% !TeX program = pdflatex %===================================================================== % XIV-F.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XIV-F.tex % pdflatex -interaction=nonstopmode -halt-on-error XIV-F.tex % % Exact release: % - the predecessor pair, six-branch germ, propagators, switches, % source representative, and partial-diagonal class are unchanged; % - exactly one route is examined: direct paired/clean composition % of the selected conormal fiber product; % - the elementary product phase is not reconstructed; % - the proposed excess-two interpretation is corrected; % - the six regular base constraints define a smooth manifold; % - the remaining cone constraint vanishes quadratically and % imposes two additional base conditions; % - the selected conormal fiber product is smooth but not clean; % - its two-dimensional tangent defect is identified explicitly; % - its covector-sum projection has constant rank fourteen and % one-dimensional local fibers, not two-dimensional fibers; % - its local image is isotropic of dimension fourteen, not a % Lagrangian submanifold of T^*W; % - a nondegenerate quadratic form on the tangent-defect plane % does not supply the missing clean-composition theorem; % - no joint principal-symbol section, remainder theorem, % characteristic prescription, or quantum output is constructed. % % Ending: % Unresolved at F2, within the inherited E1/D2/J2 stop. % F3--F5 are not entered as constructions. % J3--J5 and the quantum Ward-defect question remain unopened. % % Principal specification correction: % The two components of alpha*v-beta*w measure a failure of clean % tangency. They are not, merely by that calculation, an excess-two % integration fiber. Clean excess is defined only after the relevant % clean matching and projection hypotheses have been established. % % The exclusion proved here concerns the direct identification of % the stated fiber product with a clean excess-two construction. % It is not a nonexistence theorem for every paired representation, % every characteristic prescription, or the complete six-branch sum. % % 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{\MT}{M_T} \newcommand{\MTo}{M_T^\circ} \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{\Pmass}{P_m} \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{\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{\Dobs}{\mathcal D_{\mathrm{obs}}} \newcommand{\Mcal}{\mathcal M} \newcommand{\Mret}{\mathcal M_{\mathrm{ret}}} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\Fmatch}{\mathscr F_{\mathrm{match}}} \newcommand{\Tlin}{\mathscr T_{\mathrm{lin}}} \newcommand{\Defect}{\mathscr D_{\mathrm{tan}}} \newcommand{\Ssix}{S_6} \newcommand{\Ucan}{U_{\mathrm{can}}} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\sd}{sd} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Contr}{Contr} \DeclareMathOperator{\EL}{EL} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Int}{int} \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 XIV-F}\\[0.4em] Paired composition on a proposed excess bundle\\[0.4em] \large The clean-fiber-product audit, the excess correction,\\ and the exact F2 stop} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We continue XIV-E's unresolved characteristic composition problem without changing the pair, six-branch germ, source, or ambiguity contract. Exactly one route is examined: direct paired composition of the selected conormal fiber product, with the proposed two-dimensional excess identified with the relative transverse directions of XIV-E. The required geometric hypotheses fail. The two cone equations and the observable attachment diagonal give six regular constraints. On their smooth zero set, the remaining cone equation is a positive quadratic expression in two relative angular directions. Its zero set has two additional base conditions although its first differential vanishes there. Consequently the selected conormal fiber product is smooth but not clean: its actual tangent space has dimension fifteen, while the linearized matching space has dimension seventeen. The two-dimensional quotient is a tangent defect, not an established clean excess bundle. Moreover, the covector-sum projection has constant rank fourteen with one-dimensional local fibers. Its local image is isotropic, not a sixteen-dimensional Lagrangian submanifold of the cotangent bundle of the common ambient space. A nondegenerate quadratic form on the defect plane is identified, but it does not repair the failed tangent-space equality or define a symbol integral. No joint symbol, complete remainder estimate, or characteristic prescription is constructed. The ending is Unresolved at F2, within the inherited E1/D2/J2 stop. The result excludes the direct clean-excess-two identification, not every possible paired construction or cancellation of the full sum. J3--J5 and the Ward-defect question remain unopened. \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) {Proposed excess bundle}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Direct paired composition}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Another route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {No other route is attempted.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Only the direct paired route is attempted. 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-F} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Proposed excess bundle}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Another route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {No other route is attempted.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Only the direct paired route is attempted. 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-F} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Proposed excess bundle}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Direct paired composition}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Another route}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {No other route is attempted.}; \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-F} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Proposed excess bundle}; \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) {Another route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Direct paired composition. No other route is attempted.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Proposed excess bundle is not Another route. 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-F} \end{figure} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Proposed excess bundle}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Direct paired composition}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Another route}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {No other route is attempted.}; \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-F} \end{figure} Figure~\ref{fig:split-XIV-F} records one view of the same separation. Figure~\ref{fig:chain-XIV-F} records one view of the same separation. Figure~\ref{fig:pair-XIV-F} records one view of the same separation. Figure~\ref{fig:gate-XIV-F} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XIV-F} records that separation. The drawing adds no hypothesis. \scopebox{Release and specification correction}{% \textbf{Selected route:} direct paired composition on the proposed excess bundle. No other route is attempted. \smallskip \textbf{Correction:} the two relative transverse directions are not automatically an excess-two integration fiber. They measure failure of clean tangency for the specified matching problem. \smallskip \textbf{Proved here:} the regular six-constraint stage; the exact quadratic remaining constraint; the non-clean conormal matching; the rank-two tangent-defect bundle; and the rank-fourteen covector-sum projection with one-dimensional local fibers. \smallskip \textbf{Ending: Unresolved at F2.} No symbol bundle for the complete grouped coefficient is constructed. F3--F5 are not entered as constructions. \smallskip The pair, $\Cpart$, source coefficients, and inverse $-1$ are unchanged. No $\Rquant$, characteristic subtraction, Ward defect, or sharp-width limit is released.} \tableofcontents %===================================================================== \section{F0: copied analytic inputs} \label{sec:F0} %===================================================================== This paper refines the inherited stop; it does not declare J2, D2, or E1 discharged. The missing object remains the grouped characteristic contraction of the complete retained six-branch germ. The statements quoted below are inputs from the supplied manuscripts. Their local equation labels are new. No incidence ledger, predecessor phase theorem, or branch fiber calculation is re-proved. \subsection{XIV-E: the exact successor package} \begin{inputstatement}[Successor package, copied from XIV-E] \label{input:package} ``A successor receives: \begin{enumerate} \item the unchanged pair and six-branch germ; \item the exact regrouping; \item the complete root formula; \item the intermediate tested-kernel equality and balanced attachment bound; \item the two characteristic lifts and their codimensions; \item the non-clean phase theorem and quadratic defect identity; \item the unchanged degree, ambiguity, and wavefront requirements; \item the exact E1 stop. \end{enumerate}'' ``It receives neither a nonzero full symbol section nor permission to infer one from $Q\lambda$. It receives no $\Rquant$'' \cite{XIVE}. \end{inputstatement} The cited formulas and geometric statements are copied next. \begin{inputstatement}[Exact regrouping and root] \label{input:regroup} With \[ X_G=X[G],\qquad X_h=X[h],\qquad r_h=\rD'[G]h=\PP\ER\bigl(\JD'[G]h\bigr), \] put \begin{equation} C_G^\mu{}_\nu =f\,\partial_\nu X_G^\mu -\delta^\mu_\nu\partial_\rho(fX_G^\rho), \label{eq:CG} \end{equation} and \begin{equation} \begin{split} B^{\alpha\mu}[G;h] ={}& a_1^{\alpha\beta}[h]\partial_\beta X_G^\mu +a_1^{\alpha\beta}[G]\partial_\beta X_h^\mu\\ &+\dot a_2^{\alpha\mu}[G;h]+a_1^{\alpha\mu}[r_h]. \end{split} \label{eq:Btensor} \end{equation} The exact six-branch identity is \begin{equation} \begin{split} \Mret[G;h,\psi] ={}& \int C_G^\mu{}_\nu X_h^\nu\partial_\mu\psi\,\dd^4x\\ &+\int f\,\ER\!\left( \partial_\alpha(\sig B^{\alpha\mu}[G;h]) \right)\partial_\mu\psi\,\dd^4x\\ &+\int f\,p[r_h;\psi]\,\dd^4x. \end{split} \label{eq:regroup} \end{equation} For $F=\EAm f$, the last term is \begin{equation} \Mcal_9[G;h,\psi] =\int g\left( a_1^{\mu\nu}[r_h]\partial_\mu F\,\partial_\nu\psi -m^2b_1[r_h]F\psi\right)\dd^4x. \label{eq:root} \end{equation} ``This identity retains all six branches. It involves no characteristic composition and no regularization.'' ``Both the derivative and mass terms are retained'' \cite{XIVE}. \end{inputstatement} \begin{inputstatement}[Tested kernels] \label{input:tested} For $G\in C_c^\infty(Z;\Sym^2T^*M)$, the residual-slot pairing gives an attachment kernel $L_G$ satisfying \[ \Mret[G;h,\psi]=\pair{L_G(a,b)}{h(a)\psi(b)}. \] It equals the tested sum of $K_4,\ldots,K_9$, is compact, and has \[ \WF(L_G)\subset Z_2^{\mathrm{att}} =\{(a,\xi;b,\eta):\xi+\eta=0\}\setminus0. \] ``This is an intermediate attachment-kernel bound, not admission of the final quantum coefficient.'' ``Nor does testing each fixed $G$ supply the uniform estimates needed to reconstruct a joint characteristic distribution in all four common variables'' \cite{XIVE}. \end{inputstatement} \begin{inputstatement}[Lifts, phase exclusion, and quadratic defect] \label{input:geometryE} In \[ W=A\times B\times Z\times Y, \qquad P(a,b,z,y)=(b,a,y), \] the two characteristic lifts are \[ \Nbar=P^{-1}(\Nchar),\qquad \Dobs=\{a=z\}\cap W,\qquad \Nlift=\Nbar\cap\Dobs. \] The copied codimensions are \[ \codim_W\Nbar=4,\qquad \codim_W\Dobs=4,\qquad \codim_W\Nlift=8. \] Their intersection is transverse. The elementary product phase of XIV-E has rank-six auxiliary equations and a codimension-eight critical set. It is neither nondegenerate nor clean. On the affine transverse slice \[ b-a=\beta\ell_{\mathbf n}+(0,v),\qquad y-a=\alpha\ell_{\mathbf n}+(0,w), \qquad v,w\in\mathbf n^\perp, \] the exact identity is \begin{equation} -\frac{q_R}{\beta} +\frac{q_1}{\alpha} +\frac{q_2}{\beta-\alpha} = -\frac{|\alpha v-\beta w|^2} {\alpha\beta(\beta-\alpha)}. \label{eq:copied-quadratic} \end{equation} ``Equation~\eqref{eq:copied-quadratic} is a geometric identity, not a scaling degree of a distribution.'' The ending is ``Unresolved at E1, within D2/J2'' \cite{XIVE}. \end{inputstatement} No elementary phase is reconstructed in this paper. \subsection{XIV-D and XIV-C: the retained instance} \begin{inputstatement}[Pair, local incidence, and branch diagnostic] \label{input:D} The frozen pair has \[ \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi),\qquad \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD, \] and \[ \Gret=-\PP\ER,\qquad \Ctot\subseteq\Cpart. \] On $A\times B\times Z$, \[ K_\Sigma=K_4+K_5+K_6+K_7+K_8+K_9. \] The kernels $K_6,K_8,K_9$ are supported on $a=z$. ``The complete mixed germ is therefore a six-branch sum. In particular, neither the coordinate gravitational leg $B_8$ nor the scalar gravitational leg $B_9$ may be omitted.'' The species expression has its inherited mixed factor $i$ and scalar--scalar factor $i/2$. The scalar--scalar channel vanishes only on the retained output-location patch. For $p^2=0$ and $r=p\otimes k+k\otimes p$, \[ p_\alpha p_\mu\dot a_2^{\alpha\mu}[G;r] =G(p^\sharp,p^\sharp)(p\cdot k)=Q\lambda. \] ``The proposition computes only one branch's leading differential numerator. It supplies no sum identity or sum nonvanishing theorem for that complete expression.'' The first unresolved step is D2, within J2 \cite{XIVD}. \end{inputstatement} \begin{inputstatement}[Characteristic geometry and contact rigidity] \label{input:C} The retained points are \[ u_0=(t_*,0,0,0),\quad y_0=(t_*+L,L,0,0),\quad x_0=(t_*+2L,2L,0,0), \] where \[ \eps0$ remain fixed. The carrier is \[ \Ealg= (\Fpol[[\hbar,\kappa]],\star_H)/ \Nsol[[\hbar,\kappa]]. \] The frozen lines satisfy \[ \HT=H+i\GA,\qquad \Gret=-\PP\ER, \] and metric--scalar blocks vanish. The interaction remains \[ \Vcal=-\frac12\int g\,\gamma_{\mu\nu} T_0^{\mu\nu}[\phi]\,\dd^4x. \] The full observable retains the scalar-sourced slot once: \[ \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi). \] The target is the retained grouped mixed coefficient \begin{equation} \begin{split} \mathcal T_{\mathrm{form}} =i\,\Contr^{\gamma\phi}\Bigl( &\Vcal''_{\gamma\phi},\Kret;\ \HT_{\gamma\gamma}(y,a)\otimes\HT_{\phi\phi}(y,b)\\ &\hspace{22mm} -H_{\gamma\gamma}(y,a)\otimes H_{\phi\phi}(y,b) \Bigr), \qquad \Kret=\sum_{j=4}^{9}K_j . \end{split} \label{eq:target} \end{equation} The species factor is unchanged. The cross-line difference stays grouped; no separate mixed $H$--$\GA$ product is declared to exist. The following endings are fixed before the new calculation: \begin{enumerate} \item \textbf{Constructed}: a common representation of the complete coefficient with a joint principal-symbol section and a remainder controlling every omitted term, yielding the required cancellation; or one named prescription with its ambient space, reference domain, degree, normalization, ambiguity, and wavefront theorem. \item \textbf{Proved obstructed}: a demonstrated incompatibility excludes every map in a class specified before the test. \item \textbf{Unresolved}: the common representation, symbol, remainder, or prescription theorem is missing. \end{enumerate} These endings apply to \eqref{eq:target}, not to the success or failure of an individual geometric shortcut. For the cancellation ending, disappearance of a leading section must be accompanied by the control actually needed for the remaining grouped distribution. Merely lowering a formal order does not define a product that was undefined. \subsection{The six missing symbol data} XIV-E's list is retained: \begin{enumerate} \item the relevant canonical or characteristic relation in the chosen ambient space; \item the amplitude orders and the common symbol bundle; \item the maps accounting for additional internal vertices in $K_5,K_8,K_9$; \item the observable attachment diagonal and its derivatives; \item the complete tensor coefficients and the prescribed grouping of the cross-line difference; \item a remainder estimate that controls every term not represented by the proposed leading section. \end{enumerate} Inherited wavefront inclusions do not supply these data. There is no cutoff of $\Nchar$, no restoring limit, and no promotion of $Q\lambda$ to a joint symbol. %===================================================================== \section{F1: one selected composition route} \label{sec:F1} %===================================================================== \[ \boxed{\text{Selected route: direct paired composition on the proposed excess bundle.}} \] The proposed construction is to form the fiber product of the three selected conormal relations and the attachment relation, then reduce it by the copied linear dependence. Its intended excess-two directions are the two components of $\alpha v-\beta w$. This is a new intrinsic fiber-product audit. It does not reconstruct XIV-E's elementary phase or assert that the rejected phase has become clean. \begin{definition}[Scope of the direct route] \label{def:route} The construction examined here uses the original common ambient space $W$ and the selected conormal incidences. Before a symbol is assigned, it requires: \begin{enumerate}[label=\textup{(\roman*)}] \item a clean matching fiber product; \item a constant-rank reduction to a stated cotangent relation; \item identification of genuine integration fibers and their dimension; \item the Lagrangian or paired-Lagrangian geometry required by the proposed calculus; \item admissible internal compositions for $K_5,K_8,K_9$ and the complete grouped amplitudes. \end{enumerate} No new base constraints, resolved ambient space, or additional amplitude-cancellation theorem is silently supplied. \end{definition} A paired-distribution framework is not obtained merely by naming an excess. The selected route must establish its particular geometric matching before a paired symbol theorem can be used. The Melrose--Uhlmann framework is the intended comparison, not an extension theorem whose hypotheses are assumed \cite{MU}. \subsection{Unentered alternatives} The following remain named but unentered: \begin{enumerate} \item a common joint-symbol calculus working directly from \eqref{eq:regroup} and the conormal relation without constructing a common phase; \item a continuation of XIII-E's common-frequency family, requiring a new characteristic convergence theorem. \end{enumerate} If the selected route fails its geometric test, the paper stops. It does not switch to either alternative. \subsection{Clean matching and excess} For smooth maps $f_i:L_i\to W$, a matching fiber product is \emph{clean} when it is a smooth manifold and its tangent space is exactly the space of tangent tuples with equal images under the differentials $\dd f_i$. For canonical composition, clean matching and the appropriate projection hypotheses precede the symbol integral. The clean composition excess is the dimension of the local fibers of that projection, not an arbitrary difference between two rank counts \cite[Section 4.1]{GS}\cite[Section 9]{Wunsch}. The test below is precisely this tangent-space test. No clean-composition theorem is applied unless it passes. %===================================================================== \section{F2: ambient space and the intrinsic conormal matching} \label{sec:F2} %===================================================================== \subsection{One ambient space} All new matching calculations take place in \[ W=A\times B\times Z\times Y,\qquad \dim W=16, \] with coordinate order $(a,b,z,y)$. The three loci remain distinct: \[ \Nchar\subset B\times A\times Y,\quad \codim=4, \] \[ \Nbar=P^{-1}(\Nchar)\subset W,\quad \codim=4, \] \[ \Nlift=\Nbar\cap\Dobs\subset W,\quad \codim=8. \] No three-vertex degree or codimension is substituted for a common-kernel degree or codimension. We work on the already retained chronological neighborhood \[ y^0-a^0>0,\qquad b^0-y^0>0. \] Write \[ d=a-z. \] Define the six-constraint stage and the full incidence set by \[ \Ssix=\{q_1=q_2=0,\ d=0\}\subset W, \qquad N=\{q_R=q_1=q_2=0,\ d=0\}. \] By the copied zero-locus result, \[ N=\Nlift. \] \subsection{First new calculation: the regular stage and its last equation} \begin{proposition}[Six regular constraints and a quadratic final constraint] \label{prop:regular} Near the retained point, $\Ssix$ is a smooth codimension-six submanifold of $W$. It has the local parameterization \begin{equation} \begin{split} z&=a,\\ y&=a+\alpha(1,\mathbf n_1),\\ b&=y+\delta(1,\mathbf n_2), \end{split} \qquad \alpha>0,\quad \delta>0,\quad \mathbf n_1,\mathbf n_2\in S^2. \label{eq:Sparameter} \end{equation} On this manifold, \begin{equation} q_R=\alpha\delta\,|\mathbf n_1-\mathbf n_2|^2. \label{eq:qR-on-S} \end{equation} Consequently $N$ is the angular diagonal $\mathbf n_1=\mathbf n_2$ in $\Ssix$, with $\beta=\alpha+\delta$. \end{proposition} \begin{proof} The differentials $\dd q_1,\dd q_2$ are independent at the retained points. The four components of $\dd(a-z)$ are independent of them because the cone equations have no $z$ component. Thus the six equations define a smooth codimension-six manifold. The chronology and null equations uniquely write the two successive nonzero future null segments as $\alpha(1,\mathbf n_1)$ and $\delta(1,\mathbf n_2)$. Their time components and normalized spatial directions give a smooth inverse to \eqref{eq:Sparameter}, so this is a local parameterization. The sum of the two null segments has Minkowski square \[ 2\alpha\delta(1-\mathbf n_1\cdot\mathbf n_2) =\alpha\delta|\mathbf n_1-\mathbf n_2|^2. \] It is zero exactly when their unit spatial directions agree. \end{proof} \begin{corollary}[The last equation is not a redundant regular constraint] \label{cor:notredundant} At $N$, the differential of $q_R|_{\Ssix}$ vanishes, but its zero set has codimension two in $\Ssix$. In particular, the copied linear dependence of cone conormals does not permit the equation $q_R=0$ to be deleted while retaining the same base incidence. \end{corollary} \begin{proof} Equation~\eqref{eq:qR-on-S} is a positive quadratic expression in the relative angular variables. Its first derivative vanishes on the angular diagonal. That diagonal has codimension two in $S^2\times S^2$. Deleting $q_R=0$ leaves all of $\Ssix$, of dimension ten, rather than $N$, of dimension eight. \end{proof} This is the central distinction in the proposed route: the ``six constrained directions'' stage exists, but the remaining two directions are additional base conditions imposed quadratically. They are not freely varying integration parameters of the full matching manifold. \subsection{The selected conormal fiber product} The four selected conormal incidences are \begin{align*} \Lambda_R^-&=\{(w,-t_R\dd q_R(w)): q_R(w)=0,\ t_R>0\},\\ \Lambda_1^+&=\{(w,t_1\dd q_1(w)): q_1(w)=0,\ t_1>0\},\\ \Lambda_2^+&=\{(w,t_2\dd q_2(w)): q_2(w)=0,\ t_2>0\},\\ \Lambda_D&=\{(w,\kappa_\mu\dd d^\mu): d(w)=0,\ \kappa\in\R^4\}. \end{align*} Here $w$ denotes a point of $W$, not the transverse displacement in \eqref{eq:copied-quadratic}. The symbol $\kappa_\mu$ in this section is an attachment covector coordinate, not the interaction coupling. The zero attachment covector is included to accommodate the copied product-test selection. The conclusions below also hold on the punctured part $\kappa\ne0$; they do not depend on including zero in that factor. Each cone hypersurface is smooth near the retained points: its separation vector is nonzero, so its defining differential is nonzero. The attachment conormal is likewise a smooth bundle. We therefore have smooth individual geometric relations before their matching is examined. Let \begin{equation} \Fmatch =\Lambda_R^-\times_W\Lambda_1^+ \times_W\Lambda_2^+\times_W\Lambda_D. \label{eq:fiberproduct} \end{equation} In the common incidence coordinates, \[ \Fmatch=N\times\Theta,\qquad \Theta=(0,\infty)^3\times\R^4. \] This parameterization names the set. It does not establish clean matching or a distributional product. \begin{theorem}[The selected conormal matching is not clean] \label{thm:nonclean} The fiber product \eqref{eq:fiberproduct} is a smooth manifold of dimension fifteen. Its linearized matching space has dimension seventeen at every retained point. Hence the matching is not clean. \end{theorem} \begin{proof} The copied smoothness and codimension of $N=\Nlift$ give \[ \dim\Fmatch=\dim N+\dim\Theta=8+7=15. \] A tangent vector to the product of the four separate conormal relations satisfies linearized base matching precisely when its common base component $u\in T_wW$ obeys \[ \dd q_R(u)=\dd q_1(u)=\dd q_2(u)=0,\qquad \dd d(u)=0. \] The seven frequency variations are unrestricted in this linearized base matching. Define \[ \Tlin_w =\ker\dd(q_R,q_1,q_2,d)_w. \] The copied cone rank is two. The four attachment differentials are independent of the cone differentials, so the full differential has rank six. Therefore \[ \dim\Tlin_w=16-6=10. \] The linearized matching space has dimension $10+7=17$. The actual tangent space is $T_wN\times T_\theta\Theta$, of dimension $8+7=15$. It is strictly smaller than the linearized matching space. This violates the tangent-space equality in the definition of clean matching. \end{proof} This is an intrinsic matching calculation for the proposed relations, not another phase test. No amplitude or principal symbol has been written. \subsection{Identification of the rank-two tangent defect} Let \[ \ell_{\mathbf n}=(1,\mathbf n),\qquad p_{\mathbf n}=\eta(\ell_{\mathbf n},\cdot). \] At a point of $N$, write a base variation as \[ u=(\delta a,\delta b,\delta z,\delta y). \] For a four-vector $\xi$, denote by $\xi_\perp$ the component of its spatial part in $\mathbf n^\perp$. \begin{proposition}[The tangent-defect bundle] \label{prop:defect} The quotient \[ \Defect_w=\Tlin_w/T_wN \] has dimension two. It is identified with $\mathbf n^\perp$ by \begin{equation} \mathfrak d_w(u) =\alpha(\delta b-\delta a)_\perp -\beta(\delta y-\delta a)_\perp. \label{eq:defectmap} \end{equation} Locally over $N$, these quotients form a smooth rank-two bundle. They are not established clean-composition fibers. \end{proposition} \begin{proof} A vector belongs to $\Tlin_w$ exactly when \[ \delta z=\delta a,\qquad p_{\mathbf n}(\delta b-\delta a)=0,\qquad p_{\mathbf n}(\delta y-\delta a)=0. \] The third cone equation follows from the copied linear relation. The last two equations allow the decompositions \[ \delta b-\delta a =\delta\beta\,\ell_{\mathbf n}+(0,v), \qquad \delta y-\delta a =\delta\alpha\,\ell_{\mathbf n}+(0,w), \qquad v,w\in\mathbf n^\perp. \] The tangent vectors to $N$ are precisely those with \[ v=\beta\,\delta\mathbf n,\qquad w=\alpha\,\delta\mathbf n \] for one $\delta\mathbf n\in\mathbf n^\perp$. Thus \[ \ker(\mathfrak d_w|_{\Tlin_w})=T_wN. \] The map is onto: for any $d_\perp\in\mathbf n^\perp$, choose $v=d_\perp/\alpha$ and $w=0$. Its coefficients vary smoothly along $N$, its rank is two, and both $\Tlin$ and $TN$ have constant ranks. The quotient is consequently a smooth local bundle. \end{proof} The two components in \eqref{eq:defectmap} are exactly the relative transverse components appearing in XIV-E. Their meaning is now identified: they are linearized matching directions which are not tangents to actual matched configurations. \subsection{The quadratic form is nondegenerate, but not an excess theorem} \begin{proposition}[Quadratic control on the defect plane] \label{prop:quadratic} The copied quadratic expression induces the negative-definite form \begin{equation} \mathfrak q_w(d_\perp) =-\frac{|d_\perp|^2} {\alpha\beta(\beta-\alpha)} \label{eq:defectquadratic} \end{equation} on $\Defect_w$. Its Hessian is \[ \operatorname{Hess}\mathfrak q_w =-\frac{2}{\alpha\beta(\beta-\alpha)}\,\id \] under the Euclidean identification with $\mathbf n^\perp$. On compact tangential sets inside $0<\alpha<\beta$, its nondegeneracy is uniform. On the regular manifold $\Ssix$, the Hessian of $q_R$ in the relative angular directions is positive definite. \end{proposition} \begin{proof} The copied expression \eqref{eq:copied-quadratic} depends on $v,w$ only through $d_\perp=\alpha v-\beta w$. Its kernel is the common angular tangent direction, so it descends to the quotient identified in Proposition~\ref{prop:defect}. Differentiation gives the displayed Hessian. On a compact subset of $0<\alpha<\beta$, the positive denominator has positive lower and finite upper bounds. The Hessian eigenvalues are therefore bounded away from zero in absolute value. For the last assertion, differentiate \eqref{eq:qR-on-S} twice at $\mathbf n_1=\mathbf n_2$: for tangent angular variations, \[ \operatorname{Hess}(q_R|_{\Ssix}) =2\alpha\delta\, |\delta\mathbf n_1-\delta\mathbf n_2|^2. \] It is positive definite on the two relative angular components and zero on the common angular tangent. \end{proof} \begin{remark}[Why this does not rescue clean composition] A nondegenerate quadratic form on these base-transverse directions does not make them fibers of $\Fmatch\to\Gamma$ for a canonical relation $\Gamma$. They are absent from $T\Fmatch$, not excess directions inside it. A construction that integrates or resolves these base directions would have to specify a different representation, the map back to $W$, the amplitudes, and the resulting distributional theorem. None follows from the Hessian calculation. Such a construction is not silently substituted for the direct route. \end{remark} \subsection{The actual covector-sum projection} Define the geometric map \begin{equation} \begin{split} J:\Fmatch&\longrightarrow T^*W,\\ J(w,t_R,t_1,t_2,\kappa) &=\left(w, -t_R\dd q_R+t_1\dd q_1+t_2\dd q_2 +\kappa_\mu\dd d^\mu\right). \end{split} \label{eq:sum-map} \end{equation} This is a sum of cotangent vectors. It is not an oscillatory phase or a principal-symbol definition. \begin{theorem}[Projection rank and the excess correction] \label{thm:projection} The map $J$ has constant rank fourteen on the retained matching manifold. Its local fibers have dimension one. Its local image is an isotropic fourteen-dimensional immersed submanifold of $T^*W$, contained in $N^*N$ with the zero section allowed. It is not a Lagrangian submanifold of $T^*W$. Accordingly, neither the rank-two tangent defect nor the fibers of $J$ justify the specification's clean-excess-two claim. \end{theorem} \begin{proof} For fixed $w\in N$, the differential of $J$ in the seven frequency variables has image \[ \operatorname{span} \{\dd q_R,\dd q_1,\dd q_2,\dd d^0,\ldots,\dd d^3\}, \] of dimension six. The base projection of $J$ is the identity on $N$. Its differential maps the eight base tangent directions injectively onto $T_wN$ in the base component of $T(T^*W)$. These directions are independent of the six purely vertical image directions. Thus \[ \rank\dd J=8+6=14. \] The rank is constant. A vector in $\ker\dd J$ must have zero base component. The attachment frequency variation must then be zero, because its $z$ component cannot be canceled by cone covectors. The remaining kernel is the one-dimensional cone-frequency relation \[ (\delta t_R,\delta t_1,\delta t_2) =c\left( \frac1{2\beta}, \frac1{2\alpha}, \frac1{2(\beta-\alpha)} \right). \] The constant-rank theorem gives one-dimensional local fibers. Every covector in \eqref{eq:sum-map} annihilates $T_wN$. Hence the image lies in the conormal bundle of $N$. Let $\lambda_{\mathrm{can}}$ be the tautological one-form on $T^*W$. For every tangent vector to $\Fmatch$, its base component belongs to $T_wN$, so \[ J^*\lambda_{\mathrm{can}}=0. \] Differentiating gives $J^*\dd\lambda_{\mathrm{can}}=0$. Since $J$ has constant rank, its local image is isotropic. A Lagrangian submanifold of $T^*W$ has dimension $\dim W=16$. The image has dimension fourteen, so it is not Lagrangian. \end{proof} The constant-rank requirement passes for this map, but the clean-matching requirement already fails. Passing one requirement does not imply the others. The relation \eqref{eq:conormal-relation} yields a one-dimensional frequency kernel. It does not eliminate the two quadratic base conditions of Corollary~\ref{cor:notredundant}. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.46\linewidth}cX@{}} \toprule Quantity & Dimension & Meaning\\ \midrule $W$ & $16$ & Common base space.\\ $\Ssix$ & $10$ & Six regular base constraints.\\ $N=\Nlift$ & $8$ & Full matched base incidence.\\ $\Fmatch$ & $15$ & Actual incidence plus seven frequencies.\\ Linearized matching space & $17$ & Two directions not tangent to $\Fmatch$.\\ $\Defect_w$ & $2$ & Failure of clean tangency.\\ Local fiber of $J$ & $1$ & Cone-frequency redundancy.\\ Local image of $J$ & $14$ & Isotropic, not Lagrangian in $T^*W$.\\ \bottomrule \end{tabularx} \end{center} None of the numbers in this table is an assigned transverse scaling degree of \eqref{eq:target}. \subsection{Smooth reparameterization cannot change the failed equality} \begin{lemma}[Persistence under a smooth frequency reduction] \label{lem:persistence} Smooth coordinate changes in the individual conormal parameterizations preserve the failure of clean matching. Adding $k$ freely varying auxiliary parameters adds $k$ to both the actual and linearized matching dimensions. Deleting $k$ freely varying vertical parameters in a local product quotient subtracts $k$ from both dimensions. The two-dimensional tangent discrepancy persists. In particular, a local quotient by the one-dimensional frequency redundancy in Theorem~\ref{thm:projection} does not convert this matching into a clean one. \end{lemma} \begin{proof} A smooth reparameterization carries both the actual tangent space and the linearized matching space isomorphically to their new descriptions. Their equality or inequality is therefore invariant. For a free auxiliary factor, both spaces acquire its full tangent space as a direct summand. For a local product quotient by free vertical parameters, both lose the same vertical summand. Their dimension difference remains two. The frequency kernel in Theorem~\ref{thm:projection} is one-dimensional and vertical. In a local quotient chart, the two dimensions become fourteen and sixteen, respectively, and are still unequal. \end{proof} The lemma does not rule out a resolved or otherwise different representation. It rules out curing the stated defect merely by relabeling or deleting redundant frequency parameters. \subsection{Internal vertices are not collapsed} The kernels $K_5,K_8,K_9$ already exist as tree-kernel pushforwards. Using them in a paired-symbol calculus would also require their own canonical relations, clean internal matching, symbol orders, and uniform remainder estimates. No hidden vertex is set equal to $u_0$, $y_0$, or $x_0$. The present F2 audit requires none of those unproved internal symbol identifications. Since the proposed final matching has failed its necessary clean test, this paper does not attempt those internal symbol constructions. The existence of the tree kernels is retained; a new symbol theorem for their composition is not inferred from it. %===================================================================== \section{The exact F2 verdict} \label{sec:verdict} %===================================================================== \begin{theorem}[Direct excess-route audit] \label{thm:route} For the selected route of Definition~\ref{def:route}: \begin{enumerate}[label=\textup{(\roman*)}] \item the six regular base constraints are available; \item the full selected conormal matching is not clean; \item the rank-two transverse object is a tangent-defect bundle; \item the covector-sum projection has one-dimensional, not two-dimensional, local fibers; \item the stated projected image is not a Lagrangian relation on the named ambient space. \end{enumerate} Therefore the direct application of a clean excess-two composition formula to this fiber product is not justified. The selected route stops at F2. \end{theorem} \begin{proof} Proposition~\ref{prop:regular} gives (i). Theorem~\ref{thm:nonclean} gives (ii). Proposition~\ref{prop:defect} gives (iii). Theorem~\ref{thm:projection} gives (iv) and (v). The required clean and Lagrangian hypotheses of the direct route are consequently absent. Lemma~\ref{lem:persistence} shows that the proposed frequency reduction does not remove the failed tangent equality. \end{proof} \begin{remark}[What is excluded, and what is not] The theorem excludes the direct identification of this matching with a clean excess-two construction. It does not exclude every Melrose--Uhlmann-type representation, every resolved composition, or every characteristic prescription. Nor does it prove that the complete six-branch coefficient has a nonzero singular amplitude on this incidence. Such a conclusion would require the full symbol and remainder analysis, which has not been performed. \end{remark} No class of all characteristic prescriptions has been proved incompatible. The legal ending for the grouped target is therefore \textbf{Unresolved}, not Proved obstructed. %===================================================================== \section{F3 and F4: symbol and remainder not constructed} \label{sec:F3F4} %===================================================================== F3 is not entered as a construction. The common symbol bundle of \eqref{eq:target} has not been supplied. The regrouped pieces retain the following exact status: \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.19\linewidth} p{.38\linewidth}X@{}} \toprule Piece & Copied result & Still missing\\ \midrule $B_4+B_7$ & The coefficient $C_G$ after the exact explicit $\partial^2\psi$ cancellation & Common microlocal order and symbol of the remaining term.\\[0.3em] $B_5+B_6+B_8$ & One divergence of the complete tensor $B^{\alpha\mu}[G;h]$ & Internal relations, composition orders, and the grouped divergence symbol.\\[0.3em] $B_6$ inside that tensor & The branch identity $Q\lambda$ & A symbol of the full sum, not merely that numerator.\\[0.3em] $B_9$ & The full advanced root \eqref{eq:root} & Composed symbol with the scalar line, including the mass contribution and its remainder.\\ \bottomrule \end{tabularx} \end{center} The disappearance of an explicit test derivative in a smooth-slot identity does not assign a lower microlocal order. Likewise, the quadratic defect is geometry, not a distributional degree. F4 is not entered. A complete remainder estimate would have to control: \begin{enumerate} \item lower-order terms in every numerical line; \item the two relative transverse directions; \item derivatives on the observable attachment diagonal; \item the internal compositions in $K_5,K_8,K_9$; \item all terms retained in the complete source and root. \end{enumerate} Neither a vanishing joint section with adequate remainder control nor a nonzero joint section on a named open relation is asserted. \begin{proposition}[No cancellation verdict follows from F2] \label{prop:noverdict} The geometric exclusion in Theorem~\ref{thm:route} proves neither cancellation nor failure of cancellation of the grouped six-branch coefficient. \end{proposition} \begin{proof} The theorem concerns the clean matching and reduction hypotheses of one selected geometric construction. It assigns no joint amplitude or principal-symbol section to \eqref{eq:target}. It supplies no estimate for terms remaining after a proposed leading cancellation. The copied $Q\lambda$ identity concerns $B_6$ alone, and the copied regrouping is a smooth-slot identity. Neither replaces those missing amplitude and remainder data. \end{proof} %===================================================================== \section{F5: prescription interface not entered} \label{sec:F5} %===================================================================== No nonzero full joint section has been established. Accordingly, F5 is not entered as a prescription construction. The following is a copied interface, not a map. A future prescription must name: \begin{enumerate} \item its ambient graph or common-kernel space; \item a reference distribution on a proved domain; \item the exact characteristic locus $N$ across which it extends; \item agreement with the reference and grouped frozen prescription; \item a finite uniform transverse degree in stated coordinates; \item a closed conic wavefront bound retaining off-$N$ singularities; \item a normalization selecting one map, or a uniqueness theorem under the proved hypotheses. \end{enumerate} The three loci $\Nchar$, $\Nbar$, and $\Nlift$ cannot be mixed in one degree or ambiguity statement. The canonical product is known only on the inherited microlocally admissible domain $\Ucan$. No identification \[ \Ucan=W\setminus\Nbar \] is asserted. Naming separate line distributions does not prove a reference joint distribution on that complement. The copied wavefront requirement remains: \[ \WF(t)\subset \overline{\WF(t^\circ)}\cup N^*N \] is a prospective bound requiring its own hypotheses. A bound by $N^*N$ alone would erase every nonsmooth off-$N$ reference point. For a genuine codimension-four extension problem with degree at least four, smooth tangential multiples of $\delta^{(4)}(r)$ remain possible under the copied degree and full-conormal allowances unless a normalization or uniqueness theorem removes them. On the retained collision-free neighborhood, a nonzero such change is not a member of $\Cpart$. The codimension-eight common attachment intersection requires its own extension-stage analysis. No actual degree is assigned here. Brunetti--Fredenhagen and Dang are recorded extension frameworks \cite{BF,Dang}. No extension theorem from them is applied: the requisite reference distribution and uniform estimates have not been constructed. XIII-E's frequency family is not imported as this prescription. No cutoff-and-restore limit, characteristic finite part, or new ambiguity class is introduced. %===================================================================== \section{F6: downstream gates and non-identifications} \label{sec:F6} %===================================================================== J3 opens only for a distribution supplied by a successful cancellation or prescription construction. None has been supplied. The parallel-line degree eight is a collision degree. It is not a degree along $\Nchar$, $\Nbar$, or $\Nlift$, and the Hessian calculation in Proposition~\ref{prop:quadratic} does not change that distinction. J4 remains closed. No joint characteristic distribution has been glued to singular spectators or compatible partial contacts. J5 remains closed. Properness is support control for an existing distribution, not a definition of the missing grouped composition or its output wavefront set. No $\Rquant$ is released. The admitted tree datum is not substituted for it. No $s_{\mathrm{ext}}$, $s_\Psi$, or $s_0$ is applied to the absent quantum coefficient. No binary Ward defect is formed. The classical breaking density and the value $-8I$ neither define nor normalize this graph. No new gravitational interaction vertex, restriction quotient, state, numerical loop value, characteristic contact, triangle residue, or sharp-width limit is introduced. %===================================================================== \section{Decision and exact release} \label{sec:decision} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.07\linewidth} p{.28\linewidth}X@{}} \toprule Gate & Ending & Exact scope\\ \midrule F0 & Frozen & Copied analytic handoffs, unchanged pair and six-branch germ, no reconstructed incidence ledger.\\[0.35em] F1 & One route named & Direct paired composition on the proposed excess bundle. Two alternatives are recorded and unentered.\\[0.35em] F2 & Unresolved; direct identification excluded & The selected matching is not clean. The two-plane is a tangent defect; the sum projection has fiber dimension one and a non-Lagrangian image.\\[0.35em] F3 & Not entered & No joint symbol bundle or principal-symbol section of the complete coefficient.\\[0.35em] F4 & Not entered & No complete remainder estimate and no full cancellation or nonvanishing verdict.\\[0.35em] F5 & Not entered & No characteristic prescription or actual transverse degree.\\[0.35em] F6 & Recorded & J3--J5 and the Ward-defect question remain unopened.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[XIV-F mathematical release] \label{thm:release} For the frozen retained instance: \begin{enumerate}[label=\textup{(\roman*)}] \item the six regular constraints define $\Ssix$, and the remaining cone equation is the exact quadratic expression \eqref{eq:qR-on-S}; \item the specified conormal fiber product is smooth of dimension fifteen but has a seventeen-dimensional linearized matching space; \item the resulting tangent-defect quotient is a smooth rank-two bundle identified by \eqref{eq:defectmap}; \item its geometric quadratic form is nondegenerate, but the covector-sum projection has one-dimensional local fibers and a fourteen-dimensional isotropic image; \item these results do not construct a common symbol, complete remainder estimate, or characteristic prescription for \eqref{eq:target}. \end{enumerate} The complete grouped target has the ending \textbf{Unresolved at F2}, within the inherited E1/D2/J2 stop. No universal characteristic-extension obstruction, admitted $\Rquant$, or Ward defect is asserted. \end{theorem} \begin{proof} Proposition~\ref{prop:regular} and Corollary~\ref{cor:notredundant} prove (i). Theorem~\ref{thm:nonclean} proves (ii). Proposition~\ref{prop:defect} proves (iii). Proposition~\ref{prop:quadratic} and Theorem~\ref{thm:projection} prove (iv). Theorem~\ref{thm:route} identifies the failed direct construction hypothesis. Proposition~\ref{prop:noverdict} separates it from cancellation of the complete coefficient. Sections~\ref{sec:F3F4} and \ref{sec:F5} record the absent symbol, remainder, and prescription. Under the fixed release criteria, the ending is Unresolved. \end{proof} \subsection*{What has advanced} The proposed excess is no longer an unexplained rank count. The regular part of the incidence, its quadratic final constraint, its actual tangent defect, and its frequency redundancy have been distinguished. The smoothness of the matching set and the constant rank of its projection are not enough for clean composition. The tangent-space equality fails, and the stated image does not provide the required Lagrangian geometry. This diagnoses one direct construction without reopening the copied elementary phase failure. It does not solve the characteristic distribution problem. \subsection*{Successor package} A successor receives: \begin{enumerate} \item the unchanged XIV-E package and grouped target; \item the selected route and the precise correction to its excess-two wording; \item the regular manifold $\Ssix$ and exact formula $q_R=\alpha\delta|\mathbf n_1-\mathbf n_2|^2$ on it; \item the non-clean intrinsic conormal matching theorem; \item the rank-two tangent-defect bundle and its quadratic form; \item the rank-fourteen projection and one-dimensional frequency fibers; \item the persistence of the tangent discrepancy under smooth frequency reparameterization; \item the exact F2 stop, with no symbol or prescription released. \end{enumerate} A genuinely different paired or resolved construction must state its new representation and prove its own matching, amplitude, and mapping theorems. It cannot be obtained by calling the tangent defect an excess. The two other routes remain unentered. The successor receives no failure-of-cancellation theorem and no permission to infer one from $Q\lambda$. It receives no $\Rquant$. \subsection*{Conclusion} \[ \boxed{ \begin{gathered} \Kret=K_4+K_5+K_6+K_7+K_8+K_9,\\[2pt] \dim\Ssix=10,\qquad q_R|_{\Ssix} =\alpha(\beta-\alpha)|\mathbf n_1-\mathbf n_2|^2,\\[2pt] \dim\Fmatch=15,\qquad \dim(\text{linearized matching})=17,\\[2pt] \rank\Defect=2 \quad\text{is a tangent defect, not clean excess},\\[2pt] \rank\dd J=14,\qquad \dim(\text{local fiber of }J)=1,\\[2pt] \text{direct clean-excess-two identification: excluded},\\ \text{complete grouped characteristic coefficient: Unresolved at F2}. \end{gathered}} \] A quadratic normal defect is not an established clean integration fiber. A geometric relation is not a joint symbol. A collision contact is not a characteristic prescription. An undefined quantum coefficient is not an anomaly. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Specification correction.} The production specification identified the two relative transverse directions with clean excess two before checking clean matching. This identification is replaced by the proved rank-two tangent-defect statement. The frozen pair, propagators, source, and ambiguity class are not changed. \paragraph{Dependency scope.} Only XIV-E, XIV-D, XIV-C, XIII-C, XIII-B, XII-E, and XII-C are analytic manuscript inputs. Their quoted handoffs retain their predecessor scope. Recorded classical boundaries and the prior regulator attempt are not product, phase, or subtraction data. \paragraph{External calculus.} Definitions of clean matching and composition excess are used to test the stated geometric construction. No clean-composition, paired-distribution, stationary-phase, or extension theorem is invoked with unchecked hypotheses. The new results are the displayed intrinsic geometric proofs. \paragraph{Scope of exclusion.} The exclusion concerns direct clean-excess-two composition of the specified unrepaired fiber product. It is not a universal obstruction to every paired representation or characteristic prescription, and it does not prove nonvanishing of the complete six-branch amplitude. \paragraph{Source scope.} The differentiated source is the actual density-weighted Euler source. No layer coefficient or lower-order term is set to zero, and no hidden internal vertex is identified with a retained external point. \paragraph{Quantum scope.} No characteristic distribution, renormalized quantum coefficient, Ward defect, interacting differential, state, numerical loop value, 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 mathematical release consists of the displayed proofs and the explicit unresolved composition gate. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \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}. Immediate predecessor: exact regrouping, root, tested kernels, characteristic lifts, elementary phase exclusion, quadratic defect, and E1 stop. \bibitem{XIVD} N.~Cheung, \emph{Landmark XIV-D: Cancellation or prescription on the retained noncoincident characteristic stratum}. Supplied manuscript \texttt{XIV-D.tex}. Frozen pair, species split, six-branch germ, branch-level fiber identity, and D2 stop. \bibitem{XIVC} N.~Cheung, \emph{Landmark XIV-C: The joint two-line extension of one selected binary pair: noncoincident characteristic configurations, transverse estimates, and the output wavefront gate}. Supplied manuscript \texttt{XIV-C.tex}. Retained characteristic geometry, rank relation, positive conormals, contact rigidity, and J2 stop. \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}. Inventory, partial-diagonal ambiguity class, primitive extensions, and 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}. Completion formulas, slot sign, response roots, and admitted tree 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 unintegrated cubic representative and actual source. \bibitem{XIIIE} N.~Cheung, \emph{Landmark XIII-E: One joint off-collision prescription for the selected binary pair, or its obstruction in the frozen partial-diagonal class}. Supplied manuscript \texttt{XIII-E.tex}. Recorded common-frequency attempt only; no characteristic limit is imported. \bibitem{MU} R.~B.~Melrose and G.~A.~Uhlmann, ``Lagrangian intersection and the Cauchy problem,'' \emph{Communications on Pure and Applied Mathematics} \textbf{32} (1979), 483--519. Paired/intersecting Lagrangian framework; no theorem is applied to the non-clean matching here. \bibitem{GS} V.~Guillemin and S.~Sternberg, \emph{Semi-classical analysis}. Author manuscript. Sections~4.1 and 8.13: clean matching of canonical relations and clean Fourier-integral composition. \bibitem{Wunsch} J.~Wunsch, \emph{Microlocal analysis and evolution equations: Lecture notes from the 2008 CMI/ETH Summer School}, arXiv:0812.3181. Sections~8--9: Lagrangian distributions, symbols, clean intersections, and composition excess. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}, second edition, Springer, 1990; \emph{The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators}, Springer, 1985. Distributional wavefront calculus and Fourier-integral composition framework; no unchecked product theorem is invoked. \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. Section~6: recorded submanifold extension framework, not applied to the missing characteristic distribution. \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. Recorded extension wavefront and ambiguity framework; no characteristic extension is imported. \end{thebibliography} \end{document}