% !TeX program = pdflatex %===================================================================== % XIV-J.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XIV-J.tex % pdflatex -interaction=nonstopmode -halt-on-error XIV-J.tex % % Exact release: % - the frozen pair, inverse -1, source representative, switches, % width, partial-diagonal class, and residual variable are retained; % - XIV-I's internal amplitude construction is copied as a premise; % - a joint map is required to descend through internal reconstruction; % - an explicit reconstruction identity shows why an amplitude % presentation is not itself an intrinsic joint symbol; % - the supported response sector K_6+K_8+K_9 has an exact finite % normal-jet form; % - its grouped cross-line composition is constructed on W\N locally, % with an explicit normal-jet formula and wavefront bound; % - this reference construction is limited to that supported sector; % - it does not identify the canonical domain of the full six-branch % coefficient with a characteristic complement; % - a conditional coefficient-extension interface is proved; % - no extension coefficients, common joint map, or joint section % for the full six-branch coefficient are constructed. % % Ending: % Constructed for the named off-N supported-sector reference. % Unresolved at J2 for the required full joint characteristic map. % J3--J5 are not entered. No R_1^(1) is admitted. % XV-B remains closed. % % Debugging: % - reconstruction may identify distinct amplitude presentations; % - a joint map must respect these identities; % - the codimension-five response incidence is distinguished from % the codimension-eight characteristic test locus; % - its relative codimension-three flag is bookkeeping only, not a % replacement characteristic extension problem; % - no internal frequency order is used as a transverse degree; % - reference values and extension values are separated; % - no ordinary product theorem is retried at the retained cancellation. % % No successful compilation, proof-assistant verification, % independently executed symbolic computation, independent referee % review, or journal acceptance is asserted. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{inputstatement}[theorem]{Input statement} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\sig}{\sigma_\eps} \newcommand{\PP}{\mathcal P} \newcommand{\ER}{E_0^{\mathrm R}} \newcommand{\EA}{E_0^{\mathrm A}} \newcommand{\ERm}{E_m^{\mathrm R}} \newcommand{\EAm}{E_m^{\mathrm A}} \newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}} \newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\EBV}{\mathscr E_H^{\mathrm{BV}}} \newcommand{\Vcal}{\mathcal V_g} \newcommand{\Ocal}{\mathcal O} \newcommand{\Aone}{\mathcal A_1} \newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}} \newcommand{\CD}{\mathcal C_{3,\mathrm D}} \newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\rD}{r_{\mathrm D}} \newcommand{\rg}{r_g} \newcommand{\HT}{H_T} \newcommand{\Rtree}{R_1^{(0)}} \newcommand{\Rquant}{R_1^{(1)}} \newcommand{\Cpart}{\mathfrak C_{\mathrm{part}}} \newcommand{\Ctot}{\mathfrak C_{\mathrm{tot}}} \newcommand{\Nchar}{\mathcal N^\circ} \newcommand{\Nbar}{\overline{\mathcal N}_{\mathrm{att}}} \newcommand{\Nlift}{\widetilde{\mathcal N}_{\mathrm{att}}} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\Mret}{\mathcal M_{\mathrm{ret}}} \newcommand{\Ucan}{U_{\mathrm{can}}} \newcommand{\Tform}{\mathcal T_{\mathrm{form}}} \newcommand{\Rker}{R} \newcommand{\Bker}{B} \newcommand{\Amap}{\mathsf A} \newcommand{\bmap}{\mathsf b} \newcommand{\Lresp}{\Lambda_{\mathrm{resp}}} \newcommand{\Sresp}{S_{\mathrm{resp}}} \newcommand{\Ksup}{K_{\mathrm{supp}}} \newcommand{\Cross}{\mathscr C_{\mathrm{cross}}} \newcommand{\Kint}{\mathscr K_{\mathrm{int}}} \newcommand{\Ppres}{\mathscr P_{\mathrm{int}}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Mmap}{\mathsf M} \newcommand{\Jjet}{\mathsf J} \newcommand{\Ejet}{\mathsf E} \newcommand{\PV}{\operatorname{PV}} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Contr}{Contr} \DeclareMathOperator{\EL}{EL} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Int}{int} \newcommand{\scopebox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \title{\textbf{Landmark XIV-J}\\[0.4em] The joint characteristic map on the unchanged carrier\\[0.4em] \large Reconstruction descent, a supported-sector normal-jet reference,\\ and the exact J2 stop before quantum admission} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We enter the joint-map obligation left unresolved by XIV-I. The internal retarded amplitude module, including both $K_5$ placements, is a copied prerequisite. It is not identified with the required composition against the grouped quantum cross-lines. Two additional interfaces are established. First, a joint map on reconstructed kernels must annihilate the kernel of internal reconstruction. An explicit normal-distribution identity shows that the highest polynomial degree of an amplitude presentation need not descend to an intrinsic symbol of its reconstructed kernel. Second, the supported response sector $K_6+K_8+K_9$ is put into an exact finite normal-jet form on the codimension-five response incidence. Its grouped cross-line composition is constructed on the local complement of the codimension-eight characteristic locus. The construction retains every normal derivative and both terms of the scalar root, and has an explicit wavefront bound. It is a reference theorem for that sector, not a composition theorem for the full six-branch sum. A conditional normal-jet extension interface identifies the additional coefficient distributions that a supported realization would require. None is selected or constructed across the characteristic locus. No generalized joint composition for the full sum, complete joint section, transverse estimate, contact gluing, or output admission is supplied. The first unresolved gate remains J2. No $R_1^{(1)}$ is admitted in $\mathscr E_H$, and XV-B remains closed. \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) {Normal-jet response}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Reference composition off the locus}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Limit on the aligned locus}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The off-locus reference is not the limit on the aligned-null locus.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The reference composition stays off the null locus. 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-XIV-J} \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) {Normal-jet response}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Limit on the aligned locus}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The off-locus reference is not the limit on the aligned-null locus.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The reference composition stays off the null locus. 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-XIV-J} \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) {Normal-jet response}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Reference composition off the locus}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Limit on the aligned locus}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The off-locus reference is not the limit on the aligned-null locus.}; \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-XIV-J} \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) {Normal-jet response}; \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) {Limit on the aligned locus}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Reference composition off the locus. The off-locus reference is not the limit on the aligned-null locus.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Normal-jet response is not Limit on the aligned locus. 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-XIV-J} \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: Normal-jet response}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Reference composition off the locus}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Limit on the aligned locus}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The off-locus reference is not the limit on the aligned-null locus.}; \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-XIV-J} \end{figure} Figure~\ref{fig:split-XIV-J} records one view of the same separation. Figure~\ref{fig:chain-XIV-J} records one view of the same separation. Figure~\ref{fig:pair-XIV-J} records one view of the same separation. Figure~\ref{fig:gate-XIV-J} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XIV-J} records that separation. The drawing adds no hypothesis. \scopebox{Exact release}{% \textbf{Constructed:} the reconstruction-descent criterion; the exact normal-jet representation of the supported response sector; its grouped reference composition away from $N$; and a conditional normal-jet extension interface. \smallskip \textbf{First missing gate: J2.} The required map for the complete reconstructed six-branch sum on tests meeting $N$ has not been supplied. \smallskip \textbf{Not released:} a joint principal-symbol section, a complete quantum remainder, extension obligations, or an admitted $R_1^{(1)}$. \smallskip \textbf{XV-B remains closed.} The free carrier recorded in XV-A cannot replace the missing analytic coefficient.} \tableofcontents %===================================================================== \section{J0: copied premises and their domains} \label{sec:J0} %===================================================================== The manuscript inputs below are used with their stated domains. Their inclusion is not an independent verification of every predecessor claim. No pole identity, incidence inventory, elementary phase, or transition geometry is re-proved. \subsection{XIV-I's eight handoff items} \begin{inputstatement}[XIV-I successor package] \label{input:Ipackage} A successor receives: \begin{enumerate} \item the unchanged XIV-H package; \item the finite retarded-pole module and its exact normalization; \item the pole identity, uniform estimates, and reconstruction map; \item the two complete $K_5$ amplitude maps with $z$ retained; \item their exact finite lower-degree remainder and internal wavefront bounds; \item the common precomposition sum and its canonical-domain value; \item the precise missing I3 joint operator map; \item the still closed I4--I6 and J3--J5 gates. \end{enumerate} The internal amplitude ending is Constructed at I2. The full joint characteristic map is Unresolved at I3. No transverse degree, cancellation theorem, or admitted $\Rquant$ is handed over \cite{XIVI}. \end{inputstatement} \subsection{The internal pole module} \begin{inputstatement}[XIV-I amplitude and reconstruction] \label{input:Imodule} In the Fourier convention \[ \widehat u(\omega,\xi) =\int e^{-i(t\omega+\mathbf x\cdot\xi)} u(t,\mathbf x)\,\dd t\,\dd^3x, \] write \[ \rho=|\xi|,\qquad p(\omega,\xi)=\omega^2-\rho^2, \qquad \lambda_s=\omega-s\rho,\quad s\in\{+1,-1\}. \] On a conic neighborhood \[ \rho>0,\qquad |\lambda_s|a^0>\eps,\qquad b^0>y^0. \] No characteristic configuration is removed. \begin{inputstatement}[XIII-C and XIII-B] \label{input:BC} XIII-C retains four obligations: \begin{enumerate} \item a joint characteristic prescription or complete-pair cancellation; \item uniform transverse estimates at the three- and four-vertex collision strata and their intersections; \item compatible partial contacts and singular spectators; \item output wavefront control after proper pushforward. \end{enumerate} Its parallel temporal kinetic primitive has collision scaling degree eight in codimension four \cite{XIIIC}. The retarded convention is \begin{equation} R_1(\Vcal;\Ocal) =\frac{i}{\hbar} \left(\mathsf T_2^{\mathrm{ren}}(\Vcal,\Ocal) -\Vcal\star_H\Ocal\right). \label{eq:Rconvention} \end{equation} Strict full-support ordering gives zero or the scaled commutator. The selected tree datum is admitted. Properness controls a distributional pushforward after the distribution exists; it does not define a missing joint product \cite{XIIIB}. \end{inputstatement} \begin{inputstatement}[XII-E roots and XII-C source] \label{input:EC} The normalization and filled slots remain \[ \Gret=-\PP\ER,\qquad \rD=\PP\ER\JD,\qquad \rg[\phi]=-\frac12\PP\ER(gT_0[\phi]). \] The roots are \[ \Aone[f,g](r;\phi) =\int\alpha_{f,g}[\phi]\cdot r\,\dd^4x, \qquad \beta_{f,g}[\phi]=-\EA(\PP\alpha_{f,g}[\phi]). \] The scalar-sourced slot is included once \cite{XIIE}. The actual source is \begin{equation} \begin{split} \JD^{\mu\nu} &=\EL_\gamma(\sig\ell_3)^{\mu\nu} =\sig J_3^{\mu\nu}+\mathcal D_\sig^{\mu\nu},\\ \mathcal D_\sig^{\mu\nu} &=2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sig -2M_2^{\mu\nu\rho}\partial_\rho\sig . \end{split} \label{eq:source} \end{equation} The unintegrated cubic representative is fixed before weighting, and the selected action weights only that cubic coefficient \cite{XIIC}. \end{inputstatement} XV-A is recorded at the exit only. XIV-A and XIV-B are not analytic inputs. The value $-8I$ is not a normalization. XIII-E and a resolved paired construction remain unentered. %===================================================================== \section{J1: one joint-map class, fixed before the calculation} \label{sec:J1} %===================================================================== \[ \boxed{ \text{A joint characteristic operator map for the reconstructed six-branch sum against the grouped cross-lines.}} \] The frozen spacetime, inverse, and carrier are \[ M=\R^4,\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta_{\mathbf x}, \] \[ \Gret=-\PP\ER,\qquad \Ealg=(\Fpol[[\hbar,\kappa]],\star_H)/\Nsol[[\hbar,\kappa]]. \] All switches, observations, masses, and widths retain their predecessor values. The pair is \[ \Vcal=-\frac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}[\phi]\,\dd^4x, \qquad \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi), \] with \[ \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD. \] The retained formal target remains \begin{equation} \begin{split} \Tform =i\,\Contr^{\gamma\phi}\Bigl( \Vcal''_{\gamma\phi},\Kret;\ &\HT_{\gamma\gamma}(y,a)\otimes\HT_{\phi\phi}(y,b)\\ &-H_{\gamma\gamma}(y,a)\otimes H_{\phi\phi}(y,b) \Bigr). \end{split} \label{eq:target} \end{equation} \begin{definition}[The joint-map class] \label{def:mapclass} Let $\Kint$ be the space of actual reconstructed internal kernels obtained from the finite presentations in the copied precomposition package, with their tensor indices retained. The class considered here consists of linear operator maps \[ \Mmap:\Kint\longrightarrow \mathcal D'(W;\text{residual species}) \] that: \begin{enumerate}[label=\textup{(\roman*)}] \item on the named reconstructed sum give the specified contraction with the grouped cross-lines and mixed vertex Hessian; \item retain both $K_5$ placements and the residual variable $z$; \item agree with the canonical composition on its proved domain; \item are defined on arbitrary compact tests meeting $N$; \item have a stated and proved complete-graph wavefront bound appropriate to their composition. \end{enumerate} The product data and $\Cpart$ are unchanged. The domain consists of reconstructed kernels, not arbitrary amplitude arrays. No representative-independent operation on all free-shell quotient classes is asserted. A generalized composition belongs to this class only after its relation, amplitude, and remainder hypotheses are supplied for the present target. No common product phase is introduced. \end{definition} \begin{definition}[Endings fixed before the test] The endings are attached to the object at the gate reached: \begin{enumerate} \item \textbf{Constructed}: that object has its domain, amplitude or degree, and proved wavefront control. \item \textbf{Proved obstructed}: an incompatibility excludes every map in Definition~\ref{def:mapclass}. \item \textbf{Unresolved}: the joint map, joint section, prescription datum, or extension estimate is missing. \end{enumerate} The already-copied factorwise failure is not a universal obstruction to this class. \end{definition} The continuation after a successful joint map is fixed as the joint-section and complete-remainder exit. The prescription alternative is not entered. Only the eventual admission line $Q_H\Rquant\in\Ealg$ could open XV-B. %===================================================================== \section{J2a: reconstruction descent is a necessary interface} \label{sec:descent} %===================================================================== \subsection{Presentations and actual kernels} Let $\Ppres$ denote the finite direct sum of the internal coefficient presentation spaces. Its reconstruction map, including summation into the common tensor-valued kernel, is \[ \Rec:\Ppres\longrightarrow\mathcal D'(W). \] Thus \[ \Kint=\operatorname{im}\Rec. \] This is a space of already defined internal distributions. On $\Ucan$, canonical contraction gives a linear map \[ \mathsf C_{\mathrm{can}}: \Kint\longrightarrow\mathcal D'(\Ucan). \] Every identity between reconstructed kernels is respected by this canonical map on its proved domain. \begin{proposition}[Descent criterion for a coefficient-level construction] \label{prop:descent} A linear candidate \[ \widetilde{\Mmap}:\Ppres\longrightarrow\mathcal D'(W) \] defines a map on reconstructed kernels if and only if \begin{equation} \ker\Rec\subseteq\ker\widetilde{\Mmap}. \label{eq:descent} \end{equation} When this holds, the induced map $\Mmap(\Rec A)=\widetilde{\Mmap}(A)$ is unique. Agreement of $\widetilde{\Mmap}$ with canonical contraction on $\Ucan$ proves \eqref{eq:descent} there only. It does not prove that the same identity holds across $N$. \end{proposition} \begin{proof} If $\widetilde{\Mmap}=\Mmap\Rec$, a presentation in $\ker\Rec$ has image zero. Conversely, if two presentations have the same reconstruction, their difference belongs to $\ker\Rec$. Condition \eqref{eq:descent} therefore makes their images equal, so the induced map is well defined and unique. For a null presentation, canonical contraction of its zero reconstruction is zero on $\Ucan$. That statement only determines the restriction of its proposed global image. A distribution supported outside $\Ucan$ is not detected by this restriction. \end{proof} \subsection{An exact null presentation} The following identity concerns the copied internal module. It introduces no cross-line product. \begin{lemma}[A normal identity invisible to polynomial-degree bookkeeping] \label{lem:null} In an adapted $K_5$ chart, \begin{equation} \Rec\bigl((q\,i\theta+1)M_{\mathrm R}(\kappa)\bigr)=0. \label{eq:null} \end{equation} Nevertheless the two reconstructed summands are separately nonzero: \[ \Rec(q\,i\theta M_{\mathrm R}) =-\delta(q)\Rker(r), \qquad \Rec(M_{\mathrm R})=\delta(q)\Rker(r). \] Consequently the polynomial-degree splitting of a presentation does not by itself define an intrinsic joint-symbol splitting on $\Kint$. \end{lemma} \begin{proof} For a one-dimensional test $\psi$, \[ \langle q\delta'(q),\psi\rangle =-\partial_q(q\psi(q))\big|_{q=0} =-\psi(0). \] Hence $q\delta'=-\delta$. Tensoring with the existing nonzero distribution $\Rker(r)$ and using \eqref{eq:Rec} gives all displayed identities. \end{proof} \begin{remark}[The precise debugging correction] The copied internal module retains its proved reconstruction and frequency estimates. Lemma~\ref{lem:null} does not invalidate them. It does show that its polynomial degree is a degree of a presentation. It cannot be promoted to the principal symbol of the reconstructed quantum composition without a descent theorem and the missing joint calculus. In particular, no leading term is selected from an amplitude array before its reconstruction identities are accounted for. \end{remark} This is not an obstruction to every joint map. It is an explicit compatibility condition for any proposed coefficient-level realization of such a map. %===================================================================== \section{J2b: the supported response sector as finite normal jets} \label{sec:normaljets} %===================================================================== \subsection{One local chart in the unchanged common space} Define \[ \Sresp=\{q_R=0,\ a=z\}\subset W, \qquad \Ksup:=K_6+K_8+K_9. \] This is the supported response sector, not the full six-branch kernel. Work locally in a coordinate neighborhood $U\subset W$ of a retained point. The response incidence is regular there by the copied separated-endpoint geometry. Choose coordinates \[ (s,n)\in S_0\times B_5, \qquad n=(q_R,a^0-z^0,\ldots,a^3-z^3), \qquad \Sresp\cap U=\{n=0\}. \] Here $s$ has eleven components and includes the four independent $y$ coordinates. All local distributions are represented relative to the chosen coordinate density. The smooth density conversion from the original flat variables is included in their coefficients. The characteristic set remains \[ N_U=N\cap U. \] It is a submanifold of $\Sresp$ of relative codimension three, since $\dim\Sresp=11$ and $\dim N=8$. \scopebox{The ambient problem is unchanged}{% The flag $N_U\subset\Sresp\subset U$ has codimensions three and five, whose sum is eight. \smallskip This bookkeeping does not replace the target by a codimension-three or codimension-four numerical extension. Every reconstructed distribution still lives in the sixteen-dimensional common space with $z$ retained. \smallskip No transverse degree or extension threshold is assigned using this flag.} \subsection{Normal-jet insertion} For a distribution $b$ on $S_0$ and a multiindex $\alpha\in\mathbb N^5$, define \begin{equation} \langle \Ejet_\alpha b,\varphi\rangle :=(-1)^{|\alpha|} \left\langle b(s),\partial_n^\alpha\varphi(s,0)\right\rangle. \label{eq:Ejet} \end{equation} Thus $\Ejet_\alpha b=b(s)\partial_n^\alpha\delta^{(5)}(n)$ in the chosen coordinates. \begin{lemma}[Continuity and independence of normal jets] \label{lem:jetinsert} For each $\alpha$, \eqref{eq:Ejet} defines a continuous map \[ \Ejet_\alpha:\mathcal D'(S_0)\longrightarrow\mathcal D'(U). \] For any finite $m$, the map \[ (b_\alpha)_{|\alpha|\le m} \longmapsto\sum_{|\alpha|\le m}\Ejet_\alpha b_\alpha \] is injective. If $b_\alpha$ is supported in a set $L\subset S_0$, its insertion is supported in $L\times\{0\}$. \end{lemma} \begin{proof} The map $\varphi\mapsto\partial_n^\alpha\varphi(s,0)$ is continuous from compactly supported tests in $U$ to compactly supported tests in $S_0$. Transposition defines \eqref{eq:Ejet} continuously and gives its support. For independence choose $\chi(n)$ compactly supported and equal to one near zero, and use the tests \begin{equation} \Jjet_\gamma\psi(s,n) =\psi(s)\frac{n^\gamma}{\gamma!}\chi(n). \label{eq:Jjet} \end{equation} For every $|\alpha|\le m$, \[ \partial_n^\alpha\Jjet_\gamma\psi(s,0) =\begin{cases} \psi(s),&\alpha=\gamma,\\ 0,&\alpha\ne\gamma. \end{cases} \] These tests recover each $b_\gamma$ separately. \end{proof} \begin{proposition}[Exact finite normal form of the response sector] \label{prop:normalform} In the chart above, the supported response sector has an exact componentwise representation \begin{equation} \Ksup=\sum_{|\alpha|\le3}\Ejet_\alpha v_\alpha, \qquad v_\alpha\in C^\infty(S_0). \label{eq:normalform} \end{equation} The coefficient array is unique in the fixed coordinates. Every term of $K_6,K_8,K_9$ is retained, including derivatives of their smooth coefficients, the actual source coefficients, and the mass term of $K_9$. No equality or vanishing of these coefficients is assumed. \end{proposition} \begin{proof} The copied response formulas express $K_8,K_9$ as finite differential expressions of order at most three in smooth coefficients times $\delta(q_R)\delta^{(4)}(a-z)$. Formula~\eqref{eq:K6} has the same seed and differential order at most two. Change to the regular coordinates $(s,n)$. The chain rule expresses every derivative as a finite sum of smooth-coefficient tangential and normal derivatives. Tangential derivatives act on coefficients. The remaining normal derivatives of the seed have order at most three. For a smooth coefficient $c(s,n),$ the finite identity \begin{equation} c(s,n)\partial_n^\alpha\delta(n) = \sum_{\beta\le\alpha} (-1)^{|\beta|}\binom{\alpha}{\beta} \bigl(\partial_n^\beta c\bigr)(s,0) \partial_n^{\alpha-\beta}\delta(n) \label{eq:smoothjetproduct} \end{equation} follows by applying the Leibniz rule to its test-function pairing. It brings every coefficient into the stated normal form without dropping any derivative. Summing the three kernels gives \eqref{eq:normalform}. Uniqueness is Lemma~\ref{lem:jetinsert}. \end{proof} \subsection{Wavefront control of insertion} For $b\in\mathcal D'(S_0)$ let \[ \WF_0(b):=\WF(b)\cup\{(s,0):s\in\supp b\}. \] Define \begin{equation} \mathcal L(b) =\left\{ (s,0;\eta,\nu): (s,\eta)\in\WF_0(b),\ \nu\in\R^5,\ (\eta,\nu)\ne0 \right\}. \label{eq:liftcone} \end{equation} \begin{lemma}[Insertion wavefront bound] \label{lem:insertWF} For every $\alpha$, \[ \WF(\Ejet_\alpha b)\subseteq\mathcal L(b). \] For a finite normal-jet sum, the corresponding union is a wavefront upper bound. \end{lemma} \begin{proof} The tensor product $b(s)\otimes\delta^{(5)}(n)$ is defined. Its tensor-product wavefront bound is exactly the inclusion in \eqref{eq:liftcone}, with zero covectors adjoined on the supports. Normal differentiation does not enlarge this bound. Finite sums preserve the asserted upper bound \cite[Section 8.2]{Hormander}. \end{proof} These are bounds for actual supported internal kernels. They do not yet involve the quantum cross-lines. %===================================================================== \section{J2c: a grouped reference for the supported sector away from \(N\)} \label{sec:reference} %===================================================================== \subsection{The grouped cross-line distribution} In independent relative variables define \[ \HT_{\gamma\gamma}\otimes\HT_{\phi\phi} -H_{\gamma\gamma}\otimes H_{\phi\phi}. \] It is a tensor-product difference of defined distributions. Its pullback by \[ (a,b,z,y)\longmapsto(y-a,y-b) \] is defined because this map is a submersion. Denote this grouped pullback by $\Cross$. This construction does not multiply $\Cross$ by an internal kernel at $N$. The mixed vertex Hessian is \[ \Vcal''_{\gamma\phi}[h,\psi] =-\int g\,h_{\mu\nu}T^{\mu\nu}(\phi,\psi)\,\dd^4y, \] where $T$ is the inherited polarized stress tensor. Its finite expansion gives: \begin{enumerate} \item constant fiber contractions; \item derivatives on the crossed scalar slot; \item a smooth factor $g(y)$; \item an unused scalar field or its first derivative at $y$. \end{enumerate} Derivatives on crossed slots are applied in the two independent relative variables before the submersion pullback. For each such finite differential operator $D$, write $\Cross_D$ for the resulting grouped cross-line distribution. An eventual transfer of an unused scalar derivative to the output kernel is a $y$ derivative. Since the normal coordinates $n$ are independent of $y$, this is tangential to $\Sresp$. All assertions below therefore apply componentwise to the complete finite vertex expansion. \subsection{The required elementary line information} The proof uses the following property of the frozen lines: at separated endpoints their singularities lie on the null cone and their wavefront covectors are conormal to that cone. Finite differentiation preserves that inclusion. The massive time-ordered line may have smooth interior terms; these do not add singular support. This is the line-wavefront input already used in the predecessor graph calculus. It is also the standard separated line description in the functional framework \cite[Sections 6--7 and Appendix A]{FR}. No replacement Hadamard matrix is chosen. \begin{lemma}[Grouped normal traces off the characteristic locus] \label{lem:traces} On $\Sresp\cap(U\setminus N_U)$, every finite normal trace \begin{equation} c_{D,\beta} :=i^*\bigl(\partial_n^\beta\Cross_D\bigr), \qquad i:s\longmapsto(s,0), \label{eq:traces} \end{equation} is defined. At a singular point of such a trace its tangential wavefront is bounded by a nonzero multiple of $i^*\dd q_1$ or $i^*\dd q_2$. No point in this domain has both $q_1=q_2=0$. \end{lemma} \begin{proof} On $\Sresp$, $q_R=0$ and $a=z$. By the copied zero-locus statement \eqref{eq:zerolocus}, simultaneous vanishing of $q_1,q_2$ would put the point in $N_U$. Thus away from $N_U$ at most one of the two cross-line factors is singular. If neither is singular, the grouped distribution is smooth locally, and all normal traces exist. If only $q_1=0$, the other line is smooth locally. The grouped distribution and its derivatives have wavefront bounded by nonzero multiples of $\dd q_1$. The $y$ component of $\dd q_1$ is nonzero because $y\ne a$. Every conormal of $\Sresp$ has zero $y$ component, since $q_R$ and $a-z$ do not depend on $y$. The restriction normal-set condition therefore holds. The case with only $q_2=0$ is identical, using $y\ne b$. Differentiation does not enlarge the line-wavefront bound. The restriction theorem gives \eqref{eq:traces} and the tangential upper bound \cite[Section 8.2]{Hormander}. \end{proof} This proof is confined to points away from $N_U$. It is not a retry of the ordinary product theorem at the retained opposite covectors. \subsection{Exact normal-jet multiplication formula} \begin{theorem}[Supported-sector reference distribution] \label{thm:reference} For every vertex component $D$, the product \[ \Cross_D\Ksup \] is defined on $U\setminus N_U$. In the fixed normal coordinates its exact normal form is \begin{equation} t_D^\circ = \sum_{\substack{|\alpha|\le3\\\beta\le\alpha}} (-1)^{|\beta|}\binom{\alpha}{\beta} \Ejet_{\alpha-\beta} \bigl(v_\alpha c_{D,\beta}\bigr). \label{eq:reference} \end{equation} Smooth vertex weights, fiber contractions, and output $y$ derivatives may be applied to this distribution. The reference agrees with the canonical grouped composition where that composition was already proved. It includes every normal derivative and both scalar-root terms. Its support is contained in $\Sresp$, subject to the existing weights, and its wavefront set is bounded by \begin{equation} \begin{split} \Gamma_{\mathrm{supp}}^\circ ={}& N^*\Sresp\big|_{U\setminus N_U}\setminus0\\ &\cup \bigcup_{i=1}^2 \left\{ (w,\lambda\,\dd q_i+\omega): \begin{array}{l} w\in\Sresp\setminus N_U,\ q_i(w)=0,\\ \lambda\ne0,\quad\omega\in N_w^*\Sresp \end{array} \right\}\setminus0 . \end{split} \label{eq:referenceWF} \end{equation} This is a local reference theorem for $\Ksup$, not for the full $\Kret$. \end{theorem} \begin{proof} At a point of $\Sresp\setminus N_U$, the proof of Lemma~\ref{lem:traces} bounds the cross-line wavefront by a conormal with nonzero $y$ component. The wavefront of $\Ksup$ is contained in $N^*\Sresp$, whose covectors have zero $y$ component. There is therefore no opposing pair at that point. Off $\Sresp$ the internal supported distribution is locally zero, so the product is zero there. These local products glue on $U\setminus N_U$. For smooth cross-line coefficients the exact formula is \eqref{eq:smoothjetproduct}. For the present traceable distribution the same identity follows from the restriction and product calculus. Locally, one may regularize the single singular relative line in its admissible wavefront topology. Its normal traces converge by the restriction theorem; the products with the normal-jet distribution converge by the product theorem. The smooth identity therefore passes to the limit. This argument is entirely within the domain of Lemma~\ref{lem:traces}; no convergence at $N_U$ is claimed. The normal traces have the tangential bounds in that lemma. Multiplication by $v_\alpha$ adds no wavefront, and Lemma~\ref{lem:insertWF} adds arbitrary response-normal covectors. This gives \eqref{eq:referenceWF}. The finite vertex operations preserve the bound. Every operation agrees with the canonical distributional operations on their common domain, proving agreement. The normal form was obtained from the complete response sector in Proposition~\ref{prop:normalform}, so no source or mass term has been removed. \end{proof} \begin{remark}[Extent of the advance] Theorem~\ref{thm:reference} proves a reference on the local complement of $N_U$ for the supported sector only. It does not prove \[ \Ucan=W\setminus\Nbar \] for the complete six-branch sum. In particular, it supplies no characteristic composition of $K_{47}$ or either $K_5$ placement with $\Cross$. Those kernels have different support geometry and remain in the target. \end{remark} \subsection{Collected coefficient form} Collect the terms of \eqref{eq:reference} with the same normal derivative. For $|\gamma|\le3$, put \begin{equation} b_{D,\gamma}^\circ = \sum_{\substack{\beta\in\mathbb N^5\\|\gamma+\beta|\le3}} (-1)^{|\beta|} \binom{\gamma+\beta}{\beta} v_{\gamma+\beta}\,c_{D,\beta}. \label{eq:bcoeff} \end{equation} Then \begin{equation} t_D^\circ =\sum_{|\gamma|\le3}\Ejet_\gamma b_{D,\gamma}^\circ \quad\hbox{on }U\setminus N_U. \label{eq:collected} \end{equation} The $b_{D,\gamma}^\circ$ are distributions on $\Sresp\setminus N_U$. They need not be smooth. Their complete sum may have cancellations. No coefficient is assumed nonzero or zero. The reference estimates on compact subsets away from $N_U$ do not provide uniform estimates as those compact subsets approach $N_U$. %===================================================================== \section{J2d: a conditional extension interface, not an extension map} \label{sec:interface} %===================================================================== The preceding reference is not a prescription across $N_U$. The following theorem names one exact interface for a possible supported realization. It does not impose this realization on every map in Definition~\ref{def:mapclass}. \subsection{Normal order as an algebraic test condition} For $m\ge0$, let $\mathcal I_m$ be the space of compact tests $\varphi$ whose normal jets satisfy \[ \partial_n^\alpha\varphi(s,0)=0 \qquad\text{for all }|\alpha|\le m. \] A distribution has normal order at most $m$ in this section when it annihilates $\mathcal I_m$. This is a fixed-coordinate normal-jet condition. It is not a transverse scaling-degree assertion. \begin{proposition}[Finite normal-jet characterization] \label{prop:characterization} A distribution $t\in\mathcal D'(U)$ annihilates $\mathcal I_m$ if and only if it has a unique expression \begin{equation} t=\sum_{|\alpha|\le m}\Ejet_\alpha b_\alpha, \qquad b_\alpha\in\mathcal D'(S_0). \label{eq:generalnormal} \end{equation} The coefficients are recovered continuously by \begin{equation} \langle b_\alpha,\psi\rangle =(-1)^{|\alpha|} \langle t,\Jjet_\alpha\psi\rangle. \label{eq:extraction} \end{equation} In particular such a distribution is supported on $\Sresp$. \end{proposition} \begin{proof} An expression \eqref{eq:generalnormal} plainly annihilates $\mathcal I_m$. Its coefficients are unique by Lemma~\ref{lem:jetinsert}. Conversely, choose the same cutoff $\chi$ as in \eqref{eq:Jjet} and subtract the finite normal Taylor jet of a test: \[ \varphi(s,n) -\chi(n)\sum_{|\alpha|\le m} \frac{n^\alpha}{\alpha!}\partial_n^\alpha\varphi(s,0) \in\mathcal I_m. \] If $t$ annihilates this difference, its value is determined by the finite displayed jet. Defining the coefficients by \eqref{eq:extraction} therefore gives \eqref{eq:generalnormal}. Continuity follows from continuity of the test-extension maps $\Jjet_\alpha$. A test supported off $\Sresp$ has every jet zero, so $t$ vanishes on that test. \end{proof} \begin{remark}[Supported-distribution conventions] The normal-jet description is the local supported-distribution structure in explicitly fixed coordinates. The general microlocal distinction between distributional tangential coefficients and smooth conormal coefficients is discussed in \cite[Section 8.2]{Dang}. Here the finite characterization and extraction formula are proved directly; no extension theorem is applied. \end{remark} \subsection{The coefficient-extension equivalence} \begin{theorem}[Conditional supported realization] \label{thm:conditionalextension} Fix a vertex component $D$. The following two assertions are equivalent: \begin{enumerate}[label=\textup{(\roman*)}] \item There exists $t_D\in\mathcal D'(U)$ of normal order at most three such that \[ t_D|_{U\setminus N_U}=t_D^\circ. \] \item There exist distributions $b_{D,\gamma}\in\mathcal D'(S_0)$, $|\gamma|\le3$, whose restrictions to $\Sresp\setminus N_U$ are the coefficients \eqref{eq:bcoeff}. \end{enumerate} Given (ii), the corresponding supported realization is \[ t_D=\sum_{|\gamma|\le3}\Ejet_\gamma b_{D,\gamma}. \] Its wavefront satisfies \[ \WF(t_D) \subseteq \bigcup_{|\gamma|\le3}\mathcal L(b_{D,\gamma}). \] No existence, degree, or characteristic normalization of the coefficient extensions is asserted by this equivalence. \end{theorem} \begin{proof} If (i) holds, Proposition~\ref{prop:characterization} extracts its unique coefficient distributions. Tests $\Jjet_\gamma\psi$ with $\psi$ supported away from $N_U$ compare the extracted coefficients with \eqref{eq:collected}; uniqueness shows that their restrictions equal \eqref{eq:bcoeff}. If (ii) holds, the insertion maps are defined for arbitrary coefficient distributions. Their finite sum has normal order at most three and has the required restriction. The wavefront statement is Lemma~\ref{lem:insertWF}. \end{proof} \begin{remark}[What is, and is not, required universally] A supported, normal-order-preserving realization of this sector would face exactly the coefficient interface above. A generalized joint map in the full class might organize terms differently, or prove cancellations only after the entire six-branch sum is assembled. Theorem~\ref{thm:conditionalextension} does not claim to classify all such maps and is not a universal obstruction. No coefficient extension is chosen. The prescription alternative remains unentered. \end{remark} \subsection{The full ambient conormal is still the relevant one} Locally choose further coordinates $(\tau,\eta)$ on $\Sresp$ with $N_U=\{\eta=0,n=0\}$ and $\eta\in\R^3$. If a coefficient extension were conormal to $N_U$ inside $\Sresp$, its insertion would have covectors \[ (\tau,0,0;0,\zeta,\nu), \qquad \zeta\in\R^3,\quad\nu\in\R^5, \quad(\zeta,\nu)\ne0. \] These are precisely full ambient conormal covectors of $N_U$ in $U$. Thus the normal-jet interface retains all eight ambient normal directions. It does not discard the attachment directions, and it assigns no degree from the relative three-dimensional notation. No conormal landing condition or uniform estimate for the coefficient extensions is proved here. %===================================================================== \section{J2e: reference agreement, ambiguity, and the exact missing map} \label{sec:missing} %===================================================================== \subsection{Agreement has a domain} \begin{proposition}[Support of differences detected by agreement] \label{prop:agreement} If two distributions agree on an open domain $V\subset U$, their difference is supported in $U\setminus V$. For the supported-sector reference of Theorem~\ref{thm:reference}, two extensions agreeing on $U\setminus N_U$ differ by a distribution supported in $N_U$. For the full target, agreement on $\Ucan$ alone gives support in $W\setminus\Ucan$, not automatically in $N$. \end{proposition} \begin{proof} The difference vanishes on every test compactly supported in $V$, which is exactly the asserted support statement. The two applications use their respective agreement domains. \end{proof} This elementary distinction prevents an unproved identification of a canonical domain with an entire characteristic complement. \subsection{Characteristic-supported differences are not collision contacts} \begin{proposition}[A conditional nonuniqueness diagnostic in \(W\)] \label{prop:ambiguity} In a chart $(\tau,\eta,n)$ as above, a nonzero smooth compactly supported $a(\tau)$ defines \begin{equation} A_N=a(\tau)\delta^{(3)}(\eta)\delta^{(5)}(n). \label{eq:AN} \end{equation} It is supported in $N_U$ and has wavefront contained in $N_U^*U\setminus0$. It is not a member of the inherited interaction collision-contact class on this retained separated patch. Accordingly, agreement off $N_U$ and a wavefront allowance containing the full ambient conormal would not, by themselves, select a unique extension. This statement is conditional on such an allowance and does not assert an actual degree of the target. \end{proposition} \begin{proof} The delta in the eight normal coordinates is a defined distribution. Its wavefront is conormal to their common zero set, and multiplication by the smooth tangential coefficient preserves that upper bound. Every permitted interaction collision contact restricts to zero on the retained patch by the copied rigidity statement. The nonzero distribution \eqref{eq:AN} does not, so it is not such a contact. Adding \eqref{eq:AN} to an existing extension would preserve agreement off $N_U$ and any wavefront upper bound that already permits its conormal. No extension is being constructed or modified here. \end{proof} No degree of \eqref{eq:AN} is used to infer the degree of $\Tform$, and no characteristic term is added to $\Cpart$. \subsection{The complete joint map remains missing} The supported-sector reference is not the required value of $\Mmap$ on \[ K_{47}+K_5+K_6+K_8+K_9. \] It does not include: \begin{enumerate} \item a characteristic composition for $K_{47}$; \item a characteristic composition for either $K_5$ placement; \item coefficient extensions of the supported-sector reference; \item a complete-graph wavefront theorem after summing all tensor and branch contributions; \item a proof that a coefficient-level construction descends through every internal reconstruction identity. \end{enumerate} The inherited internal pole estimates do not give any of these by themselves. They control a different operation before the new cross-line composition. No attempt is made to multiply the complete sum by $\Cross$ at the retained opposite covectors using the ordinary product theorem. \begin{theorem}[Exact J2 verdict] \label{thm:J2} For the class fixed in Definition~\ref{def:mapclass}: \begin{enumerate}[label=\textup{(\roman*)}] \item reconstruction descent is a necessary condition, made explicit by Proposition~\ref{prop:descent}; \item the supported sector has the exact normal form \eqref{eq:normalform}; \item its grouped composition on $U\setminus N_U$ is Constructed by Theorem~\ref{thm:reference}; \item a supported normal-order-preserving extension has the conditional coefficient interface of Theorem~\ref{thm:conditionalextension}; \item no full joint characteristic map on tests meeting $N$ is constructed. \end{enumerate} The ending for the required full map is \textbf{Unresolved at J2}, within the inherited I3/G2/F2/E1/D2/J2 analytic boundary. No incompatibility excluding every map in the fixed class has been proved. \end{theorem} \begin{proof} The cited propositions and theorems establish (i)--(iv). Their domains stop short of a characteristic composition for the full six-branch kernel. The conditional interface produces no extension coefficients, and the reference theorem treats only $\Ksup$ away from $N_U$. Neither gives a value for the full reconstructed sum on a test crossing $N$. No generalized composition theorem for that operation has been supplied, proving (v) and the unresolved ending. The missing construction is not a converse failure theorem for all possible constructions, so the obstructed ending is not warranted. \end{proof} \scopebox{The first-missing-gate rule}{% A reference for $K_6+K_8+K_9$ is not the joint map for $K_4+\cdots+K_9$. \smallskip A finite normal-jet formula away from $N$ is not a coefficient extension across $N$. \smallskip A conditional extension interface is not a prescription. \smallskip The first missing gate remains \textbf{J2}.} %===================================================================== \section{J3: joint section and complete remainder, not entered} \label{sec:J3} %===================================================================== The selected exit remains the joint-section exit. Because J2 is Unresolved, J3 is not entered as a construction. A successful theorem would have to control: \begin{enumerate} \item lower terms of every numerical line; \item internal vertices and their actual kernel maps; \item attachment-diagonal derivatives; \item the two defect directions as geometric constraints, not integration fibers; \item every coefficient of the actual source; \item both terms of the scalar root; \item the grouped cross-line difference; \item the complete tensor and branch summation; \item independence from internal null presentations. \end{enumerate} The copied internal frequency remainder is not this theorem. The exact normal-jet reference formula is not this theorem. Neither a zero section nor a nonzero section is assumed. The branch numerator $Q\lambda$ supplies no section of the complete sum. \begin{proposition}[No cancellation conclusion] \label{prop:nocancellation} XIV-J proves neither cancellation nor failure of cancellation of $\Tform$. \end{proposition} \begin{proof} The new theorems concern reconstruction identities and a proper sub-sector on a domain avoiding $N$. No joint characteristic distribution or principal-symbol section of the entire sum is constructed there. The source-sector coefficients could cancel among themselves or with other reconstructed branches in a later valid composition. Conversely, their existence away from $N$ does not prove such cancellation. Neither conclusion follows without the missing joint map and its complete remainder. \end{proof} The prescription alternative is not entered. Its seven data remain obligations of a separately stated construction. No finite part, new characteristic contact, resolved base space, or XIII-E limit is substituted. %===================================================================== \section{J4: the remaining extension obligations stay closed} \label{sec:J4} %===================================================================== A J3 distribution is required before the remaining obligations of XIII-C can be entered for that distribution. \subsection{Uniform transverse estimates} No uniform transverse scaling theorem is proved for $\Tform$, the three- and four-vertex collision strata, or their intersections. The parallel-line value eight remains a collision degree. The internal orders $3/2$, one, and $1/2$ remain orders of their copied charts. The internal pole-module frequency degree remains a degree of an amplitude presentation. The normal order at most three in the supported-sector reference is an algebraic jet bound before characteristic extension. It is not a transverse degree of the full quantum coefficient. Finite-order control on each compact subset away from $N$ does not give uniform control as the subset approaches $N$. Extension frameworks require their additional growth and wavefront hypotheses \cite{BF,Dang}; none is inferred here merely from the finite jet count. \subsection{Partial contacts and singular spectators} No joint contact gluing is performed. A later theorem must define each spectator composition, retain the inherited subgraph normalizations, and prove agreement on overlaps. The frozen bound on permitted collision derivatives does not supply those compositions. No contact is chosen after the present calculation. \subsection{Output wavefront theorem} After a defined joint distribution and actual proper pushforward, the required output is a compact degree-two residual-field kernel. A sufficient target bound is \[ \WF(t^{(1)}) \subset Z_2= \{(z,p;y,q):p+q=0\}\setminus0 \] under the fixed Minkowski cotangent identifications. This is an output bound on the residual field variables, not an intermediate attachment-kernel bound. Equation~\eqref{eq:referenceWF} is not that output theorem: its attachments have not undergone the missing joint construction and pushforward. Properness of the inherited tree datum supplies support control only. It cannot establish this wavefront theorem for an absent coefficient. The branches and species channels outside the retained germ also remain part of the global pair. A successful local characteristic construction would still require the appropriate global compatibility before admission of the full coefficient. %===================================================================== \section{J5: admission, the only exit to XV-B} \label{sec:J5} %===================================================================== An admission line would have to identify $\Rquant$ as the pushed-forward coefficient of the constructed joint distribution, with the normalization \eqref{eq:Rconvention}, and prove \[ Q_H\Rquant\in\Ealg. \] No such line is asserted. The admitted tree datum is not substituted for $\Rquant$. \begin{corollary}[No quantum admission in this release] \label{cor:noadmission} XIV-J releases no admitted $\Rquant$ and does not open XV-B. \end{corollary} \begin{proof} The full joint map is Unresolved at J2. The joint section, remaining extension obligations, and output theorem have consequently not been constructed for the required coefficient. Their absence prevents the stated admission conclusion. The exit contract for XV-B requires precisely that conclusion, not one of the partial reference results. \end{proof} No differential is applied to the absent coefficient: neither $\widehat s_0$, $s_{\mathrm{ext}}$, $s_\Psi$, nor $s_0$ appears in a calculation of $\Rquant$ here. No binary Ward defect is formed. %===================================================================== \section{J6: the exit record} \label{sec:J6} %===================================================================== XV-A's stated free-carrier release is recorded only as an exit prerequisite \cite{XVA}. It is not an analytic input to any theorem above. If a later paper supplied the missing admission line, XV-B could attempt one relative interacting differential through one explicitly stated order in $\kappa$ and $\hbar$ on that free carrier. The relative classical complex would remain a comparison object, not an identified quantum subalgebra. The recorded carrier uses the odd-sector antifield coupling \[ \sum_A(-1)^{|A|}\Phi_A^*s\Phi^A. \] This coupling normalizes the carrier differential, not the characteristic graph. Even after admission, XV-B would not automatically be a quantum master-equation paper, an anomaly classification, or a promotion of $\mathcal C_\xi$. Those are later gates. Here the admission prerequisite is absent. The exit is therefore recorded as closed. %===================================================================== \section{Decision, release, and successor package} \label{sec:decision} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.07\linewidth} p{.30\linewidth}X@{}} \toprule Gate & Ending in XIV-J & XV-B consequence\\ \midrule J0--J1 & Copied inputs and frozen route & No exit. Pair, inverse, switches, width, contacts, and residual variable unchanged.\\[0.35em] J2 & Partial constructions; full map Unresolved & Reconstruction descent and a supported-sector off-$N$ reference are proved. No full map on tests meeting $N$.\\[0.35em] J3 & Not entered & No joint section or complete remainder.\\[0.35em] J4 & Not entered & Uniform transverse estimates, contact gluing, and output wavefront theorem are not discharged.\\[0.35em] J5 & Not Constructed & No $Q_H\Rquant\in\Ealg$. XV-B may not open.\\[0.35em] J6 & Closed exit recorded & The free carrier does not replace the missing coefficient.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[XIV-J mathematical release] \label{thm:release} For the unchanged retained instance and the class fixed before the calculation: \begin{enumerate}[label=\textup{(\roman*)}] \item a coefficient-level joint construction descends to reconstructed kernels exactly under \eqref{eq:descent}; \item the internal module has the explicit null presentation \eqref{eq:null}, so presentation degree is not by itself an intrinsic joint-symbol filtration; \item the supported response sector has the unique finite normal form \eqref{eq:normalform}; \item its complete grouped reference composition on $U\setminus N_U$ is \eqref{eq:reference}, with the wavefront bound \eqref{eq:referenceWF}; \item supported realizations preserving that normal order have the conditional coefficient-extension interface of Theorem~\ref{thm:conditionalextension}; \item no full joint characteristic map, joint section, or admitted $\Rquant$ is constructed. \end{enumerate} The ending for the full required map is \textbf{Unresolved at J2}. The named partial reference construction is \textbf{Constructed on its stated domain}. XV-B remains closed. No universal obstruction to the fixed joint-map class is asserted. \end{theorem} \begin{proof} Proposition~\ref{prop:descent} proves (i). Lemma~\ref{lem:null} proves (ii). Proposition~\ref{prop:normalform} proves (iii). Lemma~\ref{lem:traces} and Theorem~\ref{thm:reference} prove (iv). Theorem~\ref{thm:conditionalextension} proves (v). Theorem~\ref{thm:J2} identifies the missing complete map. Sections~\ref{sec:J3}--\ref{sec:J5} record the unentered subsequent gates, and Corollary~\ref{cor:noadmission} gives (vi) and the exit consequence. \end{proof} \subsection*{What has advanced} The internal amplitude interface now has an explicit descent test. A construction on coefficient arrays must respect their exact reconstruction identities; it cannot declare an intrinsic leading section merely by retaining their highest polynomial degree. For the local supported sector, the agreement problem is also more precise. Its reference is a finite array of normal-jet coefficients on $\Sresp\setminus N_U$, and the complete grouped reference distribution is constructed there. A supported normal-order-preserving realization would require actual extensions of that array, with their wavefront and compatibility data. Neither advance supplies the complete characteristic composition. Both $K_5$ placements, the regrouped $K_{47}$ term, and all response coefficients remain in the required target. \subsection*{Successor package} A successor receives: \begin{enumerate} \item the unchanged XIV-I package; \item the fixed joint-map class and its reconstruction-descent criterion; \item the explicit null internal presentation; \item the supported-sector finite normal-jet form; \item the grouped off-$N$ reference and its precise domain; \item the reference wavefront bound and collected coefficient array; \item the conditional supported realization interface; \item the explicit J2 stop and still closed admission gates. \end{enumerate} It receives no transverse degree of $\Tform$, no cancellation theorem, and no permission to call either seventeen-dimensional count an excess. It receives no $\Rquant$. To advance the same route, a successor must supply the full joint characteristic map on reconstructed kernels, prove its descent and wavefront properties, and then enter the joint-section and remaining extension gates. The chain remains \[ \text{joint map} \longrightarrow \text{joint section and complete remainder} \longrightarrow \text{uniform transverse estimates} \longrightarrow \text{contact compatibility} \longrightarrow \text{output wavefront theorem} \longrightarrow Q_H\Rquant\in\Ealg . \] \subsection*{Conclusion} \[ \boxed{ \begin{gathered} \Rec((q\,i\theta+1)M_{\mathrm R})=0,\\[2pt] \Ksup=\sum_{|\alpha|\le3} v_\alpha(s)\partial_n^\alpha\delta^{(5)}(n),\\[2pt] t_D^\circ =\sum_{\alpha,\ \beta\le\alpha} (-1)^{|\beta|}\binom{\alpha}{\beta} \Ejet_{\alpha-\beta}(v_\alpha c_{D,\beta}) \quad\text{on }U\setminus N_U,\\[2pt] \codim_WN=8,\qquad z\text{ retained},\\[2pt] \text{full joint characteristic map: Unresolved at J2},\\ Q_H\Rquant\in\Ealg:\ \text{not established},\\ \text{XV-B remains closed}. \end{gathered}} \] A null presentation must remain null under a joint map. A supported-sector reference is not the full composition. A normal-jet order is not a quantum transverse degree. An undefined quantum coefficient is not an anomaly. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Submission scope.} The mathematical claims proved here are the reconstruction descent, normal-jet representation, supported-sector reference, wavefront bound, and conditional realization interface. The manuscript is not a claimed proof of the unresolved full joint coefficient. \paragraph{Analytic scope.} All new distributional constructions have explicitly stated domains. The supported-sector reference avoids the characteristic locus. No ordinary product theorem is invoked at the retained opposite covectors. \paragraph{Grouped-product scope.} The quantum line difference is kept as the difference of two defined tensor products and their submersion pullback. It is not expanded into undeclared mixed products. \paragraph{Reconstruction debugging.} The internal amplitude estimates of XIV-I concern presentations. The exact identity $q\delta'+\delta=0$ shows why their polynomial degree alone cannot define an intrinsic joint symbol. A later coefficient-level construction must prove descent through the reconstruction kernel. \paragraph{Normal-coordinate scope.} The response incidence has five normal coordinates. Its characteristic submanifold has three additional relative normal coordinates within that incidence. All distributions remain on the original common space, and the characteristic locus still has ambient codimension eight. No degree is inferred from the flag. \paragraph{Source and branch scope.} Every source coefficient and both scalar-root terms are retained in the finite normal-jet form. The supported sector is explicitly distinguished from the full six-branch sum. Neither $K_5$ placement is removed. \paragraph{Ambiguity scope.} No characteristic-supported distribution is added to $\Cpart$. The displayed supported ambiguity is a conditional diagnostic, not a selected extension or a counterterm. \paragraph{Dependency scope.} XIV-I and the analytic predecessors supply the copied internal and graph data. XV-A is recorded only at the exit. XIV-A and XIV-B, the value $-8I$, XIII-E's family, and a resolved paired construction are not analytic inputs. \paragraph{Quantum scope.} No admitted $\Rquant$, interacting differential, Ward defect, quantum master-equation defect, anomaly, reduced algebra, state, numerical loop amplitude, new interaction vertex, or sharp-width limit is constructed. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant verification, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The claimed results are the displayed proofs. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{XIVI} N.~Cheung, \emph{Landmark XIV-I: Uniform internal retarded amplitudes at the diagonal--flow-out transition}. Manuscript source \texttt{XIV-I.tex} supplied in the present development. Internal amplitude module, exact reconstruction, precomposition sum, and I3 stop. \bibitem{XIVH} N.~Cheung, \emph{Landmark XIV-H: Internal kernel relations on the operator route}. Supplied manuscript \texttt{XIV-H.tex}. Source incidence, exact internal maps, regular charts, and transition geometry. \bibitem{XIVG} N.~Cheung, \emph{Landmark XIV-G: Operator and pullback composition of the regrouped retained germ}. Supplied manuscript \texttt{XIV-G.tex}. Residual-testing limitation and the limited factorwise audit. \bibitem{XIVF} N.~Cheung, \emph{Landmark XIV-F: Paired composition on a proposed excess bundle}. Supplied manuscript \texttt{XIV-F.tex}. Tangent-defect and projection geometry; recorded as boundaries, not composition relations. \bibitem{XIVE} N.~Cheung, \emph{Landmark XIV-E: Joint symbol of the six-branch retained germ, or one named characteristic prescription}. Supplied manuscript \texttt{XIV-E.tex}. Exact regrouping, complete root, characteristic lifts, and local cone zero locus. \bibitem{XIVD} N.~Cheung, \emph{Landmark XIV-D: Cancellation or prescription on the retained noncoincident characteristic stratum}. Supplied manuscript \texttt{XIV-D.tex}. Six-branch germ, local incidence, species split, and branch-limited numerator. \bibitem{XIVC} N.~Cheung, \emph{Landmark XIV-C: The joint two-line extension of one selected binary pair}. Supplied manuscript \texttt{XIV-C.tex}. Retained stratum, positive conormal relation, and collision-contact rigidity. \bibitem{XIIIC} N.~Cheung, \emph{Landmark XIII-C: The joint two-line extension of one selected binary pair, on the unchanged carrier}. Supplied manuscript \texttt{XIII-C.tex}. Partial-diagonal class, primitive collision extensions, and the four successor obligations. \bibitem{XIIIB} N.~Cheung, \emph{Landmark XIII-B: One binary causal datum for the displayed coefficient, on the unchanged carrier}. Supplied manuscript \texttt{XIII-B.tex}. Tree datum, retarded convention, ordered values, and properness boundary. \bibitem{XIIE} N.~Cheung, \emph{Landmark XII-E: Assembly of the second-order relational coefficient after admission of the density-weighted cubic insertion}. Supplied manuscript \texttt{XII-E.tex}. Slots, normalization, roots, and admitted completion kernels. \bibitem{XIIC} N.~Cheung, \emph{Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question}. Supplied manuscript \texttt{XII-C.tex}. Fixed cubic representative and actual density-weighted source. \bibitem{XVA} N.~Cheung, \emph{Landmark XV-A: The free quantum BV carrier for the ramp Einstein--scalar multiplet}. Manuscript source \texttt{XV-A.tex} supplied in the present development. Exit prerequisite only; not an analytic input. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}, second edition, Springer, 1990. Section~8.2: restrictions, tensor products, distribution products, and wavefront bounds. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}, arXiv:1208.1428v2, 2013. Separated propagator wavefronts and the domains of Hadamard and time-ordered products. \bibitem{Dang} N.~V.~Dang, ``The extension of distributions on manifolds, a microlocal approach,'' \emph{Annales Henri Poincar\'e} \textbf{17} (2016), 819--859, arXiv:1412.2808. Supported-distribution structure and microlocal extension hypotheses. No extension theorem is applied to the absent full characteristic target. \bibitem{BF} R.~Brunetti and K.~Fredenhagen, ``Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds,'' \emph{Communications in Mathematical Physics} \textbf{208} (2000), 623--661, arXiv:math-ph/9903028. Uniform microlocal scaling and extension framework; recorded for the unopened extension obligations. \end{thebibliography} \end{document}