% XVI-H.tex % % Single-file pdfLaTeX document. % Compile twice: % pdflatex XVI-H.tex % pdflatex XVI-H.tex % % No external bibliography, figures, or included files. % % Version-aware revision of the supplied XVI-H-v3.tex. % Uses the supplied revised XVI-E.tex, revised XVI-F.tex, % and current XVI-G.tex. % % Mathematical scope: % - real polarization of the stated diagonal completion; % - compact three-slot kernel representation, under explicit % inherited root/kernel assumptions; % - finite-order continuity on smooth configuration spaces; % - witness compatibility and partial contraction to j_lambda; % - a conditional audit of the inherited switch representative. % % Full-kernel wavefront/admission, free-shell comparison, % BRST/BV observability, and quantum-product construction % are not established. % % A successful compiler execution is not claimed. \documentclass[11pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathrsfs} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Landmark XVI-H: Compact polarized kernel and witness recovery}, pdfsubject={Kernel representation, finite-order continuity, witness contraction, and the admission boundary} } \setlength{\parindent}{0pt} \setlength{\parskip}{0.4em} \setlength{\emergencystretch}{2em} \setlist[itemize]{ leftmargin=*, itemsep=0.2em, topsep=0.3em } \setlist[enumerate]{ leftmargin=*, itemsep=0.25em, topsep=0.3em } \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\WF}{WF} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\Dprime}{\mathcal D'} \newcommand{\Eprime}{\mathcal E'} \newcommand{\mc}{\mathscr F_{\mathrm{mc,pol}}} \newcommand{\Isol}{\mathcal I_{\mathrm{sol}}} \newcommand{\Unresolved}{\textbf{Unresolved}} \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 } \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{assumption}[theorem]{Inherited input} \newtheorem{definition}[theorem]{Definition} \newtheorem{problem}[theorem]{Open obligation} \numberwithin{equation}{section} \title{Landmark XVI-H\\[0.3em] \large Compact polarized kernel and witness recovery\\ \normalsize Revised proof manuscript\\ Kernel representation and witness matching established;\\ free-shell admission and quantum programme unresolved} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We represent the real polarization of the stated single-metric prepared completion by a compact distributional kernel with two symmetric-tensor slots and one scalar slot. The construction is relative to explicit inherited source and compact-root inputs. It is implemented by transferring derivatives from diagonal source deltas onto general test functions and then pairing with the root kernel. This gives a distribution without invoking an unverified product of arbitrary distributions. The compact kernel yields joint continuity on the compact-set smooth topology, with finite but unspecified derivative orders. We identify the polarization conventions with those of revised XVI-E and XVI-F and import the current XVI-G witness theorem. The resulting complexified kernel, partially evaluated on the specified tensor witness pair, is exactly the compact smooth scalar density $j_\lambda$. This closes the former witness compatibility gap without identifying the scalar density with the full kernel or determining a free-shell class from one witness family. We also give a concise conditional audit of the inherited past-to-future switch representative. No wavefront set of the full kernel or its trisolution is calculated. No cubic admission, BRST/BV observability, or quantum-product construction is established. The regulator width is fixed throughout. \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) {Polarized compact kernel}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Witness compatibility}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Unresolved wavefront test}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The kernel theorems do not close the wavefront obligation.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Kernel recovery leaves the wavefront obligation 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-XVI-H} \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) {Polarized compact kernel}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Unresolved wavefront test}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The kernel theorems do not close the wavefront obligation.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{Kernel recovery leaves the wavefront obligation 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-XVI-H} \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) {Polarized compact kernel}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Witness compatibility}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Unresolved wavefront test}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The kernel theorems do not close the wavefront obligation.}; \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-XVI-H} \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) {Polarized compact kernel}; \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) {Unresolved wavefront test}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Witness compatibility. The kernel theorems do not close the wavefront obligation.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Polarized compact kernel is not Unresolved wavefront test. 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-XVI-H} \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: Polarized compact kernel}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Witness compatibility}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Unresolved wavefront test}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The kernel theorems do not close the wavefront obligation.}; \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-XVI-H} \end{figure} Figure~\ref{fig:split-XVI-H} records one view of the same separation. Figure~\ref{fig:chain-XVI-H} records one view of the same separation. Figure~\ref{fig:pair-XVI-H} records one view of the same separation. Figure~\ref{fig:gate-XVI-H} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XVI-H} records that separation. The drawing adds no hypothesis. \scopebox{Scope and release status.}{% \textbf{Established under the stated inherited inputs:} the real polarized completion, its compact three-slot kernel, finite-order continuity, the witness compatibility chain, and partial contraction of that kernel to $j_\lambda$. \textbf{Not established:} the full-kernel wavefront/admission condition, comparison with the frozen XI-G free-shell class, BRST/BV observability, or a prepared quantum product. The sharp-width limit, ultraviolet completion, and all-orders finite-parameter renormalizability are outside the scope of this note, not open problems assigned to it. The admission and quantum programme remains \Unresolved. } \section{Provenance and proof dependencies} \label{sec:provenance} This manuscript revises the supplied \texttt{XVI-H-v3.tex}. The current witness references are: \begin{itemize} \item revised \texttt{XVI-E.tex}, ``Polarized root representation and witness-level completion compatibility''; \item revised \texttt{XVI-F.tex}, ``Auxiliary/root compatibility and direct witness completion''; \item current \texttt{XVI-G.tex}, ``Frequency-dependent witness testing and resonant asymptotics.'' \end{itemize} The supplied XVI-H-v3 reproduces additional XII-D source-kernel and root-kernel statements, and XI-G switch identities. The complete XII-D and XI-G manuscripts are not independently available here. Accordingly, those statements are formulated below as explicit inherited inputs. Their use is not represented as a full-file verification. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.25\linewidth}Y Y@{}} \toprule Item & Provenance & Role here\\ \midrule Prepared source coefficients & XII-D statements reproduced in XVI-H-v3 & Explicit input; degree audit proved below\\ \addlinespace Diagonal completion and response & XII-D interface reproduced in revised XVI-D/E/F & Definition and domain input\\ \addlinespace Compact root and its joint kernel & XII-D statements reproduced in XVI-H-v3 & Explicit kernel/support assumptions\\ \addlinespace Witness compatibility & Revised XVI-E/F and current XVI-G & Imported with matching conventions and hypotheses\\ \addlinespace Kernel construction and continuity & Present manuscript & Proved from the stated root/source inputs\\ \addlinespace Partial contraction to $j_\lambda$ & Present manuscript plus current XVI-G & Proved as a distributional equality\\ \addlinespace Switch representative & XI-G identities reproduced in XVI-H-v3 & Conditional consistency audit, not admission\\ \bottomrule \end{tabularx} \end{center} The current XVI-G cites an earlier XVI-H only as a record of subsequent obligations, not as a premise of its witness proofs. No result of the present kernel construction is used to prove the imported witness chain. Thus witness matching below does not depend circularly on this revision. \section{Setting and inherited interfaces} \label{sec:inputs} \subsection{Geometry and fixed data} Let \[ M=\mathbb R^4,\qquad M_T=[0,T]\times\mathbb R^3,\qquad M_T^\circ=(0,T)\times\mathbb R^3, \] with \[ \eta=\operatorname{diag}(1,-1,-1,-1), \qquad \Box=\partial_t^2-\Delta, \qquad P_m=\Box+m^2. \] Fix \[ 0<\varepsilon0. \] Write \[ K_f=\supp f,\qquad C_\delta=J^-(K_f)\cap\{t\ge-\delta\}, \] and \begin{equation} O_{f,g} =K_f\cup\bigl(\supp g\cap J^-(K_f)\bigr). \label{eq:Oset} \end{equation} \begin{assumption}[Compact root and joint root kernel] \label{input:root} There is a real linear compact smooth root $a_{f,g}[\phi]$, with the inherited collar and normalization, such that \begin{equation} \mathcal C_{3,\mathrm D}(f,g)[\gamma,\phi] = \int_Ma_{f,g}[\phi]_{\mu\nu}(y) \mathcal J_{\mathrm D}^{\mu\nu}[\gamma](y)\dd^4y \label{eq:root-identity} \end{equation} for the real smooth inputs considered here. In a fixed source-component frame, indexed by $A$, the root has a real distributional kernel $B_A(y,x)$ satisfying \begin{equation} \left\langle B_A,q^A\otimes\phi\right\rangle = \int_Mq^A(y)a_{f,g}[\phi]_A(y)\dd^4y. \label{eq:B-root} \end{equation} The joint kernel is compactly supported, with \begin{equation} \supp B\subset C_\delta\times O_{f,g}. \label{eq:B-support} \end{equation} The root-theorem domain includes the real cosine and sine witness inputs and their required sums, with the same $f,g,\sigma,m,\vartheta$ used in revised XVI-E/F and current XVI-G. \end{assumption} The compactness in \eqref{eq:B-support} is a joint root-kernel input. It is not inferred merely from compact support of the root's output for each scalar. The supplied XVI-H-v3 attributes these statements to XII-D, \texttt{lem:root-kernel}, \texttt{eq:B-kernel}, \texttt{thm:root}, and \texttt{eq:t-bounds} \cite{XVIHvThree,XIID}. All the kernel theorems below are relative to Inputs~\ref{input:source}--\ref{input:root}. They do not purport to re-prove those predecessor interfaces. \section{Degree audit and real polarization} \label{sec:degree} \subsection{Displayed switch-correction coefficients} To avoid confusing expansion coefficients with response fields or completion coefficients, write the quoted source coefficients temporarily as $\mathsf u_j$ and $\mathsf b_j$: \begin{equation} \begin{aligned} \mathsf u_0^{\mu\nu}&=\eta^{\mu\nu},& \mathsf u_1^{\mu\nu}&=-\gamma^{\mu\nu},& \mathsf u_2^{\mu\nu}&=\gamma^\mu{}_\ell\gamma^{\ell\nu},\\ \mathsf b_0&=1,& \mathsf b_1&=\frac12\tr_\eta\gamma,& \mathsf b_2&=\frac18(\tr_\eta\gamma)^2 -\frac14\gamma_{\mu\nu}\gamma^{\mu\nu}. \end{aligned} \label{eq:source-coefficients} \end{equation} The remaining quoted coefficients are \begin{equation} \begin{aligned} C_r^{\mu\nu\alpha\beta} &=\sum_{i+j+k=r}\mathsf b_i \bigl(\mathsf u_j^{\mu\nu}\mathsf u_k^{\alpha\beta} -\mathsf u_j^{\mu(\alpha}\mathsf u_k^{\beta)\nu}\bigr), \qquad r=0,1,2,\\ \Gamma_n{}^\rho{}_{\alpha\beta} &=\frac12\mathsf u_{n-1}^{\rho\ell} \bigl(\partial_\alpha\gamma_{\beta\ell} +\partial_\beta\gamma_{\alpha\ell} -\partial_\ell\gamma_{\alpha\beta}\bigr), \qquad n=1,2,\\ M_2^{\mu\nu\rho} &=C_0^{\mu\nu\alpha\beta} \Gamma_2{}^\rho{}_{\alpha\beta} +C_1^{\mu\nu\alpha\beta} \Gamma_1{}^\rho{}_{\alpha\beta}. \end{aligned} \label{eq:CM} \end{equation} These are the formulas reproduced in XVI-H-v3. Their homogeneity, rather than a new source derivation, is audited here. \begin{proposition}[Monomial audit] \label{prop:monomials} Every nonzero displayed monomial of $\mathcal D_\sigma$ has exactly two metric-jet factors. There is no constant or linear term. Consequently, for every real $s$, \begin{equation} \mathcal D_\sigma[s\gamma]=s^2\mathcal D_\sigma[\gamma]. \label{eq:Dscale} \end{equation} Using the separately imported bulk premise gives \begin{equation} J_3[s\gamma]=s^2J_3[\gamma], \qquad \mathcal J_{\mathrm D}[s\gamma] =s^2\mathcal J_{\mathrm D}[\gamma]. \label{eq:Jscale} \end{equation} The real source polarization \begin{equation} \mathcal J_{\mathrm D}(a,c) =\frac12\bigl( \mathcal J_{\mathrm D}[a+c] -\mathcal J_{\mathrm D}[a] -\mathcal J_{\mathrm D}[c]\bigr) \label{eq:Jpol} \end{equation} is symmetric and bilinear. \end{proposition} \begin{proof} A derivative of a metric factor still has field degree one. The coefficients $\mathsf u_j$ and $\mathsf b_j$ have field degree $j$. Each summand of $C_r$ has degree $r$, and $\Gamma_n$ has degree $n$. Thus $C_2$, $C_0\Gamma_2$, and $C_1\Gamma_1$ all have degree two. Multiplication by fixed derivatives of $\sigma$ does not change field degree. This proves \eqref{eq:Dscale}. The first statement in \eqref{eq:Jscale} is the imported bulk premise, not a curvature expansion performed here. Adding the two source contributions proves the second statement. More than homogeneity is used for bilinearity. Each component of the source is a finite sum of products $L(\gamma)M(\gamma)$ of linear differential monomials. For one such product, \[ \frac12\bigl(L(a+c)M(a+c)-L(a)M(a)-L(c)M(c)\bigr) =\frac12\bigl(L(a)M(c)+L(c)M(a)\bigr). \] This is symmetric and bilinear. Summing proves the assertion. Degree-two homogeneity of an arbitrary nonlinear map would not justify this step. \end{proof} The field degree and the total field-derivative order are different counts. The latter is at most two by Input~\ref{input:source}; it does not mean that a linear field term is present. \begin{proposition}[Response and completion homogeneity] \label{prop:Chom} Under Input~\ref{input:completion}, \begin{equation} u_{\mathrm D}[s\gamma]=s^2u_{\mathrm D}[\gamma], \qquad \mathcal C_{3,\mathrm D}(f,g)[s\gamma,\phi] =s^2\mathcal C_{3,\mathrm D}(f,g)[\gamma,\phi]. \label{eq:Cscale} \end{equation} The response polarization \begin{equation} B_{\mathrm{resp}}(a,c) =\frac12\bigl( u_{\mathrm D}[a+c]-u_{\mathrm D}[a]-u_{\mathrm D}[c]\bigr) \label{eq:Bresp} \end{equation} is symmetric and real bilinear. \end{proposition} \begin{proof} The two fields $u_{\mathrm D}[s\gamma]$ and $s^2u_{\mathrm D}[\gamma]$ satisfy the same linear equation and zero initial data by \eqref{eq:Jscale}. Smooth zero-data uniqueness proves their equality. Linearity of $\mathcal A_1$ in the response proves the second identity in \eqref{eq:Cscale}. The field $B_{\mathrm{resp}}(a,c)$ solves \[ LB_{\mathrm{resp}}(a,c)=\mathcal J_{\mathrm D}(a,c) \] with zero initial data. Each bilinearity defect solves the homogeneous zero-data equation, since the right-hand side is bilinear. Uniqueness makes every defect zero. \end{proof} \section{The polarized completion and witness compatibility} \label{sec:functional} \begin{definition}[Real polarized completion] \label{def:Opol} For real smooth $a,c,\phi$, define \begin{equation} \begin{aligned} \mathcal O_{\mathrm{pol}}(a,c;\phi) =\frac12\bigl( &\mathcal C_{3,\mathrm D}(f,g)[a+c,\phi]\\ &-\mathcal C_{3,\mathrm D}(f,g)[a,\phi] -\mathcal C_{3,\mathrm D}(f,g)[c,\phi]\bigr). \end{aligned} \label{eq:Opol} \end{equation} This is a polarization of three diagonal completion values, not a definition beginning with an arbitrary polarized response slot. \end{definition} \begin{theorem}[Polarization, complexification, and witness matching] \label{thm:Opol} Under Inputs~\ref{input:source}--\ref{input:root}: \begin{enumerate} \item $\mathcal O_{\mathrm{pol}}$ is symmetric in its tensor arguments. \item It is real linear in the scalar. \item The source, response, and diagonal completion have the degree-two homogeneity proved in Propositions~\ref{prop:monomials} and~\ref{prop:Chom}. \item $\mathcal O_{\mathrm{pol}}$ is real bilinear in its tensor arguments. \item Its unique complex-bilinear extension in the tensor arguments, with the scalar kept real, is \begin{equation} \begin{aligned} \mathcal O_{\mathbb C}(a+\ii b,c+\ii d;\phi) ={}&\mathcal O_{\mathrm{pol}}(a,c;\phi) -\mathcal O_{\mathrm{pol}}(b,d;\phi)\\ &+\ii\mathcal O_{\mathrm{pol}}(a,d;\phi) +\ii\mathcal O_{\mathrm{pol}}(b,c;\phi). \end{aligned} \label{eq:OC} \end{equation} \item With the same data and domain conventions as revised XVI-E/F and current XVI-G, the specified witness pair satisfies, for each $\lambda>0$ and real smooth scalar evaluated there, \begin{equation} \begin{aligned} \mathcal O_{\mathbb C} (\gamma_1^\lambda,\gamma_2^\lambda;\phi) &=T^{\mathbb C} (\gamma_1^\lambda,\gamma_2^\lambda;\phi)\\ &=Q_\lambda[\phi] =I_\lambda[\phi]\\ &=\mathcal A_1[f,g](R_\lambda;\phi). \end{aligned} \label{eq:witness-chain} \end{equation} \item On the common real response domain, \begin{equation} \mathcal O_{\mathrm{pol}}(a,c;\phi) =\mathcal A_1[f,g](B_{\mathrm{resp}}(a,c);\phi). \label{eq:O-A} \end{equation} The kernel construction in Section~\ref{sec:kernel} does not require this response-slot formula. \end{enumerate} \end{theorem} \begin{proof} Symmetry follows from the definition. Scalar linearity follows from the scalar linearity of the three diagonal values. Homogeneity was proved above. For bilinearity, Proposition~\ref{prop:Chom} gives the bilinear response polarization. Linearity of $\mathcal A_1$ on its stated common domain allows the finite response combination in \eqref{eq:Opol} to be taken inside $\mathcal A_1$, proving \eqref{eq:O-A} and real bilinearity. Alternatively, polarizing the root identity gives \begin{equation} \mathcal O_{\mathrm{pol}}(a,c;\phi) =\int_Ma_{f,g}[\phi]_A(y) \mathcal J_{\mathrm D}^{A}(a,c)(y)\dd^4y, \label{eq:O-root} \end{equation} which proves the same bilinearity from the root interface alone. Real bilinearity verifies \eqref{eq:OC}; complex bilinearity forces that formula in any extension, proving uniqueness. For item~6, the definition of $\mathcal O_{\mathrm{pol}}$ is exactly the definition of $T_D$ in revised XVI-E, \texttt{eq:TD}, and revised XVI-F, \texttt{eq:TD}, on their common real domain. The polarization factor, diagonal insertion, source convention, and observation data are the same. Their four-term complex extension is exactly \eqref{eq:OC}. Therefore $\mathcal O_{\mathbb C}=T^{\mathbb C}$ on that complexified domain. The remaining equalities are the witness compatibility theorem of revised XVI-F, \texttt{thm:compatibility}, consolidated in revised XVI-E, \texttt{thm:compatibility}, and stated with the pointwise scalar-family quantifier in current XVI-G, \texttt{prop:import} \cite{XVIErev,XVIFrev,XVIG}. Their operator and root-domain hypotheses are retained in the inputs of this note. \end{proof} \subsection{Witness notation and the exact quantifier} Fix $|\mathbf n|=1$ and use \[ k_1=(1,\mathbf n),\qquad k_2=(1,-\mathbf n),\qquad \gamma_a^\lambda=\eta e^{\ii\lambda k_a\cdot y}. \] The source and response are those of revised XVI-B and XVI-D: \[ S_\lambda^{\mu\nu} =\bigl(\lambda^2\sigma\mathsf q^{\mu\nu} +\lambda\mathfrak d_1^{\mu\nu}\bigr)e^{2\ii\lambda t}, \] \[ V_\lambda^{\mu\nu}(t) =\int_0^t(t-s)S_\lambda^{\mu\nu}(s)\dd s, \qquad R_\lambda=-\mathcal P V_\lambda =U_D[\gamma_1^\lambda,\gamma_2^\lambda]. \] Here \[ \mathsf q^{00}=-2,\qquad \mathsf q^{0i}=0,\qquad \mathsf q^{ij}=n^in^j-3\delta^{ij}, \] and \[ \mathfrak d_1^{\mu\nu} =-\ii\sigma' \bigl(\eta^{\mu\nu}+2\delta^\mu_0\delta^\nu_0\bigr). \] For reference, define the independently checked witness slot \[ \mathcal O_{\mathrm{wit},\lambda}(\phi) =\mathcal A_1[f,g](R_\lambda;\phi). \] Equation~\eqref{eq:witness-chain} identifies this scalar functional with the witness restriction of $\mathcal O_{\mathbb C}$. The equality holds for arbitrary assignments $\lambda\mapsto\phi_\lambda$ of real smooth scalars, at each frozen frequency. No regularity in $\lambda$ is needed, and no decay is implied by the equality alone. There is no need for another comparison of coordinate prescriptions to prove this witness equality: the definitions match, and the supplied compatibility theorem supplies the operator comparison on the witness slot. This closes the former open item in XVI-H-v3. The result does not: \begin{itemize} \item insert a pair as one metric argument of $\mathcal C_{3,\mathrm D}$; \item construct a metric $\gamma$ with $u_{\mathrm D}[\gamma]=R_\lambda$; \item establish a general extension of a retarded operator to arbitrary distributions. \end{itemize} The diagonal insertion and its polarization remain differently defined objects with different input structures. \section{Construction of the compact three-slot kernel} \label{sec:kernel} \subsection{Source normal form} Choose a global component frame for the input tensor bundle, with indices $I,J$. The local polynomial structure of the source yields a finite normal form for its polarized kernel: \begin{equation} j_{\mathrm D}^{A}{}_{IJ}(y;y_1,y_2) = \sum_{|\alpha|+|\beta|\le2} c^{A}_{IJ;\alpha\beta}(y) \partial_{y_1}^{\alpha}\partial_{y_2}^{\beta} \bigl(\delta(y_1-y)\delta(y_2-y)\bigr). \label{eq:j-delta} \end{equation} The coefficient tensors are smooth and ambient. The source identity is \begin{equation} \left\langle j_{\mathrm D}^{A}(y;\cdot,\cdot),a\otimes c \right\rangle =\mathcal J_{\mathrm D}^{A}(a,c)(y). \label{eq:j-id} \end{equation} This is the source-kernel structure attributed to XII-D in the supplied XVI-H-v3. It also follows abstractly from the finite local quadratic-polynomial premise: a monomial $q(y)(\partial^\alpha a_I)(\partial^\beta c_J)$ is represented by the corresponding differentiated deltas with coefficient $(-1)^{|\alpha|+|\beta|}q(y)$. The normal form may be symmetrized without changing \eqref{eq:j-id}, so we choose \begin{equation} c^{A}_{IJ;\alpha\beta} =c^{A}_{JI;\beta\alpha}. \label{eq:c-sym} \end{equation} Its coefficient functions can be chosen to vanish for $y^0\le0$, since every source contribution contains $\sigma$ or a positive-order derivative of $\sigma$. No coefficient values of the unexpanded $J_3$ are fabricated. \subsection{Definition on arbitrary test functions} Let $F_{IJ}(y_1,y_2,x)$ be an arbitrary compactly supported smooth test field of the appropriate component type on $M^3$. Define \begin{equation} \begin{aligned} \langle t_{\mathrm D},F\rangle := \sum_{\substack{A,I,J\\|\alpha|+|\beta|\le2}} \Big\langle B_A(y,x),\, &c^{A}_{IJ;\alpha\beta}(y) (-1)^{|\alpha|+|\beta|}\\ &\times \bigl(\partial_{y_1}^{\alpha} \partial_{y_2}^{\beta}F_{IJ}\bigr)(y,y,x) \Big\rangle. \end{aligned} \label{eq:t-test} \end{equation} The right-hand side pairs a distribution with a smooth function. It is well defined because $B$ is compactly supported. Any auxiliary cutoff equal to one near $\supp B$ gives the same value. For clarity, the corresponding distribution before integrating out $y$ can be defined on a general test $G(y,y_1,y_2,x)$ on $M^4$: \begin{equation} \begin{aligned} \langle \mathcal K,G\rangle := \sum_{\substack{A,I,J\\|\alpha|+|\beta|\le2}} \Big\langle B_A(y,x),\, &c^{A}_{IJ;\alpha\beta}(y) (-1)^{|\alpha|+|\beta|}\\ &\times \bigl(\partial_{y_1}^{\alpha} \partial_{y_2}^{\beta}G_{IJ}\bigr)(y,y,y,x) \Big\rangle. \end{aligned} \label{eq:lifted-test} \end{equation} Thus derivatives on the source deltas are transferred to the test before evaluation on the diagonal. They are not silently applied to a singular root kernel. With \[ \iota(y,x)=(y,y,y,x), \qquad \pi_y(y,y_1,y_2,x)=(y_1,y_2,x), \] the distribution $\mathcal K$ is a finite sum of derivatives in $y_1,y_2$ of the pushforwards $\iota_*(cB)$. Its support is compact, so $\pi_y$ is proper on that support. The notation \begin{equation} t_{\mathrm D} =\pi_{y*}\mathcal K =\pi_{y*}\bigl[B_{f,g}(y,x)\,j_{\mathrm D}(y;y_1,y_2)\bigr] \label{eq:t-kernel} \end{equation} is understood through \eqref{eq:lifted-test}. It is this particular diagonal construction, not a claim that arbitrary pairs of distributions may be multiplied. \begin{theorem}[Compact kernel representation] \label{thm:trep} Under Inputs~\ref{input:source}--\ref{input:root}, \eqref{eq:t-test} defines a real compactly supported distributional kernel on ambient $M^3$, symmetric under interchange of its tensor slots. For real smooth $a,c,\phi$, \begin{equation} \mathcal O_{\mathrm{pol}}(a,c;\phi) =\langle t_{\mathrm D},a\otimes c\otimes\phi\rangle. \label{eq:O-t} \end{equation} The pairing is meaningful without compact support of the configuration fields. The representation uses the root identity and does not require the response-slot formula \eqref{eq:O-A}. \end{theorem} \begin{proof} A derivative of a smooth test followed by evaluation at $(y_1,y_2)=(y,y)$ is smooth. For tests supported in a fixed compact set, each derivative seminorm of this evaluated function is bounded by a finite derivative seminorm of the original test, by the chain rule. Smooth multiplication by the fixed coefficients preserves these estimates. Continuity of $B$ therefore proves that \eqref{eq:t-test} is a distribution. The definition can only detect test germs on the image of $\supp(cB)$ under $(y,x)\mapsto(y,y,x)$. That image is compact by \eqref{eq:B-support}. Thus $t_{\mathrm D}$ is compactly supported. Formula \eqref{eq:c-sym} and exchange of $\alpha,\beta$ in the finite sum prove tensor-slot symmetry. For a decomposable test, \[ \bigl(\partial_{y_1}^{\alpha} \partial_{y_2}^{\beta} (a_I(y_1)c_J(y_2)\phi(x))\bigr)(y,y,x) =(\partial^\alpha a_I)(y) (\partial^\beta c_J)(y)\phi(x). \] Consequently \eqref{eq:j-id} and \eqref{eq:B-root} give \[ \begin{aligned} \langle t_{\mathrm D},a\otimes c\otimes\phi\rangle &=\langle B_A, \mathcal J_{\mathrm D}^{A}(a,c)\otimes\phi\rangle\\ &=\int_Ma_{f,g}[\phi]_A(y) \mathcal J_{\mathrm D}^{A}(a,c)(y)\dd^4y. \end{aligned} \] The last expression is \eqref{eq:O-root}. Compactness supplies a cutoff-independent pairing with the smooth, possibly noncompact fields. This justifies the same calculation for all inputs in the stated real smooth domain. \end{proof} \subsection{Ambient support and the initial surface} Let \[ C_0=J^-(K_f)\cap\{t\ge0\}. \] The coefficient support and \eqref{eq:B-support} imply \begin{equation} \supp t_{\mathrm D} \subset \{y_1=y_2\in C_0\}\times O_{f,g}. \label{eq:t-support} \end{equation} Both sets on the right are compact in ambient $M$. Indeed, if $K_f$ lies in $\{t\le b,\ |\mathbf x|\le R\}$, then \[ C_\delta\subset \{-\delta\le t\le b,\ |\mathbf x|\le R+b+\delta\}. \] The causal past of a compact set is closed in Minkowski space, so $C_\delta$, and hence $C_0$, is compact. The set $O_{f,g}$ is compact by its definition. The right side of \eqref{eq:t-support} is the image of $C_0\times O_{f,g}$ under $(y,x)\mapsto(y,y,x)$. The support may reach $t=0$. Ambient compactness does not require support bounded away from that surface. The following statements remain separate: \begin{itemize} \item $t_{\mathrm D}$ is defined on ambient $M^3$; \item its support is compact there; \item its regularity or singularity at $t=0$ has not been determined. \end{itemize} Flatness of the switch guarantees smooth coefficient functions; it is not a theorem removing boundary singularities of the resulting kernel. \section{Finite-order continuity on smooth configurations} \label{sec:topology} The real smooth tensor and scalar spaces carry their compact-set derivative Fr\'echet topologies. For a compact set $K\subset M$, define componentwise \[ p_{r,K}(u) =\max_{|\alpha|\le r}\sup_K|\partial^\alpha u|. \] The test space $C_c^\infty$ carries its usual LF topology, not the subspace topology inherited from $C^\infty$. \begin{proposition}[Joint seminorm estimate] \label{prop:estimate} There are compact sets $K_1,K_2,K_3$, finite orders $r_1,r_2,r_3$, and a constant $C$ such that \begin{equation} |\mathcal O_{\mathrm{pol}}(a,c;\phi)| \le C\,p_{r_1,K_1}(a)\, p_{r_2,K_2}(c)\, p_{r_3,K_3}(\phi). \label{eq:target} \end{equation} Thus $\mathcal O_{\mathrm{pol}}$ is jointly continuous on the chosen smooth configuration spaces. The orders are finite but unspecified. One may take $r_1=r_2=r_3=r$ for a finite distributional order bound $r$ of $t_{\mathrm D}$. If $N$ is a finite order bound for $B$ on a suitable compact neighborhood, the construction provides an order bound at most $N+2$ for $t_{\mathrm D}$. No numerical or optimal value of $N$ or $r$ is asserted. The constant may depend on the fixed observation data, switch, mass, collar, coefficient functions, and root kernel. No estimate uniform as $\varepsilon\downarrow0$ is claimed. \end{proposition} \begin{proof} A compactly supported distribution has finite order. Choose compact neighborhoods $K_1,K_2$ of $C_0$ and $K_3$ of $O_{f,g}$. There are $r,C$ such that \[ |\langle t_{\mathrm D},F\rangle| \le C\,p_{r,K_1\times K_2\times K_3}(F) \] for every smooth $F$, using the cutoff-independent extension of the pairing. For $F=a\otimes c\otimes\phi$, differentiation in the three independent variables gives \[ p_{r,K_1\times K_2\times K_3}(F) \le C_r\,p_{r,K_1}(a)\, p_{r,K_2}(c)\, p_{r,K_3}(\phi). \] Together with \eqref{eq:O-t}, this proves \eqref{eq:target}. Alternatively, an order-$N$ bound for $B$ applied to \eqref{eq:t-test} differentiates the evaluated test at most $N$ additional times. The source deltas have at most two derivatives. The chain and product rules therefore require at most $N+2$ derivatives of $F$; derivatives of the fixed coefficients are absorbed into the constant. The product seminorm estimate proves joint continuity of the trilinear map. \end{proof} The source derivative order two is not the order of the entire kernel. The root distribution contributes as well. This continuity result is also distinct from the sharper fixed-witness scalar estimate of XVI-G. Neither estimate alone proves a microcausal cone exclusion. \section{Partial witness contraction and recovery of the density} \label{sec:matching} Let $t_{\mathrm D}^{\mathbb C}$ denote the complex-linear extension of the real distribution $t_{\mathrm D}$ to complex test fields. Its evaluation on complex tensor arguments and a real scalar implements the four-term extension \eqref{eq:OC}. Current XVI-G, \texttt{prop:kernel-slot}, supplies \begin{equation} I_\lambda[\phi]=\int_Mj_\lambda(x)\phi(x)\dd^4x, \label{eq:j-pair} \end{equation} where \begin{equation} \begin{aligned} j_\lambda ={}&-\partial_\nu \bigl(gV_\lambda^{\mu\nu}\partial_\mu v_f\bigr) -m^2g v_f b_1[R_\lambda]\\ &+\partial_\mu(fX_{\mathrm{cand}}^\mu), \qquad v_f=E_m^{\mathrm A}f. \end{aligned} \label{eq:j} \end{equation} Here \[ X_{\mathrm{cand}}^\mu(t) =-\int_0^t\sigma(s)V_\lambda^{0\mu}(s)\dd s, \] and \[ j_\lambda\in C_c^\infty(M_T^\circ;\mathbb C), \qquad \supp j_\lambda\subset\supp f\cup\supp g. \] Its extension to ambient space is smooth because the support is compact-interior. \begin{proposition}[Kernel matching on the specified witness pair] \label{prop:tmatch} For each $\lambda>0$, partial evaluation of the compact kernel on the two witness tensors defines a compact scalar distribution $q_\lambda$ by \begin{equation} \langle q_\lambda,\phi\rangle := \left\langle t_{\mathrm D}^{\mathbb C}, \gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi \right\rangle. \label{eq:q-partial} \end{equation} It satisfies \begin{equation} q_\lambda=j_\lambda \quad\text{in }\Eprime(M;\mathbb C), \label{eq:q-equals-j} \end{equation} with the fixed volume convention. In particular, \begin{equation} \left\langle t_{\mathrm D}^{\mathbb C}, \gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi \right\rangle =\int_Mj_\lambda(x)\phi(x)\dd^4x. \label{eq:t-to-j} \end{equation} \end{proposition} \begin{proof} The witness tensors are smooth, although not compactly supported. Compactness of $t_{\mathrm D}$ makes \eqref{eq:q-partial} meaningful. More explicitly, choose smooth compact cutoffs $\rho_1,\rho_2$ equal to one near the first and second support projections of $t_{\mathrm D}$. Then \[ \langle q_\lambda,\phi\rangle = \left\langle t_{\mathrm D}^{\mathbb C}, (\rho_1\gamma_1^\lambda) \otimes(\rho_2\gamma_2^\lambda)\otimes\phi \right\rangle. \] This is independent of those cutoffs and continuous in $\phi$. Equivalently, multiply $t_{\mathrm D}^{\mathbb C}$ by the smooth witness coefficients, contract its tensor indices, and push forward to the scalar slot. The pushforward is proper because the kernel is compact. By Theorem~\ref{thm:trep} and algebraic complexification, the left side of \eqref{eq:t-to-j} equals $\mathcal O_{\mathbb C} (\gamma_1^\lambda,\gamma_2^\lambda;\phi)$. Theorem~\ref{thm:Opol}(6) identifies this with $I_\lambda[\phi]$. Equation~\eqref{eq:j-pair} proves equality for every real smooth scalar. For a complex compact test, apply that equality separately to its real and imaginary parts. Complex linearity proves \eqref{eq:q-equals-j} as a distributional equality. \end{proof} The cutoffs in this proof only implement pairing with a compact distribution. They do not replace the witness by a different metric configuration. \paragraph{Exact limitation of the matching.} Equation~\eqref{eq:q-equals-j} identifies a partial evaluation of $t_{\mathrm D}$, not the full kernel with $j_\lambda$. It is not a reconstruction or uniqueness theorem for a three-slot kernel from one witness family. It is not an all-free-triples comparison. The full kernel representation is established by Theorem~\ref{thm:trep}; the witness identities are used only to identify its specified partial contraction. \subsection{Imported scalar estimates and resonant evaluations} For a fixed compact set $L\subset M_T^\circ$ containing $\supp f\cup\supp g$, current XVI-G proves \begin{equation} |I_\lambda[\phi]| \le C_{k,\varepsilon}\lambda^{2-k} p_{\max(k,2),L}(\phi), \qquad k\in\mathbb N_0,\quad\lambda\ge1. \label{eq:G-estimate} \end{equation} The constant is independent of $\lambda,\phi$ at fixed observation and background data. By \eqref{eq:t-to-j}, this estimate applies to the same witness contraction of $t_{\mathrm D}^{\mathbb C}$. It is imported from XVI-G, not derived from the unspecified finite-order estimate \eqref{eq:target}. The established rapid-decay cases are: \begin{enumerate} \item one real smooth scalar independent of $\lambda$; \item sets bounded in every $p_{r,L}$, with decay uniform on the stated set; \item families satisfying $p_{r,L}(\phi_\lambda)\le C_r\lambda^{\rho r+a}$ with $0\le\rho<1$. \end{enumerate} They are current XVI-G, \texttt{prop:fixeddecay}, \texttt{prop:bounded}, and \texttt{prop:slow} \cite{XVIG}. For real fixed $b\in C_c^\infty(M_T^\circ)$, set \[ \Upsilon[b] =\int_{M_T}\sigma g b\,\partial_tv_f\dd^4x. \] The resonant theorem of XVI-G gives \begin{align} \left\langle t_{\mathrm D}^{\mathbb C}, \gamma_1^\lambda\otimes\gamma_2^\lambda \otimes b\cos(2\lambda t)\right\rangle &=\frac{\lambda}{2\ii}\Upsilon[b]+O(1), \label{eq:res-cos}\\ \left\langle t_{\mathrm D}^{\mathbb C}, \gamma_1^\lambda\otimes\gamma_2^\lambda \otimes b\sin(2\lambda t)\right\rangle &=\frac{\lambda}{2}\Upsilon[b]+O(1). \label{eq:res-sin} \end{align} The order-$\lambda$ contribution is scalar-leg only; the coordinate contribution is bounded. No integrals or remainder estimates are recomputed here. When $\Upsilon[b]\ne0$, these are nonzero leading linear terms. When it vanishes, only $O(1)$ is concluded. The number $\Upsilon[b]$ is not a wavefront covector and does not decide admission. \section{Free-shell class and the unresolved admission test} \label{sec:admission} The compact kernel permits a free-shell class to be defined. This does not determine its admission or its relation to the frozen XI-G class. Let $\mathcal K$ be the matching space of compact three-slot kernels with the stated tensor symmetry. Define \begin{equation} \begin{aligned} \Isol:=\{z\in\mathcal K:\ &\langle z,a\otimes c\otimes\phi\rangle=0\\ &\text{for all smooth } \Box a=\Box c=0,\quad P_m\phi=0\}. \end{aligned} \label{eq:Isol} \end{equation} Since $\mathcal P$ is invertible and constant, $-\Box\mathcal P a=0$ is equivalent to $\Box a=0$. Compactness makes these pairings with smooth solutions well defined. The class of the constructed kernel is \[ [t_{\mathrm D}]_{\mathrm{sol}}=t_{\mathrm D}+\Isol. \] Comparison with a matching frozen kernel $u_0$ requires determining whether \[ t_{\mathrm D}-u_0\in\Isol. \] If that condition holds, equality of the affine classes follows by linearity; the applicable frozen obstruction then transfers. A different-class proof would remove only that transfer, not establish admission \cite{XVIA,XVIB}. The inherited target $\mc$ uses compact graded-symmetric polynomial kernels satisfying \begin{equation} \WF(u_n)\cap \bigl(\widetilde V_+^{\,n}\cup\widetilde V_-^{\,n}\bigr) =\varnothing. \label{eq:mc} \end{equation} Where the XI-G separation theorem applies with its complete hypotheses, it tests the trisolution \[ \mathsf W_u =(\mathsf E_0\otimes\mathsf E_0\otimes\mathsf E_m)u \] through \begin{equation} \WF(\mathsf W_u)\cap \bigl(\widetilde V_+^{\,3}\cup\widetilde V_-^{\,3}\bigr) =\varnothing. \label{eq:criterion} \end{equation} The propagators here are ambient causal propagators, not the slab retarded inverses by themselves \cite{XIG}. Under the inherited compact-distribution propagator interface, $\mathsf W_{t_{\mathrm D}}$ is an abstractly defined distributional trisolution. No explicit expression, wavefront estimate, or exclusion test for it is supplied in this note. Neither $\WF(t_{\mathrm D})$ nor $\WF(\mathsf W_{t_{\mathrm D}})$ is calculated. \scopebox{Admission boundary.}{% The compact kernel and its witness matching are established under the stated root/kernel assumptions. The existence gate is no longer left open for this polarized completion. The full-kernel wavefront test and the comparison with the frozen free-shell class remain \Unresolved. The XI-G admission criterion is not applied. No cubic admission or obstruction is asserted. } The supplied XII-E handoff reports kernel and admission results for its stated insertion and admissibility sheet. That report is not rescinded. Its complete theorem and hypotheses are not reproduced here as an admission premise. The XII-D kernel representation used in this note is not, by itself, a verification of every hypothesis on the XII-E admissibility sheet \cite{XIIE}. The scalar partial evaluation is different: $j_\lambda$ is compact smooth, so $\WF(j_\lambda)=\varnothing$. It satisfies the degree-one scalar kernel exclusion. That fact does not imply the cubic exclusion \eqref{eq:mc} for $t_{\mathrm D}$. \section{Conditional audit of the inherited switch representative} \label{sec:shell} This section concerns the representative formula reproduced from XI-G in XVI-H-v3. It is not needed for the kernel construction, continuity, or witness matching. The following are inherited operator inputs: \begin{enumerate} \item The ambient causal propagator is \[ \mathsf E_\mu =\mathsf E_\mu^{\mathrm R}-\mathsf E_\mu^{\mathrm A}. \] Both Green inverses invert $P_\mu=\Box+\mu^2$ on the appropriate compact distributional inputs. Hence $\mathsf E_\mu P_\mu r=0$ for $r\in\Eprime(M)$. \item The distributional slot convention is \[ \langle\mathsf E_{\mu,i}w,\varphi\rangle =-\langle w,\mathsf E_{\mu,i}\varphi\rangle. \] This is the transpose convention for the causal propagator, not an additional sign in $\mathsf E_\mu P_\mu r=0$. \item The temporal functions $\zeta_i$ are past-to-future switches: \[ \zeta_i(t)=0\quad(t\le-T_0), \qquad \zeta_i(t)=1\quad(t\ge T_0). \] They are not compactly supported. \item The inherited slot calculus supplies the Green identities, causal support, and commuting distinct-slot operations used below for compact kernels. \end{enumerate} These are the reproduced interfaces attributed to XI-G, \texttt{lem:spcompact} and \texttt{lem:kernel}. The complete XI-G source is not independently checked here \cite{XVIHvThree,XIG}. \begin{proposition}[Trisolution-preserving representative] \label{prop:switch} Let $u$ be a compact kernel in that inherited slot calculus, and put \[ P_1=\Box_1,\qquad P_2=\Box_2,\qquad P_3=P_{m,3}, \qquad \mathsf W_u=\mathsf E_{0,1}\mathsf E_{0,2}\mathsf E_{m,3}u. \] Choose the switches with transition times covering the relevant compact support projections. Then \begin{equation} \widetilde u =P_1P_2P_3 [\zeta_1\zeta_2\zeta_3\mathsf W_u] \label{eq:tilde} \end{equation} is compactly supported and satisfies \begin{equation} \mathsf W_{\widetilde u}=\mathsf W_u. \label{eq:W-preserved} \end{equation} The intermediate switched trisolution need not be compact. These conclusions alone are not an admission result. \end{proposition} \begin{proof} For a compact input $v$ in one slot, the inherited Green identities give \[ \zeta_i\mathsf E_i v =\mathsf E_i^{\mathrm R}v-r_i, \qquad r_i=(1-\zeta_i)\mathsf E_i^{\mathrm R}v +\zeta_i\mathsf E_i^{\mathrm A}v. \] The causal support bounds and the opposite temporal truncations make $r_i$ compact: the retarded contribution is truncated above, and the advanced contribution below, while the source is compact. Therefore \[ T_i v:=P_i[\zeta_i\mathsf E_i v] =v-P_i r_i \] is compact, and $\mathsf E_iT_i v=\mathsf E_i v$ because $\mathsf E_iP_i r_i=0$. There is no additional minus sign in this equality. Apply the three operations successively to compact inputs. The inherited distinct-slot commutations identify their composition with \eqref{eq:tilde}. Equivalently, telescoping the three differences gives \[ u-\widetilde u =P_1r_1+P_2r_2+P_3r_3 \] for compact corrections at the successive stages. Applying all three causal propagators annihilates each term, proving \eqref{eq:W-preserved}. Compactness follows already from the successive compact representatives. It can also be read from \[ P_1P_2P_3 [\zeta_1\zeta_2\zeta_3\mathsf W_u] =[P_1,\zeta_1][P_2,\zeta_2][P_3,\zeta_3]\mathsf W_u, \] using the trisolution equations. Each commutator is supported in its slot's transition region, and the inherited causal support controls the spatial projections there. \end{proof} The switch convention is essential. If the entire intermediate product were compact and the formula were read merely as $P_1P_2P_3$ applied to a compact distribution, its trisolution would vanish by the annihilation identity. Past-to-future switches avoid that mistaken compact-intermediate reading. The representative can participate in an admission argument only after the needed wavefront hypotheses are separately established. Preservation of the trisolution does not prove those hypotheses. No such argument is performed for $t_{\mathrm D}$ here. \section{On-shell, gauge, and quantum boundaries} \label{sec:quantum} \subsection{The resonant probes are not an on-shell calculation} With $\omega=2\lambda$, \[ \begin{aligned} P_m(b\cos\omega t) &=(P_mb-\omega^2b)\cos\omega t -2\omega(\partial_tb)\sin\omega t,\\ P_m(b\sin\omega t) &=(P_mb-\omega^2b)\sin\omega t +2\omega(\partial_tb)\cos\omega t. \end{aligned} \] Current XVI-G supplies these exact identities and explains off-shell behavior for fixed nonzero $b$ at sufficiently large frequency. The scalar probes in \eqref{eq:res-cos}--\eqref{eq:res-sin} are not the free scalar solutions quantified in \eqref{eq:Isol}. No replacement by free solutions is shown to preserve $\Upsilon[b]$. Thus witness matching transfers the stated off-shell scalar evaluation, not an on-shell asymptotic. \subsection{BRST/BV observability} No BRST/BV differential acting on this completed cubic functional is specified in the present manuscript. No descent equation, closure calculation, or gauge-invariance theorem is proved. Microcausal admission, BRST/BV observability, and renormalized quantum observability are distinct conclusions. The completed classical kernel does not settle the latter two. \subsection{A scalar pairing is definable once its data are chosen} For a specified scalar two-point distribution $\mathsf H_2\in\Dprime(M^2;\mathbb C)$ and fixed frequencies $\lambda,\mu$, the numerical expression \begin{equation} \langle\mathsf H_2,j_\lambda\otimes j_\mu\rangle \label{eq:scalar-pairing} \end{equation} is well defined: the tensor product is a compact smooth test density in the fixed volume convention. No product of two singular distributions is required for this particular scalar pairing. This observation is a definability statement, not a chosen quantum contraction. No $\mathsf H_2$ is selected; no Hadamard condition, state, positivity requirement, or time-ordering prescription is imposed here. It also does not construct contractions of the full cubic kernel with interaction kernels. No prepared quantum product, singular locus, extension, scaling degree, counterterm, or Ward defect is constructed. Those tasks require specified quantum data and their own distributional proofs. The sharp-width limit, ultraviolet completion, and all-orders finite-parameter renormalizability remain outside scope. \section{Consolidated release theorem} \label{sec:release} \begin{theorem}[Compact polarized completion and witness recovery] \label{thm:main} Retain the fixed data, inherited source and completion interfaces, compact root-kernel assumptions, and witness hypotheses stated in this manuscript. Then: \begin{enumerate} \item $\mathcal O_{\mathrm{pol}}$ is the real symmetric polarization of the diagonal completion, bilinear in the tensor slots and linear in the scalar. \item The explicit diagonal test-function construction \eqref{eq:t-test} defines a compact ambient kernel $t_{\mathrm D}$ representing that polarization. \item The functional is jointly continuous on the chosen smooth configuration spaces, with some finite derivative orders. \item Its complex extension agrees with the revised XVI-E/F polarization convention and with the current XVI-G witness value at each frozen frequency. \item Partial contraction of $t_{\mathrm D}^{\mathbb C}$ on the specified tensor witness pair equals $j_\lambda$ as a scalar distribution. \end{enumerate} These conclusions establish neither a full-kernel wavefront exclusion nor admission of the cubic free-shell class. They establish no BRST/BV observability or quantum product. \end{theorem} \begin{proof} Theorem~\ref{thm:Opol} proves items~1 and~4. Theorem~\ref{thm:trep} proves item~2. Proposition~\ref{prop:estimate} proves item~3. Proposition~\ref{prop:tmatch} proves item~5. The unperformed tests and unspecified quantum data are recorded in Sections~\ref{sec:admission} and~\ref{sec:quantum}. \end{proof} \section{Status ledger and debugging corrections} \label{sec:status} \small \renewcommand{\arraystretch}{1.12} \begin{longtable}{@{}p{0.30\linewidth}p{0.65\linewidth}@{}} \toprule Result & Current status\\ \midrule \endfirsthead \toprule Result & Current status, continued\\ \midrule \endhead \bottomrule \endfoot Switch-correction degree audit & Established from the displayed monomials: Proposition~\ref{prop:monomials}. The bulk $J_3$ homogeneity and local-polynomial structure remain explicitly imported premises.\\ \addlinespace Real polarized completion & Established on the stated inherited completion domain: Theorem~\ref{thm:Opol}. Defined by three diagonal completion evaluations.\\ \addlinespace Compact kernel $t_{\mathrm D}$ & Established under the stated root/kernel inputs: Theorem~\ref{thm:trep}. Defined on general tests by \eqref{eq:t-test}; no arbitrary distributional product is presumed.\\ \addlinespace Ambient support and tensor symmetry & Established for this kernel: \eqref{eq:t-support} and Theorem~\ref{thm:trep}. Reaching $t=0$ is compatible with ambient compactness. Boundary regularity is not determined.\\ \addlinespace Finite-order seminorm estimate & Established for some finite orders: Proposition~\ref{prop:estimate}. The root order contributes; the source order two is not substituted for the full distributional order.\\ \addlinespace Complex extension and witness equality & Established: Theorem~\ref{thm:Opol}(5)--(6). The definitions coincide with revised XVI-E/F. The supplied current XVI-G chain closes the old witness compatibility item.\\ \addlinespace Witness contraction to $j_\lambda$ & Established as a scalar distributional equality: Proposition~\ref{prop:tmatch}. Partial evaluation only, not identification of the full kernel with a one-variable density.\\ \addlinespace Witness decay and resonance & Imported from current XVI-G and transferred by \eqref{eq:t-to-j}. Their original fixed-width and scalar-family hypotheses remain unchanged.\\ \addlinespace Free-shell class & Defined for the constructed compact kernel. Comparison with the matching frozen XI-G class remains unresolved; no all-free-triples equality is proved.\\ \addlinespace Full-kernel wavefront/admission & Unresolved. Neither the full kernel's wavefront set nor its trisolution's wavefront set is calculated. The admission criterion is not applied.\\ \addlinespace Switch representative & Conditional consistency deduction: Proposition~\ref{prop:switch}. Uses the explicitly reproduced XI-G slot identities. Trisolution preservation is not admission.\\ \addlinespace BRST/BV observability & Not established. No differential or closure/descent calculation is specified.\\ \addlinespace Scalar two-point pairing & Definable for specified $\mathsf H_2$ by \eqref{eq:scalar-pairing}. No two-point distribution is selected or evaluated.\\ \addlinespace Prepared quantum product & Not constructed. No Hadamard choice, time-ordered product, extension, scaling degree, or counterterm is supplied.\\ \addlinespace Sharp-width limit and ultraviolet claims & Outside scope. Not open problems assigned to this note.\\ \end{longtable} \normalsize The principal corrections to XVI-H-v3 are: \begin{enumerate} \item \textbf{Witness compatibility is closed.} Equality with $T_D$ follows from identical definitions, and the same four-term extension gives $\mathcal O_{\mathbb C}=T^{\mathbb C}$ on the common complexified domain. The existing witness theorem then supplies the remaining chain. \item \textbf{Kernel specialization is qualified, not refused.} Partial contraction to $j_\lambda$ is proved. It is not mistaken for a full-kernel identity or a uniqueness theorem from witness data. \item \textbf{Kernel existence and admission are separated.} The compact representation is established under the stated inputs; full-kernel wavefront/admission remains unresolved. \item \textbf{General tests are used in the construction.} Formula \eqref{eq:t-test}, not only a pairing on decomposable tests, defines the distribution. Compact support and properness are stated explicitly. \item \textbf{Bulk and switch premises are separated.} The displayed switch monomials are audited. The unexpanded bulk coefficient is not silently derived. \item \textbf{Derivative indices do not reuse frequency.} $\lambda$ denotes frequency, while repeated spacetime indices have separate names. \item \textbf{The switch audit is bounded.} The inherited signs and hypotheses are reproduced explicitly. A full XI-G source verification is not claimed, and the representative is not presented as admission by itself. \item \textbf{Layout is made multipage-safe.} Defined theorem environments, a width-aware scope box, and a longtable replace the overlong one-page ledger. \end{enumerate} \section{Genuine remaining obligations and final statement} \label{sec:remaining} \begin{problem}[Full-kernel free-shell comparison and admission] For the constructed $t_{\mathrm D}$, specify the matching frozen comparison kernel and establish the all-free-triples comparison where claimed. Determine the required wavefront information for its trisolution and verify all hypotheses of the selected admission criterion. Compactness, finite-order continuity, and witness matching do not discharge this obligation. Status: \Unresolved. \end{problem} \begin{problem}[Gauge and quantum specifications] Specify the differential and domain needed for a BRST/BV observability statement. Independently, select the quantum two-point and interaction data, product, formal order, extension problem, and normalization conditions before asserting a quantum-product theorem. Neither the scalar pairing's definability nor the classical kernel construction supplies those choices or proofs. Status: not established. \end{problem} No admission or obstruction for the full cubic prepared observable is claimed. No repair of X-G or XI-G is claimed. No all-free-triples comparison is inferred from the resonant witness. No on-shell resonant asymptotic or general distributional Green-operator extension is proved. The coefficient tensors of the bulk $J_3$ normal form are not expanded. No quantum coefficient is constructed, and the independent retarded-slot obligations of XVI-A and the characteristic quantum-composition programme are not discharged. No successful compiler execution or exhaustive corpus verification is represented. \scopebox{Final release statement.}{% The real polarized completion has a compact ambient three-slot kernel under the stated inherited root/kernel assumptions. Its finite-order continuity is proved. The current witness compatibility chain identifies the complexified polarization with the independently checked witness evaluation. Partial contraction of the full kernel on that pair is exactly $j_\lambda$. The old witness matching gap is closed. The full-kernel wavefront/admission test, frozen free-shell comparison, BRST/BV observability, and prepared quantum-product construction remain unestablished. The broader admission and quantum programme is \Unresolved. } \begin{thebibliography}{99} \bibitem{XVIHvThree} \emph{Landmark XVI-H}. Supplied historical baseline \texttt{XVI-H-v3.tex}, ``Full-kernel construction, free-shell admission, and the first quantum contraction.'' Locators: \texttt{prop:monomials}, \texttt{prop:Chom}, \texttt{thm:Opol}, \texttt{thm:trep}, \texttt{prop:estimate}, and \texttt{prop:switch}. Its reproduced XII-D kernel inputs and XI-G switch interfaces are made explicit in the present revision. Its open witness-compatibility verdict is superseded. \bibitem{XVIA} \emph{Landmark XVI-A}. Supplied revised \texttt{XVI-A.tex}. Used for normalization/sign separation, conditional free-shell class transfer, and the independent retarded-slot obligations. \bibitem{XVIB} \emph{Landmark XVI-B}. Supplied revised \texttt{XVI-B.tex}. Source-level locators: \texttt{prop:trace}, \texttt{prop:pair}, \texttt{eq:JDeval}, and \texttt{cor:tf}. Class-transfer locator: \texttt{prop:transfer}. Historical later-sheet status summaries are not used as current witness compatibility verdicts. \bibitem{XVID} \emph{Landmark XVI-D}. Supplied revised \texttt{XVI-D.tex}, ``Prepared witness response and completion domains.'' Locators: \texttt{lem:uniqueness}, \texttt{lem:realB}, \texttt{lem:BC}, \texttt{lem:identify}, \texttt{lem:scalar}, and \texttt{lem:Xcand}. Used for smooth zero-data uniqueness, response polarization, and witness-response identification. \bibitem{XVIErev} \emph{Landmark XVI-E}. Supplied revised \texttt{XVI-E.tex}, ``Polarized root representation and witness-level completion compatibility.'' Locators: \texttt{eq:TD}, \texttt{lem:TD}, \texttt{lem:TC}, \texttt{prop:root-compatibility}, \texttt{thm:compatibility}, and \texttt{eq:compatibility-chain}. Used for exact matching of polarization conventions and the consolidated witness compatibility theorem. \bibitem{XVIFrev} \emph{Landmark XVI-F}. Supplied revised \texttt{XVI-F.tex}, ``Auxiliary/root compatibility and direct witness completion.'' Locators: \texttt{eq:TD}, \texttt{lem:polarized-root}, \texttt{prop:ER}, \texttt{prop:Icand}, \texttt{prop:compat}, \texttt{thm:compatibility}, and \texttt{cor:family-pointwise}. This revision, rather than a superseded XVI-F residual stop, is the current auxiliary/root and direct-completion reference. \bibitem{XVIG} \emph{Landmark XVI-G}. Supplied current \texttt{XVI-G.tex}, ``Frequency-dependent witness testing and resonant asymptotics.'' Locators: \texttt{prop:import}, \texttt{eq:pointwise}, \texttt{eq:family-pointwise}, \texttt{prop:kernel-slot}, \texttt{prop:semi}, \texttt{prop:fixeddecay}, \texttt{prop:bounded}, \texttt{prop:slow}, \texttt{prop:remainder}, and \texttt{prop:resonant}. Used for pointwise compatibility, the smooth density $j_\lambda$, and the delimited scalar estimates and resonant evaluations. No full-kernel admission theorem is imported from that manuscript. \bibitem{XIID} \emph{Manuscript XII-D}, version 1. Supplied identifier \texttt{e3e3f67d}. Reported locators: \texttt{eq:source-input}, \texttt{eq:source-coefficients}, \texttt{thm:response}, \texttt{eq:completion-legs}, \texttt{eq:Aone}, \texttt{def:completed}, \texttt{thm:root}, \texttt{eq:root-identity}, \texttt{lem:root-kernel}, \texttt{eq:B-kernel}, \texttt{eq:t-kernel}, and \texttt{eq:t-bounds}. Used through the interfaces reproduced in the supplied XVI-H-v3 and revised XVI-D/E/F. The complete source is not independently audited here. \bibitem{XIIE} \emph{Manuscript XII-E}, version 1. Supplied identifier \texttt{26f86eb3}. Reported locator: \texttt{input:XIID}, ``XII-D: the six handed objects.'' Its reported kernel/admission handoff retains its own admissibility sheet and hypotheses; it is not used here as a fully reproduced admission premise. \bibitem{XIG} \emph{Manuscript XI-G}, version 1. Supplied identifier \texttt{3cae2be9}. Reported locators: \texttt{def:domain}, \texttt{def:class}, \texttt{eq:Ecaus}, \texttt{lem:spcompact}, \texttt{lem:kernel}, \texttt{thm:criterion}, and \texttt{thm:obstruction}. The switch and slot identities are used through their reproduction in XVI-H-v3. The criterion and frozen obstruction retain their complete hypotheses. No full-source sign or domain audit is asserted. \end{thebibliography} \end{document}