% !TEX program = pdflatex %===================================================================== % IX-Free.tex % % Standalone pdfLaTeX source; no external bibliography is required. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error IX-Free.tex % pdflatex -interaction=nonstopmode -halt-on-error IX-Free.tex % % RELEASE OUTCOME: UNRESOLVED. % % Completed new results: % 1. Concrete presentation-relation kernels. % 2. Full distributional support of the fixed mixed kernel K. % 3. Injectivity of the two partial-contraction presentation maps. % 4. Evaluation of the algebraic scalar-descent set: C_desc = R. % 5. Existence and admission of all three canonical two-edge outputs. % 6. Second-stage coefficient maps on the degree-two intermediate % blocks, with reversal and fixed-support test-family continuity. % % Remaining required gates: % * descent of the prescribed degree-four second-stage coefficients; % * agreement on actual-pair and old-product overlaps. % % Accordingly, neither complete sharp association is released. % No universal obstruction, global quantum algebra, or IX-D opening % is claimed. % % The source finishes a partial-result manuscript, not an unsupported % completed factorization theorem. The missing gates are explicit. % % Compilation, proof-assistant verification, and independent review % have not been executed or claimed. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \numberwithin{equation}{section} \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \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{requirement}[theorem]{Requirement} \newtheorem{assumption}[theorem]{Assumption} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\supp}{\operatorname{supp}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\Span}{\operatorname{span}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\kerop}{\operatorname{ker}} \newcommand{\imop}{\operatorname{im}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\Rel}{\mathrm{Rel}} \newcommand{\eps}{\varepsilon} \newcommand{\Dtest}{\mathcal D} \newcommand{\Eprime}{\mathcal E'} \newcommand{\Padm}{\mathscr P_\Sigma} \newcommand{\Pdist}{\mathscr P_{\mathrm{dist}}} \newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}} \newcommand{\taus}{\tau_\Sigma} \newcommand{\Slin}{\mathscr S_1} \newcommand{\Mlin}{\mathscr M_1} \newcommand{\Qlin}{\mathscr Q_\Sigma} \newcommand{\Ddir}{\mathrm D} \newcommand{\Ezero}{\mathsf E_0} \newcommand{\GA}{G_{\mathrm A}^{0}} \newcommand{\Charge}{\mathsf b} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\Bspace}{\mathscr B} \newcommand{\Wspace}{\mathscr W} \newcommand{\Lspace}{\mathscr L} \newcommand{\Rspace}{\mathscr R} \newcommand{\PLspace}{\mathscr L_4} \newcommand{\QLspace}{\mathscr L_2} \newcommand{\PRspace}{\mathscr R_4} \newcommand{\QRspace}{\mathscr R_2} \newcommand{\Vmod}{\mathscr V_\Sigma^{[3]}} \newcommand{\Cal}{\mathcal N_c} \newcommand{\Graph}{\mathsf G} \newcommand{\Tspace}{\mathcal T} \newcommand{\EL}{\mathcal E_L} \newcommand{\ER}{\mathcal E_R} \newcommand{\cl}{\operatorname{cl}} \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 IX-Free}\\[0.4em] Actual-functional binary factorization\\ of one sharp ordered word} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We advance the actual-functional factorization problem for \(B_f,W_g,B_{f'}\), retaining the IX-A carrier and the IX-C interfaces. We identify the concrete presentation-relation kernels and prove that the fixed mixed kernel \(K\) has full distributional support. A compact-primitive argument then proves injectivity of both partial-contraction presentation maps. Consequently the algebraic scalar relation tests impose no restriction on the real parent parameter: \(\mathcal C_{\mathrm{desc}}=\R\). An explicit leaf-covector argument constructs the three canonical two-edge distributions. Their compact polynomial kernels exist, and both metric-quadratic outputs satisfy the full carrier admission test, including directional descendants. This constructs the second-stage coefficient maps on the degree-two intermediate blocks. Reversal and fixed-support test-family continuity are proved on these blocks. The release remains unresolved. Descent of the prescribed degree-four second-stage coefficients and agreement on actual-pair and old-product overlaps are not established. Thus neither complete sharp association is released. The triangle calibration remains conditional on the explicitly copied analytical premise reported in IX-C. No universal obstruction, global quantum algebra, or IX-D construction is claimed. \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) {Partial free-product steps}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Proved local statements}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Degree-four descent}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The release remains unresolved. Naming the gap does not close it.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Partial results leave the descent open. 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-IX-Free} \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) {Partial free-product steps}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Degree-four descent}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The release remains unresolved. Naming the gap does not close it.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Partial results leave the descent open. 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-IX-Free} \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) {Partial free-product steps}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Proved local statements}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Degree-four descent}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The release remains unresolved. Naming the gap does not close it.}; \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-IX-Free} \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) {Partial free-product steps}; \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) {Degree-four descent}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Proved local statements. The release remains unresolved. Naming the gap does not close it.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Partial free-product steps is not Degree-four descent. 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-IX-Free} \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: Partial free-product steps}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Proved local statements}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Degree-four descent}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The release remains unresolved. Naming the gap does not close it.}; \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-IX-Free} \end{figure} Figure~\ref{fig:split-IX-Free} records one view of the same separation. Figure~\ref{fig:chain-IX-Free} records one view of the same separation. Figure~\ref{fig:pair-IX-Free} records one view of the same separation. Figure~\ref{fig:gate-IX-Free} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-IX-Free} records that separation. The drawing adds no hypothesis. \scopebox{Release and proof boundary}{% \textbf{Release: Unresolved.} This manuscript proves the new partial results stated above. It does not supply the remaining degree-four descent and overlap proofs by declaring a common graph sum to be a binary product. A missing map is not a nonexistence theorem. The results below are relative to the explicitly stated inherited premises. The full IX-A, IX-B, and VII-D proofs are not independently certified here.} \tableofcontents %===================================================================== \section{Contract, inherited inputs, and domains} \label{sec:contract} %===================================================================== The selected word is \[ A=B_f,\qquad W=W_g,\qquad C=B_{f'}. \] The target is binary factorization through order \(\hbar^3\) on explicitly named actual-functional subspaces, with constant joint-test kernel \(u_\mu^\rho+cr\) for one fixed \(c\in\R\). A release is \emph{Constructed} only after every required interface is discharged. It is \emph{Proved obstructed} only if an explicit contradiction excludes every construction satisfying those interfaces. Otherwise it is \emph{Unresolved}. These alternatives refer to this sector and contract, not to every enlargement or normalization. \subsection{Geometric setting} Let \[ M=\R^{1+3},\qquad M_+=\{t>0\},\qquad X=M_+^3, \qquad \eta=\operatorname{diag}(-1,1,1,1). \] Write \(x=(t,\mathbf x)\), \(y=(s,\mathbf y)\), and \(x'=(t',\mathbf x')\). Set \[ \Box=-\partial_t^2+\Delta_{\mathbf x},\qquad (Ch)^\mu=\partial_\nu h^{\mu\nu} -\frac12\partial^\mu(\eta^{\alpha\beta}h_{\alpha\beta}), \qquad L=\Ezero C. \] The operator \(\Ezero\) is the zero-value, zero-first-time-derivative Cauchy inverse at \(t=0\). Write \(w=Lh\). Configurations \(\psi=(h,\phi)\) and configuration directions are smooth and off shell. The selected functionals are \begin{equation} B_f=-\int f(x)w^\mu(x)\partial_\mu\phi(x)\dd^4x, \qquad W_g=\int g(y)h_{00}(y)^2\dd^4y. \label{eq:generators} \end{equation} Repeated bundle indices are summed. We use the ordinary smooth mapping properties of these fixed Cauchy and Green operators. Smooth compact sources give smooth solutions; restriction to a fixed compact set is continuous in the source; and finite propagation over bounded time intervals gives compact propagation regions. In particular, applying \(L\) in one variable of a smooth joint function is continuous in the smooth topology, including derivatives in its parameter variables. These are smooth linear PDE properties, not singular-extension assumptions. \subsection{Prerequisite ledger} The supplied source is IX-C \cite{IXC}. Its statements concerning IX-A, IX-B, and VII-D are used below in the following explicit form. They are premises, not newly verified companion proofs. \begin{assumption}[Carrier, trace, and inherited closure] \label{ass:carrier} The carrier consists of actual finite polynomial functionals with compact symmetric distributional kernels satisfying \begin{equation} \Padm= \left\{F\in\Pdist: (\Ddir_{\boldsymbol\eta}F)'_h(\psi)\in\Slin \text{ for every smooth \(\psi\) and finite \(\boldsymbol\eta\)}\right\}. \label{eq:admission} \end{equation} Here \[ \Slin=\Mlin+\Qlin,\qquad q_a=L^*a=C^*(\theta\GA a),\qquad \Qlin=\{q_a:a\in C_c^\infty(M_+;T^*M)\}. \] The space \(\Mlin\) contains compact smooth linear metric kernels and has no nonzero nonspacelike wavefront covectors. The topology \(\taus\) records strong compact-distribution coefficients and every intrinsic trace coordinate \[ \Charge(m+q_a)=(\GA a)|_\Sigma. \] Classical multiplication, configuration-direction differentiation, and conjugation preserve \(\Padm\). Classical multiplication is separately continuous. No quantum multiplication is included. The generating test maps are injective. Their fixed-support test-family continuity and the fixed-support first-stage continuity reported in IX-C are inherited. For nonzero real \(f\), \[ B_{f,\eps}\not\longrightarrow B_f\quad\text{in }\taus. \] The sharp adjoint identity is \begin{equation} C^*(\theta b) =\theta C^*b+\delta(t)\mathcal B(b|_\Sigma). \label{eq:boundary} \end{equation} \end{assumption} \begin{assumption}[Calibration and already constructed outputs] \label{ass:calibration} For the prescribed regulator placement, \[ T_\eps^\rho =2K_\eps(x,y)\overline{K_\eps(x',y)}D_\eps(x-x'). \] There are distributions \(u_\mu^\rho,r\in\Dtest'(X)\) with \[ u_\mu^\rho=U_m+(\kappa_\rho+\log\mu)r,\qquad u_\mu^{\rho_a}-u_\mu^{\rho_b}=c_{ab}r,\quad c_{ab}\in\R. \] For each fixed compact \(K_0\subset X\), \begin{equation} \left| \pair{T_\eps^\rho-\log((\mu\eps)^{-1})r-u_\mu^\rho}{\Phi} \right| \le C_{K_0}\eps(1+|\log\eps|)p_{K_0,64}(\Phi). \label{eq:rate} \end{equation} The estimate includes compacts meeting \(\Rel\cap\Delta_{13}\). Parent conversion is exactly \(c_{ab}r\), without an additional normal derivative, nonconstant weight, or intersection term. The parent distributions satisfy dagger symmetry. Both regulated associations equal the same regulated word and have the same subtracted constant limit. The following first stages are already constructed: \begin{align} B_f\star_1W_g&=B_fW_g+\hbar\mathcal Q_{f,g}, & \mathcal Q_{f,g} &=-2\int f(x)g(y)K(x,y) \partial_t\phi(x)h_{00}(y)\dd^4x\dd^4y, \label{eq:Q}\\ W_g\star_1B_{f'}&=W_gB_{f'}+\hbar\mathcal Q^R_{g,f'}, & \mathcal Q^R_{g,f'} &=-2\int g(y)f'(x')\overline{K(x',y)} h_{00}(y)\partial_{t'}\phi(x')\dd^4y\dd^4x'. \label{eq:QR} \end{align} The three one-edge outputs in \eqref{eq:oneedges} are also already constructed and admitted. \end{assumption} Fix \(\rho,\mu\) once and define \begin{equation} U_c=u_\mu^\rho+cr,\qquad \Cal(f,g,f')=\pair{U_c}{f\otimes g\otimes f'}, \qquad c\in\R. \label{eq:cal} \end{equation} The same \(c\) is used for every test, presentation, association, and declared observation derivative. The scalar class \(\{U_c:c\in\R\}\) is not an ambiguity classification for nonconstant outputs. \begin{requirement}[Copied IX-C interfaces] \label{req:interfaces} A completed construction must satisfy: \begin{enumerate}[label=\textbf{C-I\arabic*.}] \item Every output belongs to \(\Padm\), with all required metric derivatives and directional descendants. \item Order zero is classical pointwise multiplication. \item Every specified old-bulk overlap agrees with \(\star_H\). \item The first stages are locked to \eqref{eq:Q} and \eqref{eq:QR}. \item The constant kernel is \(U_c\), with one real parent parameter transported by declared observation compatibility. \item No subtraction changes an already fixed old bulk product. \item Unchanged approximants need not converge sharply in \(\taus\). \item Every finite relation between actual presentations is respected. \item Every continuity claim has a named domain and proof. Test-family continuity is not carrier-stage hypocontinuity. \end{enumerate} \end{requirement} The inherited preservation obligations, where a product has actually been defined, are \[ \iota(A\star_HB)=\iota(A)\star_\Sigma\iota(B),\qquad C_j^\Sigma=C_j^{\mathrm{old}} \text{ on specified common regular domains},\qquad \cl(A\star_\Sigma B)=\cl(A)\cl(B). \] Here \(\cl\) denotes the order-zero projection. These are obligations, not conclusions about an unconstructed product. \subsection{Actual stages and the domain diagram} Let \[ \Tspace=C_c^\infty(M_+;\C),\qquad \Bspace=\Span\{B_f\},\qquad \Wspace=\Span\{W_g\}, \] and set \[ \PLspace=\Span\{B_fW_g\},\qquad \QLspace=\Span\{\mathcal Q_{f,g}\}, \] \[ \PRspace=\Span\{W_gB_{f'}\},\qquad \QRspace=\Span\{\mathcal Q^R_{g,f'}\}. \] Define \[ \Lspace=\PLspace\oplus\QLspace,\qquad \Rspace=\PRspace\oplus\QRspace,\qquad \Vmod=\Padm\otimes_\C\C[\hbar]/(\hbar^4). \] The direct sums are by actual homogeneous field degree, four and two. The underlying spaces \(\PLspace,\PRspace\) coincide, but their ordered roles differ. All presentation tensor products are algebraic. Put \[ \EL=\Tspace\otimes\Tspace,\qquad \ER=\Tspace\otimes\Tspace \] and define \[ p_L(f\otimes g)=B_fW_g,\quad q_L(f\otimes g)=\mathcal Q_{f,g}, \] \[ p_R(g\otimes f')=W_gB_{f'},\quad q_R(g\otimes f')=\mathcal Q^R_{g,f'}. \] The factorization diagram is \begin{equation} \begin{array}{ccccc} \kerop v\otimes\Tspace &\longrightarrow&E\otimes\Tspace &\xrightarrow{\ \lambda_j\ }&\Padm\\ &&\downarrow\,v\otimes B&&\uparrow\,\ell_j\\ &&\imop(v)\otimes\Bspace &\dashrightarrow&\Padm , \end{array} \label{eq:diagram} \end{equation} where \(v=p_L\) or \(q_L\). There is a reversed diagram on the right. The dashed arrow exists exactly when the labelled prescription annihilates the displayed relation space. Overlap conditions occur inside \(\Padm\otimes\Padm\), on \[ D_L=\Lspace\otimes\Bspace,\qquad D_R=\Bspace\otimes\Rspace,\qquad D_{\mathrm{old}}=\Pmc\otimes\Pmc. \] No presentation map is assumed to be a topological quotient. %===================================================================== \section{F1: concrete presentation-relation kernels} \label{sec:relations} %===================================================================== The fixed mixed kernel is \begin{equation} \begin{split} K(x,y) ={}&\int_{\R^3}\frac{\dd^3p}{(2\pi)^3} e^{ip\cdot(\mathbf x-\mathbf y)} \bigg[ \left(-\frac{t}{4|p|}+\frac{i}{8|p|^2}\right) e^{-i|p|(t-s)} \\ &\hspace{11em} -\frac{i}{8|p|^2}e^{i|p|(t+s)} \bigg]. \end{split} \label{eq:K} \end{equation} The low-frequency amplitudes are locally integrable in three spatial dimensions. The high-frequency terms are oscillatory distributions. For \(v=\sum_i\alpha_i f_i\otimes g_i\in\EL\), write \[ v(x,y)=\sum_i\alpha_i f_i(x)g_i(y). \] This identifies the algebraic tensor product with a subspace of \(C_c^\infty(M_+^2)\). Injectivity follows by choosing a representation with linearly independent first factors and applying linear functionals on their finite-dimensional span to isolate each second factor. \subsection{Mixed degree-two relations} Because \(v\) has support compactly inside \(M_+^2\), distributional observation-variable integration by parts gives \begin{equation} q_L(v)=2\pair{\partial_t(vK)}{\phi(x)h_{00}(y)}. \label{eq:qkernel} \end{equation} A cutoff equal to one on the coefficient support is understood when the configuration is not compactly supported. \begin{proposition}[Exact mixed relation equations] \label{prop:relations} For \(v\in\EL\), \begin{equation} q_L(v)=0 \quad\Longleftrightarrow\quad \partial_t(vK)=0 \quad\text{in }\Dtest'(M_+^2). \label{eq:leftrelation} \end{equation} For \(v_R\in\ER\), the right equation is \begin{equation} q_R(v_R)=0 \quad\Longleftrightarrow\quad \partial_{t'} \bigl(v_R(y,x')\overline{K(x',y)}\bigr)=0. \label{eq:rightrelation} \end{equation} \end{proposition} \begin{proof} The scalar slot and the \(00\)-metric slot are independent. Differentiating the mixed polynomial once in each field direction recovers its coefficient distribution on product tests, with the normalization \(2\) in \eqref{eq:qkernel}. Finite sums of product tests are dense in the joint test space, so these pairings determine the distribution. The converse is immediate. The right argument is identical. \end{proof} There is no surface term in \eqref{eq:qkernel}, since its observation coefficient is supported away from \(t=0\). This does not authorize omission of the delta term in the different operation \eqref{eq:boundary}. \subsection{Degree-four relations} Let \(J^\mu_{\alpha\beta}(x,z)\) denote the distribution kernel of \(L\): \[ w^\mu(x)=\int J^\mu_{\alpha\beta}(x,z) h_{\alpha\beta}(z)\dd^4z. \] Let \(e_{00}^{\alpha\beta}h_{\alpha\beta}=h_{00}\). For \(v\in\EL\), define \begin{equation} \begin{split} \mathsf P[v](x,z_1,z_2,z_3) ={}&\Sym_{(z_i,\alpha_i,\beta_i)} \partial_\mu^x \Big[ v(x,z_2)J^\mu_{\alpha_1\beta_1}(x,z_1) \\ &\qquad\quad{}\times \delta(z_2-z_3)e_{00}^{\alpha_2\beta_2} e_{00}^{\alpha_3\beta_3} \Big]. \end{split} \label{eq:pkernel} \end{equation} The symmetrization is the average over the three metric slots. Before smooth multiplication, \(J(x,z_1)\) and \(\delta(z_2-z_3)\) are tensor factors in disjoint variables. Thus \eqref{eq:pkernel} is a defined distribution. Observation support, finite propagation, and the delta relation give a common compact support for its field variables. \begin{proposition}[Exact degree-four relation equations] \label{prop:prelations} With the coefficient convention \[ p_L(v) =\pair{\mathsf P[v]} {\phi(x)h(z_1)h(z_2)h(z_3)}, \] one has \[ p_L(v)=0\quad\Longleftrightarrow\quad \mathsf P[v]=0. \] For \(v_R(y,x')\), let \(\widetilde v(x',y)=v_R(y,x')\). Then \[ p_R(v_R)=0\quad\Longleftrightarrow\quad \mathsf P[\widetilde v]=0. \] \end{proposition} \begin{proof} Insert the kernel of \(L\) in \eqref{eq:generators}, multiply by \(W_g\), and integrate the scalar observation derivative by parts. The original insertion sign changes the coefficient to the positive derivative in \eqref{eq:pkernel}. Differentiating once in the scalar direction and three times in independent metric directions recovers the symmetric coefficient kernel, multiplied by \(3!\). Polarization and density of product tests prove uniqueness. The right formula uses only pointwise commutativity of the original classical factors. \end{proof} This characterizes \(\kerop p_L,\kerop p_R\) concretely. It does not prove that either kernel is zero. %===================================================================== \section{F2: evaluated scalar descent} \label{sec:scalar} %===================================================================== The next argument evaluates the actual scalar relation tests. It does not merely repeat their affine classification. \begin{lemma}[Full support of the mixed kernel] \label{lem:fullsupport} The distribution in \eqref{eq:K} satisfies \[ \supp K=M_+^2. \] \end{lemma} \begin{proof} Write \(a=t-s\) and \(\mathbf z=\mathbf x-\mathbf y\). The Fourier normalization is \[ \Delta_0^+(a,\mathbf z) =\int\frac{\dd^3p}{(2\pi)^3\,2|p|} e^{ip\cdot\mathbf z-i|p|a}. \] Applying \(\Box_x\) to \eqref{eq:K} is legitimate distributionally. The two terms without the explicit factor \(t\) are homogeneous wave solutions. Moreover, \[ (-\partial_t^2-|p|^2) \bigl(t e^{-i|p|(t-s)}\bigr) =2i|p|e^{-i|p|(t-s)}. \] Consequently \begin{equation} \Box_xK(x,y) =-\frac{i}{2}\int\frac{\dd^3p}{(2\pi)^3} e^{ip\cdot\mathbf z-i|p|a} =\partial_a\Delta_0^+(a,\mathbf z). \label{eq:boxK} \end{equation} For completeness, damping the radial integral by \(e^{-\delta|p|}\), \(\delta>0\), and performing its angular integration gives \[ \Delta_{0,\delta}^+(a,\mathbf z) =\frac{1}{4\pi^2} \frac{1}{|\mathbf z|^2-(a-i\delta)^2}. \] On the open set \(a\ne0\), \( |\mathbf z|^2\ne a^2\), its limiting derivative is \[ \partial_a\Delta_0^+ =\frac{a}{2\pi^2(|\mathbf z|^2-a^2)^2}, \] which is nonzero. That open set is dense. Its inverse image in \(M_+^2\) is also dense, since the difference map is open. Thus the right-hand side of \eqref{eq:boxK} has full support on \(M_+^2\). A differential operator cannot enlarge distributional support, so \(K\) has full support. \end{proof} \begin{lemma}[Compact distributions have no constant primitive] \label{lem:primitive} Let \(I=(0,\infty)\), let \(N\) be a manifold, and let \(T\in\Eprime(I\times N)\). If \(\partial_tT=0\), then \(T=0\). \end{lemma} \begin{proof} A distribution with zero derivative in a connected interval is constant in that variable. Here is the product-test argument. Choose \(\chi\in C_c^\infty(I)\) with \(\int\chi=1\). For every \(\eta\in C_c^\infty(I)\), \(\eta-(\int\eta)\chi\) has integral zero and is the derivative of a compactly supported smooth function in \(I\). Hence \[ \pair{T}{\eta\otimes b} =\left(\int\eta\right)\pair{T}{\chi\otimes b}. \] Choose \(\eta\) with integral one and support in an interval disjoint from the time projection of \(\supp T\). The left-hand side is zero, so every pairing on the right is zero. Product-test density gives \(T=0\). \end{proof} \begin{theorem}[Injective partial-contraction presentations] \label{thm:qinjective} The maps \[ q_L:\EL\longrightarrow\QLspace,\qquad q_R:\ER\longrightarrow\QRspace \] are linear isomorphisms onto their actual-functional images. \end{theorem} \begin{proof} If \(q_L(v)=0\), Proposition~\ref{prop:relations} gives \(\partial_t(vK)=0\). The distribution \(vK\) is compactly supported in \(M_+^2\). Lemma~\ref{lem:primitive} gives \(vK=0\). If the smooth function \(v\) were nonzero at a point, it would have a smooth reciprocal on a nonempty neighborhood. There \(K=v^{-1}(vK)=0\), contradicting Lemma~\ref{lem:fullsupport}. Thus \(v=0\), and the algebraic tensor embedding proves injectivity. Surjectivity onto the named image is its definition. For the right map, use \eqref{eq:rightrelation}; conjugation and exchange of variables preserve full support. \end{proof} Define the combined scalar relation space \[ \mathcal Z= (\kerop q_L)\otimes\Tspace \ \oplus\ \Tspace\otimes(\kerop q_R). \] Pairing its joint tests with \(u_\mu^\rho,r\) defines \(\alpha,\beta\), respectively. \begin{corollary}[Evaluation of the parent-parameter test] \label{cor:cdesc} For the actual presentation maps of this manuscript, \[ \mathcal Z=\{0\},\qquad \alpha=\beta=0,\qquad \mathcal C_{\mathrm{desc}} =\{c\in\R:\alpha+c\beta=0\}=\R. \] For every fixed real \(c\), the calibrated left and right scalar maps exist uniquely as algebraic bilinear maps: \begin{align} L_{2,c}(\mathcal Q_{f,g},B_{f'}) &=\Cal(f,g,f')\,1, \label{eq:leftscalar}\\ R_{2,c}(B_f,\mathcal Q^R_{g,f'}) &=\Cal(f,g,f')\,1. \label{eq:rightscalar} \end{align} \end{corollary} \begin{proof} Theorem~\ref{thm:qinjective} makes the two relation kernels zero. The sharp generating map is injective by the inherited premise. Multilinearity of distributional pairing therefore defines unique bilinear maps on the actual images. Constants belong to \(\Padm\). \end{proof} The conclusion is \emph{algebraic scalar descent}, not permission to use every \(c\) in a completed bulk-relative product. Old-product overlap conditions may impose additional restrictions. No such restriction is evaluated here. %===================================================================== \section{F3: construction of the three two-edge distributions} \label{sec:trees} %===================================================================== Put \[ S_{\mu\nu}(x,x') =\partial_\mu^x\partial_\nu^{x'}\Delta_m^+(x-x'), \qquad m\ge0. \] We use the standard product, submersion-pullback, and proper pushforward calculus for distributions \cite{Hormander,Demanet}. The graph-specific covector checks are proved below. \subsection{Endpoint covectors} \begin{lemma}[Nonzero comparable endpoint covectors] \label{lem:endpoints} Every wavefront element of \(K\), \(\overline K\), or \(S_{\mu\nu}\) has nonzero covectors in both endpoint slots. For the Euclidean norm on coordinate covectors, the two endpoint norms are equal. \end{lemma} \begin{proof} Split each Fourier integral into low and high frequencies with a smooth radial cutoff. The low-frequency part of \(K\) is smooth: its amplitudes are locally integrable, and every coordinate derivative inserts an integrable bounded-frequency factor. The same observation applies to \(\Delta_m^+\) and its finite derivatives. The high-frequency phases of \(K\) are \[ p\cdot(\mathbf x-\mathbf y)-|p|(t-s),\qquad p\cdot(\mathbf x-\mathbf y)+|p|(t+s). \] Their endpoint covectors are, respectively, \[ (-|p|,p;\ |p|,-p),\qquad (|p|,p;\ |p|,-p). \] They are nonzero at both endpoints and have equal norms. The usual integration-by-parts estimate for oscillatory integrals excludes covectors outside these phase-generated cones. Smooth amplitudes and differentiation do not enlarge the resulting wavefront bound. For the massive scalar kernel the high-frequency phase is \[ p\cdot(\mathbf x-\mathbf x') -\sqrt{|p|^2+m^2}(t-t'). \] Its conic limiting endpoint directions are opposite nonzero null covectors with equal norms. Differentiation to obtain \(S_{\mu\nu}\) does not enlarge this bound. Conjugation reverses covectors and preserves the assertion. \end{proof} The comparison of endpoint norms also excludes spurious one-slot directions arising from taking closures of sums: if one endpoint contribution tends to zero, so does the other contribution of that edge. \begin{lemma}[Two-edge multiplication and single-vertex smoothing] \label{lem:forest} Pull back two endpoint kernels satisfying Lemma~\ref{lem:endpoints} to the three-vertex variables of a two-edge tree. Their distributional product exists. After multiplication by a compact smooth joint coefficient, pushforward integrating all vertices except any one vertex is smooth. \end{lemma} \begin{proof} Each of the two edges has a leaf belonging to no other edge. A nonzero covector from that edge has a nonzero component at that leaf, whereas the other edge has zero component there. Thus the two pulled-back wavefront covectors cannot sum to zero. The distributional product criterion applies. For the second assertion, the pushforward wavefront bound requires zero covectors in all integrated slots. Remove an integrated leaf different from the retained vertex. Its covector forces its incident edge to be inactive, meaning that the edge contributes the zero covector in the augmented wavefront bound. Repeat on the remaining edge. Both edges must be inactive, so no nonzero covector remains at the retained vertex. If the retained vertex is the central vertex, remove the two integrated leaves immediately. If it is a leaf, remove the other leaf first and then the integrated central vertex. The endpoint norm comparison justifies the same reasoning for closed conic limits. Compact multiplication makes the pushforward proper. Its wavefront set is empty, so it is smooth. \end{proof} This is the leaf-slot argument required by the forest structure. The statement ``the graph is a tree'' alone would not suffice. \subsection{The individual outputs} Let \(V=f\otimes g\otimes f'\). Define compact distributions on \(X\): \begin{align} E_{12,23}[V] &=2V K(x,y)\overline{K(x',y)},\label{eq:E1223}\\ E_{12,13;\nu}[V] &=2V K(x,y)S_{0\nu}(x,x'),\label{eq:E1213}\\ E_{13,23;\mu}[V] &=2V S_{\mu0}(x,x')\overline{K(x',y)}.\label{eq:E1323} \end{align} \begin{proposition}[Actual two-edge distributions] \label{prop:existence} Each product in \eqref{eq:E1223}--\eqref{eq:E1323} exists canonically as a distribution on all of \(X\). Each has assignment multiplicity \(2\), positive insertion sign, and residual field degree two. \end{proposition} \begin{proof} For \eqref{eq:E1223}, the leaves \(x,x'\) give the product criterion. For \eqref{eq:E1213}, the leaves are \(y,x'\). For \eqref{eq:E1323}, they are \(x,y\). Lemma~\ref{lem:forest} applies in each case. Both sharp insertions contribute a minus sign, whose product is positive. There are two assignments of the middle metric slots, hence the displayed factor \(2\), not \(4\). Two edges remove four slots from the initial six. \end{proof} Let \(\pi_{ab}\) denote projection onto the indicated vertices. The outputs are the actual quadratic functionals \begin{align} \Graph_{12,23} &=\pair{E_{12,23}[V]} {\partial_t\phi(x)\partial_{t'}\phi(x')}, \label{eq:G1223}\\ \Graph_{12,13} &=\pair{E_{12,13;\nu}[V]} {h_{00}(y)w^\nu(x')}, \label{eq:G1213}\\ \Graph_{13,23} &=\pair{E_{13,23;\mu}[V]} {w^\mu(x)h_{00}(y)}. \label{eq:G1323} \end{align} For \eqref{eq:G1223}, the compact scalar coefficient kernel is \[ \Sym_{x,x'}\partial_t\partial_{t'} (\pi_{13})_*E_{12,23}[V]. \] There is no sign change from two scalar integrations by parts. For \eqref{eq:G1213}, first form the compact residual kernel \[ R_\nu(y,x')=(\pi_{23})_*E_{12,13;\nu}[V], \] then transpose \(L\) in its second variable and symmetrize the two metric slots. For \eqref{eq:G1323}, use \[ R_\mu(x,y)=(\pi_{12})_*E_{13,23;\mu}[V] \] and transpose \(L\) in its first variable. These transpositions are defined on compact distributions. Indeed, applying \(L\) to a smooth joint test function in one variable is continuous in the joint smooth topology. Its transpose therefore acts on compact distributions. Finite propagation supplies a common compact field support. This construction does not complete any presentation space or invoke a quotient topology. \begin{remark}[What has not been classified] The products just constructed need no extension across an excluded set: their wavefront products exist on all of \(X\). This selects a canonical graph prescription. It does not classify all nonconstant terms permitted by the copied interfaces, nor impose a new prohibition on alternative contact terms. \end{remark} The following failure language remains in force: \begin{enumerate} \item Failure of a sufficient wavefront criterion means that proof route fails. \item An unestablished regulated limit is unresolved. \item Failure of one extension excludes that extension. \item Nonexistence requires a separate argument. \item Failure of one graph prescription is not, by itself, incompatibility with all copied interfaces. \end{enumerate} %===================================================================== \section{F4: admission by finite-degree reduction} \label{sec:admission} %===================================================================== \begin{theorem}[Admission of all three two-edge outputs] \label{thm:treeadmission} The functionals \eqref{eq:G1223}--\eqref{eq:G1323} belong to \(\Padm\), including every metric admission test after configuration-direction differentiation. \end{theorem} \begin{proof} The scalar-only output \(\Graph_{12,23}\) has the compact scalar kernel already constructed. Its metric derivatives and their directional descendants vanish. Consider \(G=\Graph_{12,13}\). Its polarized bilinear form is \[ b_G(h,k)=\frac12 \pair{E_{12,13;\nu}[V]} {h_{00}(y)(Lk)^\nu(x')+ k_{00}(y)(Lh)^\nu(x')}. \] Thus \(G(h)=b_G(h,h)\) and \(G'_h(\psi)[k]=2b_G(h,k)\). Define \begin{align} m_\psi(y) &=2g(y)\int f(x)K(x,y) \left[\int f'(x')S_{0\nu}(x,x')w^\nu(x')\dd^4x'\right] \dd^4x, \label{eq:m1213}\\ a_{\psi,\nu}(x') &=2f'(x')\int f(x)S_{0\nu}(x,x') \left[\int g(y)K(x,y)h_{00}(y)\dd^4y\right] \dd^4x. \label{eq:a1213} \end{align} These formulas mean the corresponding proper distributional pushforwards. By Lemma~\ref{lem:forest}, both are smooth: they integrate the compact weighted tree except for a single vertex. They are compactly supported in \(\supp g\) and \(\supp f'\), respectively. The metric derivative is therefore the actual linear kernel \begin{equation} G'_h(\psi)=e_{00}m_\psi+L^*a_\psi =e_{00}m_\psi+q_{a_\psi}. \label{eq:decomposition} \end{equation} Its first summand is compact smooth and belongs to \(\Mlin\); its second belongs to \(\Qlin\). For one configuration direction \(\zeta\), the same formulas hold with \(h\) replaced by \(\zeta_h\) and \(w\) replaced by \(L\zeta_h\). This is the configuration-independent metric derivative of \(\Ddir_\zeta G\). Two directional derivatives of \(G\) are constant functionals, whose metric derivative is zero. Higher derivatives vanish. Scalar directions have no effect. This checks every finite direction list. For \(G=\Graph_{13,23}\), the corresponding smooth compact functions are \begin{align} m_\psi(y) &=2g(y)\int f'(x')\overline{K(x',y)} \left[\int f(x)S_{\mu0}(x,x')w^\mu(x)\dd^4x\right] \dd^4x', \label{eq:m1323}\\ a_{\psi,\mu}(x) &=2f(x)\int f'(x')S_{\mu0}(x,x') \left[\int g(y)\overline{K(x',y)}h_{00}(y)\dd^4y\right] \dd^4x'. \label{eq:a1323} \end{align} Again its derivative is \(e_{00}m_\psi+q_{a_\psi}\), and the same finite-degree argument checks all descendants. \end{proof} No inference from a single nonspacelike covector was used. Membership was proved by exhibiting the required decomposition. Conversely, an admission obstruction would have to exclude every \(\Mlin+\Qlin\) decomposition, not merely one proposed decomposition. %===================================================================== \section{F5--F6: actual maps and the remaining overlap gates} \label{sec:maps} %===================================================================== The inherited one-edge outputs are \begin{equation} \Graph_{12}=\mathcal Q_{f,g}B_{f'},\qquad \Graph_{23}=B_f\mathcal Q^R_{g,f'},\qquad \Graph_{13}=\mathcal T_{f,f'}W_g, \label{eq:oneedges} \end{equation} where \[ \mathcal T_{f,f'} =\int f(x)f'(x')S_{\mu\nu}(x,x') w^\mu(x)w^\nu(x')\dd^4x\dd^4x'. \] \subsection{Completed degree-two blocks} \begin{theorem}[Degree-two intermediate coefficient maps] \label{thm:degree2maps} For each fixed real \(c\), the following prescriptions define unique bilinear maps on actual functionals: \begin{align} L_0(F,B)&=FB, & L_1(\mathcal Q_{f,g},B_{f'}) &=\Graph_{12,23}+\Graph_{12,13}, \label{eq:Ldegree2}\\ L_2(\mathcal Q_{f,g},B_{f'}) &=\Cal(f,g,f')\,1, \label{eq:Ldegree2scalar}\\ R_0(B,F)&=BF, & R_1(B_f,\mathcal Q^R_{g,f'}) &=\Graph_{12,23}+\Graph_{13,23}, \label{eq:Rdegree2}\\ R_2(B_f,\mathcal Q^R_{g,f'}) &=\Cal(f,g,f')\,1. \label{eq:Rdegree2scalar} \end{align} Their domains are \(\QLspace\times\Bspace\) and \(\Bspace\times\QRspace\), respectively, and their targets are \(\Padm\). \end{theorem} \begin{proof} Theorem~\ref{thm:qinjective} gives a unique algebraic presentation tensor for each intermediate argument. The sharp generating map gives a unique test for the other argument. The displayed graph sums are linear in the presentation tensor and in the sharp test, so they descend to bilinear maps. In particular, the coefficient sum, not merely its individual terms, is well defined. Theorem~\ref{thm:treeadmission} places the positive-order nonconstant outputs in \(\Padm\). Corollary~\ref{cor:cdesc} handles the scalar coefficient. Classical closure handles order zero. \end{proof} These are constructed coefficient maps on named blocks. They are not yet a bulk-relative binary product on the sum of all requested coefficient domains. \subsection{Uncompleted degree-four blocks} The degree-four prescriptions remain \begin{align} \lambda^L_1(f,g,f')&=\Graph_{13}+\Graph_{23}, & \lambda^L_2(f,g,f')&=\Graph_{13,23}, \label{eq:lambdaL}\\ \lambda^R_1(f,g,f')&=\Graph_{12}+\Graph_{13}, & \lambda^R_2(f,g,f')&=\Graph_{12,13}. \label{eq:lambdaR} \end{align} Their values now exist and are admitted. The unresolved question is whether they factor through \(p_L,p_R\). \begin{proposition}[Exact remaining degree-four descent tests] \label{prop:remainingdescent} The prescriptions \eqref{eq:lambdaL} define maps on \(\PLspace\times\Bspace\) exactly when \[ \lambda^L_j\bigl((\kerop p_L)\otimes\Tspace\bigr)=0, \qquad j=1,2. \] The right prescriptions define maps on \(\Bspace\times\PRspace\) exactly when \[ \lambda^R_j\bigl(\Tspace\otimes(\kerop p_R)\bigr)=0, \qquad j=1,2. \] Here the relation kernels are precisely the distributional equations in Proposition~\ref{prop:prelations}. \end{proposition} \begin{proof} Necessity follows by evaluating a proposed map at a zero actual argument. For sufficiency, choose a presentation of the actual intermediate argument. Two choices differ by the relevant relation kernel, which the hypothesis annihilates. Bilinearity and uniqueness then follow. \end{proof} These tests have not been discharged here. In particular, injectivity of \(q_L,q_R\) does not imply injectivity of \(p_L,p_R\). \subsection{Overlap diagnostics} For vector subspaces \(U,U',V,V'\), all contained in fixed ambient spaces, algebraic tensor products satisfy \begin{equation} (U\otimes V)\cap(U'\otimes V') =(U\cap U')\otimes(V\cap V'). \label{eq:tensorintersection} \end{equation} One proof is to extend bases of the intersections to bases of the sums and compare the resulting tensor coordinates. Field-degree bigrading shows that a left degree-four block and a right degree-four block have zero intersection: their argument degrees are \((4,2)\) and \((2,4)\). The mixed degree-four/degree-two intersections are also zero. The potentially nontrivial left/right degree-two intersection is \begin{equation} (\QLspace\otimes\Bspace)\cap(\Bspace\otimes\QRspace) =(\QLspace\cap\Bspace)\otimes(\Bspace\cap\QRspace). \label{eq:newoverlap} \end{equation} No vanishing of the two factors is proved here. For each new block \(U\otimes V\), its old-product overlap is \begin{equation} (U\otimes V)\cap D_{\mathrm{old}} =(U\cap\Pmc)\otimes(V\cap\Pmc). \label{eq:oldoverlap} \end{equation} Thus the old-product obligation is a concrete value-preservation test, not a convergence assertion. \begin{lemma}[Overlap gluing] \label{lem:gluing} Linear coefficient prescriptions on two subspaces glue to their sum if and only if they agree on the intersection. For three subspaces, each next prescription must agree on its intersection with the sum of the preceding domains. Pairwise agreement alone need not suffice. \end{lemma} \begin{proof} For two domains define the proposed value on \(d_1+d_2\) as the sum of the prescribed values. Independence of decomposition is equivalent to agreement on the intersection. For three domains, apply the two-domain statement to the already glued sum and the third domain. \end{proof} The last qualification prevents a hidden multi-domain relation from being omitted. The requisite agreement on \eqref{eq:newoverlap}, \eqref{eq:oldoverlap}, and the relevant sum intersections remains unresolved. %===================================================================== \section{F7: four-order coherence, conditional on the missing maps} \label{sec:coherence} %===================================================================== Write \[ P_L=AW,\quad Q_L=\mathcal Q_{f,g},\qquad P_R=WC,\quad Q_R=\mathcal Q^R_{g,f'}. \] Under the selected contraction-degree prescription, a completed candidate would have \begin{align} \mathcal A_L ={}&P_LC +\hbar\bigl(L_1(P_L,C)+Q_LC\bigr) \nonumber\\ &+\hbar^2\bigl(L_2(P_L,C)+L_1(Q_L,C)\bigr) +\hbar^3L_2(Q_L,C), \label{eq:AL}\\ \mathcal A_R ={}&AP_R +\hbar\bigl(R_1(A,P_R)+AQ_R\bigr) \nonumber\\ &+\hbar^2\bigl(R_2(A,P_R)+R_1(A,Q_R)\bigr) +\hbar^3R_2(A,Q_R). \label{eq:AR} \end{align} The residual degrees are \(6,4,2,0\). The absence of a third contraction against a degree-two sharp factor is the selected graph prescription. It is not a theorem forbidding every possible additional contact term. \begin{proposition}[Conditional completed-sector coherence] \label{prop:conditionalcoherence} Suppose the tests in Proposition~\ref{prop:remainingdescent} hold and the coefficient prescriptions glue consistently with their specified overlaps. With the same fixed real \(c\), the two associations then agree through order \(\hbar^3\). Their common coefficients are \[ \begin{array}{c|l} \hbar\text{-order}&\text{common actual output}\\ \hline 0&AWC\\ 1&\Graph_{12}+\Graph_{13}+\Graph_{23}\\ 2&\Graph_{12,23}+\Graph_{12,13}+\Graph_{13,23}\\ 3&\Cal(f,g,f')\,1. \end{array} \] \end{proposition} \begin{proof} Order zero is classical associativity. At the three positive orders, substitute \eqref{eq:Ldegree2}--\eqref{eq:Rdegree2scalar} and the descended versions of \eqref{eq:lambdaL}--\eqref{eq:lambdaR}. This gives, respectively, \begin{align*} L_1(P_L,C)+Q_LC &=R_1(A,P_R)+AQ_R,\\ L_2(P_L,C)+L_1(Q_L,C) &=R_2(A,P_R)+R_1(A,Q_R),\\ L_2(Q_L,C)&=R_2(A,Q_R). \end{align*} Each is an equality of the displayed actual polynomial functionals. Equality of all four coordinates is equality in \(\Vmod\). \end{proof} This proposition does not construct its degree-four maps. Writing the common graph sum cannot replace the missing factorization and gluing proofs. A nonconstant discrepancy cannot be absorbed into \(cr\): the order-one and order-two coordinates are separate from the constant order-three coordinate. %===================================================================== \section{F8: reversal, presentation independence, and observations} \label{sec:reversal} %===================================================================== For functionals, star means complex conjugation. For a joint distribution, dagger conjugates and exchanges \(x,x'\): \[ \pair{U^\dagger}{\Phi} =\overline{\pair U{\overline{\Phi\circ\tau}}}, \qquad \tau(x,y,x')=(x',y,x). \] \begin{proposition}[Constructed-block reversal] \label{prop:reversal} The two-edge outputs satisfy \begin{align*} \Graph_{12,23}(f,g,f')^* &=\Graph_{12,23}(\bar f',\bar g,\bar f),\\ \Graph_{12,13}(f,g,f')^* &=\Graph_{13,23}(\bar f',\bar g,\bar f). \end{align*} The constructed degree-two coefficient maps satisfy \[ L_j(F,B)^*=R_j(B^*,F^*),\qquad j=0,1,2. \] \end{proposition} \begin{proof} The free scalar kernel has \[ \overline{S_{\mu\nu}(x,x')} =S_{\nu\mu}(x',x). \] Conjugation of a defined wavefront product is the product of the conjugated factors. Exchanging the endpoint variables therefore gives the two displayed graph identities. The locked first-stage coefficients obey \[ \mathcal Q_{f,g}^* =\mathcal Q^R_{\bar g,\bar f}. \] The graph identities give reversal at order one. At order two, the inherited parent symmetry and real \(c\) give \[ \overline{\Cal(f,g,f')} =\Cal(\bar f',\bar g,\bar f). \] Order zero follows from classical conjugation. \end{proof} The degree-four labelled prescriptions also exchange under conjugation, by the inherited one-edge reversal and the two-edge identities. Their carrier-level reversal remains conditional on descent and overlap consistency. \begin{proposition}[Presentation independence on constructed blocks] \label{prop:presentation} The maps of Theorem~\ref{thm:degree2maps} preserve every rewriting that leaves their arguments unchanged as actual off-shell functionals. This includes finite linear presentation relations, kernel permutations, tensor symmetries, polarization, and integration by parts with all boundary terms retained. For a completed glued sector, the same conclusion holds provided its descent and overlap hypotheses are proved. \end{proposition} \begin{proof} A map on actual arguments assigns the same value to equal arguments. In presentation language, their difference is annihilated by descent. Gluing ensures this remains true when a rewriting moves between named prescriptions. \end{proof} Equation \eqref{eq:boundary} is an equality of actual kernels. Dropping its delta term changes the argument and is not a permitted presentation change. No equation-of-motion quotient is introduced. \begin{proposition}[Numerical observation compatibility] \label{prop:observations} For a fixed smooth multiplier \(a\) and smooth-coefficient differential operator \(D\) on \(X\), \[ aDU_c=aDu_\mu^\rho+c\,aDr. \] In particular, \[ \Cal(\partial_t^n f,g,f') =\pair{(-\partial_{x^0})^nU_c}{f\otimes g\otimes f'}. \] The inherited numerical finite-part estimate has test-order loss \(64\to64+\ord D\). \end{proposition} \begin{proof} The distributional identities follow from differentiation and linearity. The quantitative descendant estimate is the explicit IX-B transfer premise reported in IX-C, not a new ultraviolet estimate. \end{proof} These numerical operations do not identify configuration directions with observation derivatives or supply new graphs with opened functional slots. A general joint multiplier gives a numerical operation unless its carrier-level factorization is separately established. %===================================================================== \section{F9a and F9b: distinct continuity statements} \label{sec:continuity} %===================================================================== Fix compact sets \(K_1,K_2,K_3\Subset M_+\). Let \[ \Dtest_{K_i}=\{f\in C_c^\infty(M_+):\supp f\subset K_i\} \] with their usual Fr\'echet smooth seminorms. Put \(K=K_1\times K_2\times K_3\). \begin{lemma}[Fixed compact coefficient continuity] \label{lem:strongcontinuity} For each fixed two-edge distribution \(E\), the map \[ V\longmapsto VE,\qquad \Dtest_K(X)\longrightarrow\Eprime(X) \] is continuous in the strong compact-distribution topology. Fixed differentiation, proper projection, and the fixed field-slot transpositions used above preserve this continuity. \end{lemma} \begin{proof} For a bounded family \(B\) of smooth functions, a finite-order distribution estimate on the common compact gives \[ \sup_{\varphi\in B}|\pair{E}{V\varphi}| \le C_B\,p_{K,N}(V) \] for some finite \(N\). This proves strong continuity. Differentiation and projection follow by transposition of continuous maps on smooth tests. The same argument applies to the fixed \(L\)-slot operation, whose smooth joint-test map is continuous and maps bounded sets to bounded sets. \end{proof} \begin{lemma}[Continuity of the smooth derivative sources] \label{lem:sourcecontinuity} At each fixed smooth configuration and each fixed direction, the maps \[ V\longmapsto m_\psi,\qquad V\longmapsto a_\psi \] used in the metric admission proof are continuous into their fixed-support smooth topologies. \end{lemma} \begin{proof} They are linear proper pushforwards of \(VE\) multiplied by fixed smooth configuration fields. Lemma~\ref{lem:forest} shows their values are smooth. Their graphs are closed: if \(V_n\to V\) in \(\Dtest_K\) and the corresponding source functions converge smoothly, Lemma~\ref{lem:strongcontinuity} identifies the smooth limit with the distributional pushforward at \(V\). The closed graph theorem for Fr\'echet spaces gives continuity. Restriction to the fixed-support subspaces preserves the conclusion. \end{proof} \begin{theorem}[F9a for the completed coefficient families] \label{thm:F9a} On \(\Dtest_{K_1}\times\Dtest_{K_2}\times\Dtest_{K_3}\), the three two-edge output families and the positive-order degree-two coefficient families are jointly continuous with values in \((\Padm,\taus)\). The scalar family \(\Cal(f,g,f')\) is jointly continuous. The degree-two order-zero families also have fixed-support test-family continuity. \end{theorem} \begin{proof} The tensor test map \((f,g,f')\mapsto f\otimes g\otimes f'\) is jointly continuous into \(\Dtest_K(X)\). Lemma~\ref{lem:strongcontinuity} controls all compact coefficient seminorms. For the scalar-only output every metric trace coordinate is zero. For either metric-quadratic output, its derivative trace is \[ \Charge(G'_h(\psi))=(\GA a_\psi)|_\Sigma. \] Lemma~\ref{lem:sourcecontinuity} and the smooth Green-operator mapping property give continuity of this coordinate, including its smooth trace seminorms. The same proof applies after one fixed directional derivative. Later metric derivatives vanish by the finite-degree reduction. Thus every topology coordinate required by \(\taus\) is controlled. A distribution has a finite-order estimate on \(K\), so \[ |\Cal(f,g,f')| \le C\,p_{K,N}(f\otimes g\otimes f') \] and the right-hand side is bounded by a product of finitely many seminorms of the three tests. For order zero, the inherited test maps and separately continuous classical multiplication give separately continuous multilinear maps on the named Fr\'echet test spaces. The standard Fr\'echet-space uniform boundedness argument makes these finite multilinear maps jointly continuous. This is still a test-family conclusion. \end{proof} \paragraph{F9b is not claimed.} The preceding theorem does not prove continuity of \[ L_j:\QLspace\times\Bspace\longrightarrow\Padm,\qquad R_j:\Bspace\times\QRspace\longrightarrow\Padm \] in their actual-stage subspace topologies. Algebraic isomorphisms \(q_L,q_R\) need not have continuous inverses in those topologies. No quotient or topological-isomorphism identification has been established. The continuity claim discharged here is F9a on the named fixed-support test domains. F9b is a separate, unclaimed strengthening. Neither assertion gives quantum hypocontinuity on all of \(\Padm\). %===================================================================== \section{Release table, retained boundaries, and final gates} \label{sec:release} %===================================================================== \begin{theorem}[IX-Free partial construction and release] \label{thm:release} Relative to the explicit inherited premises, this manuscript constructs: \begin{enumerate}[label=\textup{(\roman*)}] \item concrete relation kernels for all four presentation maps; \item injective degree-two presentation maps and the evaluated scalar-descent set \(\mathcal C_{\mathrm{desc}}=\R\); \item all three canonical two-edge distributions and their admitted compact polynomial outputs; \item the left and right second-stage coefficient maps on the degree-two intermediate blocks; \item reversal and presentation independence on those blocks; \item fixed-support test-family continuity for the completed coefficient families. \end{enumerate} The earliest incomplete required gate is F5 on the degree-four intermediate blocks. F6 overlap agreement is also incomplete. Neither complete sharp association is constructed, and no contradiction excluding every construction satisfying Requirement~\ref{req:interfaces} is proved. The release is \emph{Unresolved}. \end{theorem} \begin{proof} The new results are Proposition~\ref{prop:relations}, Proposition~\ref{prop:prelations}, Theorem~\ref{thm:qinjective}, Corollary~\ref{cor:cdesc}, Proposition~\ref{prop:existence}, Theorem~\ref{thm:treeadmission}, Theorem~\ref{thm:degree2maps}, Proposition~\ref{prop:reversal}, Proposition~\ref{prop:presentation}, and Theorem~\ref{thm:F9a}. Proposition~\ref{prop:remainingdescent} states exact tests without proving their hypotheses. The overlap conditions are likewise not discharged. Therefore the conditional coherence proposition cannot release either full association. No violated relation or universal interface contradiction has been exhibited. \end{proof} \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth} {@{}p{.31\linewidth}p{.16\linewidth}p{.15\linewidth}X@{}} \toprule Object & Provenance & Status & Scope or missing lemma\\ \midrule \(B_f\star_1W_g\) & IX-C, inherited & Constructed & Named first-stage domain\\ \(W_g\star_1B_{f'}\) & IX-C, inherited & Constructed & Named reverse domain\\ Three one-edge outputs & IX-C, inherited & Constructed & Existence and admission\\ \(\mathcal C_{\mathrm{desc}}\) & IX-Free F2 & Constructed & Evaluated as \(\R\); algebraic test only\\ \(\Graph_{12,23}\) & IX-Free F3--F4 & Constructed & Compact scalar kernel; zero metric derivatives\\ \(\Graph_{12,13}\) & IX-Free F3--F4 & Constructed & Metric derivative \(m+q_a\) and descendants\\ \(\Graph_{13,23}\) & IX-Free F3--F4 & Constructed & Metric derivative \(m+q_a\) and descendants\\ Left scalar map & IX-Free F2, F5 & Constructed & Algebraic admitted map; overlap release pending\\ Right scalar map & IX-Free F2, F5 & Constructed & Algebraic admitted map; overlap release pending\\ Left nonconstant degree-two maps & IX-Free F5 & Constructed & Actual-functional block maps\\ Right nonconstant degree-two maps & IX-Free F5 & Constructed & Actual-functional block maps\\ Degree-four coefficient maps & IX-Free F5 & Unresolved & Annihilation of \(\kerop p_L,\kerop p_R\)\\ Common-pair and old-product gluing & IX-Free F6 & Unresolved & Agreement on intersections and sum intersections\\ \((B_f\star_1W_g)\star B_{f'}\) & IX-Free F5--F7 & Unresolved & Degree-four descent and overlap agreement\\ \(B_f\star(W_g\star_1B_{f'})\) & IX-Free F5--F7 & Unresolved & Degree-four descent and overlap agreement\\ Full-sector associativity & IX-Free F7 & Unresolved & Conditional proposition only\\ Full-sector involution & IX-Free F8 & Unresolved & Degree-four descent and compatible gluing\\ F9a, completed families & IX-Free F9a & Constructed & Named fixed-support test domains\\ F9b, actual-stage continuity & IX-Free F9b & Unresolved & Unclaimed strengthening; no inverse estimates\\ Normal-topology algebra under (E1)--(E4) & VII-C boundary, inherited & Proved obstructed & Only under the original stated hypotheses\\ IX-D and G2 & Outside this sector & Unresolved & Not begun or opened\\ \bottomrule \end{tabularx} \end{center} \subsection{Retained obstruction boundaries} The supplied IX-C records the VII-C obstruction under (E1)--(E4): preservation of the old Wick algebra, exact sharp admission, normal-topology sequential admission continuity, and hypocontinuous multiplication with continuous coefficient extraction. That conditional obstruction is retained, not independently reproved here. Likewise, the reported separated witness excludes removal of the selected logarithm by observation-diagonal counterterms. The VII-D numerical subtraction does not restore unchanged regular-product convergence or invalidate the original obstruction hypotheses. No positive-cutoff old product has been subtracted or redefined in this manuscript. No claim contradicts \(B_{f,\eps}\not\to B_f\) in \(\taus\). Numerical calibration and bulk-relative value preservation are not normal-topology admission continuity. The external \(R_{1,1}\), oscillator coefficient, and Newton--wave cubic identified in V are different objects. No theorem from V enters a proof here. No retarded, interacting, switched, gravitational, or all-graph construction is begun. \subsection{What is now required} The next construction step is no longer scalar relation evaluation or two-edge admission. It is: \begin{enumerate} \item prove that the sums \(\lambda^L_1,\lambda^R_1\) and the second coefficients \(\lambda^L_2,\lambda^R_2\) annihilate the concrete degree-four relation kernels; \item determine the actual-pair and old-domain intersections; \item prove coefficient agreement on those intersections, including intersections with already glued sums; \item invoke the four-order coherence and reversal results only after those maps exist; \item prove F9b separately only if actual-stage continuity is claimed. \end{enumerate} An explicit relation with zero actual presentation and nonzero prescribed value would obstruct that prescription. To obtain the \emph{Proved obstructed} release for the copied contract, every permitted alternative would have to be excluded. A failure of this canonical graph candidate alone would not suffice. \scopebox{Final gate statement}{% The scalar gate is evaluated: every real parent parameter passes algebraic scalar descent. All three canonical two-edge outputs exist and are admitted. The degree-two intermediate coefficient maps are actual-functional maps. The degree-four descent and overlap gates remain unresolved. Accordingly, neither complete sharp association is released, and the ternary calibration is not advertised as a completed binary product. This result does not yet close binary coherence for the declared one-word sector. Even a later successful closure of that sector would not establish the complete free-product domain required by an unspecified IX-D construction. IX-D's causal-support layer and the separate G2 gate are not begun.} %===================================================================== \appendix \section{Sign, order, and verification audit} %===================================================================== \subsection{Signs and multiplicities} Each first-stage partial coefficient has sign and multiplicity \(-2\). The complete word has two insertion signs, whose product is \(+1\). The middle-slot assignment factor is \(2\), not \(4\); the partial factor \(2\) has already counted the choice of the first middle slot. The selected order ledger is \[ \begin{array}{c|cccc} \hbar\text{-order}&0&1&2&3\\ \hline \text{residual field degree}&6&4&2&0. \end{array} \] Observation differentiation does not change contraction order. It does not introduce independent descendant calibration constants. \subsection{Analytical and algebraic distinctions} The following implications are not used: \[ \text{numerical finite part} \ \not\Rightarrow\ \text{admitted polynomial output}, \] \[ \text{admitted labelled graph} \ \not\Rightarrow\ \text{actual-functional binary map}, \] \[ \text{algebraic descent plus F9a} \ \not\Rightarrow\ \text{F9b}, \] \[ \text{failed sufficient criterion} \ \not\Rightarrow\ \text{nonexistence}. \] The proofs above address each completed implication through its separate kernel, admission, descent, or topological argument. \subsection{Verification disclosure} The new calculations are the identity \(\Box_xK=\partial_t\Delta_0^+\), full support, compact primitive reduction, graph-specific leaf checks, quadratic admission decompositions, and the completed-block descent and continuity proofs. The triangle ultraviolet estimate is an inherited premise. The full companion proofs are not independently certified. The degree-four descent and overlap tests are not claimed as verified. The source is designed for two-pass pdfLaTeX compilation. No compilation has been executed in this interface. No successful build, page count, proof-assistant check, or independent review is claimed. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Nature of the result.} This is a partial construction with an explicit unresolved full-factorization release. It is not a completed quantum algebra or a universal no-go theorem. \paragraph{Source boundary.} The supplied IX-C source reports the IX-A, IX-B, VII-C, and VII-D premises used here. Their full proofs were not provided as part of the present source text. The mathematical dependence is therefore stated explicitly rather than represented as independent verification. \paragraph{Continuity boundary.} F9a is proved on named fixed-support test domains. F9b and global quantum hypocontinuity are not asserted. \paragraph{Quantum release boundary.} No complete sharp second multiplication, global free-product domain, causal-support realization, interacting construction, or G2-compatible switched product is claimed. %===================================================================== \begin{thebibliography}{9} \addcontentsline{toc}{section}{References} \bibitem{IXC} \emph{Landmark IX-C: Actual-functional factorization and coefficientwise coherence for one sharp ordered word}. Supplied source, \texttt{IX-C.tex}. Used for the explicitly reported carrier, calibration, first-stage, one-edge, descent, and obstruction-boundary statements. \bibitem{IXA} N.~Cheung. \emph{Landmark IX-A: An off-shell admission domain and boundary-sensitive topology for sharp Cauchy relational insertions}. Companion identified in IX-C as \texttt{IX-A.tex}. Statements used here are reported through IX-C and restated in Assumption~\ref{ass:carrier}; the companion proof is not independently certified. \bibitem{IXB} \emph{Landmark IX-B: Derivative-stable triangle calibration and counterterm constraints for sharp Cauchy insertions}. Companion identified in IX-C as \texttt{IX-B-v2.tex}. Statements used here are reported through IX-C and restated in Assumption~\ref{ass:calibration}. \bibitem{VIID} \emph{Finite part of the regulated Cauchy-reflection triangle: A strong numerical distributional limit for the VII-C family}. Companion identified in IX-C as \texttt{VII-D.tex}. Used only through the explicitly restated analytical premise. \bibitem{VIIC} \emph{Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions}. Companion identified in IX-C as \texttt{VII-C.tex}. Its conditional obstruction boundary is retained as reported in the supplied source. \bibitem{Hormander} L.~H\"ormander. \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}. Springer. Standard distribution kernel and wavefront product, pullback, and pushforward calculus. \bibitem{Demanet} L.~Demanet. \emph{Waves and Imaging}. MIT course notes for 18.325, draft December 20, 2016. Chapter 8, especially Section 8.1, on operations on wavefront sets. \bibitem{Melrose} R.~B.~Melrose. \emph{Differential Analysis}. MIT lecture notes for 18.155 and 18.156. Background on distributions, kernels, and wavefront sets. \end{thebibliography} \end{document}