% !TEX program = pdflatex %===================================================================== % IX-B.tex % % Standalone source for pdfLaTeX; no external bibliography is required. % Compile twice: % pdflatex -interaction=nonstopmode -halt-on-error IX-B.tex % pdflatex -interaction=nonstopmode -halt-on-error IX-B.tex % % Source boundary: % VII-C: fixed covariances and normal-topology coefficient obstruction. % VII-D: quantified numerical finite part and complete parent conversion. % IX-A: actual-functional carrier P_Sigma and topology tau_Sigma. % % IX-B remains a calibration paper. No sharp binary product is defined. % Compilation has not been executed in this interface. %===================================================================== \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} \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{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{\ev}{\operatorname{ev}_0} \newcommand{\Rel}{\mathrm{Rel}} \newcommand{\Dtest}{\mathcal D} \newcommand{\Eprime}{\mathcal E'} \newcommand{\Ezero}{\mathsf E_0} \newcommand{\GA}{G_{\mathrm A}^{0}} \newcommand{\Padm}{\mathscr P_\Sigma} \newcommand{\Pdist}{\mathscr P_{\mathrm{dist}}} \newcommand{\Pmc}{\mathscr P_{\mathrm{mc}}} \newcommand{\Preg}{\mathscr P_{\mathrm{reg}}} \newcommand{\Slin}{\mathscr S_1} \newcommand{\Mlin}{\mathscr M_1} \newcommand{\Qlin}{\mathscr Q_\Sigma} \newcommand{\taus}{\tau_\Sigma} \newcommand{\Ddir}{\mathrm D} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\ord}{\operatorname{ord}} \newcommand{\eps}{\varepsilon} \newcommand{\diag}{\operatorname{diag}} \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-B}\\[0.4em] Derivative-stable triangle calibration and counterterm constraints\\ for sharp Cauchy insertions} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We calibrate a selected sharp-insertion triangle relative to the actual-functional carrier \(\mathscr P_\Sigma\) of IX-A. We classify the eight connected contraction skeletons through three edges and identify the unique fully contracted graph for two sharp insertions and one quadratic metric observable. Using the quantified VII-D finite-part theorem, we prove stability under fixed observation differential operators and smooth spectator multipliers, with test-order loss \(64\to64+d\). We fix the sign and multiplicity of the two-factor output already admitted by IX-A, retain the observation-diagonal obstruction, and prove that the observation-derivative orbit of the residue is linearly independent. The infinite distribution orbit is compatible with the single parent conversion parameter of VII-D. Both regularized parenthesizations have the same calibrated numerical limit, but the normal-topology obstruction of VII-C remains unchanged. The manuscript does not factor the numerical finite part \(u_\mu^\rho\) through sharp binary multiplications. \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) {IX-A carrier}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Two-factor contraction}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Finished binary product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {An inherited boundary is not a completed product.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Inherited boundaries are not a finished product. 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-B} \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) {IX-A carrier}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Finished binary product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {An inherited boundary is not a completed product.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Inherited boundaries are not a finished product. 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-B} \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) {IX-A carrier}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Two-factor contraction}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Finished binary product}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {An inherited boundary is not a completed product.}; \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-B} \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) {IX-A carrier}; \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) {Finished binary product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Two-factor contraction. An inherited boundary is not a completed product.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{IX-A carrier is not Finished binary product. 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-B} \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: IX-A carrier}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Two-factor contraction}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Finished binary product}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {An inherited boundary is not a completed product.}; \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-B} \end{figure} Figure~\ref{fig:split-IX-B} records one view of the same separation. Figure~\ref{fig:chain-IX-B} records one view of the same separation. Figure~\ref{fig:pair-IX-B} records one view of the same separation. Figure~\ref{fig:gate-IX-B} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-IX-B} records that separation. The drawing adds no hypothesis. \scopebox{Four inherited boundaries}{% \begin{enumerate}[label=\textup{(\roman*)},topsep=0pt] \item IX-A supplies the carrier \(\Padm\) and topology \(\taus\). It admits \(B_f\) and the selected two-factor contraction with \(W_g\); it does not construct or assign the triple after either sharp association. \item VII-C supplies the logarithmic divergence, the separated-support witness, and the normal-topology obstruction under its extension requirements \textup{(E1)--(E4)}. \item VII-D supplies the strong numerical finite part, the parent conversion class \(\R r\), and the rate controlled by \(p_{K,64}\). This is a numerical prescription, not a quantum algebra construction. \item None of these sources supplies the second sharp multiplication. No such multiplication is defined here. \end{enumerate} } \paragraph{Contribution and scope.} The new arguments are the skeleton classification, the transfer of the numerical estimate to observation derivatives and smooth multipliers, the residue-orbit and weighted-class results, and the explicit normalization of the admitted two-factor coefficient. An unlabelled inventory is not an existence theorem for decorated distributional graphs. Likewise, the complete conversion theorem for the fixed parent triangle does not classify ambiguities of later decorations. \tableofcontents %===================================================================== \section{Fixed data and inherited inputs} \label{sec:data} %===================================================================== \subsection{Off-shell insertion and fixed covariance} Let \[ M=\R^{1+3},\qquad M_+=\{t>0\},\qquad X=M_+^3, \qquad \eta=\diag(-1,1,1,1), \] and write \(x=(t,\mathbf x)\), \(y=(s,\mathbf y)\), \(x'=(t',\mathbf x')\). Put \[ \Box=-\partial_t^2+\Delta_{\mathbf x},\qquad (Ch)^\mu=\partial_\nu h^{\mu\nu} -\frac12\partial^\mu(\eta^{\alpha\beta}h_{\alpha\beta}), \qquad L=\Ezero C. \] The inverse \(\Ezero\) has zero value and zero first time derivative on \(\Sigma_0=\{t=0\}\), and satisfies \(\Box\Ezero j=j\). All configurations \((h,\phi)\) are smooth and off shell. The insertion and middle observable are \begin{equation} B_f[h,\phi] =-\int f(x)(Lh)^\mu(x)\partial_\mu\phi(x)\dd^4x, \qquad W_g[h]=\int g(y)h_{00}(y)^2\dd^4y, \label{eq:B} \end{equation} where \(f,g\in C_c^\infty(M_+)\). We retain the VII-C Wick-symbol convention \begin{equation} F\star_HG =\sum_{j\geq0}\frac{\hbar^j}{j!} \pair{F^{(j)}}{H^{\otimes j}G^{(j)}}. \label{eq:star} \end{equation} The sum is finite for finite polynomials; self-contractions inside a Wick symbol are excluded. The fixed even blocks are \[ H_{hh\,\alpha\beta\rho\sigma} =P_{\alpha\beta\rho\sigma}\Delta_0^+,\qquad H_{\phi\phi}=\Delta_m^+,\qquad H_{h\phi}=H_{\phi h}=0, \] where \[ P_{\alpha\beta\rho\sigma} =\eta_{\alpha(\rho}\eta_{\sigma)\beta} -\frac12\eta_{\alpha\beta}\eta_{\rho\sigma}, \] and, for fixed \(m\geq0\), \[ \Delta_m^+(x-y) =\int_{\R^3}\frac{\dd^3q}{(2\pi)^3\,2\omega_m(q)} e^{-i\omega_m(q)(t-s)+iq\cdot(\mathbf x-\mathbf y)}, \qquad \omega_m(q)=\sqrt{|q|^2+m^2}. \] The following identities are inherited, not re-derived: \begin{equation} C_x^\mu H_{hh\,\rho\sigma} =\delta^\mu_{(\rho}\partial^x_{\sigma)}\Delta_0^+, \qquad H_{ww}=0,\qquad w=Lh. \label{eq:tensor} \end{equation} They are VII-C's tensor cancellation and IX-A's fixed tensor lemma \cite[\texttt{prop:tensor}]{VIIC} \cite[\texttt{lem:tensor}]{IXA}. Both sources retain the mixed reflected branch: \begin{equation} \begin{split} K(x,y):=H_{wh}^{0}{}_{00}(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)} \\ &-\frac{i}{8|p|^2}e^{i|p|(t+s)} \bigg]. \end{split} \label{eq:K} \end{equation} In particular, \(H_{ww}=0\) does not imply \(K=0\). \subsection{Regulator placement and numerical triangle} As in VII-D, the same fixed radial mollifier is used on both slots of each insertion, and the middle observable is unsmeared \cite[\texttt{sc:placement}, \texttt{prop:positive}]{VIID}. Specifically, \(\rho\in C_c^\infty(\R^4)\) is real, nonnegative, even, Euclidean radial, and normalized by \(\int\rho=1\). Set \[ \rho_\eps(z)=\eps^{-4}\rho(z/\eps),\qquad S_\eps=\rho_\eps*,\qquad B_{f,\eps}=B_f[S_\eps h,S_\eps\phi]. \] With the Fourier convention \(\widehat v(k)=\int e^{-ik\cdot z}v(z)\dd z\), define \[ \vartheta_\rho(u)=\widehat\rho(-u,u\mathbf n),\qquad M_\eps(\lambda) =\vartheta_\rho\!\left(\eps\sqrt{\lambda^2+m^2/2}\right), \qquad \mathbf n\in S^2. \] Radiality makes the profile independent of \(\mathbf n\). Let \(K_\eps\) be \eqref{eq:K} with the multiplier \(\vartheta_\rho(\eps|p|)\), and set \[ D_\eps(z)= \int_{\R^3}\frac{\dd^3q}{(2\pi)^3} \frac{\omega_m(q)}2M_\eps(|q|)^2 e^{-i\omega_m(q)z^0+iq\cdot\mathbf z}. \] VII-D identifies the exact numerical family as \begin{equation} T_\eps^\rho(x,y,x') =2K_\eps(x,y)\overline{K_\eps(x',y)}D_\eps(x-x'). \label{eq:T} \end{equation} It is smooth for every \(\eps>0\) \cite[\texttt{prop:positive}, \texttt{eq:Texact}]{VIID}. On real product tests, its pairing is the single VII-C coefficient \begin{equation} \mathcal G_\eps(f,g,f') := \ev[\hbar^3](B_{f,\eps}\star_H W_g\star_H B_{f',\eps}) =\pair{T_\eps^\rho}{f\otimes g\otimes f'}. \label{eq:G} \end{equation} The joint-test distribution is complex-linear; conjugation in \eqref{eq:T} conjugates the kernel, not the test. \subsection{Residue and its product-test normalization} Following the VII-D residue definition, let \[ Z=\R^3\times S^2\times(0,\infty)^3, \] \[ \Psi(\mathbf z,\mathbf n,t,s,t') =\bigl((t,\mathbf z+t\mathbf n), (s,\mathbf z-s\mathbf n), (t',\mathbf z+t'\mathbf n)\bigr), \] and \[ \dd\nu=\dd^3\mathbf z\,\dd\Omega(\mathbf n)\, \dd t\,\dd s\,\dd t', \qquad \int_{S^2}\dd\Omega=4\pi. \] The inherited positive order-zero distribution is \begin{equation} r=c_0\Psi_*\nu,\qquad c_0=\frac1{64(2\pi)^3},\qquad \Rel=\Psi(Z)=\supp r. \label{eq:r} \end{equation} These are VII-D's prescribed residue and support \cite[\texttt{def:residue}, \texttt{eq:rdef}]{VIID}. For the light-ray transform \[ A_f(y,\mathbf n) =\int_\R f(t,\mathbf y+(t+s)\mathbf n)\dd t, \] the inherited product-test identity reads \begin{equation} \pair r{f\otimes g\otimes f'} =c_0\int g(y)\dd^4y\int_{S^2} A_f(y,\mathbf n)A_{f'}(y,\mathbf n)\dd\Omega(\mathbf n) =\mathcal C(f,g,f'). \label{eq:C} \end{equation} Thus this pairing is exactly the VII-C light-ray coefficient, including every numerical factor \cite[\texttt{eq:residue}]{VIIC} \cite[\texttt{eq:rjoint}, \texttt{prop:ledger}]{VIID}. \paragraph{Notation.} The symbol \(\mu>0\) denotes the subtraction scale and \(\vartheta_\rho\) the mollifier profile, as in VII-D; VII-C used \(\mu\) for the profile. Here \(r\) always denotes the residue distribution, and \(\lambda\) denotes a radial variable when needed. The reflected relation \(\Rel\) is distinct from the observation equality diagonals \(\Delta_{ij}\). \subsection{Quantified analytical input} For compact \(K\subset X\), let \[ \Dtest_K(X)=\{\Phi\in C_c^\infty(X):\supp\Phi\subset K\}, \qquad p_{K,N}(\Phi)= \max_{|\alpha|\leq N}\sup_X|\partial^\alpha\Phi|. \] Strong convergence in \(\Dtest'(X)\) means uniform convergence on bounded subsets of \(\Dtest(X)\). \begin{assumption}[VII-D finite-part and parent-conversion input] \label{ass:VIID} We use the following statements of VII-D as analytical input \cite[\texttt{thm:finitepart}, \texttt{thm:conversion}]{VIID}. There is a distribution \(U_m\), independent of \(\rho\), such that \begin{equation} u_\mu^\rho=U_m+(\kappa_\rho+\log\mu)r, \qquad \kappa_\rho= \int_0^1\frac{\vartheta_\rho(v)^4-1}{v}\dd v +\int_1^\infty\frac{\vartheta_\rho(v)^4}{v}\dd v. \label{eq:parent} \end{equation} For every compact \(K\subset X\) and every \(\Phi\in\Dtest_K(X)\), \begin{equation} \left| \pair{ T_\eps^\rho-\log((\mu\eps)^{-1})r-u_\mu^\rho }{\Phi} \right| \leq C_{K,\rho,m}\eps(1+|\log\eps|)p_{K,64}(\Phi), \qquad 0<\eps\leq\tfrac12. \label{eq:input} \end{equation} The estimate includes compacts meeting \(\Rel\cap\Delta_{13}\). For two admissible regulators, \begin{equation} u_\mu^{\rho_a}-u_\mu^{\rho_b}=c_{ab}r,\qquad c_{ab}=\int_0^\infty \frac{\vartheta_{\rho_a}(v)^4-\vartheta_{\rho_b}(v)^4}{v}\dd v \in\R. \label{eq:parentconversion} \end{equation} This is the complete parent conversion: there is no extra normal derivative, nonconstant weight, or separate term on \(\Rel\cap\Delta_{13}\). The inherited identities also give \begin{equation} c_{ab}+c_{bc}=c_{ac},\qquad u_{\mu_2}^\rho-u_{\mu_1}^\rho =\log(\mu_2/\mu_1)r. \label{eq:parentscale} \end{equation} \end{assumption} The assumption restates a theorem of the supplied source rather than introducing a binary-product hypothesis. No radial expansion, profile-integral evaluation, or wavefront estimate is re-proved here. %===================================================================== % One-page dependency ledger. %===================================================================== \clearpage \section*{Dependency ledger} \label{sec:ledger} \addcontentsline{toc}{section}{Dependency ledger} \begingroup \small \setlength{\parskip}{3pt} \noindent The source labels below refer to the delivered manuscript versions. The last column gives the local point of use. Inherited existence, topology, and ultraviolet statements are not promoted to new product theorems. \begin{center} \renewcommand{\arraystretch}{1.15} \begin{tabularx}{\linewidth}{@{}p{.15\linewidth}X p{.22\linewidth}@{}} \toprule Source & Inherited statement & Local use\\ \midrule VII-C & Tensor cancellation gives \(H_{ww}=0\); the mixed covariance \(K\) retains its reflected branch (\texttt{prop:tensor}, \texttt{eq:K}). & \eqref{eq:tensor}, \eqref{eq:K}\\ & The logarithmic coefficient is the light-ray number \(\mathcal C\), positive for separated tests (\texttt{thm:log}, \texttt{lem:separated}). & \eqref{eq:C}; Theorem~\ref{thm:diagonal}\\ & Normal kernel convergence and divergent unchanged coefficients exclude the coefficient extension and the algebra under \textup{(E1)--(E4)} (\texttt{lem:normalconv}, \texttt{thm:coefficient}, \texttt{ass:extension}, \texttt{cor:noalgebra}). & Section~\ref{sec:continuity}\\ & Observation-diagonal subtraction is insufficient. Subtraction on old regular products changes those products; a sharp-only assignment does not restore continuity (Section ``Counterterms and the scope of the conclusion''). & Theorem~\ref{thm:diagonal}; Remark~\ref{rem:subtraction}\\ \midrule VII-D & The prescribed placement yields the smooth family \(T_\eps^\rho\) (\texttt{sc:placement}, \texttt{prop:positive}). & \eqref{eq:T}, \eqref{eq:G}\\ & The residue is \(c_0\Psi_*\nu\); its product pairing equals \(\mathcal C\). The local chart has no \(\delta(t-t')\) (\texttt{def:residue}, \texttt{prop:ledger}, \texttt{eq:rlocal}). & \eqref{eq:r}, \eqref{eq:C}, \eqref{eq:density}\\ & The strong finite part has order-\(64\) control, including the reflected-diagonal intersection; complete parent conversion is \(c_{ab}r\) (\texttt{thm:finitepart}, \texttt{thm:conversion}). & Assumption~\ref{ass:VIID}\\ & Parent adjoint symmetry, cocycle, and scale dependence are fixed; the VII-C obstruction is unchanged (Sections ``Adjoint symmetry'', ``Reference profile, dilation, and cocycle'', and ``Scope audit''). & \eqref{eq:parentscale}; Section~\ref{sec:continuity}\\ \midrule IX-A & The actual-functional carrier is defined by the metric-derivative admission test; \(\taus\) records coefficients and boundary jets (\texttt{def:Padm}, \texttt{def:topology}). & Section~\ref{sec:carrier}\\ & For nonzero real \(f\), unchanged regular approximants do not converge to \(B_f\) in \(\taus\) (\texttt{thm:IX1}, \texttt{cor:compatibility}). & Section~\ref{sec:continuity}\\ & Selected two-factor outputs belong to \(\Padm\), but no subsequent triple is assigned (\texttt{prop:knownoutputs}, item 3; Section ``Preservation is recorded, not extended by fiat''). & Proposition~\ref{prop:partial}\\ & A future compact polynomial output must pass the admission test for all configurations and directional descendants (Section ``Output reservation and the admission criterion''). & \eqref{eq:admission}; Proposition~\ref{prop:missing}\\ \bottomrule \end{tabularx} \end{center} \noindent \textbf{New versus inherited.} IX-B proves the combinatorics, the continuous-operation transfer, the independent derivative orbit, and the weighted-class warning. It fixes the two-factor sign and multiplicity. It does not re-prove the carrier construction, the parent finite part, or the normal-topology obstruction. \endgroup \clearpage %===================================================================== \section{The IX-A carrier and the output boundary} \label{sec:carrier} %===================================================================== For self-contained use of the carrier name, we recall its admission criterion, not its proof. Let \(\Pdist\) denote actual finite polynomial functionals on smooth configurations, with compact symmetric distributional coefficient kernels. Let \(\Mlin\) be the space of compact linear metric kernels with no nonzero nonspacelike wavefront covectors, and put \[ q_a=L^*a=C^*(\theta\,\GA a),\qquad \Qlin=\{q_a:a\in C_c^\infty(M_+;T^*M)\}, \qquad \Slin=\Mlin+\Qlin. \] Here \(\GA\) is the massless advanced inverse. IX-A defines \begin{equation} \Padm= \left\{F\in\Pdist: \bigl(\Ddir_{\boldsymbol\eta}F\bigr)'_h(\psi)\in\Slin \text{ for every smooth configuration }\psi \text{ and finite smooth direction list }\boldsymbol\eta \right\}. \label{eq:admission} \end{equation} The empty direction list is allowed \cite[\texttt{def:Padm}]{IXA}. The topology \(\taus\) is the initial locally convex topology for the strong compact-distribution coefficient inclusion and all boundary-jet maps \[ F\longmapsto \mathsf b\!\left( \bigl(\Ddir_{\boldsymbol\eta}F\bigr)'_h(\psi) \right), \qquad \mathsf b(m+q_a)=(\GA a)|_{\Sigma_0}. \] IX-A proves that this trace is intrinsic, that \(\taus\) is Hausdorff, and that fixed-direction differentiation is continuous \cite[\texttt{lem:charge}, \texttt{def:topology}, \texttt{prop:top-basic}]{IXA}. Its multiplication theorem concerns separately continuous \emph{classical pointwise} multiplication only. IX-A admits \(B_f\), \(W_g\), and the selected two-factor coefficients. This does not supply a quantum multiplication on all of \(\Padm\), nor an assignment of either sharp triple association. A future compact polynomial output must satisfy \eqref{eq:admission}; a numerical finite part alone is not a substitute for that test. %===================================================================== \section{Contraction skeletons through three edges} \label{sec:skeletons} %===================================================================== A contraction graph has one vertex per insertion and one edge per cross-contraction. Multiple edges are allowed, but self-loops are excluded by the Wick-symbol convention. Decorations specify field slots, covariance labels, and derivatives; no slot may be used twice. \begin{theorem}[Eight unlabelled connected skeletons] \label{thm:skeletons} Every nontrivial connected loopless contraction multigraph with at most three edges has one of the following eight skeletons: \begin{enumerate} \item one edge; \item two parallel edges; \item a two-edge path; \item three parallel edges; \item a two-edge path with one edge doubled; \item a triangle; \item a three-edge path; \item a three-edge star. \end{enumerate} Every graph with at most three edges is a disjoint union of these, with total edge count at most three, together with isolated vertices. \end{theorem} \begin{proof} With one edge there is one connected graph. With two edges there are either two vertices, giving parallel edges, or three vertices, giving a path. A connected three-edge graph has at most four vertices. On two vertices it has three parallel edges. On three vertices it either joins all three pairs, giving a triangle, or joins two pairs with multiplicities two and one. On four vertices it is a tree, hence either a path or a star. The final statement follows by decomposition into connected components. \end{proof} \begin{proposition}[Unique fully contracted selected triangle] \label{prop:unique} The fully contracted order-\(\hbar^3\) graph for \[ B_{f,\eps}\star_H W_g\star_H B_{f',\eps} \] has one scalar edge between the first and third vertices and one metric edge from each sharp vertex to the middle vertex. The two assignments to the middle slots give the factor \(2\) in \eqref{eq:T}. No \(ww\)-edge occurs. \end{proposition} \begin{proof} Each sharp vertex has one scalar and one metric slot, and the middle vertex has two metric slots. Since mixed scalar--metric covariances vanish, the scalar slots must contract with each other. The remaining sharp metric slots must contract with the two middle metric slots, in two possible assignments. An edge joining the two sharp metric slots would leave the middle slots without partners, and its covariance is also zero by \eqref{eq:tensor}. The inherited tensor identity selects vector index \(0\) at both sharp vertices. Thus the scalar edge gives \(D_\eps\), and the mixed edges give the two \(K_\eps\) factors. The minus sign in the first \(B_f\) and the minus sign in the third \(B_{f'}\) multiply to \(+1\), as on the VII-D convention sheet \cite[\texttt{sc:spectral}, \texttt{prop:positive}]{VIID}. \end{proof} There are three nonzero one-edge placements in this word: the scalar first--third edge and the two sharp--middle metric edges. Choosing two of them gives three two-edge placements. These are placement statements, not sharp output-composition theorems. %===================================================================== \section{Residue geometry and the diagonal limitation} \label{sec:geometry} %===================================================================== Each observation equality diagonal \(\Delta_{ij}\subset X\) has dimension eight and codimension four; the total diagonal has dimension four. The inherited embedding \(\Psi\) parametrizes an eight-dimensional relation \(\Rel\). On it, \[ x-x'=(t-t')(1,\mathbf n). \] Consequently \(\Rel\cap\Delta_{13}=\Psi\{t=t'\}\) has dimension seven and codimension five in \(X\). Moreover, \[ \Rel\cap\Delta_{12}=\Rel\cap\Delta_{23}=\varnothing. \] These are the VII-D observation relations \cite[\texttt{eq:chartrelations}; Section ``Observation coverage' ]{VIID}. The initial Cauchy surface is an auxiliary surface, not an observation face inside \(X\). \paragraph{Recalled local density.} For the local transverse-degree calculation below, recall the VII-D chart \[ a=t+s,\qquad b=t'+s,\qquad \mathbf n=\frac{\mathbf x-\mathbf y}{|\mathbf x-\mathbf y|}, \qquad \sigma=|\mathbf x-\mathbf y|-a, \] \[ \mathbf w=\mathbf x'-\mathbf y-\frac ba(\mathbf x-\mathbf y), \qquad \mathbf z=\mathbf y+s\mathbf n. \] Its inherited density and residue representation are \begin{equation} \begin{gathered} \dd X=(a+\sigma)^2 \dd^3\mathbf z\,\dd\Omega(\mathbf n)\, \dd t\,\dd s\,\dd t'\,\dd\sigma\,\dd^3\mathbf w,\\ r=c_0a^{-2}\delta(\sigma)\delta^{(3)}(\mathbf w). \end{gathered} \label{eq:density} \end{equation} There is no \(\delta(t-t')\) \cite[\texttt{eq:density}, \texttt{eq:rlocal}]{VIID}. We do not re-prove this coordinate formula. Positivity and the separated witness use the parameter measure in \eqref{eq:r}, not this chart. \begin{theorem}[Inherited observation-diagonal limitation] \label{thm:diagonal} There are nonnegative \(f,g,f'\in C_c^\infty(M_+)\) with pairwise disjoint supports such that \[ \pair r{f\otimes g\otimes f'}=\mathcal C(f,g,f')>0. \] No family of counterterms supported on \(\Delta_{12}\cup\Delta_{23}\cup\Delta_{13}\) removes the logarithmic divergence on all joint tests. The same limitation holds for observation boundary faces avoided by these compact supports. \end{theorem} \begin{proof} Use the VII-C separated-support witness \cite[\texttt{lem:separated}]{VIIC}. Its centers are \[ y_0=(S,\mathbf0),\qquad x_0=(T,(T+S)\mathbf n_0),\qquad x'_0=(T',(T'+S)\mathbf n_0), \] with positive pairwise distinct \(S,T,T'\). The source supplies nonnegative bumps with pairwise disjoint supports and \(\mathcal C(f,g,f')>0\). Equation~\eqref{eq:C} identifies this number with the residue pairing. For any fixed joint test, Assumption~\ref{ass:VIID} gives \begin{equation} \begin{split} \pair{T_\eps^\rho}{\Phi} ={}&\log((\mu\eps)^{-1})\pair r\Phi +\pair{u_\mu^\rho}{\Phi}\\ &+O_{K,\rho,m}\!\left( \eps(1+|\log\eps|)p_{K,64}(\Phi)\right). \end{split} \label{eq:asymptotic} \end{equation} For the selected product test, the leading coefficient is therefore exactly \(\mathcal C(f,g,f')\). For fixed \(\mu\), the contribution \(-(\log\mu)\mathcal C\) is bounded, so this agrees with the VII-C form \(\mathcal C\log(\eps^{-1})+O(1)\). The joint support avoids all three diagonals. Any distribution supported on their union pairs to zero with this test, even if its coefficients depend on \(\eps\). Such a subtraction cannot remove the nonzero logarithm. This is the observation-diagonal limitation already recorded in VII-C's counterterm discussion. \end{proof} The theorem does not exclude counterterms supported on \(\Rel\), nor every possible sharp quantum algebra. %===================================================================== \section{Observation derivatives and smooth spectators} \label{sec:descendants} %===================================================================== Let \(D=\sum_{|\alpha|\leq d}b_\alpha\partial^\alpha\) be a fixed differential operator on \(X\) with smooth coefficients, and let \(a\in C^\infty(X)\). Its distributional action is defined by the formal transpose relative to the coordinate density: \[ \pair{DU}{\Phi}=\pair U{D^t\Phi},\qquad D^t\Phi=\sum_{|\alpha|\leq d}(-\partial)^\alpha(b_\alpha\Phi). \] No complex conjugation occurs in this transpose. The ordered operation is \[ \mathcal O_{a,D}U=a\,DU. \] \begin{theorem}[Derivative--spectator finite part] \label{thm:descendants} Assume~\ref{ass:VIID}. For every fixed \(a,D\), with \(d=\ord D\), \begin{equation} aDT_\eps^\rho -\log((\mu\eps)^{-1})aDr \longrightarrow aDu_\mu^\rho \quad\text{strongly in }\Dtest'(X). \label{eq:descendantlimit} \end{equation} More precisely, for \(\Phi\in\Dtest_K(X)\), \begin{equation} \begin{split} &\left| \pair{ aD(T_\eps^\rho-\log((\mu\eps)^{-1})r-u_\mu^\rho) }{\Phi} \right|\\ &\qquad\leq C_{K,a,D,\rho,m}\eps(1+|\log\eps|)p_{K,64+d}(\Phi). \end{split} \label{eq:descendantrate} \end{equation} The estimate is uniform when \(a\) and the coefficients \(b_\alpha\) range over sets bounded in the compact smooth seminorms through order \(64+d\). It includes compacts meeting \(\Rel\cap\Delta_{13}\). \end{theorem} \begin{proof} Write \(R_\eps=T_\eps^\rho-\log((\mu\eps)^{-1})r-u_\mu^\rho\). Then \[ \pair{aDR_\eps}{\Phi}=\pair{R_\eps}{D^t(a\Phi)}. \] Differential operators and multiplication preserve the test support. Leibniz's rule gives \[ p_{K,64}\bigl(D^t(a\Phi)\bigr) \leq C_{K,a,D}p_{K,64+d}(\Phi). \] All coefficient bounds used here involve at most \(64+d\) derivatives. Apply \eqref{eq:input}. Every bounded subset of \(\Dtest(X)\) has a common compact support and bounded smooth seminorms. Thus the estimate proves strong convergence and its asserted uniformity. The parent input already covers the reflected-diagonal intersection; the same Leibniz transfer covers it at order \(64+d\), without a separate intersection argument. \end{proof} \begin{corollary}[Descendant conversion and scale dependence] \label{cor:conversion} For the same operation, \[ aD(u_\mu^{\rho_a}-u_\mu^{\rho_b})=c_{ab}aDr,\qquad aD(u_{\mu_2}^{\rho}-u_{\mu_1}^{\rho}) =\log(\mu_2/\mu_1)aDr. \] The complete regulator difference satisfies \[ aD(T_\eps^{\rho_a}-T_\eps^{\rho_b}) \longrightarrow c_{ab}aDr \] strongly, with the corresponding order-\(64+d\) remainder bound. \end{corollary} \begin{proof} Apply \(aD\) to the inherited identities \eqref{eq:parentconversion}--\eqref{eq:parentscale}. For the last assertion, subtract the two estimates \eqref{eq:descendantrate}; the logarithmic terms agree. The parent cocycle is inherited from VII-D, not re-derived from profile integrals here. \end{proof} \subsection{Three derivative operations and three different claims} For product tests, \begin{equation} \pair{(-\partial_{x^0})^nT_\eps^\rho}{f\otimes g\otimes f'} = \pair{T_\eps^\rho}{(\partial_t^nf)\otimes g\otimes f'}. \label{eq:testderivative} \end{equation} This is the observation-test descendant calibration for \(B_{\partial_t^nf}\). \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.27\linewidth}X@{}} \toprule Operation & Established claim and boundary\\ \midrule Observation differentiation & Differentiation of the joint numerical distribution, equivalently transfer to its test. Theorem~\ref{thm:descendants} proves the finite-part estimate.\\ Configuration differentiation & Fixed smooth directional differentiation of an actual polynomial. IX-A proves closure and continuity on \(\Padm\) (\texttt{thm:closure}, \texttt{prop:top-basic}).\\ Functional derivative slots & An unsmeared functional derivative exposes or deletes field slots in subsequent contractions. The resulting decorated graph need not be an observation derivative of \(T_\eps^\rho\); no general extension theorem for these graphs is proved here.\\ \bottomrule \end{tabularx} \end{center} Directional derivatives are evaluations of functional derivatives on directions, but the three \emph{extension claims} above do not follow from one another. In particular, IX-A's directional closure does not place the observation residues \(r_n\) in a finite-dimensional distribution space. Conversely, the descendant finite part does not establish directional differentiation of an unconstructed sharp triple output. \subsection{Smooth spectator attachment} \begin{corollary}[Smooth spectator rule] \label{cor:spectators} Suppose a selected graph coefficient has the form \(a[h,\phi]\,DT_\eps^\rho\) after fixing smooth configurations, where \(a[h,\phi]\) is smooth on \(X\). If these multipliers depend continuously on the configurations in the compact smooth seminorms required by Theorem~\ref{thm:descendants}, its numerical finite part is \[ a[h,\phi]\,Du_\mu^\rho. \] Convergence is uniform on configuration sets for which those multiplier seminorms are bounded. \end{corollary} \begin{proof} Use Theorem~\ref{thm:descendants} with \(a=a[h,\phi]\) and its uniform coefficient bounds. \end{proof} This applies to fixed finite products of smooth field jets and to smooth Cauchy responses when the required bounds are supplied. It neither proves carrier membership of every resulting polynomial nor permits multiplication by an additional singular kernel. Disconnected factors sharing singular variables are not justified by calling them spectators. %===================================================================== \section{Parent conversion, derivative orbits, and weighted classes} \label{sec:counterterms} %===================================================================== The following distinctions prevent a confusion between the complete VII-D parent conversion and the new derivative-orbit theorem. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.30\linewidth}X@{}} \toprule Question & Answer\\ \midrule Fixed parent triangle, two regulators & The complete conversion is \(c_{ab}r\), with constant real weight. No additional normal or intersection term occurs. Inherited from VII-D.\\ Observation derivatives of the residue & For every \(n\geq1\), \(r_n=(-\partial_{x^0})^nr\) is not in \(\R r\). Proved below.\\ Unrestricted tangential weights & The weighted class is infinite-dimensional if admitted. Such weights are not admitted here and are not classified by the VII-D parent conversion theorem.\\ \bottomrule \end{tabularx} \end{center} \begin{theorem}[Infinite observation-derivative orbit] \label{thm:independence} The distributions \[ r_n=(-\partial_{x^0})^nr,\qquad n=0,1,2,\ldots, \] are linearly independent over \(\C\). No finite-dimensional subspace of \(\Dtest'(X)\) containing \(r\) is invariant under \(-\partial_{x^0}\). \end{theorem} \begin{proof} Let \[ F(x,y,x')=|\mathbf x-\mathbf y|^2-(x^0+y^0)^2. \] Then \(F=0\) on \(\Rel\), so \(Fr=0\), whereas \[ \partial_{x^0}F=-2(x^0+y^0) \] is nowhere zero on \(\Rel\). For an integer \(N\geq1\) and \(j0\), the fully contracted order-\(\hbar^3\) coefficients of \[ (B_{f,\eps}\star_H W_g)\star_H B_{f',\eps}, \qquad B_{f,\eps}\star_H(W_g\star_H B_{f',\eps}) \] agree and equal the single VII-C coefficient \[ \mathcal G_\eps(f,g,f') =\pair{T_\eps^\rho}{f\otimes g\otimes f'}. \] Their common logarithmic subtraction has the numerical limit \(\pair{u_\mu^\rho}{f\otimes g\otimes f'}\). On arbitrary joint tests the corresponding distributional convergence is strong. \end{theorem} \begin{proof} Introduce independent field variables at the three vertices. The cross-contraction operators \(\mathcal P_{ij}\) have field-independent coefficients and commute on the positive-cutoff old-domain expressions. Both parenthesizations therefore expand as \[ \exp\!\bigl(\hbar(\mathcal P_{12} +\mathcal P_{13}+\mathcal P_{23})\bigr) \] followed by identification of the field variables. The middle Wick symbol is an old bulk polynomial; the two insertion kernels are regular at positive cutoff, so the relevant old products are defined. Proposition~\ref{prop:unique} identifies the fully contracted graph and its multiplicity. Equations~\eqref{eq:T}--\eqref{eq:G} give the common coefficient, and Assumption~\ref{ass:VIID} gives the limit. Association does not split this one graph divergence into two different logarithms. \end{proof} \begin{remark}[Subtraction does not modify the old product by fiat] \label{rem:subtraction} VII-C's counterterm discussion distinguishes a numerical subtraction from an extension preserving the old Wick product. Subtracting from the positive-cutoff coefficients changes products already fixed in \(\Pmc\). Assigning a subtracted value only at the sharp kernel does not restore sequential continuity of the unsubtracted coefficient map \cite[\texttt{thm:coefficient}; Section ``Counterterms and the scope of the conclusion' ]{VIIC}. VII-D explicitly retains this obstruction \cite[Section ``Scope audit' ]{VIID}. Accordingly, the finite part here is not a counterterm inserted inside the unchanged old polynomial product. \end{remark} \subsection{The selected two-factor output in \(\Padm\)} \begin{proposition}[Sign and multiplicity of the admitted partial output] \label{prop:partial} The selected one-contraction coefficient for the ordered pair \((B_f,W_g)\) is \begin{equation} \mathcal Q_{f,g}[h,\phi] =-2\int f(x)g(y)K(x,y)\, \partial_t\phi(x)\,h_{00}(y)\dd^4x\dd^4y. \label{eq:Qpartial} \end{equation} It is an actual continuous bilinear functional with a compact distributional kernel and belongs to \(\Padm\). Its admission is supplied by IX-A; \eqref{eq:Qpartial} fixes the previously unspecified combinatorial coefficient and sign. \end{proposition} \begin{proof} IX-A's selected contraction-output proposition, item 3, establishes existence and admission of this two-factor coefficient \cite[\texttt{prop:knownoutputs}, item 3]{IXA}. It remains to identify the coefficient in that admitted expression. The metric derivative of \(W_g\) supplies the factor \(2\) from its two metric slots. The inherited identity \eqref{eq:tensor} selects the \(0\)-component of \(Lh\) when contracted with \(h_{00}\), giving \(K(x,y)\partial_t\phi(x)\). There is exactly one insertion \(B_f\) in this partial coefficient, so its single defining minus sign gives the sign \(-2\). This proves \eqref{eq:Qpartial} without re-proving admission. \end{proof} The formula is the admitted two-factor output, not an element produced by a newly defined quantum product on \(\Padm\). In the full triangle there are instead two insertion minus signs, one from \(B_f\) and one from \(B_{f'}\), and their product is \(+1\). Thus the partial sign \(-2\) and the positive assignment factor in VII-D are consistent. \subsection{Why the finite part is not a sharp binary factorization} \begin{proposition}[Missing second multiplication] \label{prop:missing} The admitted output \(\mathcal Q_{f,g}\in\Padm\), the equality of regularized associations, and the parent finite-part assignment do not determine either sharp composition \[ (\mathsf B(f)\star_\Sigma W_g)\star_\Sigma\mathsf B(f'), \qquad \mathsf B(f)\star_\Sigma(W_g\star_\Sigma\mathsf B(f')). \] Here \(\mathsf B(f)\) denotes the actual admitted \(B_f\), while \(\star_\Sigma\) is only a placeholder for a future product. Such a construction still requires: \begin{enumerate} \item binary quantum maps on specified stages of the IX-A carrier, including multiplication of partial outputs by sharp insertions; \item verification of the IX-A admission test for all resulting polynomial coefficients and directional descendants; \item graph-specific subtraction and normalization conditions compatible with the fixed old products; \item proof that those maps factor the calibration and satisfy the required coherence identities. \end{enumerate} \end{proposition} \begin{proof} IX-A supplies the carrier and selected partial outputs, not their second quantum multiplication. VII-D assigns a numerical distribution to a particular fully contracted three-vertex graph. Its arguments are joint observation tests, not arbitrary intermediate elements of \(\Padm\). Neither statement defines a binary map acting on \(\mathcal Q_{f,g}\) and \(B_{f'}\). A common subtracted limit of two existing regularized compositions specifies a ternary numerical coefficient. It does not factor that coefficient through the missing binary maps. Moreover, the constant fully contracted component is only one coefficient of a full product; it does not verify \eqref{eq:admission} for its nonconstant components. Therefore neither displayed sharp composition is determined. \end{proof} The carrier is no longer unspecified. The unresolved step is quantum multiplication on that carrier, not the naming of a space for the selected partial coefficient. %===================================================================== \section{Involution and the three continuity statements} \label{sec:continuity} %===================================================================== \subsection{Inherited parent involution and descendant exchange} Let \(\tau(x,y,x')=(x',y,x)\), and define \[ \pair{U^\dagger}{\Phi} =\overline{\pair U{\overline{\Phi\circ\tau}}}. \] VII-D proves the parent identities \begin{equation} (T_\eps^\rho)^\dagger=T_\eps^\rho,\qquad r^\dagger=r,\qquad (u_\mu^\rho)^\dagger=u_\mu^\rho. \label{eq:adjoint} \end{equation} We use these as inherited symmetry statements \cite[Section ``Adjoint symmetry' ]{VIID}. For a differential--multiplier operation \(\mathcal O\), define its exchanged-conjugate operation by \[ \mathcal O^\dagger U =\bigl(\mathcal O(U^\dagger)\bigr)^\dagger. \] Then, using \eqref{eq:adjoint}, \[ (\mathcal O u_\mu^\rho)^\dagger =\mathcal O^\dagger u_\mu^\rho. \] This is the descendant exchange rule. A first-observation derivative is exchanged with the corresponding third-observation derivative, with conjugation of smooth coefficients; it is not generally an individually self-adjoint operation. \subsection{Continuity reconciliation} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.24\linewidth}X@{}} \toprule Topology or operation & Statement and source\\ \midrule VII-C normal stages & The unchanged kernels \(b_{f,\eps}\) converge to \(b_f\), while the unchanged coefficient \(\mathcal G_\eps\) diverges for separated witness tests. Agreement with the old product and sequential continuity are incompatible.\\ IX-A admission topology \(\taus\) & For every nonzero real \(f\), \(B_{f,\eps}\not\to B_f\). The additional boundary-jet coordinates prevent the normal-stage admission convergence. This is a different topology, not a subtraction.\\ VII-D numerical subtraction & \(T_\eps^\rho-\log((\mu\eps)^{-1})r\to u_\mu^\rho\) strongly in \(\Dtest'(X)\). IX-B transfers this changed numerical operation to \(aD\); it does not infer a sharp product.\\ \bottomrule \end{tabularx} \end{center} The first row is VII-C's coefficient obstruction. Its algebra consequence assumes all of \textup{(E1)--(E4)}: retention of the old bulk Wick algebra, exact sharp admission, normal-topology sequential admission continuity agreeing on regular kernels, and bounded-set hypocontinuous multiplication with continuous field-independent coefficient extraction \cite[\texttt{ass:extension}, \texttt{cor:noalgebra}]{VIIC}. No relaxation of those requirements is silently introduced here. The second row is IX-A's explicit nonconvergence and compatibility boundary \cite[\texttt{thm:IX1}, \texttt{cor:compatibility}]{IXA}. It is consistent with the different-topology possibility that VII-C explicitly does not exclude. No claim is made that \(\taus\) is globally finer than every normal topology on every polynomial stage, or that it supports a hypocontinuous quantum product. The third row is Assumption~\ref{ass:VIID} and Theorem~\ref{thm:descendants}. As VII-D's scope audit states, this numerical subtraction leaves the VII-C obstruction unchanged. A limit in the strong topology of joint-test distributions is not admission continuity for a sharp binary algebra. %===================================================================== \section{Completed calibration sector and successor obligations} \label{sec:conclusion} %===================================================================== \begin{theorem}[Source-locked IX-B calibration package] \label{thm:package} Retain the exact off-shell insertion, fixed covariance, prescribed regulator placement, the IX-A carrier, and Assumption~\ref{ass:VIID}. Then: \begin{enumerate}[label=\textup{(\roman*)}] \item the eight unlabelled connected skeletons through three edges are exactly those in Theorem~\ref{thm:skeletons}; \item the selected fully contracted word has the unique triangle coefficient \eqref{eq:T}; \item its numerical finite part and complete conversion transfer to every fixed observation differential operator and smooth spectator multiplier, with order loss \(64\to64+d\); \item the inherited diagonal limitation is detected by the exact coefficient \(\pair r{f\otimes g\otimes f'}=\mathcal C(f,g,f')>0\); \item the residue derivatives \(r_n\) are linearly independent, while compatible conversions can retain the single parent parameter; \item the admitted two-factor output in \(\Padm\) is exactly \eqref{eq:Qpartial}; \item both regularized associations have the same coefficient \(\mathcal G_\eps\) and the same subtracted numerical limit. \end{enumerate} These results neither remove the VII-C obstruction nor construct either sharp second multiplication. \end{theorem} \begin{proof} The assertions follow respectively from Theorem~\ref{thm:skeletons}, Proposition~\ref{prop:unique}, Theorem~\ref{thm:descendants} and Corollary~\ref{cor:conversion}, Theorem~\ref{thm:diagonal}, Theorem~\ref{thm:independence} and the inherited parent conversion, Proposition~\ref{prop:partial}, and Theorem~\ref{thm:regularassociation}. Proposition~\ref{prop:missing} and Section~\ref{sec:continuity} establish the stated logical boundaries. \end{proof} \begin{center} \small \renewcommand{\arraystretch}{1.12} \begin{tabularx}{\linewidth}{@{}p{.44\linewidth}X@{}} \toprule Object & Status\\ \midrule IX-A carrier \(\Padm\) and topology \(\taus\) & Inherited; not reconstructed\\ Unlabelled skeletons through three edges & Complete\\ Parent reflected triangle & Inherited numerical calibration\\ Observation derivatives of the triangle & Proved with loss \(64\to64+d\)\\ Smooth spectator attachments & Proved under explicit smooth bounds\\ Two-factor coefficient in \(\Padm\) & Supplied by IX-A; coefficient fixed here\\ Second multiplication on that carrier & Not constructed\\ Observation-diagonal subtraction & Insufficient, as in VII-C\\ Finite-dimensional derivative-stable residue space & Excluded by linear independence\\ Parent compatible conversion freedom & Inherited one-dimensional constant-weight class\\ New independently decorated graphs & Not extended here\\ Singular spectator multiplication & Not established\\ Complete later-graph ambiguity catalogue & Not established\\ Two sharp binary parenthesizations & Not constructed\\ Associative sharp quantum algebra & Not claimed\\ Retarded, time-ordered, or switched realization & Outside scope\\ \bottomrule \end{tabularx} \end{center} \subsection{What a successor construction must supply} The carrier and its admission criterion are already specified by IX-A. A successor must work relative to that carrier and fix: \begin{enumerate} \item allowed vertex valences and independently differentiated decorations; \item bounded derivative stages or a precisely declared derivative module; \item the regulator placement for each new decorated graph; \item graph-specific extension conditions and restrictions on tangential weights, normal derivatives, and intersection terms; \item quantum maps on partial outputs and proof of output admission; \item compatibility with old products, the parent calibration, involution, presentation relations, and associativity. \end{enumerate} A binary factorization reproducing \(u_\mu^\rho+cr\) is a further construction, not a consequence of the calibration alone. \scopebox{Closed conclusion}{% IX-B supplies a source-locked derivative- and smooth-spectator-stable numerical triangle calibration. The carrier is \(\Padm\), the admitted partial coefficient is fixed by \eqref{eq:Qpartial}, and the second sharp multiplication remains undefined. The VII-D class \(\R r\) is the complete parent regulator-conversion class, not the distribution space of all descendants or all later decorations. The VII-C normal-topology obstruction remains in force. No ternary numerical limit is advertised as either sharp binary association.} %===================================================================== \appendix \section{Proof and verification boundary} %===================================================================== The analytic existence and rate of the parent finite part are the VII-D input, stated in Assumption~\ref{ass:VIID}. Its ultraviolet analysis, normalization ledger, profile integrals, and intersection estimates are not re-proved here. The carrier, intrinsic boundary traces, and nonconvergence theorem are IX-A inputs. The normal-topology obstruction and separated witness are VII-C inputs. The new analytical transfer uses only the exact identity \[ \pair{aDU}{\Phi}=\pair U{D^t(a\Phi)} \] and its finite-order Leibniz estimate. The residue-orbit proof uses the defining function \(F\) and the positive parameter measure \(c_0\Psi_*\nu\); it does not rely on a new wavefront or radial analysis. No all-graph ultraviolet estimate, quantum hypocontinuity theorem in \(\taus\), or complete coherence theorem is claimed. No second-order geometric insertion, retarded product, switched realization, or connection to a conical construction is introduced. \section{Order ledger} For the calibrated parent, \[ \begin{gathered} (\text{coupling order},\ \hbar\text{-order},\ \text{insertion count},\ \text{residual field degree})\\ =(\text{not assigned here},\ 3,\ 3,\ 0). \end{gathered} \] The field degree before full contraction is six. Observation differentiation preserves insertion count and contraction order. Smooth spectator attachment changes residual field degree but does not by itself add a contraction. A bound on \(\hbar\)-order does not bound derivative decorations, isolated spectator vertices, or word length. \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Nature of the result.} This is a calibration theorem conditional on the explicitly restated VII-D analytical input, with new combinatorial, continuous-operation, and residue-space consequences. The IX-A and VII-C boundaries are inherited and identified at their points of use. \paragraph{Verification status.} The source is designed for two-pass pdfLaTeX compilation and uses no external bibliography, figures, or code. Compilation has not been executed in this interface; no proof-assistant verification or independent review is claimed. \begin{thebibliography}{9} \addcontentsline{toc}{section}{References} \bibitem{VIIC} \emph{Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions}. Delivered manuscript, \texttt{VII-C.tex}. Relevant source labels: \texttt{prop:tensor}, \texttt{eq:K}, \texttt{eq:Gdef}, \texttt{eq:residue}, \texttt{thm:log}, \texttt{lem:separated}, \texttt{lem:normalconv}, \texttt{thm:coefficient}, \texttt{ass:extension}, and \texttt{cor:noalgebra}; also the subsection ``Counterterms and the scope of the conclusion''. \bibitem{VIID} \emph{Finite part of the regulated Cauchy-reflection triangle: A strong numerical distributional limit for the VII-C family}. Delivered manuscript, \texttt{VII-D.tex}. Relevant source labels: \texttt{sc:placement}, \texttt{sc:spectral}, \texttt{prop:positive}, \texttt{eq:Texact}, \texttt{def:residue}, \texttt{eq:rdef}, \texttt{eq:rjoint}, \texttt{prop:ledger}, \texttt{eq:chartrelations}, \texttt{eq:density}, \texttt{eq:rlocal}, \texttt{thm:finitepart}, and \texttt{thm:conversion}; also ``Scope audit'', ``Adjoint symmetry'', and ``Reference profile, dilation, and cocycle''. \bibitem{IXA} N.~Cheung. \emph{Landmark IX-A: An off-shell admission domain and boundary-sensitive topology for sharp Cauchy relational insertions}. Delivered manuscript, \texttt{IX-A.tex}. Relevant source labels: \texttt{def:Padm}, \texttt{lem:charge}, \texttt{def:topology}, \texttt{thm:closure}, \texttt{prop:top-basic}, \texttt{thm:IX1}, \texttt{cor:compatibility}, \texttt{lem:tensor}, and \texttt{prop:knownoutputs}, item 3; also ``Preservation is recorded, not extended by fiat'' and ``Output reservation and the admission criterion''. \end{thebibliography} \end{document}