% XVI-J.tex % Single-file manuscript for pdfLaTeX. % Compile twice: pdflatex XVI-J.tex % No external bibliography, included manuscript, or local script is required. % The appendix contains a reproducible exact-arithmetic verification program. % Neither compiler execution nor execution of that program is claimed here. \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{listings} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \setlength{\parindent}{0pt} \setlength{\parskip}{0.4em} \setlength{\emergencystretch}{2em} \setlist[itemize]{leftmargin=*,itemsep=0.2em} \setlist[enumerate]{leftmargin=*,itemsep=0.25em} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \DeclareMathOperator{\WF}{WF} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\sd}{sd} \newcommand{\ii}{\mathrm i} \newcommand{\dd}{\,\mathrm d} \newcommand{\Dprime}{\mathcal D'} \newcommand{\Eprime}{\mathcal E'} \newcommand{\R}{\mathbb R} \newcommand{\Pair}[2]{\left\langle #1,#2\right\rangle} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Inherited input} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \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 } \lstset{ language=Python, basicstyle=\ttfamily\footnotesize, columns=fullflexible, keepspaces=true, breaklines=true, showstringspaces=false, frame=single, numbers=left, numberstyle=\tiny, numbersep=6pt, tabsize=4 } \title{Landmark XVI-J\\[0.3em] \large A selected extension and a root-attached coefficient\\ for the \(C_2\)/constant-slot two-edge sector\\[0.3em] \normalsize Constructed under the explicitly stated root hypotheses} \author{} \date{} \usepackage{tikz} \begin{document} \maketitle \begin{abstract} We construct a selected-sector coefficient containing only the polarized source \(2C_2\,\partial\partial\sigma_\varepsilon\), the constant-slot part of the declared cubic Hessian, and two undifferentiated graviton edges. The plateau source, the \(M_2\) source term, and vertex terms differentiating contracted fields are excluded. The tensor contraction is derived from the inverse fiber pairing and is independent of the chosen symmetric-tensor basis. The Hessian and source-decorated coefficient on a specified pure-trace background are proved by explicit polynomial identities, without reliance on an unavailable script. An appendix supplies a second, exact-arithmetic verification through curvature atoms and two implementations of the source contraction. The numerical square has scaling degree four in four relative dimensions. A subtraction function and a zero-moment condition select its extension; an explicit distributional action is given. Under the imported compactness and \(Z_2\) wavefront bound for the root, pullback, multiplication, and proper pushforward construct a continuous bilinear functional of the residual metric and scalar test. Its prescription dependence is a diagonal contact term propagated through the root. This is not the complete two-edge interaction coefficient, a retarded interacting field, or a BRST/BV observable. \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) {Narrowed square}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {One selected extension}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Complete interaction}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The root-attached coefficient does not construct the retarded coefficient of the fuller graph.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One selected extension is not the full interaction. 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-J} \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) {Narrowed square}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Complete interaction}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The root-attached coefficient does not construct the retarded coefficient of the fuller graph.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One selected extension is not the full interaction. 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-J} \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) {Narrowed square}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {One selected extension}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Complete interaction}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The root-attached coefficient does not construct the retarded coefficient of the fuller graph.}; \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-J} \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) {Narrowed square}; \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) {Complete interaction}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {One selected extension. The root-attached coefficient does not construct the retarded coefficient of the fuller graph.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Narrowed square is not Complete interaction. 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-J} \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: Narrowed square}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: One selected extension}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Complete interaction}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The root-attached coefficient does not construct the retarded coefficient of the fuller graph.}; \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-J} \end{figure} Figure~\ref{fig:split-XVI-J} records one view of the same separation. Figure~\ref{fig:chain-XVI-J} records one view of the same separation. Figure~\ref{fig:pair-XVI-J} records one view of the same separation. Figure~\ref{fig:gate-XVI-J} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XVI-J} records that separation. The drawing adds no hypothesis. \scopebox{Scope and the four levels.}{% The preparation width \(0<\varepsilon0,\\ e^{\ii|\mathbf p|t}/(2|\mathbf p|),&t<0. \end{cases} \] The distribution fixed by \eqref{eq:DF} is symmetric under \(z\mapsto-z\) and satisfies \[ \Box d=-\ii\delta, \qquad D_F^{\mathsf T}=D_F, \qquad KD_F=\ii\,\mathrm{id}. \] \end{lemma} \begin{proof} For \(t<0\), \[ \frac{e^{-\ii|\mathbf p|t}}{2|\mathbf p|} +\ii\frac{\sin(t|\mathbf p|)}{|\mathbf p|} =\frac{e^{\ii|\mathbf p|t}}{2|\mathbf p|}. \] For \(t>0\), the advanced summand vanishes. The temporal derivative of the two branches has jump \(-\ii\). Hence the wave operator gives \(-\ii\delta\). Using \(\mathcal P^2=\mathrm{id}\) proves the tensor inverse identity. The two branches also prove symmetry. \end{proof} The time-ordered expression is understood through this distributional definition, not through an unspecified value of a step function at coincidence. Neither \(d\) nor \(D_F\) is extended in the later square-extension problem. \subsection{Off-origin product} Let \[ Q_{\mathrm{geom}}(z)=(z^0)^2-|\mathbf z|^2. \] On \(z\ne0\), direct Fourier evaluation gives \begin{equation} d(z)=-\frac1{4\pi^2} \bigl(Q_{\mathrm{geom}}(z)-\ii0\bigr)^{-1}. \label{eq:rational} \end{equation} At a nonzero null point, \(Q_{\mathrm{geom}}\) is a submersion. The boundary value in \eqref{eq:rational} has one, rather than both, conormal orientations. Consequently its wavefront covectors at that point cannot cancel pairwise. This is the usual off-diagonal Feynman relation; the product criterion and that relation are described in \cite[Theorem 13 and Proposition 26]{BDH}. \begin{proposition}[The numerical kernel] \label{prop:q} The distribution \begin{equation} q^\circ(z)=S(z)^2=d(z)^2, \qquad z\ne0, \label{eq:q} \end{equation} exists on \(\R^4\setminus\{0\}\), including nonzero null points. It is homogeneous of degree \(-4\), has scaling degree four at the origin, and is nonzero. \end{proposition} \begin{proof} The single conormal orientation just described satisfies H\"ormander's product criterion. At non-null points both factors are smooth. The product is therefore defined on the entire punctured domain. The massless \(d\) is homogeneous of degree \(-2\), so its defined punctured square is homogeneous of degree \(-4\). It is not zero: at a spacelike point, \[ q^\circ(z) =\frac1{16\pi^4\bigl(|\mathbf z|^2-(z^0)^2\bigr)^2}. \] A nonzero homogeneous distribution of degree \(-4\) has scaling degree four. \end{proof} Only \(q^\circ\) is translation invariant. The tensor and switch coefficients restored below are not. \section{Finite source and density definitions} \label{sec:algebra} \subsection{The source polynomial} For a symmetric matrix \(H\), define \[ N=\eta H,\qquad u_0=\eta,\qquad u_1=-\eta H\eta,\qquad u_2=\eta H\eta H\eta, \] \[ b_1=\frac12\tr N,\qquad b_2=\frac18(\tr N)^2-\frac14\tr(N^2). \] The tensor \(C_2\) can be specified compactly and completely by the coefficient identity \begin{equation} C_2^{\mu\nu\alpha\beta}(H) = [\lambda^2]\, (1+\lambda b_1+\lambda^2b_2) \left( U^{\mu\nu}U^{\alpha\beta} -U^{\mu(\alpha}U^{\beta)\nu} \right), \quad U=u_0+\lambda u_1+\lambda^2u_2. \label{eq:C2} \end{equation} Here \([\lambda^2]\) means polynomial coefficient extraction. For a symmetric covariant source jet \(J=(J_{\alpha\beta})\), set \begin{equation} Q(H;J)^{\mu\nu} =2C_2^{\mu\nu\alpha\beta}(H)J_{\alpha\beta}, \qquad \mathfrak s(X,Y;J) =\frac12\bigl(Q(X+Y;J)-Q(X;J)-Q(Y;J)\bigr). \label{eq:source} \end{equation} Thus \(Q(H;J)=\mathfrak s(H,H;J)\). A second, matrix form useful for hand calculations is the following. Write \[ V=\eta H\eta,\qquad W=\eta H\eta H\eta,\qquad c=\tr(\eta J),\quad k=\tr(VJ),\quad j=\tr(WJ). \] Then \begin{equation} \begin{aligned} \frac12Q(H;J) ={}&b_2(\eta c-\eta J\eta)\\ &+b_1(-Vc-\eta k+VJ\eta+\eta JV)\\ &+Wc+\eta j+Vk-WJ\eta-\eta JW-VJV. \end{aligned} \label{eq:Qmatrix} \end{equation} Equation~\eqref{eq:Qmatrix} follows by expanding \[ [\lambda^2]\, \sqrt{-\det(\eta+\lambda H)} \left(U_{\mathrm{exact}}\tr(U_{\mathrm{exact}}J) -U_{\mathrm{exact}}J U_{\mathrm{exact}}\right), \qquad U_{\mathrm{exact}}=(\eta+\lambda H)^{-1}. \] It agrees term by term with \eqref{eq:C2}. The actual source jet is always \[ J_{\alpha\beta}(y) =\partial_\alpha\partial_\beta\sigma_\varepsilon(y). \] \subsection{The unintegrated cubic density} The following finite formulas fix the density without integration by parts: \begin{align} \Gamma_n^\rho{}_{\mu\nu} &=\frac12u_{n-1}^{\rho a} \left(\partial_\mu h_{\nu a} +\partial_\nu h_{\mu a} -\partial_a h_{\mu\nu}\right), &&n=1,2,3, \label{eq:Gamma}\\ R_{n,\mu\nu} &=\partial_\rho\Gamma_n^\rho{}_{\mu\nu} -\partial_\nu\Gamma_n^\rho{}_{\mu\rho} \nonumber\\ &\quad+ \sum_{\substack{a+b=n\\a,b\ge1}} \left( \Gamma_a^\rho{}_{\rho\lambda} \Gamma_b^\lambda{}_{\mu\nu} -\Gamma_a^\rho{}_{\nu\lambda} \Gamma_b^\lambda{}_{\mu\rho} \right), \label{eq:Ricci}\\ r_n&=\sum_{i=0}^{n-1}u_i^{\mu\nu}R_{n-i,\mu\nu}, \qquad \ell_3=-2(r_3+b_1r_2+b_2r_1). \label{eq:ell} \end{align} The \(u_i,b_i\) in these equations are evaluated on \(h\). Every monomial is cubic in field jets and has total derivative order two. This is the supplied XII-C unintegrated representative \cite{XIIC}; changing it by a total divergence would change the constant-slot sector unless a separate equivalence proof were supplied. \begin{definition}[Constant-slot Hessian] \label{def:L} For constant tensor fields \(A,C\in V\), define \[ L_h(w)(A,C)=D^2\ell_3(h)[A,C](w). \] This is precisely the part of the Hessian with no derivatives on its two increments. \end{definition} \begin{lemma}[Dependence on the residual field] \label{lem:Llinear} The map \(h\mapsto L_h\) is a linear differential operator depending only on second derivatives of \(h\). In particular, it is continuous on smooth configurations. \end{lemma} \begin{proof} A cubic monomial of type \(h\,h\,\partial^2h\) contributes only when its two undifferentiated factors are varied. Its remaining coefficient is linear in \(\partial^2h\). A monomial of type \(h\,\partial h\,\partial h\) has only one undifferentiated factor. Varying twice in constant directions therefore gives zero. This proves the assertion directly from the total derivative count. \end{proof} \section{Fiber contraction and normalization} \label{sec:basis} Let \(B_0,\ldots,B_9\) be the constant symmetric matrices in the order \[ (00),(01),(02),(03),(11),(12),(13),(22),(23),(33), \] with the indicated entry equal to one and, for an off-diagonal pair, its symmetric partner also equal to one. \begin{lemma}[Gram matrix and inverse pairing] \label{lem:dual} The Gram matrix is \begin{equation} G=\operatorname{diag}(1,-2,-2,-2,1,2,2,1,2,1). \label{eq:Gram} \end{equation} Let \(B^i=(G^{-1})^{ij}B_j\). Then \[ b(B^i,B_j)=\delta^i_j, \qquad b^{-1}=\sum_i B^i\otimes B_i \] as a contravariant tensor on the fiber. \end{lemma} \begin{proof} A diagonal basis matrix has pairing with itself equal to one. An off-diagonal \(B_{(ab)}\) contributes two entries, giving \(2\eta^{aa}\eta^{bb}\). Distinct basis matrices have disjoint component support. This gives \eqref{eq:Gram}. The remaining statements are the defining inverse-matrix identities. \end{proof} The vertex Hessian is a covariant bilinear form on \(V\). A field derivative at either end of an edge is a covector in \(V^*\). The propagator takes such a covector to a field vector after the identification by \(b^{-1}\). Its fiber tensor, with the scalar factor \(S\) removed, is \begin{equation} \mathsf E=(-\mathcal P\otimes\mathrm{id})b^{-1} =\sum_i Z_i\otimes B_i, \qquad Z_i=-\mathcal P B^i. \label{eq:edgefiber} \end{equation} \begin{definition}[Source-decorated tensor] \label{def:K} Set \begin{equation} \mathcal K^{\mu\nu}(y,w;h) =\sum_{i,j=0}^9 L_h(w)(B_i,B_j)\, \mathfrak s^{\mu\nu}(Z_i,Z_j;J(y)). \label{eq:Kdef} \end{equation} The reduced selected-sector distribution on \(y\ne w\) is \[ \mathcal Q^{\circ,\mu\nu}(y,w;h) =\chi_{\mathrm I}(w)\mathcal K^{\mu\nu}(y,w;h) q^\circ(y-w). \] \end{definition} \begin{proposition}[Basis independence and graph factor] \label{prop:invariant} Equation~\eqref{eq:Kdef} is independent of the basis of \(V\). It is the contraction of the source polarization and vertex Hessian by the two fiber edge tensors \eqref{eq:edgefiber}. For the ordinary, non-divided second derivatives used here, its numerical graph factor is \(2/2!=1\). \end{proposition} \begin{proof} Applying the source bilinear map to the first vectors of two copies of \(\mathsf E\), and the vertex bilinear form to their second vectors, gives exactly \eqref{eq:Kdef}. For an arbitrary invertible basis change \(\widetilde B_i=B_jA_{ji}\), \[ \widetilde B^i=(A^{-1})_{ij}B^j, \qquad \widetilde L=A^{\mathsf T}LA. \] The two inverse matrices from the dual vectors cancel the two matrices from the Hessian. The double sum is unchanged. Equivalently, the fiber tensor \(\mathsf E\) is intrinsically defined. The second tensor derivative of the quadratic insertion is \(2\mathfrak s\), whereas the vertex derivative is the ordinary Hessian \(L_h\). The second term of the cross-contraction exponential has coefficient \(1/2!\). Thus the product of these two numerical factors is one. No additional division for symmetric spacetime indices is made: the Gram inverse already accounts for the independent components. The identical-edge convention is the one displayed in \cite[Eq.~(82)]{FR}. \end{proof} In particular, \[ -\mathcal P B^{(00)} =\operatorname{diag}(-1/2,-1/2,-1/2,-1/2), \qquad -\mathcal P B^{(01)}=\tfrac12 B_{(01)}. \] Neither is obtained by treating the field basis as self-dual. Trace reversal acts on these fiber vectors, not on \(q^\circ\). \section{Independent polynomial proofs of the numerical tables} \label{sec:tables} \subsection{Closed formula for the Hessian} Consider the test background \[ h_*(w)=(w^1)^2\eta. \] By Lemma~\ref{lem:Llinear}, its constant-slot Hessian is independent of \(w\). \begin{lemma}[Closed Hessian formula] \label{lem:closedL} For \(N_A=\eta A\), \(N_C=\eta C\), \begin{equation} L_{h_*}(A,C) =\tr N_A\,\tr N_C -2\tr(N_A N_C) -4(N_A N_C+N_C N_A)^1{}_1. \label{eq:Lclosed} \end{equation} Equivalently, if \(H=\sum_i z_iB_i\), its quadratic polynomial is \begin{equation} F(H) =-4\bigl((\eta H)^2\bigr)^1{}_1 -\tr\bigl((\eta H)^2\bigr) +\frac12\bigl(\tr(\eta H)\bigr)^2, \qquad L_{ij}=\partial_{z_i}\partial_{z_j}F. \label{eq:F} \end{equation} \end{lemma} \begin{proof} Since only the residual second derivative contributes, evaluate at \(w^1=0\), where the residual field and its first derivatives vanish. For \[ g_\lambda=\eta+\lambda(h_*+H), \] one has there \[ g_\lambda=\eta+\lambda H,\qquad \partial g_\lambda=0,\qquad \partial_1^2g_\lambda=2\lambda\eta. \] The Ricci formula \eqref{eq:Ricci} gives, with \(U=(\eta+\lambda H)^{-1}\), \[ R(g_\lambda) =2\lambda\left((U\eta U)^{11} -U^{11}\tr(U\eta)\right). \] Write \(N=\eta H\), \(t=\tr N\), \(v=\tr(N^2)\), and \(m=(N^2)^1{}_1\). The expression in parentheses is \[ 3+\lambda(-2N^1{}_1-t) +\lambda^2(m+tN^1{}_1+v)+O(\lambda^3). \] Multiplication by \[ \sqrt{-\det(\eta+\lambda H)} =1+\frac{\lambda t}{2} +\lambda^2\left(\frac{t^2}{8}-\frac v4\right) +O(\lambda^3) \] leaves second-order coefficient \(m+v/4-t^2/8\). The signed cubic density is therefore the quadratic expression \(-4m-v+t^2/2\) at this residual jet. Differentiating it twice proves both formulas. \end{proof} Expanding \eqref{eq:F} gives the full quadratic polynomial \begin{equation} \begin{aligned} F={}&-\tfrac12(z_0^2+z_7^2+z_9^2) -\tfrac92 z_4^2 -z_0z_4-z_0z_7-z_0z_9\\ &+z_4z_7+z_4z_9+z_7z_9\\ &+6z_1^2+2z_2^2+2z_3^2 -6z_5^2-6z_6^2-2z_8^2. \end{aligned} \label{eq:Fexpanded} \end{equation} Consequently the complete Hessian is \begin{equation} L= \begin{pmatrix} -1&0&0&0&-1&0&0&-1&0&-1\\ 0&12&0&0&0&0&0&0&0&0\\ 0&0&4&0&0&0&0&0&0&0\\ 0&0&0&4&0&0&0&0&0&0\\ -1&0&0&0&-9&0&0&1&0&1\\ 0&0&0&0&0&-12&0&0&0&0\\ 0&0&0&0&0&0&-12&0&0&0\\ -1&0&0&0&1&0&0&-1&0&1\\ 0&0&0&0&0&0&0&0&-4&0\\ -1&0&0&0&1&0&0&1&0&-1 \end{pmatrix}. \label{eq:Ltable} \end{equation} This proves every entry, including every zero, without a script or a numerical approximation. \begin{corollary}[Hand checks at the Hessian level] \label{cor:Hchecks} For \(A=\eta=B_0-B_4-B_7-B_9\), \[ L_{h_*}(A,B_4) =-1-(-9)-1-1=6. \] Also \[ L_{h_*}(B_0,B_2) =L_{h_*}(B_1,B_5) =L_{h_*}(B_2,B_4)=0. \] \end{corollary} \begin{proof} The first identity follows from \eqref{eq:Ltable}. Each zero follows either from \eqref{eq:Lclosed} or the absence of the corresponding mixed monomial in \eqref{eq:Fexpanded}. Thus the value six is checked independently of the other entries and is not used to certify them. \end{proof} \subsection{A second contraction method} The following auxiliary operation is finite-dimensional polynomial algebra, not a probability expectation: \[ \mathbb E_L[p(a)] =\frac12\sum_{i,j}L_{ij} \left.\partial_{a_i}\partial_{a_j}p(a)\right|_{a=0}. \] Let \[ H(a)=\sum_i a_i Z_i. \] Then \begin{equation} \mathcal K(J)=\mathbb E_L[Q(H(a);J)]. \label{eq:secondmethod} \end{equation} Indeed, the coefficient of \(a_i a_j\) in the quadratic source is its polarization on \(Z_i,Z_j\). The diagonal components of \(H(a)\) are \[ \begin{pmatrix}H_{00}\\H_{11}\\H_{22}\\H_{33}\end{pmatrix} =\frac12 \begin{pmatrix} -1&-1&-1&-1\\ -1&-1&1&1\\ -1&1&-1&1\\ -1&1&1&-1 \end{pmatrix} \begin{pmatrix}a_0\\a_4\\a_7\\a_9\end{pmatrix}. \] Its off-diagonal components are \[ (H_{01},H_{02},H_{03},H_{12},H_{13},H_{23}) =\tfrac12(a_1,a_2,a_3,-a_5,-a_6,-a_8). \] Thus the diagonal quadratic contraction table, in order \(00,11,22,33\), is \begin{equation} \bigl(\mathbb E_L[H_{\mu\mu}H_{\nu\nu}]\bigr) = \begin{pmatrix} -3&-3&1&1\\ -3&-3&3&3\\ 1&3&-3&-1\\ 1&3&-1&-3 \end{pmatrix}. \label{eq:covdiag} \end{equation} The off-diagonal squares are \begin{equation} \begin{array}{c|rrrrrr} \mu\nu&01&02&03&12&13&23\\ \hline \mathbb E_L[H_{\mu\nu}^2]&3&1&1&-3&-3&-1 \end{array} \label{eq:covoff} \end{equation} and all mixed contractions involving two different off-diagonal components, or one diagonal and one off-diagonal component, vanish. These tables follow by multiplying the displayed rational change-of-basis matrix with \eqref{eq:Ltable}. They provide a contraction calculation independent of the double polarization sum. \begin{proposition}[Complete decorated coefficient] \label{prop:Ktable} For \(h=h_*\) and a general symmetric jet \(J\), the tensor is \begin{align} \mathcal K^{00}&=14J_{11}+10J_{22}+10J_{33}, \nonumber\\ \mathcal K^{01}&=-14J_{01}, & \mathcal K^{02}&=-10J_{02}, & \mathcal K^{03}&=-10J_{03}, \nonumber\\ \mathcal K^{11}&=14(J_{00}-J_{22}-J_{33}), \nonumber\\ \mathcal K^{12}&=14J_{12}, & \mathcal K^{13}&=14J_{13}, \label{eq:Ktable}\\ \mathcal K^{22}&=10J_{00}-14J_{11}-10J_{33}, \nonumber\\ \mathcal K^{23}&=10J_{23}, \nonumber\\ \mathcal K^{33}&=10J_{00}-14J_{11}-10J_{22}. \nonumber \end{align} The remaining entries follow by symmetry. \end{proposition} \begin{proof} Apply \eqref{eq:covdiag}--\eqref{eq:covoff} to \eqref{eq:Qmatrix}. A compact form of the resulting identity is \begin{equation} \begin{aligned} \mathbb E_L[Q(H;J)] ={}&-10\bigl(\eta\,\tr(\eta J)-\eta J\eta\bigr)\\ &+4\bigl(E\,\tr(\eta J)+\eta J_{11} -EJ\eta-\eta JE\bigr), \end{aligned} \label{eq:Kcompact} \end{equation} where \(E\) is the contravariant matrix with \(E^{11}=1\) and all other entries zero. Both sides are linear in \(J\). Substitution of the ten symmetric component jets, using the complete quadratic contraction tables, proves this polynomial identity. Expanding its diagonal and off-diagonal entries gives \eqref{eq:Ktable}. For additional explicit arithmetic checks, the next lemma evaluates three component jets directly. \end{proof} \begin{lemma}[Independent component checks] \label{lem:Kchecks} The contraction gives \[ [\mathcal K^{00}]_{J_{11}=1}=14,\qquad [\mathcal K^{01}]_{J_{01}=J_{10}=1}=-14,\qquad [\mathcal K^{22}]_{J_{00}=1}=10. \] All other jet components are zero in each check. \end{lemma} \begin{proof} The contraction tables give \[ \mathbb E_L[b_2]=9,\qquad \mathbb E_L[W_{00}]=-8,\quad \mathbb E_L[W_{11}]=12,\quad \mathbb E_L[W_{22}]=8. \] Writing \(t=\tr(\eta H)\), they also give \[ \mathbb E_L[tH_{00}]=-2,\quad \mathbb E_L[tH_{11}]=-6,\quad \mathbb E_L[tH_{22}]=2. \] For \(J_{11}=1\), equation~\eqref{eq:Qmatrix} yields \[ \frac12Q^{00} =-b_2+b_1(H_{00}-H_{11}) -W_{00}+W_{11}+H_{00}H_{11}-H_{01}^2. \] Its contraction is \[ -9+2+8+12-3-3=7, \] so \(\mathcal K^{00}=14\). For \(J_{01}=J_{10}=1\), \[ \begin{aligned} \frac12Q^{01} ={}&b_2+b_1(-H_{00}+H_{11})\\ &+H_{00}^2+H_{11}^2-H_{00}H_{11} -H_{01}^2-H_{02}^2-H_{03}^2 +H_{12}^2+H_{13}^2. \end{aligned} \] The three groups contract to \(9,-2,-14\), respectively. Thus the result is \(2(9-2-14)=-14\). For \(J_{00}=1\), \[ \frac12Q^{22} =-b_2+b_1(H_{00}-H_{22}) +W_{22}-W_{00}+H_{22}H_{00}-H_{02}^2. \] Its contraction is \[ -9-2+8+8+1-1=5, \] so \(\mathcal K^{22}=10\). \end{proof} \subsection{Nonvanishing at the appropriate level} \begin{corollary}[Actual ramp and qualified nonvanishing] \label{cor:nonzero} On \(h_*\), the time-only preparation jet gives \[ \mathcal K^{11}=14\sigma_\varepsilon'', \qquad \mathcal K^{22}=\mathcal K^{33}=10\sigma_\varepsilon'', \] with every other component zero. This tensor is not identically zero. \end{corollary} \begin{proof} Only \(J_{00}\) survives, so the substitution in \eqref{eq:Ktable} gives the displayed formula. If \(\sigma_\varepsilon''\) were identically zero, the smooth ramp would be affine. Its constant value on \(t\le0\) would then force it to be constant everywhere, contradicting its value one for \(t\ge\varepsilon\). \end{proof} \begin{center} \begin{tabularx}{\linewidth}{@{}p{0.32\linewidth}Y@{}} \toprule Level & Nonvanishing conclusion\\ \midrule Hessian & Nonzero; the specified mixed value is six\\ Formal source jet & Nonzero linear map \eqref{eq:Ktable}\\ Actual time-only ramp & Nonzero tensor on \(h_*\)\\ After the interaction switch & Not asserted for arbitrary \(\chi_{\mathrm I}\), which may be zero\\ After root attachment and smearing & Not asserted\\ \bottomrule \end{tabularx} \end{center} \section{An operational extension prescription} \label{sec:extension} \subsection{Action on tests vanishing at the origin} Choose an auxiliary smooth radial cutoff \(\zeta\in C_c^\infty(\R^4)\) equal to one near zero. For \(\psi\in C_c^\infty(\R^4)\) with \(\psi(0)=0\), define \begin{equation} T_0(\psi) =\lim_{\delta\downarrow0} \Pair{q^\circ}{(1-\zeta(\,\cdot\,/\delta))\psi}. \label{eq:T0} \end{equation} For each positive \(\delta\), the test vanishes near the origin, so this pairing is already defined on the punctured domain. This cutoff limit defines the action on a larger test subspace; it is not a change of the preparation width. \begin{lemma}[Convergence and independence of the auxiliary cutoff] \label{lem:T0} The limit \eqref{eq:T0} exists, is independent of the radial cutoff used, and is a continuous linear functional on compact tests satisfying \(\psi(0)=0\). \end{lemma} \begin{proof} Use Euclidean polar coordinates \(z=\rho\omega\), with \(\omega\in S^3\). Homogeneity and \eqref{eq:rational} give the polar form \begin{equation} T_0(\psi) =\int_0^\infty\frac{\dd\rho}{\rho} \Pair{A}{\psi(\rho\,\cdot)}, \qquad A=\frac1{16\pi^4} \bigl((\omega^0)^2-|\boldsymbol\omega|^2-\ii0\bigr)^{-2}. \label{eq:angular} \end{equation} The angular distribution is well defined: at its zeros the argument is a submersion on \(S^3\). The compact sphere has a finite-order distributional estimate for \(A\). Since \(\psi(0)=0\), the function \(\psi(\rho\omega)\) and each fixed finite set of its angular derivatives are \(O(\rho)\), uniformly on the sphere. The integral is therefore absolutely convergent in its radial distributional estimate near zero. It vanishes for sufficiently large \(\rho\). The omitted radial interval contributes \(O(\delta)\), with constants controlled by finitely many test seminorms. Consequently all such cutoffs give the same limit, and the same estimates prove continuity. \end{proof} \subsection{The selected extension} Fix, as part of the prescription, a real \(\varpi\in C_c^\infty(\R^4)\) with \(\varpi(0)=1\). Define \[ W_\varpi\psi=\psi-\psi(0)\varpi. \] \begin{definition}[Selected extension] \label{def:extension} The prescription \(\mathfrak R=\mathfrak R_\varpi\) is \begin{equation} \boxed{\quad \Pair{\bar q^{\mathfrak R_\varpi}}{\psi} =T_0(W_\varpi\psi) =\int_0^\infty\frac{\dd\rho}{\rho} \Pair{A}{ \psi(\rho\,\cdot)-\psi(0)\varpi(\rho\,\cdot)} \quad} \label{eq:extension} \end{equation} for every compact smooth test \(\psi\). \end{definition} \begin{proposition}[Moment, scaling degree, and uniqueness] \label{prop:extension} The formula defines an extension of \(q^\circ\) with scaling degree four, satisfying \[ \Pair{\bar q^{\mathfrak R_\varpi}}{\varpi}=0. \] It is the unique scaling-degree-four extension with this moment. Every other extension of scaling degree at most four has the form \[ \bar q^{\mathfrak R_\varpi}+c\delta,\qquad c\in\mathbb C. \] \end{proposition} \begin{proof} Continuity follows from Lemma~\ref{lem:T0} and continuity of \(W_\varpi\). If a test is supported away from zero, its value at zero vanishes and \eqref{eq:extension} returns the original punctured distribution. Also \(W_\varpi\varpi=0\), giving the stated moment. The radial estimate gives a logarithmic scaling correction: under shrinking of the test, the subtracted action is bounded by a constant times \(\lambda^{-4}(1+|\log\lambda|)\) in fixed test seminorms. Thus its scaling degree is at most four. Its nonzero homogeneous restriction already has degree four, so the degree is exactly four. The difference of any two such extensions is supported at zero. A point-supported distribution is a sum of derivatives of \(\delta\), and \(\sd(\partial^\alpha\delta)=4+|\alpha|\). Only a multiple of \(\delta\) is permitted. The moment fixes that multiple because \(\varpi(0)=1\). This also agrees with the point-extension theorem \cite[Theorem 5.3]{BF}. \end{proof} \begin{proposition}[Changing the subtraction function] \label{prop:changevarpi} For two allowed functions \(\varpi_1,\varpi_2\), \begin{equation} \bar q^{\mathfrak R_{\varpi_2}} -\bar q^{\mathfrak R_{\varpi_1}} =T_0(\varpi_1-\varpi_2)\,\delta. \label{eq:changevarpi} \end{equation} \end{proposition} \begin{proof} Their difference vanishes at zero. Subtracting the two instances of \eqref{eq:extension} proves the identity on every test. \end{proof} A fixed subtraction function is an actual normalization datum. It may encode a coordinate length or shape. No Lorentz covariance, scaling covariance, or compatibility with a separately selected derivative extension follows merely from the zero-moment condition. Derivatives of \(q^\circ\) are not extended here. The extension concerns one point in four relative dimensions. After smooth tangential coefficients are restored, this prescription is a specific extension of the selected sector; it is not a classification of all extensions with arbitrary tangential coefficient functions. \section{Root attachment and the selected-sector functional} \label{sec:attachment} Use throughout the variable order \[ (y,x,w)\in X:=M_y\times M_x\times M_w. \] Define \[ p(y,x,w)=(y,x), \qquad r(y,x,w)=y-w, \qquad \pi_x(y,x,w)=x. \] \begin{theorem}[Pullback, product, and proper pushforward] \label{thm:attachment} Under the root hypotheses \eqref{eq:rootinput}: \begin{enumerate} \item The pullbacks \(p^*B_{f,g,\mu\nu}\) and \(r^*\bar q^{\mathfrak R}\) exist. \item Their products exist on all \(X\), including \(y=w\). \item For smooth \(h\), the distribution \begin{equation} U^{\mathfrak R}(h) =\sum_{\mu,\nu} p^*B_{f,g,\mu\nu}\, r^*\bar q^{\mathfrak R}\, \chi_{\mathrm I}(w)\mathcal K^{\mu\nu}(y,w;h) \label{eq:U} \end{equation} is compactly supported, and \(\pi_x\) is proper on its support. Thus \((\pi_x)_*U^{\mathfrak R}(h)\in\Eprime(M)\) is defined. \end{enumerate} \end{theorem} \begin{proof} Both maps \(p,r\) are submersions, so their pullbacks exist for every distribution on their respective target spaces. The lifted root covectors and relative-kernel covectors have the respective forms \[ (\xi,-\xi,0),\qquad(\kappa,0,-\kappa). \] For the root, \(\xi\ne0\) follows from its \(Z_2\) bound. For the relative kernel, \(\kappa\ne0\), with no restriction on its orientation needed. Their sum is \[ (\xi+\kappa,-\xi,-\kappa). \] Vanishing of the \(x\)- and \(w\)-components forces \(\xi=\kappa=0\), impossible for opposing nonzero wavefront covectors. H\"ormander's product criterion therefore applies, including at coincidence. This argument uses the root wavefront premise, not its compactness. The tensor coefficient is smooth. Multiplication by it and finite component contraction preserve existence. The support is contained in \begin{equation} K_*= \{(y,x,w):(y,x)\in\supp B_{f,g},\ w\in\supp\chi_{\mathrm I}\}, \label{eq:support} \end{equation} which is compact. Compactness in \((y,x)\) comes from the root; compactness in \(w\) comes from the interaction switch. The projection is therefore proper on the support, proving the pushforward claim. \end{proof} The lifted shapes are enough; no complete diagonal wavefront calculation for \(\bar q^{\mathfrak R}\) is required. The root's singular support is retained. Its internal composition is not independently multiplied again. \begin{theorem}[Selected-sector coefficient and continuity] \label{thm:main} Assume Input~\ref{input:all}, choose the ambient propagator \eqref{eq:DF}, and fix \(\mathfrak R_\varpi\). For the \(C_2\)/constant-slot sector only, \begin{equation} \mathcal C_{2e}^{\mathfrak R}(h,\varphi) = \Pair{(\pi_x)_*U^{\mathfrak R}(h)}{\varphi} = \Pair{U^{\mathfrak R}(h)}{1_y\otimes\varphi_x\otimes1_w} \label{eq:C} \end{equation} is defined for every real smooth residual metric \(h\) and real smooth scalar \(\varphi\). It is jointly continuous and bilinear for their \(C^\infty\) topologies. Its values are generally complex. There are compact sets \(K_h,K_\varphi\), a finite integer \(m\), and a constant \(C\) such that \begin{equation} |\mathcal C_{2e}^{\mathfrak R}(h,\varphi)| \le C\,p_{m+2,K_h}(h)\,p_{m,K_\varphi}(\varphi). \label{eq:continuity} \end{equation} The integer \(m\) is not evaluated. The coefficient agrees with the original punctured sector when localized away from \(y=w\). Nonvanishing of \eqref{eq:C} is not asserted. \end{theorem} \begin{proof} Existence and the pairing with noncompact smooth functions follow from Theorem~\ref{thm:attachment} and compact support. Constants may equivalently be replaced by cutoffs equal to one on that support. Fix the compactly supported component distributions \[ T_{\mu\nu} =\chi_{\mathrm I}(w)\, p^*B_{f,g,\mu\nu}\,r^*\bar q^{\mathfrak R}. \] They have some common finite distributional order \(m\) on a compact neighborhood of their supports. Lemma~\ref{lem:Llinear} shows that \(h\mapsto\mathcal K(y,w;h)\) is a linear differential operator of order two with fixed smooth coefficients. Applying the order-\(m\) estimate to \(\mathcal K(y,w;h)\varphi(x)\), and using the product rule, gives \eqref{eq:continuity}. The compact sets may be taken as the corresponding \(w\)- and \(x\)-projections of that neighborhood. The finite-order estimate proves joint continuity and bilinearity. It does not identify distributional order with the pointwise order-two residual jet: the additional \(m\) comes from the fixed attached distributions. Finally, the extension restricts to \(q^\circ\) away from the diagonal, so the localized pairings agree. \end{proof} \section{Propagated ambiguity and limits of the construction} \label{sec:ambiguity} \begin{proposition}[Propagated contact term] \label{prop:ambiguity} If the numerical extension changes by \(c\delta\), then \begin{equation} \Delta\mathcal C_{2e}^{\mathfrak R}(h,\varphi) = c\Pair{B_{f,g,\mu\nu}(y,x)} {\chi_{\mathrm I}(y) \mathcal K^{\mu\nu}(y,y;h)\varphi(x)}. \label{eq:ambiguity} \end{equation} In particular, the change of subtraction function is obtained by taking \(c=T_0(\varpi_1-\varpi_2)\). \end{proposition} \begin{proof} The product of the pulled-back root and \(\delta(y-w)\) satisfies the same noncancellation test as Theorem~\ref{thm:attachment}. Proper integration in \(w\) evaluates the smooth coefficient on \(w=y\), leaving the distributional root pairing in \eqref{eq:ambiguity}. \end{proof} This is a contact ambiguity at a composite source vertex, propagated through the inherited root. It is not identified with a local bulk gravitational counterterm. Locality of the resulting functional is not established. The construction specifies one numerical normalization but does not assemble a retarded interacting coefficient. No Ward identity, interacting BRST differential, BV antibracket, master equation, physical state, or gauge observability statement is obtained. The different frozen class excluded by XI-G is not repaired by this selected-sector construction. No shell-class comparison is needed or asserted here. \section{Verification checklist and status} \label{sec:status} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{0.33\linewidth}Y@{}} \toprule Check & Result and locator\\ \midrule Propagator normalization & \(KD_F=\ii\,\mathrm{id}\): Lemma~\ref{lem:DF}\\ Polarization and edge counting & Ordinary Hessians, insertion factor two, one \(1/2!\): Proposition~\ref{prop:invariant}\\ Fiber signature and off-diagonal factors & Explicit Gram inverse and basis-independent contraction: Lemma~\ref{lem:dual}\\ Hessian table & Closed polynomial proof of every entry: \eqref{eq:Fexpanded}--\eqref{eq:Ltable}\\ Decorated coefficient & Double-sum and quadratic-contraction methods agree: \eqref{eq:secondmethod}--\eqref{eq:Kcompact}\\ Independent component arithmetic & Values \(14,-14,10\): Lemma~\ref{lem:Kchecks}\\ Extension moment & \(W_\varpi\varpi=0\): Proposition~\ref{prop:extension}\\ Prescription change & Explicit constant multiplying \(\delta\): Proposition~\ref{prop:changevarpi}\\ Root product & Conditional on \(Z_2\), not on compactness alone: Theorem~\ref{thm:attachment}\\ Proper pushforward & Root compactness plus compact interaction switch: \eqref{eq:support}\\ Continuity & Distributional order kept separate from residual jet: \eqref{eq:continuity}\\ Nonvanishing & Established only at the stated algebraic levels: Corollary~\ref{cor:nonzero}\\ \bottomrule \end{tabularx} \end{center} \scopebox{Release statement.}{% The \(C_2\)/constant-slot selected-sector coefficient is constructed under the stated root compactness and wavefront hypotheses, for a fixed subtraction function and moment. Its central tensor tables have explicit algebraic proofs and an included exact-arithmetic reproduction procedure. No unavailable local script is a premise. The complete two-edge interaction, its retarded interpretation, and BRST/BV observability remain outside the proved result. } \appendix \section{Exact-arithmetic reproduction of the tables} \label{app:code} \subsection{What is verified} The program below uses symbolic rational arithmetic. It contains no floating-point constants. It verifies the following independent paths: \begin{enumerate} \item The Hessian is computed from the finite curvature atoms at the residual jet \(h=0\), \(\partial h=0\), \(\partial_1^2h=2\eta\). It is compared with the closed trace polynomial \eqref{eq:F}. \item The source is computed both through the four-index coefficient \(C_2\) and through the matrix expression \eqref{eq:Qmatrix}. \item The decorated tensor is contracted both by direct double polarization and by differentiating the quadratic source after the dual-edge substitution. \end{enumerate} The origin-jet calculation suffices for the entire background \(h_*\) because Lemma~\ref{lem:Llinear} proves that the constant-slot Hessian depends only on its constant second derivative. The program is not an additional mathematical assumption: the preceding polynomial proofs establish the tables. The listing requires Python 3 and SymPy. Its mathematical operations are rational sums, products, symbolic differentiation, and polynomial expansion. No execution of the listing is represented in this manuscript. \subsection{Complete verification program} \begin{lstlisting} from itertools import product import sympy as s R = s.Rational eta = s.diag(1, -1, -1, -1) pairs = [(i, j) for i in range(4) for j in range(i, 4)] B = [] for i, j in pairs: X = s.zeros(4) X[i, j] = 1 X[j, i] = 1 B.append(X) def tr(X): return s.trace(X) def zero_matrix(): return s.zeros(4) def lincomb(coefficients, matrices): return sum((c * X for c, X in zip(coefficients, matrices)), zero_matrix()) def P(X): return X - eta * tr(eta * X) / 2 G = s.Matrix(10, 10, lambda i, j: tr(eta * B[i] * eta * B[j])) Gi = G.inv() dual = [lincomb([Gi[i, j] for j in range(10)], B) for i in range(10)] Z = [-P(X) for X in dual] assert G == s.diag(1, -2, -2, -2, 1, 2, 2, 1, 2, 1) a = s.symbols("a0:10") H = lincomb(a, B) u = [eta, -eta * H * eta, eta * H * eta * H * eta] b1 = tr(eta * H) / 2 b2 = tr(eta * H)**2 / 8 - tr(eta * H * eta * H) / 4 # Curvature-atom computation at the residual jet: # h(0)=H, first derivatives vanish, h_{,11}=2*eta. # Since Gamma(0)=0, products of Christoffel symbols vanish. def second_jet(r, m, n, ell): return 2 * eta[n, ell] if r == 1 and m == 1 else 0 def dGamma(n, r, rho, m, v): return sum( u[n-1][rho, ell] * ( second_jet(r, m, v, ell) + second_jet(r, v, m, ell) - second_jet(r, ell, m, v) ) / 2 for ell in range(4) ) Ric = {} for n in (1, 2, 3): Ric[n] = s.Matrix(4, 4, lambda m, v: sum(dGamma(n, rho, rho, m, v) - dGamma(n, v, rho, m, rho) for rho in range(4))) rscalar = {} for n in (1, 2, 3): rscalar[n] = s.expand(sum( u[i][m, v] * Ric[n-i][m, v] for i in range(n) for m, v in product(range(4), repeat=2) )) ell_at_jet = s.expand(-2 * ( rscalar[3] + b1 * rscalar[2] + b2 * rscalar[1] )) L_atoms = s.hessian(ell_at_jet, a) N = eta * H F = s.expand(-4 * (N*N)[1, 1] - tr(N*N) + tr(N)**2 / 2) L_trace = s.hessian(F, a) assert s.expand(ell_at_jet - F) == 0 assert L_atoms == L_trace L = L_atoms L_expected = s.Matrix([ [-1,0,0,0,-1,0,0,-1,0,-1], [0,12,0,0,0,0,0,0,0,0], [0,0,4,0,0,0,0,0,0,0], [0,0,0,4,0,0,0,0,0,0], [-1,0,0,0,-9,0,0,1,0,1], [0,0,0,0,0,-12,0,0,0,0], [0,0,0,0,0,0,-12,0,0,0], [-1,0,0,0,1,0,0,-1,0,1], [0,0,0,0,0,0,0,0,-4,0], [-1,0,0,0,1,0,0,1,0,-1] ]) assert L == L_expected # A general symmetric covariant source jet. js = s.symbols("J00 J01 J02 J03 J11 J12 J13 J22 J23 J33") J = lincomb(js, B) # Source method 1: explicit four-index C2. def source_indices(X): U1 = -eta * X * eta U2 = eta * X * eta * X * eta t = tr(eta * X) c1 = t / 2 c2 = t*t / 8 - tr(eta * X * eta * X) / 4 def cross(U, V, m, n, p, q): return U[m,n]*V[p,q] - ( U[m,p]*V[q,n] + U[m,q]*V[p,n] ) / 2 def coefficient(m, n, p, q): return ( c2 * cross(eta, eta, m,n,p,q) + c1 * (cross(U1, eta, m,n,p,q) + cross(eta, U1, m,n,p,q)) + cross(U2, eta, m,n,p,q) + cross(eta, U2, m,n,p,q) + cross(U1, U1, m,n,p,q) ) return s.Matrix(4, 4, lambda m, n: s.expand(2 * sum( coefficient(m,n,p,q) * J[p,q] for p, q in product(range(4), repeat=2) ))) # Source method 2: independent matrix expression. def source_matrix(X): V = eta * X * eta W = eta * X * eta * X * eta t = tr(eta * X) c1 = t / 2 c2 = t*t / 8 - tr(eta * X * eta * X) / 4 c, k, j = tr(eta*J), tr(V*J), tr(W*J) result = 2 * ( c2 * (eta*c - eta*J*eta) + c1 * (-V*c - eta*k + V*J*eta + eta*J*V) + W*c + eta*j + V*k - W*J*eta - eta*J*W - V*J*V ) return result.applyfunc(s.expand) assert (source_indices(H) - source_matrix(H)).applyfunc( s.expand) == s.zeros(4) # Direct double polarization, with dual edge vectors. Qsingle = [source_indices(X) for X in Z] K_pair = s.zeros(4) for i in range(10): for j in range(i, 10): if L[i, j] == 0: continue polar = ( source_indices(Z[i] + Z[j]) - Qsingle[i] - Qsingle[j] ) / 2 multiplicity = 1 if i == j else 2 K_pair += multiplicity * L[i, j] * polar K_pair = K_pair.applyfunc(s.expand) # Independent contraction by a quadratic differential operator. Hedge = lincomb(a, Z) Qedge = source_matrix(Hedge) origin = {x: 0 for x in a} K_diff = s.Matrix(4, 4, lambda m, n: s.expand(sum( L[i,j] * s.diff(Qedge[m,n], a[i], a[j]).subs(origin) / 2 for i, j in product(range(10), repeat=2) ))) assert K_pair == K_diff j00,j01,j02,j03,j11,j12,j13,j22,j23,j33 = js K_expected = s.Matrix([ [14*j11+10*j22+10*j33, -14*j01, -10*j02, -10*j03], [-14*j01, 14*(j00-j22-j33), 14*j12, 14*j13], [-10*j02, 14*j12, 10*j00-14*j11-10*j33, 10*j23], [-10*j03, 14*j13, 10*j23, 10*j00-14*j11-10*j22] ]) assert K_pair == K_expected print("L:") for row in L.tolist(): print(row) print("K upper triangular:") for m, n in pairs: print((m,n), s.expand(K_pair[m,n])) print("All exact polynomial checks passed.") \end{lstlisting} \subsection{Complete expected mathematical output} The Hessian output is the full matrix \eqref{eq:Ltable}. The ten upper-triangular tensor entries are: \begin{lstlisting}[language={}] L: [-1, 0, 0, 0, -1, 0, 0, -1, 0, -1] [0, 12, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 4, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 4, 0, 0, 0, 0, 0, 0] [-1, 0, 0, 0, -9, 0, 0, 1, 0, 1] [0, 0, 0, 0, 0, -12, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, -12, 0, 0, 0] [-1, 0, 0, 0, 1, 0, 0, -1, 0, 1] [0, 0, 0, 0, 0, 0, 0, 0, -4, 0] [-1, 0, 0, 0, 1, 0, 0, 1, 0, -1] K upper triangular: (0, 0) 14*J11 + 10*J22 + 10*J33 (0, 1) -14*J01 (0, 2) -10*J02 (0, 3) -10*J03 (1, 1) 14*J00 - 14*J22 - 14*J33 (1, 2) 14*J12 (1, 3) 14*J13 (2, 2) 10*J00 - 14*J11 - 10*J33 (2, 3) 10*J23 (3, 3) 10*J00 - 14*J11 - 10*J22 All exact polynomial checks passed. \end{lstlisting} This is the expected output determined by the proved identities, not a report of a computational run performed for this revision. \section{Version and correction notes} \label{app:versions} The supplied original XVI-J cited an unavailable local script as the proof of the Hessian and decorated tensor. That dependency is removed. The closed quadratic polynomial, the complete contraction tables, and the included verification program replace it. The historical correction ledger referred to an earlier XVI-I draft. The revised XVI-I supplied in this conversation already separates numerical relative kernels from smooth coefficients, includes insertion-source derivatives in the fuller power count, and proves compressed attachment under the same root cone premise. It is therefore not described here as still stopping at an unnamed root-product domain. The present narrowing is intentional: none of those fuller derivative sectors is added to XVI-J by implication. The constructed kernel remains the undifferentiated square. Bibliographic versions of manuscript sources below refer only to the supplied text and excerpts. A companion retrieval summary is not treated as verification of an entire manuscript or as a source of extra formal premises. \begin{thebibliography}{99} \bibitem{XIIC} N.~Cheung, \emph{Landmark XII-C}, supplied manuscript, version 1. Source identifier supplied with the excerpts: \texttt{e0611fd7}. Locators: \texttt{eq:finite-curvature}, \texttt{eq:ell3}, \texttt{eq:Ccoefficients}, and \texttt{thm:actualsource}. The declared density and source are reproduced as finite definitions in this manuscript. \bibitem{XIID} N.~Cheung, \emph{Landmark XII-D}, supplied manuscript, version 1. Source identifier supplied with the excerpts: \texttt{e3e3f67d}. Root interface: \texttt{lem:root-kernel}, as quoted in the supplied XVI-J; related compact-root locator: \texttt{thm:root}. Compactness and the \(Z_2\) bound are explicit assumptions, not results independently proved here. \bibitem{XIB} N.~Cheung, \emph{Landmark XI-B}, supplied manuscript, version 1. Locators: \texttt{eq:ER}, \texttt{eq:EA}, \texttt{lem:Green}, and \texttt{thm:graviton}. The ambient prescription and its slab restriction are explicitly specified in Section~\ref{sec:propagator}. \bibitem{XIE} N.~Cheung, \emph{Landmark XI-E}, supplied manuscript, version 1. Locators: \texttt{eq:W}, \texttt{eq:h}, \texttt{eq:Heven}, and \texttt{eq:antisym}. The massless normalization \(h_0=-W_0\) is retained. \bibitem{XVIH} N.~Cheung, \emph{Landmark XVI-H}, supplied revised manuscript. Locators: the compact polarized representation and diagonal quadratic normalization. Used only for the insertion normalization stated in Input~\ref{input:all}. \bibitem{XVII} N.~Cheung, \emph{Landmark XVI-I}, revised manuscript supplied in this conversation. Locators: the tensor derivatives of the insertion, the explicit ambient propagator, and the compressed root-attachment test. No selected normalization of its full extension family is imported into the narrower sector here. \bibitem{BF} R.~Brunetti and K.~Fredenhagen, \emph{Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds}, Communications in Mathematical Physics \textbf{208} (2000), 623--661. arXiv:math-ph/9903028. Sections 5.1--5.2, especially Theorems 5.2--5.3, give the point-extension framework. The extension action used here is specified explicitly in \eqref{eq:extension}. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}, arXiv:1208.1428v2. Section 7, Theorem 7.2, and equation (82). Only the point-extension background and identical-edge factorial convention are used; a full time-ordered family is not transferred. \bibitem{BDH} C.~Brouder, N.~V.~Dang, and F.~H\'elein, \emph{A smooth introduction to the wavefront set}, Journal of Physics A: Mathematical and Theoretical \textbf{47} (2014), 443001. arXiv:1404.1778. Theorem 13 and Proposition 26 give the distributional product criterion and the Feynman wavefront description. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I}, second edition, Springer, 1990. Chapter VIII: wavefront sets, pullback under smooth maps, distributional products, and proper pushforward. \end{thebibliography} \end{document}