusals. \bibitem{Horm} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I}, 2nd ed., Springer, 1990. Used for: distributions supported at a point (Thm.~2.3.4); convolution under support conditions (\S4.2); Paley--Wiener--Schwartz (Thm.~7.3.1); pullback by submersions and its wavefront bound (Thm.~8.2.4); wavefront sets of tensor products (Thm.~8.2.9). \end{thebibliography} \end{document}