% !TeX program = pdflatex %===================================================================== % XIII-C.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XIII-C.tex % pdflatex -interaction=nonstopmode -halt-on-error XIII-C.tex % % Exact release: % - the displayed composite is decomposed into eleven branches; % - all double cross-contraction incidence matchings are inventoried; % - the total-diagonal class is replaced by a partial-diagonal class; % - elementary parallel-line scaling degrees and extensions are proved; % - a noncoincident characteristic cancellation is identified; % - the complete joint extension and output admission remain Unresolved. % % Principal debugging correction: % Removing collision diagonals does not, for this nonlocal entry, % establish the joint off-collision product. Retarded internal lines % and time-ordered cross-lines can fail the sufficient product criterion % at distinct, null-related points. This failure is not a nonexistence % theorem. It prevents an unsupported affirmative C2 release. % % No gauge-consistency defect, restriction quotient, reduced algebra, % completed observable, R_2, general T_n, interacting differential, % master equation, state, gravitational interaction insertion, % numerical loop amplitude, or sharp-width limit is constructed. % % No successful compilation, proof-assistant verification, % independently executed symbolic computation, independent referee % review, or journal acceptance is asserted. % % External analytic sources checked: % Brunetti--Fredenhagen, math-ph/9903028; % Fredenhagen--Rejzner, 1208.1428; % Dang, 1412.2808. %===================================================================== \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,longtable} \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{\MT}{M_T} \newcommand{\MTo}{M_T^\circ} \newcommand{\PP}{\mathcal P} \newcommand{\RR}{\mathcal R} \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{\GR}{G_{\mathrm{act}}^{\mathrm R}} \newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}} \newcommand{\Pmass}{P_m} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}} \newcommand{\NW}{\mathscr N_W} \newcommand{\Csol}{\mathscr C_{\mathrm{sol}}} \newcommand{\CW}{\mathscr C_W} \newcommand{\Ocal}{\mathcal O} \newcommand{\Vcal}{\mathcal V_g} \newcommand{\Aone}{\mathcal A_1} \newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}} \newcommand{\CD}{\mathcal C_{3,\mathrm D}} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}} \newcommand{\rD}{r_{\mathrm D}} \newcommand{\rg}{r_g} \newcommand{\TT}{\mathsf T_2} \newcommand{\Htime}{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{\Lie}{\mathcal L} \newcommand{\Edist}{\mathcal E'} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\sd}{sd} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\id}{id} \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 XIII-C}\\[0.4em] The joint two-line extension of one selected binary pair,\\ on the unchanged carrier\\[0.4em] \large Partial-diagonal inventory, primitive extensions,\\ and the remaining joint microlocal gate} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We examine the coefficient linear in $\hbar$ left unresolved in XIII-B. The entries are the frozen switched metric--scalar vertex and the displayed second-order composite. The carrier and the off-shell lifts remain unchanged. We replace the total-diagonal ambiguity class by a class admitting the partial collision diagonals produced by the selected pair. An eleven-branch decomposition gives a complete incidence inventory of its double cross-contractions. For the elementary parallel-line factors, we compute transverse scaling degrees, construct extensions, and identify their finite contact ambiguities. A further joint issue must not be hidden. For a nonlocal observable, removing collision diagonals does not by itself establish the off-collision distribution. Retarded internal lines and time-ordered cross-lines admit a nonzero wavefront cancellation at distinct null-related vertices. We exhibit this cancellation explicitly. It is a failure of a sufficient product criterion, not a theorem that no joint distribution exists. Consequently, the elementary extensions do not constitute a complete joint extension or an output admission theorem. The requested coefficient retains the ending Unresolved. Corrected ordered values, tree overlap, properness, and the exact missing microlocal obligations are separated. No gauge-consistency defect is opened. \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) {Frozen product data}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Partial-diagonal incidence}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Coefficient cancellation}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Recording the incidence does not cancel the singular contribution.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Incidence is not cancellation. 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-XIII-C} \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) {Frozen product data}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Coefficient cancellation}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Recording the incidence does not cancel the singular contribution.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Incidence is not cancellation. 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-XIII-C} \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) {Frozen product data}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Partial-diagonal incidence}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Coefficient cancellation}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Recording the incidence does not cancel the singular contribution.}; \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-XIII-C} \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) {Frozen product data}; \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) {Coefficient cancellation}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Partial-diagonal incidence. Recording the incidence does not cancel the singular contribution.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Frozen product data is not Coefficient cancellation. 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-XIII-C} \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: Frozen product data}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Partial-diagonal incidence}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Coefficient cancellation}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Recording the incidence does not cancel the singular contribution.}; \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-XIII-C} \end{figure} Figure~\ref{fig:split-XIII-C} records one view of the same separation. Figure~\ref{fig:chain-XIII-C} records one view of the same separation. Figure~\ref{fig:pair-XIII-C} records one view of the same separation. Figure~\ref{fig:gate-XIII-C} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XIII-C} records that separation. The drawing adds no hypothesis. \scopebox{Release and proof boundary}{% \textbf{Constructed:} the complete incidence inventory for the frozen product data; the widened partial-diagonal class; the off-collision products on their microlocally admissible open sets; and the elementary parallel-line extensions. \smallskip \textbf{Proved correction:} the sum-zero bounds and exclusion of collision diagonals do not establish the joint off-collision product for this nonlocal pair. An explicit noncoincident characteristic cancellation prevents that argument. \smallskip \textbf{Ending for the complete coefficient: Unresolved.} No universal incompatibility excludes every extension in $\Cpart$. The joint off-collision prescription, uniform transverse estimates at its intersections, and microcausal output have not all been established. \smallskip The carrier remains $\Ealg$. No quotient by $\NW$, reduced algebra, completed observable, higher retarded product, general time-ordered family, gauge defect, interacting differential, master equation, state, numerical loop amplitude, gravitational interaction insertion, or $\eps\to0$ limit is constructed.} \tableofcontents %===================================================================== \section{C0: copied release and the selected problem} \label{sec:inputs} %===================================================================== The three endings are: \begin{enumerate} \item \textbf{Constructed}: the named maps exist and satisfy their declared interfaces. \item \textbf{Proved obstructed}: an explicit incompatibility excludes every map in the previously declared class. \item \textbf{Unresolved}: an extension, scaling estimate, or output wavefront theorem required for the release has not been established. \end{enumerate} A failed sufficient product criterion belongs to the third category, unless an independent nonexistence argument is supplied. \subsection{XIII-B's release, copied} \begin{inputstatement}[Binary handoff] \label{input:B} \begin{quote} Constructed: the vertex functional, the selected tree-level binary causal composition, its properness and compact microcausal kernel, its inherited-root overlap, and the free-differential display. Proved obstructed: the literal normalization $R_1(\Vcal;\Ocal)=\Vcal\star_H\Ocal$ on ordered supports, when combined with the specified retarded definition and contraction grading. Unresolved through the requested quantum order: the joint two-line extension in the frozen counterterm class. Neither a failed wavefront criterion nor an unconstructed extension is called a no-go. The carrier remains $\Ealg$. No quotient by $\NW$ is released. \end{quote} These statements are inputs from XIII-B \cite{XIIIB}. \end{inputstatement} The constructed tree datum is not recomputed: \begin{equation} \Vcal\in\Fpol,\qquad \Rtree=-\pair{\Ocal'}{\GR\Vcal'}\in\Fpol,\qquad Q_H\Rtree\in\Ealg, \label{eq:tree-input} \end{equation} \begin{equation} \Rtree =\pair{\Ocal'_\gamma}{\rg[\phi]} +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]}. \label{eq:tree-branches-input} \end{equation} The selected convention remains \begin{equation} R_1(\Vcal;\Ocal) =\frac{i}{\hbar} \left(\TT^{\mathrm{ren}}(\Vcal,\Ocal) -\Vcal\star_H\Ocal\right). \label{eq:R-convention} \end{equation} Its ordered values are zero when the entire interaction support is later, and $(i/\hbar)[\Ocal,\Vcal]_{\star_H}$ when the entire observable support is later. The literal equality with $\Vcal\star_H\Ocal$ is closed and is not reopened. \subsection{The partial-collision sentence, copied verbatim} \begin{quote} The scalar-sourced coordinate part of $\Ocal$ contains a branch with schematic form \[ \int f(x)\,Q^\mu{}_{ab}(x,z)\,g(z)T_0^{ab}[\phi](z)\, \partial_\mu\phi(x)\,\mathrm d^4x\,\mathrm d^4z. \] Two scalar contractions can join the stress slots at $z$ to the two scalar slots of the interaction at $y$. The collision $y=z$ may occur while $x\neq z$. A counterterm supported only where $x=y=z$ does not by its support alone normalize this internal collision. \end{quote} \subsection{The evaluation boundary and the carrier} From XIII-A: \begin{quote} Under (UCP$_W$), $\Delta_\xi\Ocal|_{\CW}=0$. The coefficientwise vanishing space $\NW$ is not a two-sided ideal. No restricted quantum domain is handed off. \end{quote} Here \[ \CW=\{u\in\Csol:D^*b=0\text{ on }W\},\qquad \overline W\subset\operatorname{Int}\{g=1\} \cap\{\eps0$ are fixed. The selected order is \begin{equation} R_{\lambda\Vcal}\Ocal =\Ocal+\lambda\left(\Rtree+\hbar\Rquant\right) +O(\lambda^2,\lambda\hbar^2). \label{eq:order} \end{equation} As in XIII-B, $\lambda=\kappa$ counts one additional interaction insertion beyond the displayed $\kappa^2$ coefficient. The copied XII-F value \begin{equation} \Delta_\xi\Ocal(u_*) =-8\int f(t,\mathbf x)\int_0^t(t-s)^2a(s)\, \dd s\,\dd^4x<0 \label{eq:witness} \end{equation} remains a boundary of the evaluation claim \cite{XIIF}. It is not a binary anomaly and is not used to choose counterterms. %===================================================================== \section{The widened class, frozen before extension} \label{sec:class} %===================================================================== The total-diagonal class of XIII-B is not inherited as the complete ambiguity space. \begin{definition}[Collision sets] For each graph in the inventory below, let $y$ be the interaction vertex and let $x,u,v$ denote its displayed observable vertices when present. For a connected vertex subset $I$ containing $y$, with $|I|\ge2$, let \[ \Delta_I=\{x_a=x_b\text{ for every }a,b\in I\}. \] Include intersections of these loci and the loci obtained after contracting such subsets. Diagonals involving only a pre-existing observable subgraph retain their inherited normalization. \end{definition} These sets include the parallel-line collision $y=v$ while the observation point $x$ remains free. They also include the three-vertex and four-vertex collisions of the characteristic cycles listed below. A connected-subset convention may include harmless forest collisions; their inherited derivative normalization remains an additional compatibility condition. \begin{definition}[Partial-diagonal ambiguity class] \label{def:Cpart} The class $\Cpart$ consists of finite sums of graphwise contact changes at order $\hbar^2$ in $\TT$, hence order $\lambda\hbar$ in \eqref{eq:order}. Before pushforward they have the form \begin{equation} D_\Gamma\left[ w_\Gamma\, \partial_r^\alpha\delta(r_I)\, S_{\Gamma/I}\right], \qquad |\alpha|\le6, \label{eq:partial-counterterm} \end{equation} where: \begin{enumerate}[label=\textup{(\roman*)}] \item $r_I$ is a system of relative coordinates transverse to $\Delta_I$; \item $S_{\Gamma/I}$ is an inherited spectator graph coefficient, with the two remaining field slots retained; \item $w_\Gamma$ is built from $g,\partial g,\sig,\sig',\sig''$ and the smooth external graph weights already present; \item $D_\Gamma$ is the specified finite graph differential operator; \item the change is even, has ghost number zero and field degree two, and is supported in the existing compact graph closure; \item it leaves the tree datum and the inherited observable subgraph normalizations unchanged. \end{enumerate} Only well-defined products and pushforwards of \eqref{eq:partial-counterterm} are members of the class. The bound six is a conservative frozen bound; an established scaling degree may impose a smaller bound on a particular locus. The subclass supported on the total graph diagonal is denoted $\Ctot\subseteq\Cpart$. \end{definition} This widening is a new definition. It does not retroactively modify XIII-B. Nor does it assert that every formal contact expression defines a distribution. The spectator coefficient in a partial-diagonal change can remain singular in its tangential variables. For example, a contact at $y=v$ can retain a retarded coefficient linking $v$ to $x$. Calling all such coefficients smooth would exclude the graph that motivated the widening. The smoothness requirement concerns the external weights, not the entire spectator distribution. Differentiating a diagonal distribution can differentiate its multiplying weights upon pushforward. This is not permission to add arbitrary independent higher-jet coefficient functions. %===================================================================== \section{C1: complete incidence inventory for the selected pair} \label{sec:inventory} %===================================================================== The interaction is the functional \begin{equation} \Vcal=-\tfrac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}[\phi]\,\dd^4x, \qquad T_0^{\mu\nu} =\partial^\mu\phi\partial^\nu\phi -\tfrac12\eta^{\mu\nu}\left((\partial\phi)^2-m^2\phi^2\right). \label{eq:vertex} \end{equation} Its kinetic monomials have one undifferentiated metric slot and two scalar slots with one derivative each. Its mass monomials have no field derivatives. The vertex is not an Euler operator. The observable is \[ \Ocal=\Asc[f,g](\gamma,0;\phi)-\CD +\Aone[f,g](\rg[\phi];\phi). \] Its formula and admission are inherited from XII-E \cite{XIIE}. For inventory purposes only, abbreviate \[ X=X_1[\gamma],\qquad p=p_1[\gamma;\phi], \qquad \int_f A=\int fA\,\dd^4x. \] Put \[ U_\sig[\gamma]q =\partial_\alpha(\sig a_1^{\alpha\beta}[\gamma] \partial_\beta q), \] \[ V_{2,g}[\gamma,0]\phi =\partial_\mu(g a_2^{\mu\nu}[\gamma]\partial_\nu\phi) +m^2g b_2[\gamma]\phi. \] Here $a_1,b_1,a_2,b_2$ have their inherited density-expansion definitions. \begin{lemma}[Eleven-branch decomposition] \label{lem:branches} The displayed lift is exactly $\Ocal=\sum_{j=1}^{11}B_j$, where \begin{align} B_1&=-\int_f\ERm\bigl(V_{1,g}[\gamma]p\bigr),& B_2&=-\int_f\ERm\bigl(V_{2,g}[\gamma,0]\phi\bigr),\\ B_3&=-\int_fX^\mu\partial_\mu p,& B_4&=\int_fX^\nu\partial_\nu X^\mu\partial_\mu\phi,\\ B_5&=\int_f\ER\bigl(U_\sig[\gamma]X^\mu\bigr)\partial_\mu\phi,& B_6&=\int_f\ER\bigl(\partial_\alpha(\sig a_2^{\alpha\mu}[\gamma])\bigr)\partial_\mu\phi,\\ B_7&=\tfrac12\int_fX^\mu X^\nu\partial_\mu\partial_\nu\phi,& B_8&=-\int_fX_1^\mu[\rD]\partial_\mu\phi,\\ B_9&=\int_fp_1[\rD;\phi],& B_{10}&=-\int_fX_1^\mu[\rg]\partial_\mu\phi,\\ B_{11}&=\int_fp_1[\rg;\phi]. \label{eq:branches} \end{align} \end{lemma} \begin{proof} Expand the two parts of $p_2[\gamma,0;\phi]$, the two parts of $-X_2[\gamma,0]\cdot\partial\phi$, and the other three terms of the second completion. The affine second-slot identity supplies $B_8+B_9=\Aone(\rD;\phi)=-\CD$. The two legs of the scalar-sourced slot give $B_{10}+B_{11}$. \end{proof} No source coefficient is suppressed in $B_8,B_9$. The finite source prescription of XII-D gives monomials \[ c_{\alpha\beta}(\sig,\sig',\sig'') \,\partial^\alpha\gamma\,\partial^\beta\gamma, \qquad |\alpha|+|\beta|\le2. \] This notation covers every constant fiber coefficient in the declared source, including the layer terms \cite{XIID}. \subsection{Derivative placement} Move each displayed divergence onto its incoming retarded kernel. Leave derivatives of uncontracted fields on those fields. Let $e=1$ for a kinetic scalar monomial and $e=0$ for a mass monomial. Write $\Gamma(a)$ for a metric slot at $a$ and $\varphi(a)$ for a scalar slot there. \begin{center} \scriptsize \begin{tabularx}{\linewidth}{@{}c p{.29\linewidth} p{.29\linewidth} X@{}} \toprule Branch & Three field-slot locations & Internal retarded edges & Maximum total field/line derivative count\\ \midrule $B_1$ & $\Gamma(u),\Gamma(v),\varphi(v)$ & $x-u,\ u-v$ & $4$\\ $B_2$ & $\Gamma(u),\Gamma(u),\varphi(u)$ & $x-u$ & $2$\\ $B_3$ & $\Gamma(u),\Gamma(v),\varphi(v)$ & $x-u,\ x-v$ & $4$\\ $B_4$ & $\Gamma(u),\Gamma(v),\varphi(x)$ & $x-u,\ x-v$ & $4$\\ $B_5$ & $\Gamma(u),\Gamma(v),\varphi(x)$ & $x-u,\ u-v$ & $4$\\ $B_6$ & $\Gamma(u),\Gamma(u),\varphi(x)$ & $x-u$ & $2$\\ $B_7$ & $\Gamma(u),\Gamma(v),\varphi(x)$ & $x-u,\ x-v$ & $4$\\ $B_8$ & $\partial^\alpha\Gamma(v),\partial^\beta\Gamma(v),\varphi(x)$ & $x-u,\ u-v$ & $4$\\ $B_9$ & $\partial^\alpha\Gamma(v),\partial^\beta\Gamma(v),\varphi(u)$ & $x-u,\ u-v$ & $4$\\ $B_{10}$ & $\varphi(v),\varphi(v),\varphi(x)$ & $x-u,\ u-v$ & $4$\\ $B_{11}$ & $\varphi(v),\varphi(v),\varphi(u)$ & $x-u,\ u-v$ & $4$\\ \bottomrule \end{tabularx} \end{center} The exact cross-line derivative order is the sum of the derivative orders of its two matched field slots. Thus: \[ d_{\gamma\gamma}=0+|\alpha| \quad\text{when a differentiated source metric is selected}, \qquad d_{\phi\phi}=e_{\mathrm{vertex}}+ d_{\mathrm{observable\ scalar}}. \] Coordinate metric slots outside the source have order zero. Scalar slots in a switched scalar or stress vertex have order $e$; observation scalar slots in $B_4,B_5,B_6,B_8,B_{10}$ have order one, and that in $B_7$ has order two. These rules, together with \eqref{eq:branches}, specify every derivative placement without imposing a tensor-component truncation. \subsection{Species and residual slots} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.29\linewidth}X@{}} \toprule Two-line channel & Status for the frozen $H$\\ \midrule Two metric--metric lines & Absent: $\Vcal$ has only one metric slot.\\ One metric--metric and one scalar--scalar line & Present for $B_1,\ldots,B_9$; the two residual slots are one metric and one scalar.\\ Two scalar--scalar lines & Present for $B_{10},B_{11}$; the residual slots are the interaction metric and the unused observable scalar.\\ An individual metric--scalar line & Zero: both $H_{\gamma\phi}$ and $(G_{\mathrm{act}}^{\mathrm A})_{\gamma\phi}$ vanish for the frozen data.\\ \bottomrule \end{tabularx} \end{center} Label the two vertex scalar slots separately. For each metric-sector branch there are two choices of observable metric slot and two choices of vertex scalar slot: four matchings. For a scalar-sector branch there are three choices of observable scalar pair and two bijections to the vertex scalar pair: six matchings. The inventory therefore has \begin{equation} 9\cdot4+2\cdot6=48 \label{eq:count} \end{equation} labelled incidence matchings, before expansion into mass, kinetic, source, and fiber monomials. The factor $1/2$ in the double derivative contraction accounts for ordered line enumeration; it is not another omitted branch. \subsection{Collision inventory} The two new edges join $y$ to the selected observable locations. They create a unique combinatorial cycle in the observable tree. The following table lists the cycle vertices; its connected collision subsets are the $\Delta_I$ used in $\Cpart$. \begin{center} \small \begin{tabularx}{\linewidth}{@{}c p{.36\linewidth}X@{}} \toprule Branch & Selected observable locations & Cycle vertices\\ \midrule $B_1$ & $(u,v)$ or $(v,v)$ & $\{y,u,v\}$ or $\{y,v\}$\\ $B_2$ & $(u,u)$ & $\{y,u\}$\\ $B_3$ & $(u,v)$ or $(v,v)$ & $\{y,u,x,v\}$ or $\{y,v\}$\\ $B_4$ & $(u,x)$ or $(v,x)$ & $\{y,u,x\}$ or $\{y,v,x\}$\\ $B_5$ & $(u,x)$ or $(v,x)$ & $\{y,u,x\}$ or $\{y,v,u,x\}$\\ $B_6$ & $(u,x)$ & $\{y,u,x\}$\\ $B_7$ & $(u,x)$ or $(v,x)$ & $\{y,u,x\}$ or $\{y,v,x\}$\\ $B_8$ & $(v,x)$ & $\{y,v,u,x\}$\\ $B_9$ & $(v,u)$ & $\{y,v,u\}$\\ $B_{10}$ & $(v,v)$ or $(v,x)$ & $\{y,v\}$ or $\{y,v,u,x\}$\\ $B_{11}$ & $(v,v)$ or $(v,u)$ & $\{y,v\}$ or $\{y,v,u\}$\\ \bottomrule \end{tabularx} \end{center} Repeated locations still carry separately labelled field slots. The scalar-sourced parallel collision is one row, not a model for all the mixed-species triangles and four-vertex cycles. %===================================================================== \section{C2: what is defined away from collisions} \label{sec:offcollision} %===================================================================== Formally, the required coefficient is \begin{equation} \frac{i}{2} \pair{\Vcal''\otimes\Ocal''} {\Htime^{\otimes2}-H^{\otimes2}}. \label{eq:formal-two-line} \end{equation} The $H$ contraction exists by the inherited product theorem. The joint time-ordered contraction requires a separate argument. Keep the difference grouped as in \eqref{eq:formal-two-line}. Expanding it into separate mixed products of $H$ and $\GA$ can introduce undefined intermediate products even when a grouped boundary-value expression exists. \begin{definition}[Microlocally admissible open set] For an inventory graph $\Gamma$, let $\mathcal U_\Gamma$ be the open set of configurations on which the product of its specified line distributions satisfies the H\"ormander sum condition: no selection of line wavefront covectors, allowing zero for unused factors, has zero sum at every graph vertex unless all selections are zero. \end{definition} \begin{proposition}[Off-collision construction on the admissible set] \label{prop:local-off} On $\mathcal U_\Gamma$, the specified graph product exists. Differentiation and multiplication by its smooth weights preserve existence. On compactly supported portions its pushforward is proper. The grouped difference agrees there with \eqref{eq:formal-two-line}. \end{proposition} \begin{proof} This is the distribution product criterion applied to the complete graph variables. The derivative kernels have the prescribed sum-zero bounds, and the line kernels are balanced. Properness is the graphwise support statement inherited from XIII-B. Restriction to an open set does not enlarge the original compact support control \cite{XIIIB,Hormander}. \end{proof} The proposition does not identify $\mathcal U_\Gamma$ with the complement of the collision diagonals. \subsection{A noncoincident cancellation} \begin{lemma}[Characteristic triangle cancellation] \label{lem:pinch} Consider the numerical triangle \begin{equation} E_0^{\mathrm R}(x-u)\,F_0(y-u)\,F_m(y-x), \label{eq:triangle} \end{equation} where $F_\mu$ is the scalar time-ordered boundary-value line, up to its fixed nonzero action fiber factor. There are configurations with $u,y,x$ pairwise distinct for which the H\"ormander sum condition for \eqref{eq:triangle} fails. The same cancellation remains available for derivatives whose principal conormal factors are nonzero. \end{lemma} \begin{proof} Choose a future null vector $\ell$ and points \[ y=u+\ell,\qquad x=u+2\ell. \] Let $p$ be a nonzero null covector proportional to the metric dual of $\ell$, with sign chosen to match the time-ordered orientation. At $(y,u)$ the time-ordered line has a wavefront covector $(p,-p)$. At $(y,x)$ it has $(-p,p)$, because the chronological ordering is reversed. The retarded line at $(x,u)$ permits $(-p,p)$: on its nonzero forward null cone the real retarded singularity has both conormal signs. The sums at the three vertices are \[ x:\ -p+p=0,\qquad u:\ p-p=0,\qquad y:\ p-p=0. \] All line covectors are nonzero, and none of the vertices coincide. Differentiation does not remove this configuration when its principal symbol is nonzero at the selected covectors. \end{proof} The incidence occurs in $B_6$: one metric slot at $u$ and the observation scalar slot at $x$ are contracted with the metric and one scalar slot of the vertex at $y$. The remaining fields are a metric at $u$ and a scalar at $y$. The incoming differentiated retarded line and the two differentiated cross-lines have precisely the triangle structure of \eqref{eq:triangle}. For example, the local density coefficient $a_2^{00}$ contains a nonzero $(\gamma_{00})^2$ monomial. Temporal derivative components have nonzero conormal symbols on the displayed null direction. This locates the sufficient-criterion issue in the actual incidence inventory, not in an unrelated graph. It does not establish absence of cancellations after summing all tensor components and all eleven branches. \begin{corollary}[The proposed global C2 argument is insufficient] \label{cor:C2} The $Z_3$ bounds, the vertex diagonal bound, and removal of the collision loci do not prove the joint time-ordered contraction on the entire remaining configuration space. \end{corollary} \begin{proof} Lemma~\ref{lem:pinch} supplies a nonzero cancellation at distinct vertices. At each entry the two contracted covectors may be opposite while the unused covector is zero. Their sum remains zero, so the $Z_3$ bound does not exclude them. \end{proof} This is a correction to a proposed proof, not Proved obstructed for every extension. A joint boundary-value prescription, a cancellation theorem for the complete pair, or another construction could resolve it. None follows solely from widening diagonal counterterms. %===================================================================== \section{C3: scaling degrees and their exact scope} \label{sec:scaling} %===================================================================== For a distribution in normal coordinates $r\in\R^q$, its scaling degree is the infimum of $s$ such that $\rho^s t(\rho r)\to0$ distributionally as $\rho\downarrow0$. For $\Delta_I$, the transverse codimension is \[ q_I=4(|I|-1). \] The tangential variables, including an uncollapsed observation point, remain variables of the distribution. The extension theorem gives uniqueness below codimension and a finite derivative-of-delta ambiguity at or above codimension, provided the relevant distribution and scaling estimates exist. The submanifold version additionally requires tangential and microlocal control \cite[Sections 5--6]{BF,Dang}. \subsection{The parallel-line primitive} For the frozen scalar lines, \[ F_\mu(r)=c(-r^2+i0)^{-1} +\text{terms of lower short-distance degree}, \qquad c\ne0, \] with the fixed conventional fiber factors restored for each species. The leading term has scaling degree two. Away from $r=0$, products of two time-ordered lines and their derivatives exist: at a nonzero null separation their wavefront covectors have the same conormal orientation. \begin{proposition}[Primitive parallel-line degrees] \label{prop:parallel-degree} Let \[ t_{\alpha\beta}(r) =\partial^\alpha F_\mu(r)\, \partial^\beta F_\nu(r), \qquad r\ne0. \] Then \[ \sd(t_{\alpha\beta})\le4+|\alpha|+|\beta|. \] Equality holds when the leading differentiated homogeneous tensor product is nonzero. In particular, \[ \sd(F_\mu F_\nu)=4,\qquad \sd\bigl((\partial_0^2F_\mu) (\partial_0^2F_\nu)\bigr)=8. \] \end{proposition} \begin{proof} The leading mass-independent term is homogeneous of degree $-2$. Each nonvanishing derivative raises its short-distance degree by its derivative order. The lower-order massive terms do not increase that bound. For the first example the leading product is nonzero. For the second, evaluate its homogeneous leading coefficient at spacelike $r$ with $r^0=0$. Each temporal second derivative is a nonzero constant multiple of $|\mathbf r|^{-4}$, and their product is a nonzero multiple of $|\mathbf r|^{-8}$. This proves the lower bounds as well as the stated upper bounds. An identically vanishing tensor contraction, such as a wave-operator combination away from its source, is excluded from the equality assertion. \end{proof} For the stress--stress contraction in $B_{10}$ or $B_{11}$, the two scalar derivatives at the interaction and the two at the stress vertex give the following monomial degrees: \begin{center} \small \begin{tabular}{@{}lccc@{}} \toprule Interaction/stress monomials & Derivatives on the two lines & Primitive degree & Degree minus codimension\\ \midrule mass/mass & $0$ & $4$ & $0$\\ kinetic/mass or mass/kinetic & $2$ & at most $6$ & at most $2$\\ kinetic/kinetic & $4$ & at most $8$ & at most $4$\\ temporal kinetic component & $4$ & $8$ & $4$\\ \bottomrule \end{tabular} \end{center} A mass row disappears when its mass coefficient vanishes. These are primitive monomial statements, not a claim that tensor summation cannot lower the degree. For $y=v$, use $r=y-v$ and retain $x,u,v$ as tangential variables. Its codimension is four even when $x\ne v$. Multiplying the primitive by the independent spectator coefficient does not turn this partial diagonal into the total diagonal. Whether the complete product has a uniform transverse microlocal estimate remains a separate obligation. \subsection{Mixed channels and larger collisions} The inventory gives an exact derivative-counting rule. For a collision subset $I$, let $E_I$ be its number of internal numerical edges and $D_I$ the sum of derivative orders on those edges after the specified placement. Its superficial homogeneous exponent is \begin{equation} \omega_I^{\mathrm{sup}}=2E_I+D_I. \label{eq:superficial} \end{equation} The complete graph has at most six field/line derivatives after including the interaction. Derivatives left on unused fields do not increase the numerical exponent. \begin{center} \small \begin{tabularx}{\linewidth}{@{}lcccc@{}} \toprule Collapsed subgraph & $E_I$ & Codimension & Superficial exponent & Exponent minus codimension\\ \midrule Parallel two-vertex pair & $2$ & $4$ & $4+D_I$ & $D_I\le6$\\ Three-vertex cycle & $3$ & $8$ & $6+D_I$ & $D_I-2\le4$\\ Four-vertex cycle & $4$ & $12$ & $8+D_I$ & $D_I-4\le2$\\ Forest with $n$ vertices & $n-1$ & $4(n-1)$ & $2(n-1)+D_I$ & $D_I-2(n-1)$\\ \bottomrule \end{tabularx} \end{center} Equation~\eqref{eq:superficial} is not advertised as a proved scaling-degree bound for an undefined joint distribution. It is a finite graphwise power count. If a graph product and its uniform scaled limits are established, it supplies the expected degree bound. At the characteristic cancellation of Lemma~\ref{lem:pinch}, the prerequisite product theorem has not been established. Consequently, the table does not complete C3 for the mixed triangles, four-vertex cycles, or their intersections. Importing the degree of a different two-propagator graph would not fill that gap. %===================================================================== \section{C4: constructed primitive extensions and the joint stop} \label{sec:extension} %===================================================================== \begin{theorem}[Elementary parallel-line extension] \label{thm:primitive-extension} A primitive from Proposition~\ref{prop:parallel-degree} with finite degree $s$ extends from $\R^4\setminus\{0\}$ to $\R^4$ without increasing its scaling degree. If $s\ge4$, its ambiguity is \[ \sum_{|\alpha|\le\lfloor s-4\rfloor} c_\alpha\partial^\alpha\delta(r). \] For the temporal kinetic component the allowed derivative order is at most four. A relative-coordinate extension has a balanced two-endpoint wavefront bound. \end{theorem} \begin{proof} Choose a compact smooth function $w$ equal to one near zero and put $N=\lfloor s-4\rfloor$. Subtract the normal Taylor jet: \[ W_N\psi(r)=\psi(r) -w(r)\sum_{|\alpha|\le N} \frac{r^\alpha}{\alpha!}\partial^\alpha\psi(0). \] The remainder vanishes to order $N+1$. On a dyadic annulus of radius $2^{-j}$, the scaling estimate and Taylor remainder give a summable bound with exponent $4+N+1-s-\delta>0$ for sufficiently small $\delta>0$. Thus the primitive has a continuous extension to tests with the specified vanishing jet. Assigning the omitted finite jet defines an extension on all tests. Two such extensions differ by a distribution supported at zero. The scaling-degree bound restricts its derivative order to $\lfloor s-4\rfloor$. This is also the standard point-extension construction \cite[Section 5.2]{BF}. An extension depending only on $r=y-v$ is pulled back by the relative-coordinate submersion. Its endpoint covectors are $(p,-p)$, including those of its contact terms. Hence it is balanced. \end{proof} The constructed contacts belong to the allowed partial-diagonal form whenever their spectator product is defined. For $y=v$, the spectator can still connect $v$ to $x$. The contact is not restricted to $x=y=v$. The oriented $H$ parallel-line product is already defined by the inherited product calculus. Subtracting it from an extended time-ordered primitive therefore constructs the corresponding primitive difference. Since the difference has its prescribed values off $r=0$, any added contact changes only that collision. \begin{remark}[Primitive extension is not joint extension] Theorem~\ref{thm:primitive-extension} does not prove multiplication by every singular spectator, compatibility at intersecting diagonals, or the noncoincident prescription of Lemma~\ref{lem:pinch}. A product of an extended primitive with an undefined spectator composition is still undefined. \end{remark} \begin{theorem}[Present joint-extension verdict] \label{thm:joint-status} The results above do not construct the full $\Rquant$ for the selected pair in $\Cpart$. They also do not exclude every extension in that class. The exact ending is \textbf{Unresolved}. \end{theorem} \begin{proof} The full contraction inventory contains the triangle of Lemma~\ref{lem:pinch}. A joint distribution on its complete off-collision domain has not been constructed by the available sum-zero argument. The larger-cycle transverse estimates and intersection compatibility have likewise not been proved. The primitive extension theorem treats only the specified parallel-line factors. It does not establish those missing hypotheses. Conversely, failure of the sufficient product criterion is not a theorem that a joint distribution or a cancellation prescription cannot exist. No universal normalization or admission incompatibility has been proved. The release is therefore Unresolved. \end{proof} Widening the ambiguity class removes the artificial exclusion of partial contacts. It does not, by itself, solve an additional off-collision microlocal problem. %===================================================================== \section{C5: normalization and overlap audit} \label{sec:normalization} %===================================================================== A completed extension must retain the inherited ordered values: \begin{align} \supp\Vcal\text{ strictly later than }\supp\Ocal &\Longrightarrow R_1=0,\\ \supp\Ocal\text{ strictly later than }\supp\Vcal &\Longrightarrow R_1=\frac{i}{\hbar}[\Ocal,\Vcal]_{\star_H}. \label{eq:ordered-values} \end{align} These refer to full functional supports, not just the observation support. \begin{proposition}[Contact changes and strict ordering] \label{prop:normalization} A well-defined contact change in $\Cpart$ involving an interaction and an observable attachment cannot alter a region in which the full functional supports are strictly ordered and disjoint. The constructed primitive contacts therefore preserve the off-collision ordered values on their domains. \end{proposition} \begin{proof} Strict separation prohibits equality of any interaction attachment with an observable attachment. Every relevant contact is supported on a collision containing such an equality. Its restriction to the strictly separated region is zero. \end{proof} This proposition does not establish the joint extension outside those regions. In particular, partial chronological order inside a nonlocal entry is not full functional-support ordering. The inherited-root overlap is entirely at tree order: \[ \Rtree =\pair{\Ocal'_\gamma}{\rg[\phi]} +\pair{\Ocal'_\phi}{p_1[\gamma;\phi]}. \] Every permitted change here begins at $\hbar^2$ in $\TT$ and hence at $\hbar$ in $R_1$. It cannot alter this classical coefficient. No cubic-response root is demanded as the value of the complete degree-two quantum coefficient. The retarded condition remains an attachment condition. We do not assert that every remaining functional-support variable of $\Rquant$ must lie in $J^+(\supp\Vcal)$. %===================================================================== \section{C6: properness and the unproved output estimate} \label{sec:admission} %===================================================================== XIII-B proves properness for support-preserving extensions in the existing compact graph closure \cite{XIIIB}. The widened contacts retain that closure. Properness is therefore available once the relevant graph distributions exist. It cannot define an undefined distribution product. \begin{proposition}[Conditional pushforward admission] \label{prop:conditional-admission} Suppose the complete two-line graphs admit a joint extension in $\Cpart$, with the stated normalizations, and suppose their full graph wavefront sets have a balanced bound preserved under all specified internal compositions. Then the pushed-forward coefficient is a compact even, ghost-number-zero degree-two polynomial satisfying \[ \WF(t^{(1)})\subset Z_2=\{(z_1,p_1;z_2,p_2):p_1+p_2=0\}\setminus0. \] Consequently \[ Q_H\bigl(\Rtree+\hbar\Rquant\bigr)\in\Ealg. \] \end{proposition} \begin{proof} Properness gives a compact distributional pushforward. For an output wavefront point the internal covectors are zero. Summing the balanced complete-graph covectors leaves $p_1+p_2=0$ on the two residual field slots. A nonzero tuple lying wholly in either pointed closed causal cone cannot have zero sum, so the output is microcausal. Every double contraction leaves two even, ghost-number-zero slots. The inherited quotient map supplies the stated class. \end{proof} The hypothesis is not proved by assigning balanced covectors separately to each line. It requires a complete defined composition and an extension wavefront theorem. Extensions can introduce conormal covectors on collision loci, and their compositions with spectators must still be checked. The microlocal extension literature makes this additional wavefront obligation explicit \cite{Dang}. In the present release, neither the joint C4 construction nor that C6 estimate is complete. We therefore do not write $\Rquant$ as an admitted output. Classical evaluation on $\CW$ remains subsequent to admission. The tree zero subclasses $\{\gamma=b=0\}$ and $\{\phi=0\}$ do not imply corresponding zeros for a degree-two symbol. The sourced directions $\rg$ and $p_1$ are not used to differentiate the restriction identity. %===================================================================== \section{C7: decision and successor boundary} \label{sec:decision} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.08\linewidth} p{.24\linewidth}X@{}} \toprule Gate & Ending & Exact scope\\ \midrule C0 & Recorded & Tree datum inherited; literal factorization error closed; carrier unchanged.\\[0.35em] C1 & Inventory complete & Eleven branches and all 48 labelled incidence matchings for the frozen product data.\\[0.35em] C2 & Partly Constructed; joint gate Unresolved & Products exist on $\mathcal U_\Gamma$. A noncoincident characteristic cancellation prevents identifying that set with the entire collision complement.\\[0.35em] C3 & Primitive degrees computed; joint gate Unresolved & Parallel-line degrees proved. Larger-cycle counts are explicitly superficial, not unproved scaling theorems.\\[0.35em] C4 & Primitive extensions Constructed; joint gate Unresolved & Partial contacts are admitted as a class; the full joint extension is not constructed or universally excluded.\\[0.35em] C5 & Audit established & Contact support preserves strict ordered regions; tree overlap cannot be altered at this order.\\[0.35em] C6 & Conditional theorem only; Unresolved & Properness inherited; complete output wavefront estimate absent.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[Exact release] \label{thm:release} For the frozen displayed lifts, the fixed width, and the widened class $\Cpart$: \begin{enumerate}[label=\textup{(\roman*)}] \item The total-diagonal ambiguity class is correctly a subclass, not the complete partial-collision class. \item The incidence inventory and elementary parallel-line extensions are constructed. \item Collision exclusion and the $Z_3$ bounds do not establish the required global off-collision product. \item The complete coefficient linear in $\hbar$ in \eqref{eq:order} has the ending \textbf{Unresolved}. No universal obstruction and no admitted joint quantum output are asserted. \end{enumerate} \end{theorem} \begin{proof} Definition~\ref{def:Cpart}, Lemma~\ref{lem:branches}, the matching and collision inventory, Corollary~\ref{cor:C2}, Proposition~\ref{prop:parallel-degree}, Theorem~\ref{thm:primitive-extension}, and Theorem~\ref{thm:joint-status}. Proposition~\ref{prop:conditional-admission} identifies the remaining admission interface. \end{proof} \subsection*{What a successor must establish} The next extension attempt must supply: \begin{enumerate} \item a joint prescription across the noncoincident characteristic configurations, or a proved cancellation of those configurations in the complete selected pair; \item uniform transverse scaling and wavefront estimates for the three-vertex and four-vertex collision strata and their intersections; \item compatible extensions with the permitted partial contacts, rather than only isolated primitive extensions; \item the full output wavefront estimate after proper pushforward. \end{enumerate} If these obligations are met, the successor may release one $\Rquant$ in the unchanged carrier and only then open the binary gauge-consistency question at order $\lambda\hbar$. Until then, the free-differential display is not applied to the unconstructed coefficient. The absence of a joint construction is not an anomaly. The tree datum remains valid and is not rebranded as the quantum answer. No restriction quotient, new representative theorem, Einstein--Hilbert vertex, or sharp limit is introduced. \subsection*{Conclusion} \[ \boxed{ \begin{gathered} \Ctot\subseteq\Cpart,\qquad \text{parallel kinetic primitive: }\sd=8,\quad\codim=4,\\[2pt] \text{partial contact ambiguity: normal derivative order }\le4 \text{ for that primitive},\\[2pt] \text{distinct null-related vertices can admit a nonzero wavefront cancellation},\\[2pt] \text{complete joint two-line extension and admission: } \textbf{Unresolved}. \end{gathered}} \] The tree datum was already constructed. The relevant collision is not confined to the total diagonal. Elementary extension is not joint extension. The coefficient through $\lambda\hbar$ has one precise ending, and the gauge-consistency gate remains closed. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Dependency scope.} XIII-B supplies the tree datum, properness, convention, overlap, and unresolved binary gate. XIII-A supplies only the evaluation boundary. XII-A supplies the frozen product data and carrier. XII-D and XII-E supply the explicit source and completion branches. The incidence inventory, primitive scaling calculations, primitive extensions, and characteristic cancellation are given here. \paragraph{Meaning of the unresolved ending.} The manuscript does not claim a complete extension theorem. It identifies an additional joint microlocal issue that cannot be removed by citing a one-line proof or by allowing partial-diagonal counterterms. Failure of a sufficient criterion is not equated with failure of every possible construction. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant check, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \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 construction, normalization correction, properness, root overlap, and unresolved two-line extension. \bibitem{XIIIA} N.~Cheung, \emph{Landmark XIII-A: Classical cancellation on a fixed subclass, and the obstruction to a coefficientwise restriction quotient}. Supplied manuscript \texttt{XIII-A.tex}. Evaluation-only boundary and restriction-ideal obstruction. \bibitem{XIIF} N.~Cheung, \emph{Landmark XII-F: The gauge-fixed remainder of $\Phi^{(2)}_{\mathrm D}$}. Supplied manuscript \texttt{XII-F.tex}. Copied negative witness and its classical scope. \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}. Completion formula, affine slot identity, and admitted kernels. \bibitem{XIID} N.~Cheung, \emph{Landmark XII-D: Retarded response and carrier admission for the cubic-only density-weighted Einstein source}. Supplied manuscript \texttt{XII-D.tex}. Actual source coefficients, roots, and inherited response. \bibitem{XIIA} N.~Cheung, \emph{Landmark XII-A: One graded polynomial carrier for the finite-ramp multiplet}. Supplied manuscript \texttt{XII-A-v2.tex}. Frozen Hadamard matrix, action inverses, product, and shell quotient. \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. Sections~5--6: scaling degree and extension to points and submanifolds. \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. Microlocal control of extensions across embedded submanifolds. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}, arXiv:1208.1428. Time-ordered products, renormalization, and functional domains. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I}, second edition, Springer, 1990. Distribution products, wavefront sets, and proper pushforward. \end{thebibliography} \end{document}