% XV-B.tex % Standalone pdfLaTeX document; no external bibliography file is required. % Recommended build: pdflatex XV-B.tex, followed by pdflatex XV-B.tex. % Compilation and proof-assistant verification have not been executed here. \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} \usepackage{tikz} \setlength{\emergencystretch}{3em} \setlength{\parskip}{0.3em} \setlength{\parindent}{1.2em} \setlist[enumerate]{label=\textup{(\roman*)},leftmargin=*,itemsep=0.25em} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\D}{\mathcal D} \newcommand{\Dp}{\mathcal D'} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \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{\Sresp}{S_{\mathrm{resp}}} \newcommand{\Cross}{\mathscr C_{\mathrm{cross}}} \newcommand{\Cpart}{\mathfrak C_{\mathrm{part}}} \newcommand{\Ctot}{\mathfrak C_{\mathrm{tot}}} \newcommand{\Rquant}{R_1^{(1)}} \newcommand{\Eout}{\mathscr E_H} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Mmap}{\mathsf M} \newcommand{\Ejet}{\mathsf E} \newcommand{\Jjet}{\mathsf J} \newcommand{\eps}{\varepsilon} \newcommand{\dd}{\,\mathrm d} \newcommand{\pair}[2]{\langle #1,#2\rangle} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\PV}{PV} \title{Landmark XV-B\\ Relative Lifting and Reconstruction Descent\\ for the Full Joint Characteristic Target} \author{} \date{} \hypersetup{ pdftitle={Landmark XV-B: Relative Lifting and Reconstruction Descent}, pdfsubject={Distributional extension, reconstruction descent, and normal jets} } \begin{document} \maketitle \begin{abstract} We study the compatibility between distributional extension and reconstruction descent for linear constructions on presentations. Given a canonical map on a proved open domain, a preliminary lift, and a specified affine space of permitted corrections, we prove that a reconstruction-descending candidate exists exactly when the preliminary lift's null defect vanishes in the corresponding relative obstruction quotient. Contact-supported corrections cannot cancel a null defect detected away from their support. For a specified finite normal-jet module, we obtain an explicit normal form and a complete distributional null-relation criterion. Supported extension preserving the normal-order bound is equivalent to extension of its unique coefficients; the complete unrestricted supported ambiguity is also determined. We give a general closed-set extension criterion, distinguish universal and flat-annihilator relations from potentially jet-detected defects, and construct a simultaneous annihilator-preserving correction at a point puncture by finitely many derivatives of a delta distribution. The point-puncture result is not asserted for a positive-dimensional characteristic locus. The application retains the full six-branch target $K_{47}+K_5+K_6+K_8+K_9$, including both placements of $K_5$. The interface results do not construct its missing characteristic contraction, establish its complete-graph estimates, or admit a quantum coefficient. \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) {XIV-J continuation}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Relative reconstruction}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Joint characteristic coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Lifting language does not evaluate the coefficient.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Relative lifting is not the joint coefficient. 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-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-J continuation}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Joint characteristic coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Lifting language does not evaluate the coefficient.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Relative lifting is not the joint coefficient. 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-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-J continuation}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Relative reconstruction}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Joint characteristic coefficient}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Lifting language does not evaluate the coefficient.}; \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-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {XIV-J continuation}; \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) {Joint characteristic coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Relative reconstruction. Lifting language does not evaluate the coefficient.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{XIV-J continuation is not Joint characteristic coefficient. 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-B} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: XIV-J continuation}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Relative reconstruction}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Joint characteristic coefficient}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Lifting language does not evaluate the coefficient.}; \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-B} \end{figure} Figure~\ref{fig:split-XV-B} records one view of the same separation. Figure~\ref{fig:chain-XV-B} records one view of the same separation. Figure~\ref{fig:pair-XV-B} records one view of the same separation. Figure~\ref{fig:gate-XV-B} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XV-B} records that separation. The drawing adds no hypothesis. \section{Introduction and scope} \label{sec:introduction} \subsection{The extension--descent problem} A distributional construction on a presentation space need not depend only on the reconstructed distribution. This distinction is particularly important when different presentation degrees can cancel after reconstruction. Extending each displayed term separately does not, by itself, preserve those cancellations. The problem has three logically distinct parts: \begin{enumerate} \item extend prescribed canonical values from their proved domain; \item annihilate every reconstruction-null presentation; \item satisfy the specified support, normalization, continuity, and microlocal requirements. \end{enumerate} The first two parts are already different before any complete-graph wavefront theorem is considered. Cheung's XIV-J makes the descent condition explicit and records a null relation mixing presentation degrees \cite{CheungXIVJ}. The present paper develops an exact relative criterion for correcting such defects. It also incorporates the finite normal-jet reconstruction test supplied in XV-C \cite{CheungXVC} and develops the annihilator and point-puncture interfaces discussed in the supplied account of XV-E \cite{CheungXVE}. All abstract statements used below are proved here; no unavailable proof from a companion manuscript is required. The distributional background is standard \cite{Hormander}. Extension with prescribed scaling or wavefront control requires further hypotheses, as in the frameworks of Brunetti--Fredenhagen and Dang \cite{BF,Dang}. Our algebraic lifting criterion is not a substitute for those analytic hypotheses. \subsection{Principal results} The principal results are: \begin{enumerate} \item the exact affine relative-lifting criterion (Theorem~\ref{thm:lifting}); \item the contact-support obstruction (Theorem~\ref{thm:contact}); \item the complete null criterion for a specified finite normal-jet module (Theorem~\ref{thm:jetnull}); \item the supported coefficient-extension equivalence and complete unrestricted supported ambiguity (Theorem~\ref{thm:jetextension}); \item the local uniform finite-order criterion for extension across a general relatively closed set (Theorem~\ref{thm:closedextension}); \item the finite-jet localization of natural null defects and the simultaneous point-puncture correction (Lemma~\ref{lem:jetdefect} and Theorem~\ref{thm:pointcorrection}). \end{enumerate} The normal-jet module is explicitly defined. It is not identified with the inherited physical presentation space. Likewise, unrestricted supported ambiguities are not identified with the frozen partial-diagonal correction class. \subsection{Application boundary and numbering} The retained application asks for a map \begin{equation} \Mmap:\Kint\longrightarrow\Dp(W;E_{\mathrm{res}}) \label{eq:targetmap} \end{equation} for the full joint characteristic target inherited from XIV-I and XIV-J \cite{CheungXIVI,CheungXIVJ}. Here $W$ is the attachment domain and $E_{\mathrm{res}}$ is the residual-species bundle. A successful construction must agree with canonical composition on its proved domain, act on tests meeting the characteristic locus, retain every branch and prescribed tensor operation, respect reconstruction, and satisfy the complete-graph microlocal bounds. In this paper, \emph{XV-B} denotes the technical continuation studying these obligations. The predecessor phrase ``XV-B remains closed'' concerns its earlier downstream admission contract, not the nonexistence of a technical continuation. The admission condition \begin{equation} Q_H\Rquant\in\Eout \label{eq:admission} \end{equation} is not established here. Changing the manuscript convention neither proves \eqref{eq:admission} nor supplies a free quantum BV carrier. In particular, the model-and-domain audit cited as XV-A below is not identified with the differently described free-carrier prerequisite appearing under that label in older passages \cite{CheungXVA,CheungXIVI,CheungXIVJ}. \subsection{Source control} The inherited formulas and historical statuses in Section~\ref{sec:application} are attributed to the supplied manuscript excerpts. Bibliographic references to those manuscripts use the supplied author, version, and statement labels; no journal publication, independent corpus verification, or public availability of a complete manuscript is asserted. The additional record contains definitions and formulas absent from the original XV-B excerpt, including the coordinate-response line, the characteristic attachment locus, the grouped cross-line expression, and the partial-diagonal class. Their recovery changes the available evidence. It does not construct the full characteristic operation. The reported XII-A and XII-D candidate identifications are not used to assign values to unspecified inherited symbols. In particular, a displayed candidate tensor or response formula is not an identification of the complete package required in Section~\ref{sec:obligations}. The independently supplied XII-C and XII-D classical passages are not premises of the abstract theorems below. \subsection{Conventions} All vector spaces are over $\C$; the real versions are identical. $\Hom$ denotes algebraic linear maps unless continuity is explicitly required. Manifolds are smooth, Hausdorff, and second countable. Distributions on manifolds act on compactly supported smooth densities; in Euclidean charts we use the fixed coordinate density and write tests as functions. Bundle-valued assertions are interpreted in local trivializations of a finite-rank smooth bundle and proved componentwise. Multi-index inequalities are componentwise, and \[ \binom{\alpha}{\beta} =\prod_j\binom{\alpha_j}{\beta_j}. \] For $K\Subset X\subset\R^d$ and $M\in\N$, write \[ \|\phi\|_{K,M} =\max_{|\nu|\le M}\sup_{x\in K}|\partial^\nu\phi(x)|. \] The notation $K\Subset X$ means that $K$ is compact and contained in the open set $X$. \section{Reconstruction and the exact relative-lifting criterion} \label{sec:lifting} \subsection{The linear interface} \begin{definition}[Reconstruction interface] \label{def:interface} Let $\Pint$ be a vector space and let \[ \Rec:\Pint\longrightarrow\Kint \] be a surjective linear map. Set \[ H=\ker\Rec,\qquad \Kint=\im\Rec. \] Let $O\subset W$ be open, and put \[ V=\Dp(W;E_{\mathrm{res}}),\qquad V_O=\Dp(O;E_{\mathrm{res}}). \] Let $r:V\to V_O$ be restriction, and let \[ m^\circ:\Kint\longrightarrow V_O \] be a prescribed linear canonical map. Define \[ c=m^\circ\Rec,\qquad Y=\ker r. \] A \emph{preliminary lift} is a linear map $F_0:\Pint\to V$ satisfying \begin{equation} rF_0=c. \label{eq:prelift} \end{equation} \end{definition} No surjectivity of $r$ is assumed. A distribution on an open subset need not extend across its boundary; an explicit example appears in Proposition~\ref{prop:nonextendable}. For distributions, \begin{equation} Y=\{t\in V:\supp t\subset W\setminus O\}. \label{eq:Y} \end{equation} Indeed, zero restriction is equivalent to vanishing on all tests supported in $O$. \begin{proposition}[Descent onto the reconstructed image] \label{prop:descent} A linear map $F:\Pint\to V$ has a unique factorization \[ F=\Mmap\Rec,\qquad \Mmap:\Kint\to V, \] if and only if \begin{equation} F|_H=0. \label{eq:descent} \end{equation} If $rF=c$, the induced map satisfies $r\Mmap=m^\circ$. \end{proposition} \begin{proof} A factorization annihilates $\ker\Rec$. Conversely, suppose $F|_H=0$ and define \[ \Mmap(\Rec A)=F(A). \] If $\Rec A=\Rec B$, then $A-B\in H$, so $F(A)=F(B)$. Thus $\Mmap$ is well defined and linear. Surjectivity of $\Rec$ gives uniqueness. Finally, \[ r\Mmap(\Rec A)=rF(A)=c(A)=m^\circ(\Rec A). \] \end{proof} This is the elementary factorization underlying the corresponding XIV-J and XV-A statements \cite{CheungXIVJ,CheungXVA}. \begin{remark}[Carrier and topology] If a larger carrier is also denoted by $\Kint$, uniqueness holds only on $\im\Rec$ until a further extension is specified. If $\Pint$ is a topological vector space, $\Kint$ carries the quotient topology, and $F$ is continuous and annihilates $H$, the induced $\Mmap$ is continuous. This follows from the defining property of the quotient topology. It does not identify that topology with an inherited physical carrier topology. \end{remark} \subsection{Specified affine corrections} \begin{definition}[Relative obstruction] \label{def:obstruction} Fix a preliminary lift $F_0$ and a vector subspace \[ \mathcal A\subset\Hom(\Pint,Y). \] The candidate family is $F_0+\mathcal A$. Define \[ \partial:\mathcal A\longrightarrow\Hom(H,Y), \qquad \partial a=a|_H. \] Since $c|_H=0$, the null defect \[ d_0=F_0|_H \] takes values in $Y$. Its relative obstruction is \begin{equation} \Ob_{\mathcal A}(F_0) =[d_0]\in\Hom(H,Y)/\partial\mathcal A. \label{eq:obstruction} \end{equation} \end{definition} The symbol $\partial$ here denotes restriction of a map, not a distributional derivative. \begin{theorem}[Exact relative-lifting criterion] \label{thm:lifting} For Definition~\ref{def:obstruction}: \begin{enumerate} \item The class $\Ob_{\mathcal A}(F_0)$ is unchanged if $F_0$ is replaced by another element of $F_0+\mathcal A$. \item A reconstruction-descending member of $F_0+\mathcal A$ exists if and only if \begin{equation} \Ob_{\mathcal A}(F_0)=0. \label{eq:vanishing} \end{equation} \item An individual candidate $F_0+a$ descends if and only if \begin{equation} d_0+\partial a=0. \label{eq:cancellation} \end{equation} \item If $F_\ast$ is one descending candidate, all descending candidates are exactly \[ F_\ast+\mathcal A_0,\qquad \mathcal A_0=\{a\in\mathcal A:a|_H=0\}. \] \item Every descending candidate induces a unique map $\Mmap:\Kint\to V$ agreeing with $m^\circ$ after restriction to $O$. The induced ambiguity consists precisely of those maps $\bar a:\Kint\to Y$ for which $\bar a\Rec\in\mathcal A_0$. \end{enumerate} \end{theorem} \begin{proof} If $F_1=F_0+a_1$, then \[ F_1|_H=d_0+\partial a_1, \] which proves (i). A candidate annihilates $H$ precisely when \eqref{eq:cancellation} holds. Such an $a$ exists exactly when $d_0\in\partial\mathcal A$, because $\partial\mathcal A$ is a vector subspace. This proves (ii) and (iii) by Proposition~\ref{prop:descent}. If $F_\ast$ annihilates $H$, then $F_\ast+a$ annihilates $H$ exactly when $a|_H=0$, proving (iv). Every correction takes values in $\ker r$, so \[ r(F_0+a)=c. \] Proposition~\ref{prop:descent} gives the claimed agreement. The same proposition factors every $a\in\mathcal A_0$ uniquely through $\Rec$, and conversely any stated $\bar a$ gives such a correction. This proves (v). \end{proof} \begin{corollary}[Continuous candidate families] \label{cor:continuous} The same criterion holds when $\mathcal A$ is a specified vector space of continuous corrections and the candidate maps are continuous. Vanishing means membership in the actual image $\partial\mathcal A$, not merely its closure. \end{corollary} \begin{proof} The proof uses the exact equation $d_0+\partial a=0$ in the specified correction space. It does not replace the image of $\partial$ by a closure or completion. \end{proof} \begin{remark}[Quantifier and admissibility] Equation~\eqref{eq:vanishing} asserts existence of a descending candidate; it does not assert that every candidate descends. An application to a frozen analytic construction must establish that the selected family really is affine, that its members meet the required constraints, and, for an exclusion theorem, that it exhausts the class being excluded. Defining \eqref{eq:obstruction} does not evaluate that class for a particular graph. \end{remark} \subsection{Unrestricted algebraic choices} \begin{proposition}[Algebraic extension equivalence] \label{prop:algebraic} The following are equivalent: \begin{enumerate} \item a preliminary linear lift exists; \item every $m^\circ(k)$, $k\in\Kint$, has some extension in $V$; \item there exists a linear $\Mmap:\Kint\to V$ with $r\Mmap=m^\circ$. \end{enumerate} This equivalence imposes no continuity, locality, normalization, or wavefront condition. \end{proposition} \begin{proof} For (i)$\Rightarrow$(ii), choose $A$ with $\Rec A=k$ and use $F_0(A)$. For (ii)$\Rightarrow$(iii), choose a Hamel basis of $\Kint$, choose an extension for the canonical value of each basis element, and extend linearly. Every vector is a finite linear combination, so the resulting assignment is a linear map with the required restriction. For (iii)$\Rightarrow$(i), take $F_0=\Mmap\Rec$. \end{proof} \begin{proposition}[Unrestricted corrections cancel algebraic defects] \label{prop:unrestricted} If a preliminary lift exists and $\mathcal A=\Hom(\Pint,Y)$, then $\Ob_{\mathcal A}(F_0)=0$. \end{proposition} \begin{proof} Choose an algebraic complement $\Pint=H\oplus L$ and its projection $\pi_H:\Pint\to H$. The correction \[ a=-d_0\pi_H \] takes values in $Y$ and satisfies $a|_H=-d_0$. Apply Theorem~\ref{thm:lifting}. \end{proof} These basis and complement choices are not graph prescriptions. In particular, they need not preserve a topology, differential operator structure, covariance, contact restrictions, or microlocal bounds. \section{Contact support and domain-sensitive transfer} \label{sec:contacts} \subsection{The contact obstruction} Let $C\subset W$ be closed and set \[ V_C=\{t\in V:\supp t\subset C\}. \] \begin{theorem}[Separated null defects cannot be removed by contacts] \label{thm:contact} Suppose every $a\in\mathcal A$ takes values in $V_C$. If there exist $A\in H$ and an open set $G\subset W\setminus C$ such that \[ F_0(A)|_G\ne0, \] then no member of $F_0+\mathcal A$ descends. Equivalently, $\Ob_{\mathcal A}(F_0)\ne0$. \end{theorem} \begin{proof} Every permitted correction restricts to zero on $G$. Therefore \[ (F_0+a)(A)|_G=F_0(A)|_G\ne0 \] for every $a\in\mathcal A$. No candidate annihilates $H$. Theorem~\ref{thm:lifting} proves the obstruction statement. \end{proof} \begin{proposition}[Exact unrestricted contact-valued criterion] \label{prop:allcontacts} If $C\subset W\setminus O$ and $\mathcal A=\Hom(\Pint,V_C)$, a descending candidate exists if and only if $F_0(H)\subset V_C$. \end{proposition} \begin{proof} If $F_0+a$ annihilates $H$, then $F_0(A)=-a(A)\in V_C$ for $A\in H$. Conversely, assume $F_0(H)\subset V_C$ and choose the algebraic projection $\pi_H$ used above. Then \[ a=-F_0|_H\,\pi_H \] is contact-valued and cancels the defect. \end{proof} \begin{remark}[Support comparison, not a regulator import] Theorem~\ref{thm:contact} is the precise support-theoretic version of the observation that collision contacts cannot alter an off-collision prescription. The supplied XIII-E release records that observation \cite{CheungXIIIE}. This comparison imports no regulator, characteristic limit, selected finite part, or convergence theorem from XIII-E. \end{remark} \subsection{Restriction does not test global descent} \begin{proposition}[Invisible defects outside the canonical domain] \label{prop:restriction} If $rF_0=c$, then $rF_0(A)=0$ for every $A\in H$. This does not in general imply $F_0(A)=0$ on $W$. \end{proposition} \begin{proof} The first assertion follows from $c|_H=0$. If $p\in W\setminus O$ and the residual bundle has a nonzero fiber there, a nonzero delta distribution at $p$, with a chosen fiber coefficient, restricts to zero on $O$. Thus restriction need not detect a defect supported outside $O$. \end{proof} The counterexample concerns injectivity of restriction; it does not assert that an arbitrary delta term belongs to the frozen analytic ambiguity class. This is the domain qualification in XIV-J and XV-A \cite{CheungXIVJ,CheungXVA}. \subsection{A transfer lemma with explicit domains} \begin{proposition}[Domain-sensitive transport of descent] \label{prop:transfer} For $i=1,2$, let $\Rec_i:P_i\to K_i$ be linear and let $D_i\subset P_i$ be vector subspaces. Suppose $F_1:D_1\to V_1$ annihilates $D_1\cap\ker\Rec_1$. Let \[ \alpha:D_2\to D_1,\qquad \beta:K_2\to K_1,\qquad \tau:V_1\to V_2 \] be linear maps satisfying \begin{equation} \Rec_1\alpha=\beta\Rec_2|_{D_2}. \label{eq:transport} \end{equation} Then \[ F_2=\tau F_1\alpha:D_2\to V_2 \] annihilates $D_2\cap\ker\Rec_2$ and descends onto $\Rec_2(D_2)$. No assertion is made outside $D_2$. If all maps are continuous and the reconstructed image carries the appropriate quotient topology, the corresponding induced map is continuous. \end{proposition} \begin{proof} For $A\in D_2\cap\ker\Rec_2$, \[ \Rec_1(\alpha A)=\beta(\Rec_2 A)=0. \] Since $\alpha A\in D_1$, the assumed null condition gives $F_1(\alpha A)=0$, hence $F_2(A)=0$. Apply Proposition~\ref{prop:descent} to $\Rec_2|_{D_2}:D_2\to\Rec_2(D_2)$. The continuity assertion is the quotient-topology statement. \end{proof} This lemma makes the needed identifications explicit. A common label, an analogous formula, or an internal Fourier representation does not supply $\alpha$, $\beta$, $\tau$, or the domain inclusion $\alpha(D_2)\subset D_1$. In particular, it does not enlarge a canonical composition domain to a characteristic domain. \section{Finite normal jets and complete reconstruction-null tests} \label{sec:jets} \subsection{Insertion and extraction} Fix a product chart \[ U=X\times V_q,\qquad S=X\times\{0\}, \] where $X\subset\R^d$ and $V_q\subset\R^m$ are open and $0\in V_q$. The notation and representation are tied to these fixed normal coordinates. For $\gamma\in\N^m$ and $b\in\Dp(X)$, define \begin{equation} \pair{\Ejet_\gamma b}{\phi} =(-1)^{|\gamma|} \pair{b}{\partial_q^\gamma\phi(x,0)}, \qquad \phi\in\D(U). \label{eq:insertion} \end{equation} Thus $\Ejet_\gamma b=b\otimes\partial_q^\gamma\delta_0$ in the chosen product coordinates. Choose $\chi\in\D(V_q)$ equal to one near zero and put \begin{equation} \Jjet_\gamma\psi(x,q) =\frac{(-1)^{|\gamma|}}{\gamma!} q^\gamma\chi(q)\psi(x), \qquad \psi\in\D(X). \label{eq:testlift} \end{equation} These are continuous maps between the corresponding test-function spaces, and \begin{equation} (-1)^{|\beta|} \partial_q^\beta\Jjet_\gamma\psi(x,0) =\delta_{\beta\gamma}\psi(x). \label{eq:extractjets} \end{equation} \begin{lemma}[Coefficient extraction] \label{lem:extraction} For a finite family $b_\gamma\in\Dp(X)$, \[ \sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma=0 \quad\Longleftrightarrow\quad b_\gamma=0\ \text{for every }|\gamma|\le r. \] Moreover, testing the sum against $\Jjet_\gamma\psi$ extracts $\pair{b_\gamma}{\psi}$. \end{lemma} \begin{proof} Equation~\eqref{eq:extractjets} and the insertion convention give \[ \left\langle \sum_{|\beta|\le r}\Ejet_\beta b_\beta,\, \Jjet_\gamma\psi \right\rangle =\pair{b_\gamma}{\psi}. \] The forward implication follows for all $\psi$; the reverse is immediate. \end{proof} \begin{definition}[Supported normal order] \label{def:normalorder} A distribution $t\in\Dp(U)$ has supported normal order at most $r$ along $S$ if $\supp t\subset S$ and it annihilates every test $\phi$ satisfying \[ \partial_q^\gamma\phi(x,0)=0 \quad\text{for all }|\gamma|\le r. \] On an open subset of $U$, this condition is understood locally in product neighborhoods of the points of $S$. \end{definition} \begin{theorem}[Finite normal-jet characterization] \label{thm:normal} A distribution has supported normal order at most $r$ along $S$ if and only if it has a unique representation \begin{equation} t=\sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma, \qquad b_\gamma\in\Dp(X). \label{eq:normal} \end{equation} \end{theorem} \begin{proof} Suppose $t$ has the stated normal order. Define \[ \pair{b_\gamma}{\psi} =\pair{t}{\Jjet_\gamma\psi}. \] Continuity of $\Jjet_\gamma$ makes $b_\gamma$ a distribution. For $\phi\in\D(U)$, set \[ a_\gamma(x)=(-1)^{|\gamma|} \partial_q^\gamma\phi(x,0). \] Each $a_\gamma$ is a test on $X$, and \[ \phi-\sum_{|\gamma|\le r}\Jjet_\gamma a_\gamma \] has zero normal jets through order $r$. Consequently, \[ \pair{t}{\phi} =\sum_{|\gamma|\le r}\pair{b_\gamma}{a_\gamma}, \] which is \eqref{eq:normal}. Conversely, every insertion is supported on $S$ and depends only on its indicated jet. The finite sum therefore has the required normal order. Lemma~\ref{lem:extraction} proves uniqueness. \end{proof} This fixed-order statement is a special case of the standard local structure of distributions supported on a smooth submanifold \cite{Hormander,Dang}. A general supported distribution need only have a locally finite normal-order bound; a single global $r$ is an additional hypothesis. \subsection{Smooth multiplication and the specified module} \begin{lemma}[Exact smooth multiplication formula] \label{lem:leibniz} For $c\in C^\infty(U)$ and $v\in\Dp(X)$, \begin{equation} c\,\Ejet_\alpha v = \sum_{\beta\le\alpha} (-1)^{|\beta|}\binom{\alpha}{\beta} \Ejet_{\alpha-\beta} \left((\partial_q^\beta c)(x,0)\,v\right). \label{eq:leibniz} \end{equation} \end{lemma} \begin{proof} Pair the left side with $\phi$. The Leibniz rule gives \[ (-1)^{|\alpha|} \sum_{\beta\le\alpha}\binom{\alpha}{\beta} \left\langle v,\, (\partial_q^\beta c)(x,0) \partial_q^{\alpha-\beta}\phi(x,0) \right\rangle. \] The insertion on the right contributes the sign $(-1)^{|\beta|}(-1)^{|\alpha-\beta|} =(-1)^{|\alpha|}$, giving the formula. \end{proof} The formula concerns smooth multipliers. Applying an analogous formula to singular cross-lines requires separately proved traces and products. \begin{definition}[A specified finite normal-jet module] \label{def:jetmodule} Fix $T\in\Dp(X)$ and $r\in\N$. Let \[ \mathscr P_r^{\mathrm{jet}}(T) =\bigoplus_{|\alpha|\le r}C^\infty(U) \] and define \begin{equation} \Rec_T(a)=\sum_{|\alpha|\le r}a_\alpha\,\Ejet_\alpha T. \label{eq:jetrec} \end{equation} All products are multiplication by smooth functions. \end{definition} \begin{theorem}[Complete null criterion for the specified module] \label{thm:jetnull} For $a=(a_\alpha)\in\mathscr P_r^{\mathrm{jet}}(T)$, define the smooth functions \begin{equation} f_\gamma(a)(x) = \sum_{\substack{\alpha\ge\gamma\\|\alpha|\le r}} (-1)^{|\alpha-\gamma|} \binom{\alpha}{\gamma} (\partial_q^{\alpha-\gamma}a_\alpha)(x,0) \label{eq:fcoeff} \end{equation} and the distributions \begin{equation} B_\gamma(a)=f_\gamma(a)\,T. \label{eq:B} \end{equation} Then \begin{equation} \Rec_T(a)= \sum_{|\gamma|\le r}\Ejet_\gamma B_\gamma(a), \label{eq:jetnormal} \end{equation} and \begin{equation} \ker\Rec_T = \left\{ a:B_\gamma(a)=0\text{ in }\Dp(X) \text{ for every }|\gamma|\le r \right\}. \label{eq:completekernel} \end{equation} Thus \eqref{eq:completekernel} is a complete distributional null-relation test. \end{theorem} \begin{proof} Apply Lemma~\ref{lem:leibniz} to each summand $a_\alpha\Ejet_\alpha T$ and collect the coefficient of $\Ejet_\gamma$. This gives \eqref{eq:fcoeff}--\eqref{eq:jetnormal}. Lemma~\ref{lem:extraction} then gives \eqref{eq:completekernel}. \end{proof} This is the explicit local criterion in the supplied XV-C theorem \cite{CheungXVC}, now proved with the insertion and extraction conventions fixed in the present paper. \begin{remark}[Tangential annihilators are not discarded] The equations $f_\gamma(a)T=0$ are distributional equations. They need not imply $f_\gamma(a)=0$ pointwise. For example, $x\delta_0=0$ although $x$ is not the zero smooth function. Replacing \eqref{eq:completekernel} by pointwise coefficient vanishing would lose genuine null relations. \end{remark} \subsection{Why highest presentation degree is not intrinsic} \begin{lemma}[The elementary degree-mixing relation] \label{lem:null} In one normal variable, \[ q\delta'(q)+\delta(q)=0. \] Suppose an adapted reconstruction convention satisfies \[ \Rec(M_{\mathrm R})=\delta(q)\otimes R,\qquad \Rec(i\theta M_{\mathrm R})=\delta'(q)\otimes R, \] and multiplication by $q$ is reconstructed as multiplication by the same normal coordinate. Then \begin{equation} \Rec\bigl((q\,i\theta+1)M_{\mathrm R}\bigr)=0. \label{eq:null} \end{equation} If $R\ne0$, the degree-zero and degree-one summands reconstruct separately to nonzero opposite distributions. \end{lemma} \begin{proof} For $\psi\in\D(\R)$, \[ \pair{q\delta'}{\psi} =-\left.\frac{\dd}{\dd q}(q\psi(q))\right|_{q=0} =-\psi(0). \] Tensoring $q\delta'=-\delta$ with $R$ gives the identity. The tensor $\delta\otimes R$ is nonzero when $R\ne0$, as is seen by choosing separated tests with nonzero pairings. \end{proof} \begin{corollary}[Highest representative degree and degree projections] \label{cor:degree} In the convention of Lemma~\ref{lem:null}, the same nonzero reconstructed kernel has representatives \[ M_{\mathrm R} \quad\text{and}\quad -q\,i\theta M_{\mathrm R} \] of different highest polynomial degrees. Consequently, highest degree of a chosen presentation is not an invariant of the reconstructed kernel. Likewise, projection to the degree-one summand does not descend: it sends the null presentation in \eqref{eq:null} to a summand with nonzero reconstruction. \end{corollary} \begin{proof} Both displayed representatives reconstruct to $\delta(q)\otimes R$. The degree-one projection of the null presentation reconstructs to $-\delta(q)\otimes R\ne0$. Apply Proposition~\ref{prop:descent}. \end{proof} \begin{remark}[A quotient filtration is a different question] This does not rule out a filtration defined by reconstructed images of filtered presentation spaces, nor does it invalidate XIV-I's estimates on its named module. It rules out promoting a representative-dependent highest term or splitting to an intrinsic joint symbol without a descent theorem and a compatible joint calculus \cite{CheungXIVI,CheungXIVJ}. \end{remark} \section{Supported extension, ambiguity, and closed-set estimates} \label{sec:extension} \subsection{Extension of finite normal-jet coefficients} Let $A\subset U$ be relatively closed and put \[ Z=\{x\in X:(x,0)\in A\}. \] Then $Z$ is relatively closed in $X$. \begin{theorem}[Supported coefficient-extension equivalence] \label{thm:jetextension} Suppose $t^\circ\in\Dp(U\setminus A)$ is supported on $S\setminus A$ and locally has supported normal order at most $r$. Its unique normal-jet coefficients glue to distributions \[ b_\gamma^\circ\in\Dp(X\setminus Z), \qquad |\gamma|\le r. \] The following are equivalent: \begin{enumerate} \item there exists $t\in\Dp(U)$ supported on $S$, of supported normal order at most $r$, with \[ t|_{U\setminus A}=t^\circ; \] \item every $b_\gamma^\circ$ extends to some $b_\gamma\in\Dp(X)$. \end{enumerate} For any chosen coefficient extensions, \begin{equation} t=\sum_{|\gamma|\le r}\Ejet_\gamma b_\gamma \label{eq:jetrealization} \end{equation} is such a realization. If realizations exist, all of them form an affine space over \begin{equation} \left\{ \sum_{|\gamma|\le r}\Ejet_\gamma d_\gamma: d_\gamma\in\Dp(X),\ \supp d_\gamma\subset Z \right\}. \label{eq:jetambiguity} \end{equation} \end{theorem} \begin{proof} For every $x\in X\setminus Z$, choose a product neighborhood of $(x,0)$ disjoint from $A$. Theorem~\ref{thm:normal} gives its coefficients there. Uniqueness makes these coefficients agree on overlaps, so they glue on $X\setminus Z$. If (i) holds, extract the coefficients of $t$ by Theorem~\ref{thm:normal}. On every such smaller product neighborhood, uniqueness identifies their restrictions with the coefficients of $t^\circ$. This proves (ii). If (ii) holds, \eqref{eq:jetrealization} is a distribution supported on $S$ with the stated normal order. Near every point of $S\setminus A$, its coefficients agree with those of $t^\circ$, so the distributions agree by Theorem~\ref{thm:normal}. Away from $S$ both distributions vanish. These local identities establish equality on $U\setminus A$, proving (i). Finally, two extensions of a coefficient differ by a distribution supported on $Z$. Conversely, any such difference changes no restriction on $U\setminus A$. Coefficient uniqueness proves that \eqref{eq:jetambiguity} is the full ambiguity under the stated support and normal-order conditions. \end{proof} \begin{remark}[The concrete order-three interface] \label{rem:orderthree} For the supported-sector reference in XIV-J, the intended local substitution is \[ r=3,\qquad S=\Sresp,\qquad A=N_U, \] in a specified response-normal chart in which $N_U$ is relatively closed. The copied reference and its collected coefficients are recorded in Section~\ref{sec:reference} \cite{CheungXIVJ}. The support hypothesis is essential. An arbitrary extension may also contain terms supported in $A$ outside $S$; such terms are not described by the response-normal insertions. \end{remark} \begin{remark}[No automatic identification of ambiguity classes] The space \eqref{eq:jetambiguity} is the unrestricted supported ambiguity with the fixed response-normal bound. It is not automatically the partial-diagonal class, a normalized contact class, or a space of admissible maps on presentations. A finite normal-order bound does not make this ambiguity finite-dimensional. For example, when $d\ge1$ and $Z=\{0\}$, already at $r=0$ the distributions \[ \Ejet_0(\partial_x^\nu\delta_0), \qquad \nu\in\N^d, \] give an infinite linearly independent family. A further bound on the tangential order is needed for a finite-dimensional space. \end{remark} \subsection{A general closed-set extension criterion} \begin{theorem}[Uniform local finite-order criterion] \label{thm:closedextension} Let $X\subset\R^d$ be open, let $Z\subset X$ be relatively closed, and let $b^\circ\in\Dp(X\setminus Z)$. Then $b^\circ$ extends to a distribution in $\Dp(X)$ if and only if, for every $K\Subset X$, there exist $C_K>0$ and $M_K\in\N$ such that \begin{equation} \left|\pair{b^\circ}{\phi}\right| \le C_K\|\phi\|_{K,M_K} \label{eq:closedestimate} \end{equation} for all $\phi\in\D(X\setminus Z)$ with $\supp\phi\subset K$. The constants and order in \eqref{eq:closedestimate} are uniform as these supports approach $Z$. The criterion imposes no wavefront or prescribed scaling-degree bound. \end{theorem} \begin{proof} Necessity follows from the finite-order estimate of any extension on the fixed compact set $K$. For sufficiency, choose a locally finite cover of $X$ by relatively compact open sets $U_i$ whose closures $K_i$ are contained in $X$. On the subspace $\D(U_i\setminus Z)\subset\D(U_i)$, the functional $b^\circ$ is bounded by \[ p_i(\phi)=C_{K_i}\|\phi\|_{K_i,M_{K_i}}, \] where tests are extended by zero when needed. The complex Hahn--Banach theorem extends it to a linear functional $B_i$ on $\D(U_i)$ with the same bound. This bound proves that $B_i$ is a distribution on $U_i$. Choose a smooth locally finite partition of unity $(\eta_i)$ subordinate to the cover and define \[ \pair{b}{\phi} =\sum_i\pair{B_i}{\eta_i\phi}. \] Only finitely many terms occur for any fixed compact support, and their estimates show distributional continuity. If $\phi$ is supported in $X\setminus Z$, every term agrees with the original distribution, giving \[ \pair{b}{\phi} =\sum_i\pair{b^\circ}{\eta_i\phi} =\pair{b^\circ}{\phi}. \] Thus $b$ is an extension. \end{proof} \begin{corollary}[An explicit estimate interface for supported extension] \label{cor:jetestimate} The extension in Theorem~\ref{thm:jetextension} exists if and only if every $b_\gamma^\circ$ satisfies \eqref{eq:closedestimate} on $X\setminus Z$. Since there are finitely many $\gamma$, the orders and constants may be chosen common to all coefficients on each fixed compact set. \end{corollary} \begin{proof} Apply Theorem~\ref{thm:closedextension} to each coefficient and then Theorem~\ref{thm:jetextension}. Take the maximum of the finitely many orders and enlarge the finitely many constants. \end{proof} The criterion is applicable to a general closed locus, not only a finite puncture. It does not select an admissible characteristic prescription or provide microlocal control. Those are separate tasks; compare the additional scaling and wavefront hypotheses in \cite{BF,Dang}. \subsection{Insertion wavefront bounds do not become output bounds} \begin{proposition}[A local insertion wavefront upper bound] \label{prop:insertWF} For $b\in\Dp(X)$, \begin{equation} \begin{split} \WF(\Ejet_\gamma b)\subset \bigl\{(x,0;\xi,\eta):\;& (x,\xi)\in \WF(b)\cup(\supp b\times\{0\}),\\ &\eta\in\R^m,\quad(\xi,\eta)\ne0 \bigr\}. \end{split} \label{eq:insertWF} \end{equation} \end{proposition} \begin{proof} The insertion is $b\otimes\partial_q^\gamma\delta_0$. The tensor-product wavefront bound, with zero covectors adjoined on the supports of the factors, gives \eqref{eq:insertWF}. Differentiation does not enlarge that bound \cite[Section~8.2]{Hormander}. \end{proof} An extension of $b^\circ$ may introduce tangential wavefront directions at $Z$. Therefore coefficient extendability alone is not a complete-graph wavefront theorem, and an insertion bound is not an output bound after the missing characteristic composition and pushforward. \section{Natural null defects and point-puncture corrections} \label{sec:relations} \subsection{The punctured module and its natural lift} Retain $U=X\times V_q$, let $Z\subset X$ be relatively closed, and set \[ X^\circ=X\setminus Z,\qquad U^\circ=X^\circ\times V_q. \] Fix $T^\circ\in\Dp(X^\circ)$ and use the presentation space \[ P=\bigoplus_{|\alpha|\le r}C^\infty(U). \] Thus the coefficients are smooth across $Z$; we are not using the larger space of arbitrary smooth coefficients on $U^\circ$. Define \[ \Rec^\circ(a) =\sum_{|\alpha|\le r} a_\alpha|_{U^\circ}\,\Ejet_\alpha T^\circ, \qquad H^\circ=\ker\Rec^\circ. \] The complete criterion of Theorem~\ref{thm:jetnull} reads \begin{equation} a\in H^\circ \quad\Longleftrightarrow\quad f_\gamma(a)T^\circ=0 \text{ on }X^\circ\text{ for every }|\gamma|\le r. \label{eq:puncturednull} \end{equation} If $T_0\in\Dp(X)$ extends $T^\circ$, the natural preliminary lift is \begin{equation} F_{T_0}(a)=\sum_{|\alpha|\le r}a_\alpha\Ejet_\alpha T_0. \label{eq:naturallift} \end{equation} For a null presentation, \begin{equation} F_{T_0}(a) =\sum_{|\gamma|\le r} \Ejet_\gamma\bigl(f_\gamma(a)T_0\bigr), \qquad \supp\bigl(f_\gamma(a)T_0\bigr)\subset Z. \label{eq:naturaldefect} \end{equation} Consequently the natural null defect is supported on $Z\times\{0\}$. \begin{remark}[Why the word ``natural'' matters] The formula \eqref{eq:naturaldefect} concerns the generator-compatible lift \eqref{eq:naturallift}. An arbitrary linear preliminary lift may assign nonzero values even to universal presentation identities. The relation classification below is therefore not a theorem about every possible preliminary lift without additional compatibility assumptions. \end{remark} \subsection{Finite-jet localization of the defect} \begin{lemma}[Finite-jet dependence along a smooth missing locus] \label{lem:jetdefect} Suppose $Z$ is locally a smooth embedded submanifold of $X$. In a product coordinate neighborhood $(s,h)$ with $Z=\{h=0\}$, let $T_0$ have distributional order at most $M$ on a fixed compact coordinate neighborhood. If $f\in C^\infty(X)$ satisfies $fT^\circ=0$ off $Z$ and \begin{equation} \partial_h^\nu f(s,0)=0 \quad\text{for }|\nu|\le M, \label{eq:vanishingjet} \end{equation} then $fT_0=0$ on the interior of that neighborhood. Thus, among annihilator functions, $fT_0$ depends locally only on the transverse jet of $f$ through order $M$. In particular, an annihilator flat along $Z$ produces no natural defect. \end{lemma} \begin{proof} Let $\phi$ be a test supported in a smaller compact coordinate neighborhood. The distribution $u=fT_0$ is supported on $Z$. Choose $\rho\in\D(\R^{\dim h})$ equal to one near zero and write $\rho_\epsilon(h)=\rho(h/\epsilon)$. Since $u$ is supported on $Z$, \[ \pair{u}{\phi} =\pair{T_0}{f\rho_\epsilon\phi}. \] Taylor's theorem and \eqref{eq:vanishingjet} imply, uniformly on the fixed compact set, \[ |\partial_s^\mu\partial_h^\nu f(s,h)| \le C_{\mu,\nu}|h|^{M+1-|\nu|} \qquad (|\mu|+|\nu|\le M). \] The derivatives of $\rho_\epsilon$ contribute the corresponding powers of $\epsilon^{-1}$. The Leibniz rule therefore gives \[ \|f\rho_\epsilon\phi\|_{K,M} \le C_{f,\phi}\epsilon \] for sufficiently small $\epsilon$. The order-$M$ estimate for $T_0$ implies $\pair{u}{\phi}=0$ on letting $\epsilon\downarrow0$. If two annihilator functions have the same transverse jet through order $M$, their difference satisfies \eqref{eq:vanishingjet}, so their defects agree. Flatness implies vanishing of every finite transverse jet. \end{proof} \begin{definition}[Relation filtration for natural lifts] \label{def:relationfiltration} In the preceding smooth-locus setting, put \[ H_{\mathrm{univ}} =\{a:f_\gamma(a)=0\text{ on }X\text{ for every }\gamma\}, \] and \[ H_{\mathrm{flat}} =\{a\in H^\circ: f_\gamma(a)\text{ is flat along }Z \text{ for every }\gamma\}. \] Then \[ H_{\mathrm{univ}} \subset H_{\mathrm{flat}} \subset H^\circ. \] The successive categories are called universal relations, non-universal flat-annihilator relations, and potentially jet-detected relations. \end{definition} \begin{corollary}[Which relations can contribute to a natural defect] \label{cor:relationclassification} The null defect of $F_{T_0}$ vanishes on $H_{\mathrm{flat}}$, and hence factors algebraically through $H^\circ/H_{\mathrm{flat}}$. Locally, its coefficient defects depend on finite transverse jets of the functions $f_\gamma(a)$. These categories form a filtration, not a canonical direct-sum decomposition. A potentially jet-detected relation need not have a nonzero defect. \end{corollary} \begin{proof} Universal relations vanish for every generator-compatible lift by \eqref{eq:naturaldefect}. Flat-annihilator relations vanish locally by Lemma~\ref{lem:jetdefect}, hence globally by locality. The factorization follows. The same lemma proves finite-jet dependence. No argument asserts nonvanishing for every remaining relation. \end{proof} For a positive-dimensional $Z$, the transverse jets are functions along $Z$, and supported coefficient defects can have distributional tangential dependence. They are not generally finite-dimensional data. This formulation makes precise, for the stated module and natural lifts, the relation distinctions discussed in the supplied XV-E account \cite{CheungXVE}. It does not evaluate the inherited graph's relative obstruction. \subsection{A simultaneous correction at one point} \begin{theorem}[Point-puncture annihilator-preserving extension] \label{thm:pointcorrection} Let $X\subset\R^d$ be open with $0\in X$, and let $T^\circ\in\Dp(X\setminus\{0\})$ have a distributional extension $T_0\in\Dp(X)$. Define its smooth annihilator ideal \[ I=\{f\in C^\infty(X):fT^\circ=0 \text{ on }X\setminus\{0\}\}. \] Then there exists \begin{equation} D=\sum_{|\nu|\le M}c_\nu\partial^\nu\delta_0 \label{eq:pointD} \end{equation} for some finite $M$ such that \[ T=T_0-D \] still extends $T^\circ$ and satisfies \begin{equation} fT=0\qquad(f\in I). \label{eq:allannihilators} \end{equation} One may take $M$ to be a distributional order bound for $T_0$ on a compact neighborhood of zero. \end{theorem} \begin{proof} Fix such an order bound $M$. For every $f\in I$, the distribution $fT_0$ is supported at zero. If $j_0^M f=0$, the point case of Lemma~\ref{lem:jetdefect} gives $fT_0=0$. Let \[ J_0^M =C^\infty(X)/\{f:j_0^M f=0\} \] be the finite-dimensional algebra of jets through order $M$, and let $I_M=j_0^M(I)$ be the image ideal in that algebra. Choose $\chi\in\D(X)$ equal to one near zero and define \[ \ell(j_0^M f)=\pair{fT_0}{\chi}, \qquad f\in I. \] This is well defined: representatives with the same jet differ by an element of $I$ with vanishing $M$-jet, whose product with $T_0$ is zero. The value is independent of the chosen $\chi$, since $fT_0$ is supported at zero. Extend the linear functional $\ell$ from $I_M$ to a linear functional $\Lambda$ on the finite-dimensional vector space $J_0^M$. Define \[ \pair{D}{\phi}=\Lambda(j_0^M\phi). \] This is a distribution supported at zero. In the jet basis $e_\nu=[x^\nu/\nu!]$, it has the explicit form \[ D=\sum_{|\nu|\le M} (-1)^{|\nu|}\Lambda(e_\nu)\partial^\nu\delta_0, \] which is \eqref{eq:pointD}. For $f\in I$ and $\phi\in\D(X)$, the product $f\phi$ is also in $I$. Because $(f\phi)T_0$ is supported at zero, \[ \begin{split} \pair{fT_0}{\phi} &=\pair{(f\phi)T_0}{\chi}\\ &=\ell(j_0^M(f\phi))\\ &=\Lambda(j_0^M f\cdot j_0^M\phi)\\ &=\pair{fD}{\phi}. \end{split} \] Thus $f(T_0-D)=0$ for every $f\in I$. Subtracting $D$ changes no restriction off zero. \end{proof} \begin{corollary}[Descending natural lift for the point-punctured module] \label{cor:pointdescent} In the module of Section~\ref{sec:relations}, with $Z=\{0\}$, if $T^\circ$ has any distributional extension, it has an extension $T$ for which $F_T$ annihilates all of $H^\circ$. Hence $F_T$ descends to a map on $\im\Rec^\circ$ extending every reconstructed value. Relative to $F_{T_0}$, the correction is explicitly \begin{equation} F_T-F_{T_0}=-\Rec_D, \qquad \Rec_D(a)=\sum_{|\alpha|\le r}a_\alpha\Ejet_\alpha D, \label{eq:pointmapcorrection} \end{equation} where $D$ is given by \eqref{eq:pointD}. Its values are supported at $(0,0)$, with response-normal order at most $r$ and tangential order at most $M$. \end{corollary} \begin{proof} For $a\in H^\circ$, equation~\eqref{eq:puncturednull} says that every $f_\gamma(a)$ belongs to $I$. By Theorem~\ref{thm:pointcorrection}, $f_\gamma(a)T=0$ for every $\gamma$. Theorem~\ref{thm:jetnull} therefore gives $F_T(a)=0$. Proposition~\ref{prop:descent} gives the factorization. Equation~\eqref{eq:pointmapcorrection} follows from $T=T_0-D$. The support and order bounds follow from the insertion formula and smooth multiplication. \end{proof} \begin{remark}[Scope of the point result] The correction is simultaneous for the entire smooth annihilator ideal, not a separate choice for each null relation. It uses the finite-dimensional jet algebra of one point. The result assumes extendability of $T^\circ$; it does not create an extension for a nonextendable coefficient. It does not prove preservation of a specified scaling degree, a frozen normalization, or a restricted physical contact class. The order $M$ depends on the chosen local extension; it is not bounded by the response-normal order $r$ alone. The presentation maps in this local construction are continuous for the usual smooth coefficient topology, since they multiply fixed finite-order distributions by smooth coefficients. No identification with the physical carrier topology follows. \end{remark} \section{Nonextendability and flat multipliers} \label{sec:counterexamples} \begin{proposition}[An explicit nonextendable coefficient] \label{prop:nonextendable} For any $a>0$, the smooth function on the punctured interval \[ b_a^\circ(x)= \begin{cases} e^{a/x^2},&x>0,\\ 0,&x<0 \end{cases} \] defines a distribution on $(-1,1)\setminus\{0\}$ with no extension in $\Dp((-1,1))$. \end{proposition} \begin{proof} Suppose an extension $B$ exists. On a fixed compact neighborhood of zero, there are $C>0$ and $M\in\N$ such that \[ |\pair{B}{\phi}| \le C\max_{0\le j\le M}\|\phi^{(j)}\|_\infty \] for tests supported in that neighborhood. Choose a nonnegative nonzero $\phi\in\D((0,1))$ and set \[ \phi_\eta(x)= \phi\!\left(\frac{x-\eta}{\eta^3}\right), \qquad 0<\eta\ll1. \] Its support lies in $(\eta,\eta+\eta^3)$ and \[ \max_{0\le j\le M}\|\phi_\eta^{(j)}\|_\infty \le C_\phi\eta^{-3M}. \] For small $\eta$, the support lies in the fixed compact neighborhood and satisfies $x\le2\eta$. Since $B$ agrees there with $b_a^\circ$, \[ \begin{split} \pair{B}{\phi_\eta} &=\eta^3\int_0^1 e^{a/(\eta+\eta^3u)^2}\phi(u)\dd u\\ &\ge\eta^3e^{a/(4\eta^2)} \int_0^1\phi(u)\dd u. \end{split} \] The lower bound exceeds every fixed power of $\eta^{-1}$ as $\eta\downarrow0$, contradicting the distributional estimate. \end{proof} \begin{corollary}[Finite normal order alone does not ensure extension] \label{cor:normalnotextension} A punctured supported distribution of normal order zero need not have a supported normal-order-preserving extension. \end{corollary} \begin{proof} Take \[ U=(-1,1)\times V_q,\quad S=(-1,1)\times\{0\},\quad A=\{0\}\times V_q, \] and set $t^\circ=\Ejet_0 b_1^\circ$ on $U\setminus A$. Theorem~\ref{thm:jetextension} would turn an extension of the required support and normal order into an extension of $b_1^\circ$, contradicting Proposition~\ref{prop:nonextendable}. \end{proof} \begin{proposition}[Flat multipliers can fail to repair extension] \label{prop:flatcounterexample} Define \[ v(x)= \begin{cases} e^{-2/x^2},&x\ne0,\\ 0,&x=0. \end{cases} \] Then $v$ is smooth and flat at zero, but \[ v\,b_3^\circ=b_1^\circ \] is nonextendable. \end{proposition} \begin{proof} Every derivative of $e^{-2/x^2}$ is a finite sum of powers of $x^{-1}$ multiplied by $e^{-2/x^2}$, and tends to zero at zero. The same estimates after division by any fixed power of $x$ give the smooth flat extension. The product identity holds on the punctured interval, and Proposition~\ref{prop:nonextendable} applies. \end{proof} \begin{remark}[Flatness is not a general impossibility theorem] The correct conclusion is that flatness alone does not guarantee extendability of a product. It is false that a flat multiplier can never repair nonextendability. For the same $v$, \[ v\,b_1^\circ(x)= \begin{cases} e^{-1/x^2},&x>0,\\ 0,&x<0 \end{cases} \] has a smooth flat extension obtained by assigning zero at the origin. Flat-annihilator relations in Corollary~\ref{cor:relationclassification} concern the different hypothesis $fT^\circ=0$ together with an already existing extension. They must not be confused with multiplying a nonextendable coefficient by a flat function. \end{remark} \begin{proposition}[Bounded continuous coefficients extend distributionally] \label{prop:bounded} Every bounded continuous function on $(-1,1)\setminus\{0\}$ has an $L^\infty$ representative on $(-1,1)$ and therefore an order-zero distributional extension. A continuous extension need not exist. \end{proposition} \begin{proof} Assign any value at zero. The resulting measurable function is bounded, and \[ \phi\longmapsto\int_{-1}^1 f(x)\phi(x)\dd x \] is a distribution extending the punctured one. Its definition is independent of the value assigned at one point. The bounded smooth function $\sin(1/x)$ has no limit at zero, so it has no continuous extension although its distributional extension exists. \end{proof} These examples distinguish pointwise continuity, distributional extendability, and extendability with a specified support or microlocal condition. None evaluates a coefficient of the inherited six-branch graph. \section{The retained target and the available inherited formulas} \label{sec:application} \subsection{Full branch content and the characteristic locus} The retained internal sum is \begin{equation} \Kret=K_{47}+K_5+K_6+K_8+K_9, \qquad K_5=K_5^{[a]}+K_5^{[z]}. \label{eq:sum} \end{equation} Here $K_{47}$ groups $B_4+B_7$, while both placements of $K_5$ belong to $B_5$. Thus five named summands represent the six original branches $B_4,\ldots,B_9$. The supported sector is only \begin{equation} \Ksup=K_6+K_8+K_9. \label{eq:supported} \end{equation} It contains neither $K_{47}$ nor either $K_5$ placement \cite{CheungXIVI,CheungXVA,CheungXVE}. The supplied XIV-I record retains attachment variables \[ W=\Omega_A\times\Omega_B\times\Omega_Z\times\Omega_Y, \qquad w=(a,b,z,y), \] on an open patch of $\R^{16}$, and identifies the characteristic stratum by \begin{equation} N= \left\{ \begin{array}{l} a=z,\quad y=a+\alpha\ell_{\mathbf n},\quad b=a+\beta\ell_{\mathbf n},\\ 0<\alpha<\beta,\quad \ell_{\mathbf n}=(1,\mathbf n),\quad\mathbf n\in S^2 \end{array} \right\}. \label{eq:N} \end{equation} This is inherited geometric data \cite{CheungXIVI}. \begin{proposition}[The retained stratum is not a point puncture] \label{prop:dimension} On the stated nondegenerate patch, the stratum in \eqref{eq:N} has dimension eight and codimension eight in $W$. \end{proposition} \begin{proof} The parameters are $a\in\R^4$, the two scalars $\alpha,\beta$, and $\mathbf n\in S^2$, giving dimension $4+2+2=8$. The displayed parametrization has a smooth inverse on its image: \[ a=a,\quad \alpha=y^0-a^0,\quad \beta=b^0-a^0,\quad \mathbf n=\frac{\mathbf y-\mathbf a}{\alpha}. \] The positivity of $\alpha$ makes the inverse smooth. Thus the parametrization is an embedding on the retained patch. Since $\dim W=16$, the codimension is eight. \end{proof} There is no supplied identification reducing the relevant characteristic coefficient locus to a single point. Accordingly, Theorem~\ref{thm:pointcorrection} is not applied to this graph. The general closed-set estimate interface is the applicable abstract starting point. Any use of Theorem~\ref{thm:jetextension} must also verify that the chosen $N_U$ is relatively closed in its chart. \subsection{Internal lines and tensor formulas} The supplied coordinate-response relation is \begin{equation} L^{\mu,I}(s,a) =\sigma_\eps(a)\mathsf A^{\lambda\mu,I} \partial_{a^\lambda}R(s-a), \label{eq:L} \end{equation} with the source's fixed smooth switch and tensor map \cite{CheungXIVH,CheungXIVI}. This recovers a definition omitted from the original XV-B excerpt. It does not evaluate the tensor map by identifying it with an unrelated candidate formula. The two $K_5$ placements are \begin{align} K_5^{[a],I,J}(a,b,z) &=\partial_{b^\mu}\Bigl[ f(b)\sigma_\eps(a) \partial_{a^\alpha}R(b-a)\, \mathsf A^{\alpha\beta,I} \partial_{a^\beta}L^{\mu,J}(a,z) \Bigr], \label{eq:K5a}\\ K_5^{[z],I,J}(a,b,z) &=\partial_{b^\mu}\Bigl[ f(b)\sigma_\eps(z) \partial_{z^\alpha}R(b-z)\, \mathsf A^{\alpha\beta,J} \partial_{z^\beta}L^{\mu,I}(z,a) \Bigr]. \label{eq:K5z} \end{align} The grouped kernel and the local supported term are \begin{equation} \begin{split} K_{47}^{I,J}(a,b,z) =-\partial_{b^\mu}\Bigl[ \bigl( f(b)\partial_{b^\nu}L^{\mu,J}(b,z) -\delta^\mu_\nu\partial_{b^\rho} (f(b)L^{\rho,J}(b,z)) \bigr)L^{\nu,I}(b,a) \Bigr], \end{split} \label{eq:K47} \end{equation} \begin{equation} K_6^{I,J}(a,b,z) =\partial_{b^\mu}\Bigl[ f(b)\sigma_\eps(a)\mathsf D^{\alpha\mu,I,J} \partial_{a^\alpha}R(b-a)\delta^{(4)}(a-z) \Bigr]. \label{eq:K6} \end{equation} The tensor $\mathsf D$ is fixed by the source identity \begin{equation} \dot a_2^{\alpha\mu}[G;h] =\mathsf D^{\alpha\mu,I,J}h_I G_J. \label{eq:Didentity} \end{equation} These formulas are inherited internal formulas \cite{CheungXIVH,CheungXIVI}. Equation~\eqref{eq:K47} retains the grouped coefficient; no canceled explicit scalar-Hessian term is reintroduced. \begin{proposition}[Independent-variable tree products] \label{prop:tree} For any $R\in\Dp(\R^4)$, the products \begin{equation} R(b-a)R(a-z),\qquad R(b-z)R(z-a),\qquad R(b-a)R(b-z) \label{eq:treeproducts} \end{equation} are defined distributions in $(a,b,z)$ by diffeomorphic pullback of tensor products. The product \[ \partial^\nu R(b-a)\delta^{(4)}(a-z) \] is defined by the same independent-variable construction. Passive residual variables may be retained. \end{proposition} \begin{proof} For the first product use \[ F_1(a,b,z)=(a,b-a,a-z), \qquad F_1^{-1}(a,r,u)=(a,a+r,a-u). \] It is a diffeomorphism, and the product is $F_1^*(1\otimes R\otimes R)$. For the second use \[ F_2(a,b,z)=(z,b-z,z-a), \qquad F_2^{-1}(z,r,u)=(z-u,z+r,z). \] For the third use \[ F_3(a,b,z)=(b,b-a,b-z), \qquad F_3^{-1}(b,r,u)=(b-r,b,b-u). \] The respective tensor products exist, and all pullbacks are by diffeomorphisms. Finally, \[ F_1^*(1\otimes\partial^\nu R\otimes\delta_0) \] gives the supported product. Adding passive variables corresponds to adjoining a smooth factor $1$. \end{proof} Tensor products and diffeomorphic pullbacks are standard distributional operations \cite{Hormander}. The first two constructions agree with the inherited XIV-H proof and the XV-A audit \cite{CheungXIVH,CheungXVA}. \begin{corollary}[Finite internal differential expressions] \label{cor:internalorders} With the smooth weights and fixed tensor maps in \eqref{eq:L}--\eqref{eq:K6}, each $K_5$ placement and $K_{47}$ is a finite differential expression in its respective independent tree product, of total kernel derivative order at most four. The displayed $K_6$ expression has total kernel derivative order at most two. \end{corollary} \begin{proof} Insert \eqref{eq:L}. In a $K_5$ placement, the derivative of the incoming line has order one, the differentiated $L$ has kernel order at most two, and the outer divergence adds at most one. For $K_{47}$, the differentiated $L$ has order at most two, the other $L$ order one, and the outer divergence adds at most one. Derivatives falling on smooth weights only lower these upper kernel derivative counts. In $K_6$, the incoming derivative has order one and the outer divergence adds at most one. All products are defined by Proposition~\ref{prop:tree}. \end{proof} These counts are internal derivative counts, not transverse scaling degrees of the quantum composition. No new multiplication by singular cross-lines follows from them. \subsection{The fixed internal pole} For $s\in\{+1,-1\}$, let \[ \rho>0,\qquad \lambda_s=\omega-s\rho,\qquad |\lambda_s|