% !TeX program = pdflatex %===================================================================== % XIV-I.tex % % Standalone source for pdfLaTeX. % Build twice: % pdflatex -interaction=nonstopmode -halt-on-error XIV-I.tex % pdflatex -interaction=nonstopmode -halt-on-error XIV-I.tex % % Exact release: % - the pair, propagators, source representative, inverse -1, % switches, width, residual variable, and ambiguity class are % unchanged; % - the internal K_5 transition is treated by an explicitly defined % finite distribution-valued Fourier-amplitude module; % - the amplitude is computed from the existing retarded kernel; % no characteristic prescription for a new product is selected; % - its pole decomposition, distributional frequency estimates, % reconstruction, and finite lower-degree remainder are proved; % - both K_5 placements are reconstructed with z retained; % - the internal amplitude gate I2 is Constructed in that precisely % stated module, not in a single ordinary Lagrangian symbol class; % - a common precomposition presentation of the six-branch tree % kernel is supplied; % - the joint characteristic operator map required by I3 is not % constructed; % - I4--I6 and J3--J5 are not entered. % % Ending: % Constructed for the named internal amplitude map at I2. % Unresolved at I3 for the grouped characteristic coefficient. % No R_1^(1) is admitted. XV-B remains closed. % % Principal distinctions: % - frequency homogeneity in a distribution-valued amplitude module % is not a transverse scaling degree of the quantum target; % - the internal retarded pole is the Fourier transform of an already % fixed distribution, not a new cross-line prescription; % - exact internal reconstruction does not authorize multiplication % by the grouped cross-line distribution at the characteristic set; % - neither K_5 placement is removed; % - the clean internal diagonal/flow-out intersection is not called % non-clean, and its tangent count is not XIV-F's matching count; % - XV-A is recorded only as an exit prerequisite. % % No successful compilation, proof-assistant verification, % independently executed symbolic computation, independent referee % review, or journal acceptance is asserted. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{inputstatement}[theorem]{Input statement} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\sig}{\sigma_\eps} \newcommand{\PP}{\mathcal P} \newcommand{\ER}{E_0^{\mathrm R}} \newcommand{\EAm}{E_m^{\mathrm A}} \newcommand{\Gret}{G_{\mathrm{gr}}^{\mathrm R}} \newcommand{\GA}{G_{\mathrm{act}}^{\mathrm A}} \newcommand{\Ealg}{\mathscr E_H} \newcommand{\EBV}{\mathscr E_H^{\mathrm{BV}}} \newcommand{\Fpol}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Nsol}{\mathscr N_{\mathrm{sol}}} \newcommand{\Vcal}{\mathcal V_g} \newcommand{\Ocal}{\mathcal O} \newcommand{\Aone}{\mathcal A_1} \newcommand{\Asc}{\mathcal A_2^{\mathrm{sc}}} \newcommand{\CD}{\mathcal C_{3,\mathrm D}} \newcommand{\PhiD}{\Phi^{(2)}_{\mathrm D}} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\rD}{r_{\mathrm D}} \newcommand{\rg}{r_g} \newcommand{\HT}{H_T} \newcommand{\Rtree}{R_1^{(0)}} \newcommand{\Rquant}{R_1^{(1)}} \newcommand{\Cpart}{\mathfrak C_{\mathrm{part}}} \newcommand{\Ctot}{\mathfrak C_{\mathrm{tot}}} \newcommand{\Nchar}{\mathcal N^\circ} \newcommand{\Nbar}{\overline{\mathcal N}_{\mathrm{att}}} \newcommand{\Nlift}{\widetilde{\mathcal N}_{\mathrm{att}}} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\Mret}{\mathcal M_{\mathrm{ret}}} \newcommand{\Ucan}{U_{\mathrm{can}}} \newcommand{\Rker}{R} \newcommand{\Bker}{B} \newcommand{\Amap}{\mathsf A} \newcommand{\bmap}{\mathsf b} \newcommand{\Lresp}{\Lambda_{\mathrm{resp}}} \newcommand{\Ar}{\mathfrak A_{\mathrm R}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\Cross}{\mathscr C_{\mathrm{cross}}} \newcommand{\Tform}{\mathcal T_{\mathrm{form}}} \newcommand{\shat}{\widehat s_0} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\Contr}{Contr} \DeclareMathOperator{\EL}{EL} \DeclareMathOperator{\PV}{PV} \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-I}\\[0.4em] Uniform internal retarded amplitudes\\ at the diagonal--flow-out transition\\[0.4em] \large Exact reconstruction of both $K_5$ placements\\ and the remaining common-composition gate} \author{Ngo Cheung} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We continue the operator route at XIV-H's missing internal amplitude theorem. The frozen pair, source, inverse coefficient, switches, width, and partial-diagonal ambiguity class are unchanged. The residual metric variable is retained. For the two internal placements of $K_5$, we define a finite distribution-valued Fourier-amplitude module. It is not a classical symbol space on one ordinary Lagrangian. The fixed massless retarded multiplier has an explicit simple-pole decomposition near either nonzero characteristic sheet. This decomposition relates the noncharacteristic multiplier to the separated light-cone kernel, with its retarded normalization fixed by the already existing distribution. We prove uniform annular estimates across the pole, continuity of the reconstruction map, and an exact finite decomposition into the highest normal-frequency degree and its lower-degree remainder. Both placements are reconstructed in their actual coordinates, with uniform control on compact sets of all untested variables. This constructs the named internal amplitude map at I2. It does not construct the joint characteristic multiplication required for the grouped six-branch coefficient. A common precomposition presentation is supplied, but the operator map composing it with the grouped cross-lines on the retained characteristic locus is missing. The first unresolved gate is therefore I3. No joint section, characteristic prescription, transverse degree, gluing theorem, or output admission is released. In particular, no $R_1^{(1)}$ is admitted in $\mathscr E_H$, and XV-B remains closed. \end{abstract} \begin{figure}[ht] \centering \begin{tikzpicture}[x=1cm,y=1cm] \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Internal amplitude space}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Exact reconstruction}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {XV-B relative lift}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Distance to XV-B remains. This paper does not construct that lift.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The amplitude space is not the XV-B lift. 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-I} \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) {Internal amplitude space}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {XV-B relative lift}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Distance to XV-B remains. This paper does not construct that lift.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{The amplitude space is not the XV-B lift. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:split-XIV-I} \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) {Internal amplitude space}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Exact reconstruction}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {XV-B relative lift}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Distance to XV-B remains. This paper does not construct that lift.}; \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-I} \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) {Internal amplitude space}; \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) {XV-B relative lift}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Exact reconstruction. Distance to XV-B remains. This paper does not construct that lift.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Internal amplitude space is not XV-B relative lift. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.} \label{fig:pair-XIV-I} \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: Internal amplitude space}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Exact reconstruction}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: XV-B relative lift}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Distance to XV-B remains. This paper does not construct that lift.}; \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-I} \end{figure} Figure~\ref{fig:split-XIV-I} records one view of the same separation. Figure~\ref{fig:chain-XIV-I} records one view of the same separation. Figure~\ref{fig:pair-XIV-I} records one view of the same separation. Figure~\ref{fig:gate-XIV-I} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XIV-I} records that separation. The drawing adds no hypothesis. \scopebox{Release and distance to XV-B}{% \textbf{Constructed at I2:} an explicit internal amplitude space, its fixed pole normalization, uniform distributional frequency estimates, exact reconstruction, and complete finite lower-degree remainder for both $K_5$ placements. \smallskip \textbf{First missing gate: I3.} The construction is internal and precompositional. No justified joint characteristic operator map for $\mathcal T_{\mathrm{form}}$ is supplied. \smallskip \textbf{Not released:} a joint principal-symbol section, a characteristic prescription, J3--J5, or an admitted $R_1^{(1)}$. \smallskip \textbf{XV-B remains closed.} XV-A's free carrier is an exit prerequisite, not an analytic substitute for the missing coefficient.} \tableofcontents %===================================================================== \section{I0: copied inputs and the exact predecessor stop} \label{sec:I0} %===================================================================== All manuscript handoffs below are premises with their predecessor domains. No incidence ledger, elementary phase theorem, or clean-excess theorem is re-proved. \subsection{XIV-H's eight successor items} \begin{inputstatement}[XIV-H successor package] \label{input:Hpackage} A successor receives: \begin{enumerate} \item the unchanged XIV-G package; \item the untested source incidence and its regular relation; \item the exact coordinate-leg convolution map and its regular conormal representation; \item the separate complete scalar-leg root map; \item the exact two-placement $K_5$ tensor pullbacks; \item their regular relation charts and orders; \item the clean diagonal/flow-out intersection and the exclusion of a single ordinary Lagrangian identification; \item the precise missing uniform internal amplitude theorem and the H2 stop. \end{enumerate} It receives no transverse degree of the target, common characteristic representation, cancellation theorem, or permission to call the tangent defect an excess. It receives no $\Rquant$ \cite{XIVH}. \end{inputstatement} \subsection{The untested source and two response maps} Define constant fiber maps by \[ a_1^{\alpha\mu}[r]=\Amap^{\alpha\mu,I}r_I, \qquad b_1[r]=\bmap^I r_I, \] where $I$ indexes symmetric metric components. Let $\Rker$ be the kernel of $\ER$. \begin{inputstatement}[Source incidence and gravitational legs] \label{input:Hresponse} The full differentiated source has the finite local form \[ (\JD'[G]h)^L(v) =\sum_{|\alpha|+|\beta|\le2} q_{\alpha\beta}^{L,I,J}(v) \partial^\alpha G_J(v)\partial^\beta h_I(v), \] with every coefficient taken from the actual density-weighted source. Before residual testing its kernel is \begin{equation} Q^{L,I,J}(v;a,z) =\sum_{\alpha,\beta} q_{\alpha\beta}^{L,I,J}(v) \partial_v^\alpha\delta^{(4)}(v-z) \partial_v^\beta\delta^{(4)}(v-a). \label{eq:Q} \end{equation} This is a defined regular incidence supported on $v=a=z$. Writing the response kernel with all three species indices, \begin{equation} T^{L,I,J}(b;a,z) =\PP^L{}_M\int \Rker(b-v)Q^{M,I,J}(v;a,z)\,\dd^4v, \label{eq:T} \end{equation} the exact coordinate-leg map on the retained patch is \begin{equation} \begin{split} K_8^{I,J}(a,b,z) =-\partial_{b^\mu}\Bigl[ f(b)\Amap^{\alpha\mu}{}_L\PP^L{}_M \partial_{b^\alpha} \int \Bker(b-v)Q^{M,I,J}(v;a,z)\,\dd^4v \Bigr], \qquad \Bker=\Rker*\Rker . \end{split} \label{eq:K8} \end{equation} The complete scalar-leg root, with $F=\EAm f$, gives \begin{equation} \begin{split} K_9^{I,J}(a,b,z) ={}& -\partial_{b^\nu}\Bigl[ g(b)\partial_{b^\mu}F(b)\, \Amap^{\mu\nu}{}_L T^{L,I,J}(b;a,z) \Bigr]\\ &-m^2g(b)F(b)\bmap_L T^{L,I,J}(b;a,z). \end{split} \label{eq:K9} \end{equation} The mass term is retained. On the separated chronological patch, both kernels have regular conormal representations of upper order $3/2$ on \[ \Lambda_{\mathrm{resp}} =N^*\{(b-a)^2=0,\ a=z\}\setminus0 \subset T^*W. \] The $y$ covector is zero \cite{XIVH}. \end{inputstatement} The explicit output index $L$ in \eqref{eq:T} only removes an index suppression in the predecessor. It changes no map. \begin{inputstatement}[Existing internal convolution] \label{input:convolution} In the normalization $\Box\ER=\id$, \begin{equation} \Rker(t,\mathbf x) =\frac{\Theta(t)}{2\pi}\delta(t^2-|\mathbf x|^2), \qquad \Rker*\Rker =\frac1{8\pi}\Theta(t)\Theta(t^2-|\mathbf x|^2). \label{eq:R} \end{equation} The second equality is an identity of existing internal retarded kernels, not a cross-line product \cite{XIVH}. \end{inputstatement} \subsection{Both internal placements of \(K_5\)} Put \[ L^{\mu,I}(s,a) =\sig(a)\Amap^{\lambda\mu,I} \partial_{a^\lambda}\Rker(s-a). \] \begin{inputstatement}[Two-placement formula and regular charts] \label{input:K5} The complete untested mixed kernel of $B_5$ is \begin{equation} \begin{split} K_5^{I,J}(a,b,z) =\partial_{b^\mu}\Bigl[ &f(b)\sig(a) \partial_{a^\alpha}\Rker(b-a) \Amap^{\alpha\beta,I}\partial_{a^\beta}L^{\mu,J}(a,z)\\ +{}& f(b)\sig(z) \partial_{z^\alpha}\Rker(b-z) \Amap^{\alpha\beta,J}\partial_{z^\beta}L^{\mu,I}(z,a) \Bigr]. \label{eq:K5} \end{equation} The two underlying products are \[ \Rker(b-a)\Rker(a-z), \qquad \Rker(b-z)\Rker(z-a). \] They are defined tensor-product pullbacks, including at $a=z$. Their exact finite differential formulas have total kernel derivative order at most four. On regular separated-cone charts their upper conormal order is one. On comparable-frequency, noncharacteristic internal diagonal charts their upper order is $1/2$. For the first placement, near a nonzero internal null covector, the diagonal piece and limiting flow-out piece are distinct sixteen-dimensional Lagrangians. They intersect cleanly in dimension fifteen and have tangent sum of dimension seventeen. No single ordinary sixteen-dimensional Lagrangian contains both full local pieces there. The second placement has the corresponding reversed internal separation. The first missing theorem was a uniform internal amplitude description through this transition: \[ \boxed{\text{Unresolved at H2, within G2/F2/E1/D2/J2.}} \] No common characteristic representation was released \cite{XIVH}. \end{inputstatement} The clean intersection is not called non-clean. Its tangent-sum count is not XIV-F's linearized matching count. \subsection{Earlier analytic boundaries} \begin{inputstatement}[XIV-G and XIV-F] \label{input:GF} XIV-G's tested source is a local differential operator of order at most two; its local polynomial symbol is not either response symbol. Smooth residual testing can hide attachment-diagonal wavefront directions. Its balanced-kernel counterexample is not a model of the characteristic singularity of $\Kret$. The ordinary final product of its two individually defined numerical factors fails the sufficient product criterion at the retained covectors. That audit supplies no full-pair nonvanishing theorem \cite{XIVG}. XIV-F's rank-two object is a tangent defect. Its excluded matching has actual dimension fifteen and linearized dimension seventeen. The covector-sum map has rank fourteen, one-dimensional fibers, and an isotropic, non-Lagrangian image in $T^*W$. Its regular-stage formula is \[ q_R|_{S_6}=\alpha\delta|\mathbf n_1-\mathbf n_2|^2, \qquad \beta=\alpha+\delta. \] These are boundaries, not relations used in the present amplitude construction \cite{XIVF}. \end{inputstatement} \begin{inputstatement}[XIV-E regrouping and complete root] \label{input:E} For smooth directions, \begin{equation} \begin{split} \Mret[G;h,\psi] ={}& \int C_G^\mu{}_\nu X_h^\nu\partial_\mu\psi\,\dd^4x\\ &+\int f\,\ER\partial_\alpha \bigl(\sig B^{\alpha\mu}[G;h]\bigr) \partial_\mu\psi\,\dd^4x\\ &+\int f\,p[r_h;\psi]\,\dd^4x, \end{split} \label{eq:regroup} \end{equation} where \[ C_G^\mu{}_\nu =f\partial_\nu X_G^\mu -\delta^\mu_\nu\partial_\rho(fX_G^\rho), \] \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], \qquad r_h=\PP\ER(\JD'[G]h). \end{split} \label{eq:B} \end{equation} The complete scalar root is \begin{equation} \int f\,p[r_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, \qquad F=\EAm f . \label{eq:root} \end{equation} These are exact precomposition identities, not characteristic products. The tested attachment bound is not a uniform untested composition theorem \cite{XIVE}. \end{inputstatement} \begin{inputstatement}[XIV-D and XIV-C] \label{input:DC} On the retained patch, \[ \Kret=K_4+K_5+K_6+K_7+K_8+K_9. \] Both gravitational-response legs are retained. The numerator $Q\lambda$ belongs only to the leading differential fiber calculation for $B_6$ \cite{XIVD}. The common space is \[ W=A\times B\times Z\times Y,\qquad (a,b,z,y), \] with the residual variable $z$ retained. The characteristic test locus is \[ N=\Nlift =\{a=z,\ y=a+\alpha\ell_{\mathbf n},\ b=a+\beta\ell_{\mathbf n},\ 0<\alpha<\beta\}, \qquad \ell_{\mathbf n}=(1,\mathbf n). \] It has codimension eight in $W$. The three separated cone conormals have the copied positive relation. Defined interaction collision contacts vanish on the retained collision-free attachment neighborhood \cite{XIVC,XIVD}. \end{inputstatement} \begin{inputstatement}[XIII-C and XIII-B] \label{input:BC} XIII-C's four obligations remain: \begin{enumerate} \item a characteristic prescription or complete-pair cancellation; \item uniform transverse estimates at three- and four-vertex collision strata and their intersections; \item compatible partial contacts with singular spectators; \item output wavefront control after proper pushforward. \end{enumerate} The parallel temporal kinetic primitive has collision scaling degree eight in codimension four \cite{XIIIC}. XIII-B supplies the admitted tree datum and the convention \[ R_1(\Vcal;\Ocal) =\frac{i}{\hbar} \left(\mathsf T_2^{\mathrm{ren}}(\Vcal,\Ocal) -\Vcal\star_H\Ocal\right). \] The strictly ordered full-support values are zero or the scaled commutator. Properness does not define an absent distributional composition \cite{XIIIB}. \end{inputstatement} \begin{inputstatement}[XII-E and XII-C] \label{input:EC} The inverse, slots, and roots retain their normalization: \[ \Gret=-\PP\ER,\qquad \rD=\PP\ER\JD,\qquad \rg[\phi]=-\frac12\PP\ER(gT_0[\phi]), \] \[ \Aone[f,g](r;\phi) =\int\alpha_{f,g}[\phi]\cdot r\,\dd^4x, \qquad \beta_{f,g}[\phi]=-\EA(\PP\alpha_{f,g}[\phi]). \] The full observable contains the scalar-sourced slot once \cite{XIIE}. The actual source is \[ \JD=\EL_\gamma(\sig\ell_3) =\sig J_3+ 2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sig -2M_2^{\mu\nu\rho}\partial_\rho\sig \] in component notation, with the fixed unintegrated cubic representative of XII-C. No source coefficient is deleted \cite{XIIC}. \end{inputstatement} \subsection{XV-A is only an exit prerequisite} XV-A states a free-carrier release $(\EBV,\shat)$ and a generator-preserving embedding obstruction for the relative classical complex \cite{XVA}. Neither is used to prove an amplitude estimate or define a cross-line product here. Their stated free-carrier conclusion is recorded only when discussing the exit to XV-B. XIV-A and XIV-B are not analytic inputs. The value $-8I$ is not a normalization. XIII-E and a resolved paired construction remain unentered. %===================================================================== \section{I1: the frozen route and the amplitude class} \label{sec:I1} %===================================================================== \[ \boxed{ \text{Uniform internal amplitudes for both }K_5 \text{ placements, then a common operator representation.}} \] Use \[ \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta_{\mathbf x}. \] All inherited functions and the width $\eps>0$ remain fixed. The pair is \[ \Vcal=-\frac12\int g\,\gamma_{\mu\nu}T_0^{\mu\nu}[\phi]\,\dd^4x, \qquad \Ocal=\PhiD+\Aone[f,g](\rg[\phi];\phi), \] and \[ \PhiD=\Asc[f,g](\gamma,0;\phi)-\CD. \] The target remains \begin{equation} \begin{split} \Tform =i\,\Contr^{\gamma\phi}\Bigl( \Vcal''_{\gamma\phi},\Kret;\ &\HT_{\gamma\gamma}(y,a)\otimes\HT_{\phi\phi}(y,b)\\ &-H_{\gamma\gamma}(y,a)\otimes H_{\phi\phi}(y,b) \Bigr). \end{split} \label{eq:target} \end{equation} The difference stays grouped. No mixed $H$--$G_{\mathrm{act}}$ product is declared. \begin{definition}[Class fixed before the amplitude calculation] \label{def:class} The selected class consists of internal representations on the relations already named in XIV-H, with both $K_5$ placements and $z$ retained. It permits: \begin{enumerate}[label=\textup{(\roman*)}] \item partial Fourier transforms of the already defined internal kernels; \item explicitly defined distribution-valued normal amplitudes, with a proved reconstruction map, uniform estimates, and a stated remainder; \item regular conormal charts and named pullbacks or operator maps whose hypotheses are checked. \end{enumerate} It does not require the union of the internal diagonal and flow-out pieces to be one ordinary Lagrangian. It does not permit an unproved generalized composition with the quantum cross-lines. Such a composition is the later I3 obligation. No common product phase, resolved base space, frequency-regulator family, or characteristic prescription for a new product is introduced. \end{definition} The construction below is a finite, explicit amplitude module. It is not an assertion of a general paired-distribution calculus. \begin{definition}[Three endings] The endings are applied to the object at the gate reached: \begin{enumerate} \item \textbf{Constructed}: that object has its stated domain, amplitude or degree, and wavefront control. \item \textbf{Proved obstructed}: an incompatibility excludes every candidate in Definition~\ref{def:class}. \item \textbf{Unresolved}: a necessary amplitude, common composition, section, prescription, or J-gate estimate is missing. \end{enumerate} The earlier exclusion of a single ordinary Lagrangian is not a universal obstruction to this class. \end{definition} Only admission of $\Rquant$ in $\Ealg$ at I6 could open XV-B. An I2 amplitude theorem opens I3 only. %===================================================================== \section{I2a: the fixed retarded multiplier and its transition pole} \label{sec:multiplier} %===================================================================== \subsection{Fourier convention} For a relative variable $r=(t,\mathbf x)$, use \[ \widehat u(\omega,\xi) =\int e^{-i(t\omega+\mathbf x\cdot\xi)}u(t,\mathbf x)\,\dd t\,\dd^3x. \] Inverse Fourier transformation has the factor $(2\pi)^{-4}$. Thus $\partial_r^\beta$ has multiplier $(i\kappa)^\beta$, where \[ \kappa=(\omega,\xi),\qquad \rho=|\xi|,\qquad p(\kappa)=\omega^2-\rho^2. \] The $\kappa$ in this section is a frequency variable, not the interaction coupling. All Fourier calculations concern the already fixed distribution $\Rker$. They choose no new characteristic value for \eqref{eq:target}. \begin{lemma}[The retarded multiplier with the declared sign] \label{lem:multiplier} The tempered Fourier transform of $\Rker$ is \begin{equation} M_{\mathrm R}(\omega,\xi) =-\lim_{\epsilon\downarrow0} \frac1{(\omega-i\epsilon)^2-\rho^2}. \label{eq:MR} \end{equation} On $\rho>0$, \begin{equation} M_{\mathrm R} =-\frac1{2\rho} \left[ \frac1{\omega-\rho-i0} -\frac1{\omega+\rho-i0} \right]. \label{eq:partialfraction} \end{equation} It is homogeneous of degree $-2$ in the distributional sense. \end{lemma} \begin{proof} The spatial Fourier transform of \eqref{eq:R} is \[ \Theta(t)\frac{\sin(\rho t)}{\rho}, \] with continuous value $\Theta(t)t$ at $\rho=0$. Multiplying this existing distribution by $e^{-\epsilon t}$ on its support and taking its temporal Fourier transform gives \[ \int_0^\infty e^{-(\epsilon+i\omega)t} \frac{\sin(\rho t)}{\rho}\,\dd t =\frac1{(\epsilon+i\omega)^2+\rho^2} =-\frac1{(\omega-i\epsilon)^2-\rho^2}. \] The damped distributions converge to $\Rker$ in $\mathcal S'(\R^4)$, as follows directly from its cone formula and the rapid decay of Schwartz tests. Fourier continuity gives \eqref{eq:MR}. Partial fractions followed by the same distributional limit give \eqref{eq:partialfraction} when $\rho>0$. The cone formula has physical homogeneity degree $-2$ in dimension four. Its Fourier transform consequently has degree $-2$ as well. Alternatively the same identity follows by positive dilation in \eqref{eq:MR}. \end{proof} The damping in this proof computes the Fourier transform of an existing kernel. It is not a regulator for a graph product, and no restoring limit of the quantum coefficient is used. \subsection{A uniform pole chart on either characteristic sheet} For $s\in\{+1,-1\}$, define \[ \lambda_s=\omega-s\rho. \] Take a conic neighborhood of that nonzero sheet satisfying \[ \rho>0,\qquad |\lambda_s|0, \label{eq:homogeneity} \end{equation} as distributions. For every fixed frequency multiindex $\nu$, there are $N$ and $C$ such that \begin{equation} \left| \left\langle L^{-(d-2-|\nu|)} (\partial_{\theta,\kappa}^{\nu}A_{j\beta})(L\,\cdot), \psi \right\rangle \right| \le C\max_{|\alpha|\le N}\sup_K|\partial^\alpha\psi|, \quad L\ge1. \label{eq:uniform} \end{equation} For amplitudes \eqref{eq:amodule}, these estimates are uniform after any fixed number of coefficient derivatives on a compact subset of $V$. No separation from the pole is required. \end{theorem} \begin{proof} Lemma~\ref{lem:multiplier} gives degree $-2$ for $M_{\mathrm R}$. The polynomial has degree $d$, giving \eqref{eq:homogeneity}. A distributional frequency derivative lowers the homogeneity by one. It remains to verify finite seminorm control on $K$, including its intersection with the pole. In the coordinates $(\lambda_s,\xi)$, \eqref{eq:pole} is a smooth homogeneous coefficient times $B_-(\lambda_s)$ plus a smooth term. The change of frequency coordinates is regular where $\rho>0$. On $K$ all its coefficients and the required finitely many derivatives are bounded. Apply Lemma~\ref{lem:PVbound} in $\lambda_s$, integrate over the remaining compact variables, and include the polynomial factors and any fixed derivatives. This gives a finite-order distribution bound. The homogeneity then makes its constant independent of $L$. For parameter-dependent coefficients, the same argument is multiplied by their finite smooth seminorms on a compact subset of $V$. \end{proof} These are estimates for internal \emph{frequency distributions}. They are not normal rescaling estimates of $\Tform$ at $N$. \subsection{Exact finite remainder} For an amplitude of degree at most four, split its coefficient polynomial by degree: \begin{equation} A=A_{[4]}+A_{[\le3]}. \label{eq:split} \end{equation} Terms with no degree-four contribution have $A_{[4]}=0$. \begin{corollary}[Complete lower-degree remainder in the internal module] \label{cor:remainder} The splitting \eqref{eq:split} is exact. On normalized annular tests, \[ L^{-2}A_{[4]}(w;L\,\cdot) \] is uniformly bounded, and \[ L^{-2}A_{[\le3]}(w;L\,\cdot)=O(L^{-1}) \] in the finite-seminorm distribution topology of Theorem~\ref{thm:dyadic}, uniformly on compact parameter sets. Every lower-degree term of the finite differential formula is retained in $A_{[\le3]}$. The regular opposite-sheet term $b_s$ in \eqref{eq:pole} remains part of the full multiplier in both pieces. \end{corollary} \begin{proof} Each degree-$d$ term scales as $L^{d-2}$. For $d\le3$, division by $L^2$ gives at most $L^{-1}$. There are finitely many terms, so the uniform estimates can be summed. No term of $M_{\mathrm R}$ was dropped in defining the generators. \end{proof} This is not a smoothing remainder theorem for the quantum coefficient. It is a proved, complete lower-degree estimate for the named internal module. \subsection{Continuous reconstruction} \begin{theorem}[Internal amplitude reconstruction] \label{thm:reconstruction} The map \eqref{eq:reconstruction} is a continuous linear map from the coefficient module, with its compact-set smooth topology, to $\mathcal D'(V)$. It is also the iterated normal inverse Fourier transform of the generators \eqref{eq:generators}, followed by the specified coefficient multiplication. On compact subsets of the original $W$ chart, its estimates hold for arbitrary joint tests in $(a,b,z,y)$. No restriction to a fixed singular residual value is used. A fixed derivative in the original variables raises the finite polynomial degree by at most its differential order. \end{theorem} \begin{proof} The tensor product $\delta^{(j)}(q)\otimes\partial_r^\beta\Rker(r)$ is defined. Its pullback to the adapted chart merely adds passive variables. Multiplication by a smooth coefficient is defined. For a compactly supported joint test, the pairing is first a finite $q$ derivative evaluated at $q=0$, then a pairing with the fixed distribution $\partial_r^\beta\Rker$, and integration over the remaining compact tangential variables. The distributional finite-order estimate for $\Rker$ on the projected compact set bounds this by finitely many derivatives of the coefficient and test. This proves continuity. Separate Fourier inversion gives \[ \mathcal F^{-1}_{\theta}(i\theta)^j=\delta^{(j)}(q), \qquad \mathcal F^{-1}_{\kappa} \bigl((i\kappa)^\beta M_{\mathrm R}\bigr) =\partial_r^\beta\Rker(r). \] This proves the reconstruction identity without evaluating an oscillatory amplitude at a singular point. The adapted coordinate maps are diffeomorphisms. Pulling the estimates back therefore bounds arbitrary joint tests in the original variables, including $z$. The chain rule expresses an original-variable derivative as smooth-coefficient derivatives in the adapted coordinates. Its normal derivatives add polynomial factors and its tangential derivatives differentiate coefficients. This proves the final statement. \end{proof} Uniformity in $z$ means this joint distributional control. It does not mean that the kernel admits a pointwise smooth slice at every $z$. %===================================================================== \section{I2c: application to both \(K_5\) placements} \label{sec:apply} %===================================================================== \subsection{Finite coefficient extraction from the exact formula} In either adapted chart, the outer line is \[ \Rker(d_\pm)=\frac1{2\pi}\delta(q_\pm). \] The inner line remains $\Rker(r_\pm)$, including its vertex. The chain rule and the Leibniz rule applied to the complete formula \eqref{eq:K5} give coefficient arrays $c_{\pm,j\beta}^{I,J}$ such that \begin{equation} K_{5,\pm}^{I,J} =\sum_{j+|\beta|\le4} c_{\pm,j\beta}^{I,J}(w_\pm) \delta^{(j)}(q_\pm) \partial_{r_\pm}^{\beta}\Rker(r_\pm). \label{eq:coefficientform} \end{equation} Factors $1/(2\pi)$ and all coordinate density factors are included in the coefficients. Equation~\eqref{eq:coefficientform} is an exact coefficient extraction rule: expand the finite differential operator in \eqref{eq:K5} in the adapted coordinates, transfer its normal derivatives to the two indicated factors, and retain every derivative of every coefficient. It is not a tensor truncation. Define \begin{equation} A_{5,\pm}^{I,J} =\sum_{j+|\beta|\le4} c_{\pm,j\beta}^{I,J}(w_\pm) A_{j\beta}(\theta,\kappa) \in\Ar^2(V_\pm). \label{eq:A5} \end{equation} \begin{proposition}[Both amplitudes and all finite lower terms] \label{prop:both} The amplitudes \eqref{eq:A5} reconstruct the two placements in \eqref{eq:K5} exactly: \[ K_5=K_{5,+}+K_{5,-} =\Rec(A_{5,+})+\Rec(A_{5,-}), \] after the two adapted coordinate maps are pulled back to $W$. They satisfy the uniform transition estimates and exact lower-degree remainder of the preceding section. Both metric-slot placements and every inherited coefficient are retained. \end{proposition} \begin{proof} The copied formula differentiates the two retarded factors at most four times. In adapted coordinates, its tangential derivatives act on smooth coefficients, whereas its normal derivatives act on $\delta(q_\pm)$ or $\Rker(r_\pm)$. The finite Leibniz expansion is therefore precisely \eqref{eq:coefficientform}. Theorem~\ref{thm:reconstruction} proves the equality. Theorem~\ref{thm:dyadic} and Corollary~\ref{cor:remainder} apply to its finite coefficient arrays. No metric slot has been tested against $G$ or assigned a value. \end{proof} The amplitudes use the two original relative variables $r_+=a-z$ and $r_-=z-a$. The internal time orientation is therefore handled separately for each placement; one is not substituted for the other. \subsection{Relation to the predecessor's regular charts} The same internal module gives: \begin{enumerate} \item the ordinary noncharacteristic chart by replacing the full multiplier with its actual value $-1/p$ on an open set avoiding $p=0$; \item the separated-cone chart by inverse Fourier reconstruction of the full pole expression, giving the exact retarded cone distribution; \item the transition itself by the distributional expression \eqref{eq:pole}, with the uniform bounds proved through the pole. \end{enumerate} The copied regular conormal orders remain upper orders in their own charts. The distributional degree $2$ of the degree-four amplitude in \eqref{eq:A5} is not a replacement for either order. In particular, there is no contradiction between a regular diagonal upper order $1/2$, a separated two-cone upper order one, and the internal distributional dilation estimate. They describe different symbol presentations and different normal variables. \subsection{A joint internal wavefront bound} Let $\mathcal C_{\mathrm R}^{\,0}$ consist of: \[ \{(r,0):r\in\supp\Rker\}, \] together with \[ \{(0,\kappa):\kappa\ne0\}, \] and the nonzero forward-cone directions \[ \{(r,\lambda r^\flat): r^2=0,\ r^0>0,\ r\ne0,\ \lambda\ne0\}. \] Here $r^\flat=\eta(r,\cdot)$. This is the usual retarded-kernel wavefront upper bound, with zero covectors on its support adjoined. In either adapted chart, define \begin{equation} \begin{split} \Gamma_{5,\pm} =\Bigl\{& \bigl(w_\pm,\theta\,\mathrm dq_\pm +\kappa_\mu\mathrm dr_\pm^\mu\bigr): q_\pm=0,\ (r_\pm,\kappa)\in \mathcal C_{\mathrm R}^{\,0},\\ &\theta\in\R,\quad (\theta,\kappa)\ne0 \Bigr\}. \end{split} \label{eq:Gamma5} \end{equation} The covectors are pulled back to $T^*W$ by the actual adapted coordinate maps. \begin{proposition}[Wavefront control of internal reconstruction] \label{prop:WF} The reconstructed kernels satisfy \[ \WF(K_{5,\pm})\subset\Gamma_{5,\pm}, \qquad \WF(K_5)\subset\Gamma_{5,+}\cup\Gamma_{5,-}. \] The bounds retain the internal diagonal and the separated flow-out, as well as the directions where one tensor factor has zero covector. They are uniform bounds on the untested kernel, not merely bounds after smooth residual testing. \end{proposition} \begin{proof} At a nonzero point of the retarded cone, \eqref{eq:R} is delta-type on its smooth defining hypersurface, so its wavefront is conormal there. At the vertex the asserted upper bound includes every nonzero covector and is therefore valid. The tensor-product wavefront bound, with zero covectors adjoined on supports, gives \eqref{eq:Gamma5} for the two independent factors in \eqref{eq:coefficientform}. The coordinate pullbacks are diffeomorphisms. Differentiation and smooth multiplication do not enlarge these bounds. The finite sum gives the result \cite[Section 8.2]{Hormander}. \end{proof} The $b$ covector in the outer cone component is not set to zero. The balanced-kernel testing example of XIV-G is not used to model these directions. \begin{theorem}[The internal amplitude gate is Constructed] \label{thm:I2} In the finite retarded-pole module of Definition~\ref{def:amodule}, the I2 amplitude map for both $K_5$ placements is Constructed: \begin{enumerate}[label=\textup{(\roman*)}] \item one explicit module contains their regular diagonal, separated flow-out, and internal transition presentations; \item \eqref{eq:pole} fixes the relation between the noncharacteristic multiplier and the retarded cone amplitude; \item the untested reconstruction has uniform compact-set control with $z$ retained; \item every finite lower-degree term is included in the exact remainder \eqref{eq:split}; \item the reconstructed internal kernels have the wavefront bounds \eqref{eq:Gamma5}. \end{enumerate} This is not a single-Lagrangian principal-symbol theorem. It supplies no joint characteristic product with the quantum cross-lines. It opens I3 only. \end{theorem} \begin{proof} Proposition~\ref{prop:pole} proves the transition identity. Theorems~\ref{thm:dyadic} and \ref{thm:reconstruction} prove the uniform estimates and reconstruction. Corollary~\ref{cor:remainder} proves the complete finite lower-degree statement. Proposition~\ref{prop:both} applies these results to both exact placements, and Proposition~\ref{prop:WF} supplies the untested wavefront control. \end{proof} %===================================================================== \section{I3: the common precomposition sum and the missing joint map} \label{sec:I3} %===================================================================== I2 is now supplied in its named amplitude module. I3 is therefore entered, but its target is stronger than internal reconstruction. \subsection{The other precomposition pieces} The regrouped $B_4+B_7$ kernel can be written directly with the existing coordinate-response line $L$: \begin{equation} \begin{split} K_{47}^{I,J}(a,b,z) =-\partial_{b^\mu}\Bigl[ \bigl( f(b)\partial_{b^\nu}L^{\mu,J}(b,z) -\delta^\mu_\nu\partial_{b^\rho} (f(b)L^{\rho,J}(b,z)) \bigr)L^{\nu,I}(b,a) \Bigr]. \label{eq:K47} \end{split} \end{equation} This retains the complete $C_G$ term. It does not reintroduce the canceled explicit $\partial^2\psi$ term. Let $\mathsf D^{\alpha\mu,I,J}$ be the constant bilinear fiber map determined by \[ \dot a_2^{\alpha\mu}[G;h] =\mathsf D^{\alpha\mu,I,J}h_I G_J. \] The local $B_6$ piece has kernel \begin{equation} K_6^{I,J}(a,b,z) =\partial_{b^\mu}\Bigl[ f(b)\sig(a)\mathsf D^{\alpha\mu,I,J} \partial_{a^\alpha}\Rker(b-a) \delta^{(4)}(a-z) \Bigr]. \label{eq:K6} \end{equation} \begin{proposition}[A common presentation before cross-line composition] \label{prop:precomposition} On the retained patch, \begin{equation} \Kret=K_{47}+K_5+K_6+K_8+K_9 \label{eq:presum} \end{equation} has a defined finite operator presentation with $z$ retained. The pieces have the following internal descriptions: \begin{enumerate}[label=\textup{(\roman*)}] \item $K_{47}$ is a finite derivative of the two independent separated retarded factors $\Rker(b-a)\Rker(b-z)$; \item $K_5$ has the two constructed internal amplitude maps of Theorem~\ref{thm:I2}; \item $K_6,K_8,K_9$ have finite conormal amplitudes on $\Lambda_{\mathrm{resp}}$. \end{enumerate} On regular conormal charts their upper orders are, respectively, one for $K_{47}$, $1/2$ for $K_6$, and $3/2$ for $K_8,K_9$. These are internal orders only. \end{proposition} \begin{proof} Formula~\eqref{eq:K47} is obtained by inserting the kernel of $X$ into the copied coefficient $C_G$ and transferring the scalar derivative to the resulting kernel. Its two relative variables are independent under the diffeomorphism \[ (a,b,z,y)\mapsto(b,b-a,b-z,y). \] Both endpoints are separated on this patch. The two cone differentials are independent, and the total derivative count is at most four. In dimension sixteen and codimension two the copied conormal convention gives upper order one. For \eqref{eq:K6}, transfer the divergence to the incoming retarded line and then transfer the scalar derivative. The two signs cancel. The actual bilinear local vertex forces $a=z$. Its two normal derivatives give upper order $2-(16-10)/4=1/2$. The $K_5$ statement is Theorem~\ref{thm:I2}; the two response statements are Input statement~\ref{input:Hresponse}. The copied regrouping then gives \eqref{eq:presum}. \end{proof} This proposition constructs a common \emph{precomposition} presentation of the observable Hessian. It does not construct the operator map required for \eqref{eq:target} at $N$. \subsection{The grouped cross-line distribution is not the joint product} The two-line tensor \[ \HT_{\gamma\gamma}\otimes\HT_{\phi\phi} -H_{\gamma\gamma}\otimes H_{\phi\phi} \] is a defined distribution in its two independent relative variables. Its pullback by \[ (a,b,z,y)\longmapsto(y-a,y-b,z,y) \] is defined, since this map is a diffeomorphism onto its image. Denote the resulting grouped distribution by $\Cross$. Only this grouped pullback is used. It is not expanded into mixed $H$--$\GA$ products, and it is not replaced by the numerical factor of XIV-G's audit. The local mixed Hessian of $\Vcal$ supplies its inherited finite fiber contraction and differential operators. Applying those operators to a defined distribution is legitimate. It does not define multiplication of that distribution by $\Kret$ at a characteristic covector cancellation. \subsection{The actual missing I3 map} Let $\mathscr K_{\mathrm{int}}$ denote the finite direct-sum coefficient spaces in Proposition~\ref{prop:precomposition}. Their reconstruction map is defined on the full retained patch. A successful I3 construction must supply a map of the form \begin{equation} \mathsf M: \mathscr K_{\mathrm{int}} \longrightarrow \mathcal D'(W;\text{residual species}), \label{eq:jointmap} \end{equation} whose value on the reconstructed six-branch sum is the specified contraction with $\Cross$ and the mixed vertex Hessian, and which: \begin{enumerate} \item agrees with the canonical composition on its proved domain; \item is defined on tests meeting $N$; \item retains the prescribed grouped cross-lines and all tensor coefficients; \item has a proved complete-graph wavefront bound appropriate to that composition. \end{enumerate} The internal Fourier-amplitude module proves none of these requirements for multiplication by $\Cross$. Its estimates are estimates on the internal amplitude before this new operation. \begin{proposition}[Canonical-domain representation] \label{prop:canonical} On the inherited microlocally admissible domain $\Ucan$, the reconstructed sum \eqref{eq:presum}, composed with $\Cross$ and the specified vertex operators, equals the canonical grouped expression \eqref{eq:target}. No equality \[ \Ucan=W\setminus\Nbar \] follows from this representation. \end{proposition} \begin{proof} Each reconstruction is an equality of already defined tree distributions. Restrict those equalities to the open domain where the subsequent products and pullbacks have been proved. The ordinary distributional operations there respect equality, giving the first assertion. The internal reconstruction proves no new normal-set condition for that subsequent product, so it cannot identify a larger domain. \end{proof} The copied opposite-covector audit prevents declaring the ordinary final product theorem automatically applicable at $N$. It is not retried here. A complete-pair cancellation or a generalized composition could in principle provide \eqref{eq:jointmap}. Neither is established by the exact internal pole decomposition. \begin{theorem}[First unresolved gate: I3] \label{thm:I3} I2 is Constructed for the named internal amplitude map. I3 has a defined common precomposition presentation and a canonical-domain representation. The joint characteristic operator map \eqref{eq:jointmap} on tests meeting $N$ is not constructed. No complete-pair cancellation theorem or alternative composition theorem is supplied. The ending for the grouped characteristic target is therefore \textbf{Unresolved at I3}, within G2/F2/E1/D2/J2. No incompatibility excluding every representation in Definition~\ref{def:class} is proved. \end{theorem} \begin{proof} Theorem~\ref{thm:I2} discharges the stated internal amplitude gate. Proposition~\ref{prop:precomposition} supplies the exact internal sum, and Proposition~\ref{prop:canonical} supplies its composition only on the inherited admissible domain. Those maps do not act on a test crossing $N$ by the required characteristic composition. The uniform frequency estimates control the internal retarded pole, not its product with the grouped cross-lines. No theorem defining that latter operation has been proved. This is precisely the missing I3 requirement. Nothing in the internal construction proves that all possible such maps fail. The ending is accordingly Unresolved, not Proved obstructed. \end{proof} \scopebox{The distinction at I3}{% The internal amplitude is no longer the missing H2 object. Its pole, reconstruction, uniform estimates, and finite remainder have been supplied. \smallskip The missing object is now the joint characteristic operator map for the complete grouped coefficient. Calling the precomposition sum that map would skip a different product hypothesis. \smallskip No quantum coefficient is released by moving the first missing gate from H2 to I3.} %===================================================================== \section{I4: joint section or prescription, not entered} \label{sec:I4} %===================================================================== Because I3 is Unresolved, I4 is not entered as a construction. If I3 were later supplied on the same route, the selected continuation would be the \emph{joint-section and complete-remainder} exit. No characteristic prescription is selected here. That later theorem must control: \begin{enumerate} \item lower terms of every numerical line, including the massive cross-line contributions; \item every internal vertex and its actual operator map; \item attachment-diagonal derivatives; \item the two geometric defect directions without treating them as integration fibers; \item the complete actual source; \item the derivative and mass terms of the scalar root; \item the complete grouped cross-line difference; \item tensor and branch summation on one justified common representation. \end{enumerate} The finite internal remainder of Corollary~\ref{cor:remainder} is not that theorem. Its frequency degree is not a transverse degree of $\Tform$. No zero joint section is assumed, and no nonzero joint section is inferred from the $B_6$ numerator $Q\lambda$. \begin{proposition}[No full cancellation verdict] \label{prop:nocancellation} The results of XIV-I prove neither cancellation nor failure of cancellation of $\Tform$. \end{proposition} \begin{proof} The new amplitude reconstruction concerns $K_5$ before cross-line composition. The other new formulas identify the common internal sum. Neither constructs \eqref{eq:jointmap} at $N$ or assigns a joint principal-symbol section there. No complete remainder for that absent joint operation is supplied. The internal homogeneous-degree splitting cannot replace it. \end{proof} \subsection{The unentered prescription interface} The alternative prescription route remains a different construction. It would have to name all seven data: \begin{enumerate} \item ambient space; \item reference distribution on a proved domain; \item exact extension locus; \item agreement with the grouped frozen prescription; \item uniform transverse degree; \item a closed wavefront bound retaining off-locus singularities; \item normalization or a uniqueness theorem. \end{enumerate} The present locus is still the codimension-eight $N=\Nlift\subset W$. It is not converted into a codimension-four numerical problem. No Brunetti--Fredenhagen or Dang extension theorem is applied to the missing target \cite{BF,Dang}. XIII-E and a resolved paired construction stay unentered. There is no new contact class. %===================================================================== \section{I5: J3--J5 remain unopened} \label{sec:I5} %===================================================================== J3 requires a distribution supplied by I4 before uniform transverse estimates can be assigned to it. No such distribution is supplied. The parallel-line scaling degree eight remains a collision degree. Neither that value, the copied internal conormal orders, nor the distributional frequency degree in Theorem~\ref{thm:dyadic} is used as a transverse degree of $\Tform$. J4 would require composition with the singular spectators and permitted partial contacts, together with overlap agreement and the inherited normalizations. No joint gluing is performed. J5 would require the complete output wavefront theorem after the actual proper pushforward. The residual-field target is a compact microcausal degree-two kernel, for example with the sufficient bound \[ \WF(t^{(1)}) \subset Z_2=\{(z,p;y,q):p+q=0\}\setminus0 \] under the fixed flat cotangent identifications. This residual-field output bound is different from an intermediate attachment-kernel bound. Properness of the existing tree datum cannot supply it. The internal wavefront bound of Proposition~\ref{prop:WF} is not the J5 bound: its attachment variables have not been composed and pushed forward in the required quantum graph. %===================================================================== \section{I6: admission and the XV-B exit} \label{sec:I6} %===================================================================== An admission line would have to identify $\Rquant$ as the pushed-forward coefficient of the constructed, normalized joint distribution and prove \[ Q_H\Rquant\in\Ealg \] with XIII-B's retarded convention. No such line is asserted in this paper. The tree coefficient is not substituted for $\Rquant$. \begin{corollary}[XV-B remains closed] \label{cor:XVB} Under the stated series contract, XIV-I does not open XV-B. \end{corollary} \begin{proof} XV-B requires both the stated free carrier of XV-A and an admitted $\Rquant$ in $\Ealg$. The latter has not been supplied: I3 is Unresolved and I4--I6 have not been discharged. The conjunction of exit prerequisites therefore fails. \end{proof} If a later paper supplied the admission line, XV-B could attempt one relative interacting differential through one explicit order in $\kappa$ and $\hbar$ on the free carrier. The relative classical complex would remain a comparison object, not an identified quantum subalgebra. XV-A's stated odd-sector antifield coupling \[ \sum_A(-1)^{|A|}\Phi_A^*s\Phi^A \] belongs to that carrier prerequisite. It does not normalize any amplitude in this paper. A quantum master-equation defect, anomaly classification, or promotion of $\mathcal C_\xi$ would still belong to subsequent gates, not automatically to XV-B. No $\shat$, $s_{\mathrm{ext}}$, $s_\Psi$, or $s_0$ is applied to the absent coefficient. No Ward defect is formed. %===================================================================== \section{Decision and exact release} \label{sec:decision} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.07\linewidth} p{.29\linewidth}X@{}} \toprule Gate & Ending in XIV-I & XV-B consequence\\ \midrule I0--I1 & Frozen inputs and route & No exit. No model or ambiguity change.\\[0.35em] I2 & Constructed internal amplitude map & Both placements, exact fixed pole, uniform distributional estimates, finite remainder, and internal wavefront bounds. Opens I3 only.\\[0.35em] I3 & Unresolved & Common precomposition sum exists; the joint characteristic operator map is missing. No exit.\\[0.35em] I4 & Not entered & No joint section with complete remainder and no prescription.\\[0.35em] I5 & Not entered & J3--J5 not discharged for a joint distribution.\\[0.35em] I6 & Not Constructed & No admitted $\Rquant$ in $\Ealg$. XV-B may not open.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[XIV-I mathematical release] \label{thm:release} For the unchanged frozen pair and the route fixed in I1: \begin{enumerate}[label=\textup{(\roman*)}] \item the fixed internal retarded multiplier has the exact pole decomposition \eqref{eq:pole}; \item the finite amplitude module has the uniform estimates, continuous reconstruction, and exact lower-degree remainder proved in Section~\ref{sec:amplitudespace}; \item both $K_5$ placements have the amplitude maps \eqref{eq:A5}, reconstructing their complete untested kernels with the stated wavefront bounds; \item the retained six-branch tree kernel has the defined precomposition presentation \eqref{eq:presum}; \item no common characteristic operator map for the grouped target on tests meeting $N$ is constructed; \item no $\Rquant$ is admitted in $\Ealg$. \end{enumerate} The named internal amplitude ending is \textbf{Constructed at I2}. The grouped characteristic coefficient remains \textbf{Unresolved at I3}, within G2/F2/E1/D2/J2. XV-B remains closed. No universal obstruction to the class fixed in I1, full cancellation theorem, or admitted quantum output is asserted. \end{theorem} \begin{proof} Lemma~\ref{lem:multiplier} and Proposition~\ref{prop:pole} prove (i). Theorems~\ref{thm:dyadic} and \ref{thm:reconstruction}, with Corollary~\ref{cor:remainder}, prove (ii). Propositions~\ref{prop:both} and \ref{prop:WF} prove (iii), summarized in Theorem~\ref{thm:I2}. Proposition~\ref{prop:precomposition} proves (iv). Theorem~\ref{thm:I3} identifies (v). Sections~\ref{sec:I4}--\ref{sec:I6} record the undischarged subsequent gates and establish (vi). Corollary~\ref{cor:XVB} gives the exit consequence. \end{proof} \subsection*{What has advanced} The internal amplitude is now more than the assertion that a tree kernel exists. Its frequency singularity is explicitly a fixed retarded simple pole plus its regular companion. Its estimates hold on tests crossing that pole, and its inverse transform reconstructs both the diagonal chart and the separated cone chart. Both placements and every finite lower differential term are retained with $z$ untested. The remaining problem has changed, but it has not disappeared: compose the complete reconstructed sum with the grouped cross-lines on the characteristic domain. No internal Fourier identity supplies that missing product theorem. \subsection*{Successor package} A successor receives: \begin{enumerate} \item the unchanged XIV-H package; \item the finite retarded-pole module and its exact normalization; \item the pole identity, uniform estimates, and reconstruction map; \item the two complete $K_5$ amplitude maps with $z$ retained; \item their exact finite lower-degree remainder and internal wavefront bounds; \item the common precomposition sum and its canonical-domain value; \item the precise missing I3 joint operator map; \item the still closed I4--I6 and J3--J5 gates. \end{enumerate} It receives no transverse degree of $\Tform$, no cancellation theorem, and no permission to call either seventeen-dimensional count an excess. It receives no $\Rquant$. The remaining chain is \[ \text{I3 joint representation} \longrightarrow \text{I4 joint section and remainder} \longrightarrow \text{J3} \longrightarrow \text{J4} \longrightarrow \text{J5} \longrightarrow Q_H\Rquant\in\Ealg . \] Aiming at XV-B does not replace any arrow by a name. \subsection*{Conclusion} \[ \boxed{ \begin{gathered} M_{\mathrm R} =-\frac{s}{2\rho} (\omega-s\rho-i0)^{-1} +\frac{s}{2\rho(\omega+s\rho)},\\[2pt] A_{5,\pm}\in\Ar^2,\qquad \Rec(A_{5,+})+\Rec(A_{5,-})=K_5,\\[2pt] \text{uniform internal amplitude: Constructed at I2},\\ \text{joint characteristic operator map: Unresolved at I3},\\ Q_H\Rquant\in\Ealg:\ \text{not established},\\ \text{XV-B remains closed}. \end{gathered}} \] An internal pole amplitude is not a cross-line prescription. A reconstructed tree sum is not its characteristic product. A frequency remainder is not the quantum remainder theorem. An undefined quantum coefficient is not an anomaly. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Analytic scope.} The new construction is an explicit finite distribution-valued internal amplitude module. It includes a proved pole identity, uniform annular estimates, reconstruction, finite lower-degree remainder, and internal wavefront bounds. No general intersecting-Lagrangian calculus is claimed. \paragraph{No common product phase.} Separate Fourier transformations represent existing independent internal tensor factors. No sum of the retarded and quantum cross-line phases is formed or declared nondegenerate. \paragraph{No new prescription.} The boundary value in the internal multiplier is fixed by the already declared retarded kernel. Its Fourier computation does not select a value for the absent characteristic graph product. No characteristic ambiguity is added to $\Cpart$. \paragraph{Residual-variable scope.} The variable $z$ is retained in the adapted coordinates and in the reconstructed kernels. Uniformity is joint distributional control on compact sets, not a pointwise restriction at a singular residual value. \paragraph{Remainder scope.} The finite lower-degree estimate concerns the internal normal-frequency module only. It controls every lower term of the finite $K_5$ differential formula, but not the later complete quantum graph. \paragraph{Relation and degree scope.} The internal diagonal and flow-out intersection remains clean. It is not XIV-F's excluded matching. No internal conormal order or amplitude homogeneity is used as a transverse scaling degree of $\Tform$. \paragraph{Dependency scope.} The analytic manuscript inputs are those listed in I0. XV-A is used only to state the exit prerequisite. XIV-A and XIV-B, the auxiliary value $-8I$, XIII-E's family, and resolved paired constructions supply no analytic map here. \paragraph{Quantum scope.} No $\Rquant$, interacting differential, Ward defect, quantum master-equation defect, anomaly, state, numerical loop amplitude, new vertex, or sharp-width limit is constructed. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant verification, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The claimed mathematical results are the explicit proofs above. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{XIVH} N.~Cheung, \emph{Landmark XIV-H: Internal kernel relations on the operator route}. Supplied manuscript \texttt{XIV-H.tex}. Immediate predecessor: exact internal maps, regular charts, transition geometry, and the H2 amplitude stop. \bibitem{XIVG} N.~Cheung, \emph{Landmark XIV-G: Operator and pullback composition of the regrouped retained germ}. Supplied manuscript \texttt{XIV-G.tex}. Testing limitation, factorwise audit, and G2 boundary. \bibitem{XIVF} N.~Cheung, \emph{Landmark XIV-F: Paired composition on a proposed excess bundle}. Supplied manuscript \texttt{XIV-F.tex}. Tangent-defect and projection geometry, recorded only as boundaries. \bibitem{XIVE} N.~Cheung, \emph{Landmark XIV-E: Joint symbol of the six-branch retained germ, or one named characteristic prescription}. Supplied manuscript \texttt{XIV-E.tex}. Exact regrouping, complete root, tested-kernel bound, and characteristic lifts. \bibitem{XIVD} N.~Cheung, \emph{Landmark XIV-D: Cancellation or prescription on the retained noncoincident characteristic stratum}. Supplied manuscript \texttt{XIV-D.tex}. Six-branch germ, actual local incidences, and branch-limited fiber numerator. \bibitem{XIVC} N.~Cheung, \emph{Landmark XIV-C: The joint two-line extension of one selected binary pair}. Supplied manuscript \texttt{XIV-C.tex}. Retained characteristic stratum, positive relation, and contact rigidity. \bibitem{XIIIC} N.~Cheung, \emph{Landmark XIII-C: The joint two-line extension of one selected binary pair, on the unchanged carrier}. Supplied manuscript \texttt{XIII-C.tex}. Four successor obligations and primitive collision extensions. \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}. Slot sign, roots, and existing completion kernels. \bibitem{XIIC} N.~Cheung, \emph{Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question}. Supplied manuscript \texttt{XII-C.tex}. Fixed density representative and actual source. \bibitem{XVA} N.~Cheung, \emph{Landmark XV-A: The free quantum BV carrier for the ramp Einstein--scalar multiplet}. Supplied manuscript \texttt{XV-A.tex}. Recorded free-carrier prerequisite and comparison boundary only; not an analytic input to the coefficient. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}, second edition, Springer, 1990. Fourier distributions, tensor products, pullbacks, and wavefront calculus. \bibitem{Wunsch} J.~Wunsch, \emph{Microlocal analysis and evolution equations: Lecture notes from the 2008 CMI/ETH Summer School}, arXiv:0812.3181v3, 2023. Section~8: individual conormal and Lagrangian conventions. No general transition theorem is imported from these notes. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, arXiv:0806.1036. Causal Green operators and support conventions. The internal multiplier used here is computed explicitly. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}, arXiv:1208.1428v2, 2013. Functional products and their domain distinctions. No renormalized product for the present nonlocal pair is imported. \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. Recorded extension framework only; not applied to the missing characteristic target. \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 microlocal extension framework only; no characteristic prescription is imported. \end{thebibliography} \end{document}