\documentclass{article} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1in]{geometry} \usepackage{longtable} \usepackage{listings} \lstset{basicstyle=\ttfamily\scriptsize,breaklines=true,columns=fullflexible,keepspaces=true} \newtheorem{proposition}{Proposition} \newtheorem{theorem}[proposition]{Theorem} \newtheorem{lemma}[proposition]{Lemma} \newtheorem{definition}[proposition]{Definition} \theoremstyle{remark} \newtheorem{remark}[proposition]{Remark} \newcommand{\scopebox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{0.92\textwidth} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \newcommand{\RK}{\mathfrak R_{\mathrm K}} \newcommand{\RL}{\mathfrak R_{\mathrm L}} \newcommand{\RLmu}{\mathfrak R_{\mathrm L,\mu}} \newcommand{\RLmuref}{\mathfrak R_{\mathrm L,\mu_0}} \newcommand{\Pc}{\mathcal P} \newcommand{\sk}{\mathsf k} \newcommand{\JD}{\mathcal J_{\mathrm D}} \newcommand{\Ds}{\mathcal D_\sigma} \newcommand{\Eu}{\mathcal E} \newcommand{\sigeps}{\sigma_\varepsilon} \title{Landmark XVI-L\\ \large Complete-source two-edge renormalization, local collision counterterms, and scale dependence\\ \large Draft --- Release Status: XVI-L constructs the complete-source contribution for the specified two-edge graph, conditional on the imported root hypotheses, using a coherent extension on the fixed admissible numerical span. It characterizes coherent collision freedom abstractly and constructs a smooth subtraction-scale family. All-sector contraction is computationally verified as reported on the stated residual-metric background. L-D is not claimed. L-A/L-C Constructed; L-B abstract finite-dimensional characterization} \author{} \date{} \usepackage{tikz} \begin{document} \setlength{\emergencystretch}{3em} \maketitle \scopebox{Status (XVI-L, this draft).}{% Overall release: \textbf{XVI-L constructs the complete-source contribution for the specified two-edge graph, conditional on the imported root hypotheses, using a coherent extension on the fixed admissible numerical span. It characterizes coherent collision freedom abstractly and constructs a smooth subtraction-scale family. All-sector contraction is computationally verified as reported on the stated residual-metric background. L-D is not claimed.} L-A and L-C of Section~\ref{sec:ladder}: \textbf{Constructed}, conditional on (R1)--(R3). L-B: \textbf{abstract finite-dimensional characterization} (all numerical-kernel relations; selected constraints exhibited; raw contact counts are not dimensions). L-D is \textbf{not} claimed. L.0: \textbf{Recorded}. L.1--L.2: \textbf{constructed-as-stated} with complete-graph contraction computationally verified (Sections~\ref{sec:l1}--\ref{sec:l2}). L.3--L.5: \textbf{constructed-as-stated} with inventory/coherence repairs as recorded (Sections~\ref{sec:l3}--\ref{sec:l5}). L.6 (renormalization scale): \textbf{constructed-as-stated} --- scale family $\RLmu$, first scale theorem, infinitesimal generator $A_\mu$, prescription versus scale-variation-coefficient distinction \textbf{Proved locally}; explicit nonzero scaling residue (anomaly form) \textbf{Unresolved} (Section~\ref{sec:l6}). L.7 (root attachment and exact regressions): \textbf{constructed-as-stated} --- $U^{\RL}_{\mathrm L}$, $\mathcal C^{\RL}_{\mathrm L}$, source-sector K regression and separate restriction $\RL|_{\mathcal V_{\mathrm K}}=\RK$, propagated $\Delta\mathcal C_{\mathrm L}$ and scale variation through $B_{f,g}$ \textbf{Proved locally}; truncation regressions \textbf{Computationally verified} (Section~\ref{sec:l7}). L.8 (bridge to gravitational EFT): \textbf{Recorded as matching questions only} --- short matching section and BV boundary; L-D remains \textbf{not} claimed (Section~\ref{sec:l8}). } \scopebox{Scope (XVI-L).}{% Ambient \(M=\mathbb R^4\), \(\eta=\mathrm{diag}(1,-1,-1,-1)\), \(\Pc r=r-\tfrac12\eta\,\mathrm{tr}_\eta r\), \(\Pc^2=\mathrm{id}\); fixed \(0<\varepsilon] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One complete source is not the one-loop correction. 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-L} \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) {Imported cubic source}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Complete one-loop correction}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Completing the source does not complete the diagrammatic content of the theory.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{One complete source is not the one-loop correction. 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-L} \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) {Imported cubic source}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {One two-edge graph}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Complete one-loop correction}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Completing the source does not complete the diagrammatic content of the theory.}; \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-L} \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) {Imported cubic source}; \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 one-loop correction}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {One two-edge graph. Completing the source does not complete the diagrammatic content of the theory.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Imported cubic source is not Complete one-loop correction. 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-L} \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: Imported cubic source}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: One two-edge graph}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Complete one-loop correction}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Completing the source does not complete the diagrammatic content of the theory.}; \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-L} \end{figure} Figure~\ref{fig:split-XVI-L} records one view of the same separation. Figure~\ref{fig:chain-XVI-L} records one view of the same separation. Figure~\ref{fig:pair-XVI-L} records one view of the same separation. Figure~\ref{fig:gate-XVI-L} records one view of the same separation. The drawings add no hypothesis. Figure~\ref{fig:scope-XVI-L} records that separation. The drawing adds no hypothesis. \section{L.0 Frozen XVI-K dependency and target ledger} \label{sec:l0} \subsection{What is imported from XVI-K, and how} XVI-K.tex is treated as a frozen dependency: it is cited, not edited. Its release is ``Constructed, conditional on the imported root hypotheses, for the \(C_2\) sector with full cubic vertex'' (XVI-K.tex, title and Section sec:status). The following items are imported \emph{as stated there}; nothing beyond the statements is used. \begin{enumerate} \item \emph{Edge kernel.} \(S:=h_0+iE_0^{\mathrm A}=-\Delta_F^{\mathrm{FR}}\), \(D_F=-\Pc S\) (XVI-K, eq:S), a globally defined distribution of \(z=y-w\), homogeneous of degree \(-2\) (XVI-K, lem:homog). \item \emph{Full cubic Hessian.} XVI-K, Proposition prop:hess, eq:Lpq: \[ D^2\ell_3(h)[A,C](w)=\sum_{i,j=0}^{9}\sum_{|p|+|q|\le2}L^{pq}_{ij}(w;h)\,\partial^pA^i(w)\,\partial^qC^j(w), \] with \(L^{pq}_{ij}\) linear in \(\partial^rh\), \(|r|=2-|p|-|q|\), and \(L^{pq}_{ij}=L^{qp}_{ji}\). \item \emph{Placement sign.} XVI-K, eq:sign: \(\partial_w^p[u(y-w)]=(-1)^{|p|}(\partial^pu)(y-w)\), \(\partial_y^p[u(y-w)]=(\partial^pu)(y-w)\). \item \emph{Extension map.} XVI-K, Definition def:RK and Proposition prop:RK: the linear map \(\RK:\mathcal V_{\mathrm K}\to\mathcal D'(\mathbb R^4)\) built with the \(\varpi\)-projections \(W_\rho\) (XVI-K, eq:Wrho) and zero declared moments (XVI-K, eq:moments). \item \emph{Root attachment.} XVI-K, Lemma lem:attachK and Theorem thm:main, conditional on (R1)--(R3). \item \emph{Propagated ambiguity.} XVI-K, Lemma lem:amb, eq:deltaCK. \end{enumerate} \subsection{Required closure items, recorded} \paragraph{(a) Coherent versus placementwise changes.} XVI-K, Remark rem:coherent, distinguishes two levels, and XVI-L keeps them apart: \emph{placementwise} families (a separate same-degree extension for each labelled kernel \(q^\circ_{\sk}\); XVI-K, eq:deltaCK holds at that level), and \emph{coherent} changes \(\RK'-\RK\), linear on the actual span \(\mathcal V_{\mathrm K}\), for which every constant-coefficient relation \(\sum_{\sk}c_{\sk}q^\circ_{\sk}=0\) forces \(\sum_{\sk}c_{\sk}C_{\sk\gamma}=0\) for every \(\gamma\). Only coherent changes will be called renormalization-prescription changes in XVI-L (target L.4--L.5). Within coherent changes, XVI-K distinguishes those altering the XVI-J component (\(C_{\{0,0\},0}\ne0\)) from those preserving it (\(C_{\{0,0\},0}=0\)). \paragraph{(b) Declared subtraction orders.} XVI-K, Remark rem:N: \(N_{\sk}:=|p|+|q|\) is a \emph{declared} order, not defined through the scaling degree; \(\mathrm{sd}(q^\circ_{\sk})\le4+N_{\sk}\), with equality for nonzero \(q^\circ_{\sk}\), and a vanishing kernel costs nothing. XVI-L adopts the same convention for the new kernels: the declared order of a numerical kernel is its total numerical derivative count, fixed before any question of vanishing is decided (target L.3). \paragraph{(c) Computational artifact.} The XVI-K artifact is the folder \texttt{XVI-K-artifacts/} (\texttt{hessian\_full.py}, \texttt{xvi\_common.py}, \texttt{recompute\_K\_dual.py}, \texttt{hessian\_full\_out.txt}); the shared routines (\texttt{negP}, \texttt{polar\_s}, \texttt{Q\_of}, \texttt{C2\_component}, \texttt{ell3\_of}) are named functions of \texttt{xvi\_common.py} (XVI-K, Appendix app:script). XVI-L does not modify that folder. The XVI-L artifact is the folder \texttt{XVI-L-artifacts/}, which contains a copy of \texttt{xvi\_common.py} (byte-identical; scripts check this when the original is present), \texttt{source\_full.py} with \texttt{source\_full\_out.txt}/\texttt{source\_full\_table.txt} (L.1), \texttt{two\_edge\_full.py} with \texttt{two\_edge\_full\_out.txt} (L.2), \texttt{kernel\_inventory.py} with \texttt{kernel\_inventory\_out.txt} (L.3), \texttt{counterterm\_map.py} with \texttt{counterterm\_map\_out.txt} (L.4--L.5), and \texttt{regression\_L.py} with \texttt{regression\_L\_out.txt} (L.6--L.7) (Appendix~\ref{app:script}). \paragraph{(d) Verification boundary.} Three kinds of evidence are distinguished throughout, as in XVI-K, Appendix app:script: \emph{symbolic constructions} (exact polynomial identities in independent jet coordinates, valid for all smooth fields); \emph{coefficientwise structural checks} (properties verified monomial by monomial on such a symbolic polynomial); \emph{finite exact tests} (exact rational evaluation on specified test data; agreement there is not a coefficientwise equivalence proof). A statement is called ``Proved locally'' only if it has a written argument in this note; a statement supported only by the script is called ``Computationally verified'' and its kind of evidence is named. \paragraph{(e) Archive.} Run commands (from any working directory; package versions are printed in the first line of each output): \texttt{/workspace/tmp/venv/bin/python XVI-L-artifacts/source\_full.py > XVI-L-artifacts/source\_full\_out.txt}; \texttt{/workspace/tmp/venv/bin/python XVI-L-artifacts/two\_edge\_full.py > XVI-L-artifacts/two\_edge\_full\_out.txt}; \texttt{/workspace/tmp/venv/bin/python XVI-L-artifacts/kernel\_inventory.py > XVI-L-artifacts/kernel\_inventory\_out.txt}; \texttt{/workspace/tmp/venv/bin/python XVI-L-artifacts/counterterm\_map.py > XVI-L-artifacts/counterterm\_map\_out.txt}; \texttt{/workspace/tmp/venv/bin/python XVI-L-artifacts/regression\_L.py > XVI-L-artifacts/regression\_L\_out.txt} (Appendix~\ref{app:out}). \subsection{No hidden transfer} No BV, BRST, Ward-identity, or all-orders renormalization result is transferred to XVI-L. The locally covariant BV framework and the EFT reading of gravity are background orientation only; neither is a premise of any statement below, and neither promotes one graph coefficient to a theory. No statement of XVI-K about the \(C_2\) sector is upgraded by XVI-L: XVI-K's release stays exactly as stated there. \subsection{Target ledger and release ladder} \label{sec:ladder} \begin{center} \begin{longtable}{p{0.10\textwidth}p{0.50\textwidth}p{0.30\textwidth}} \hline Level & Release claim & Status in this draft\\ \hline L-A & Complete-source, full-vertex selected extension and root-attached coefficient constructed & \textbf{Constructed}, conditional on (R1)--(R3)\\ L-B & Finite-dimensional coherent contact freedom characterized abstractly by all numerical-kernel relations; selected constraints exhibited & \textbf{Abstract characterization} (raw counts 14321/321/14000 are not dimensions)\\ L-C & Scale-dependent prescription family and local infinitesimal scale variation constructed & \textbf{Constructed}, conditional on (R1)--(R3) (explicit nonzero scaling residue Unresolved)\\ L-D & Particular counterterms matched to local gravitational or insertion functionals --- only where proved & \textbf{Not claimed} (L.8 records matching questions only; no match proved)\\ \hline \end{longtable} \end{center} \begin{center} \begin{longtable}{p{0.08\textwidth}p{0.52\textwidth}p{0.30\textwidth}} \hline Gate & Content & Status\\ \hline L.0 & Frozen XVI-K dependency and target ledger & Recorded (this section)\\ L.1 & Complete source as a jet polynomial; derivative budgets & Done: Proved locally (structure) and Computationally verified (Section~\ref{sec:l1})\\ L.2 & Complete graph decomposition \(Q^{\circ\,\mu\nu}_{\mathrm L}=\sum_\lambda a^{\mu\nu}_\lambda q^\circ_\lambda\); term ledger; normalization gate & Done / constructed-as-stated (Section~\ref{sec:l2})\\ L.3 & Product existence, power counting, kernel inventory \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\) & Done / constructed-as-stated (Section~\ref{sec:l3})\\ L.4 & Coherent extension \(\RL\) with \(\RL|_{\mathcal V_{\mathrm K}}=\RK\); defect ledger & Constructed-as-stated (Section~\ref{sec:l4})\\ L.5 & Classification of local collision counterterms before attachment & Constructed-as-stated (Section~\ref{sec:l5})\\ L.6 & Renormalization scale and infinitesimal scale variation & Constructed-as-stated (Section~\ref{sec:l6}); residue Unresolved\\ L.7 & Root attachment, exact K/J regressions, propagated counterterms & Constructed-as-stated (Section~\ref{sec:l7})\\ L.8 & Bridge to gravitational EFT (matching questions only) & Recorded (Section~\ref{sec:l8}); L-D not claimed\\ \hline \end{longtable} \end{center} %===================================================================== \section{L.1 The complete source as a jet polynomial} \label{sec:l1} No source derivative count is assumed in this section. The printed formulas are quoted, implemented atom by atom, and the derivative budget is then \emph{derived}. \subsection{The printed formulas, quoted} XII-C.tex, Section sec:density (lines above eq:finite-curvature): with \(\tau=\mathrm{tr}_\eta\gamma\), \[ u_0^{\mu\nu}=\eta^{\mu\nu},\quad u_1^{\mu\nu}=-\gamma^{\mu\nu},\quad u_2^{\mu\nu}=\gamma^\mu{}_\lambda\gamma^{\lambda\nu},\qquad b_0=1,\quad b_1=\tfrac12\tau,\quad b_2=\tfrac18\tau^2-\tfrac14\gamma_{\mu\nu}\gamma^{\mu\nu}. \] XII-C.tex, eq:finite-curvature: for \(n=1,2,3\), \begin{align} \Gamma_n^\rho{}_{\mu\nu}&=\tfrac12u_{n-1}^{\rho\lambda}\bigl(\partial_\mu\gamma_{\nu\lambda}+\partial_\nu\gamma_{\mu\lambda}-\partial_\lambda\gamma_{\mu\nu}\bigr),\notag\\ R_{n,\mu\nu}&=\partial_\rho\Gamma_n^\rho{}_{\mu\nu}-\partial_\nu\Gamma_n^\rho{}_{\mu\rho} +\sum_{\substack{a+b=n\\a,b\ge1}}\bigl(\Gamma_a^\rho{}_{\rho\lambda}\Gamma_b^\lambda{}_{\mu\nu}-\Gamma_a^\rho{}_{\nu\lambda}\Gamma_b^\lambda{}_{\mu\rho}\bigr),\qquad r_n=\sum_{i=0}^{n-1}u_i^{\mu\nu}R_{n-i,\mu\nu}. \label{eq:L-fc} \end{align} XII-C.tex, eq:ell3: \begin{equation} \ell_3=-2(r_3+b_1r_2+b_2r_1). \label{eq:L-ell3} \end{equation} XII-C.tex, the definitions printed before eq:J3, and eq:J3: \begin{equation} V_{1,\mu\nu}=R_{1,\mu\nu}-\tfrac12\eta_{\mu\nu}r_1,\qquad V_{2,\mu\nu}=R_{2,\mu\nu}-\tfrac12\eta_{\mu\nu}r_2-\tfrac12\gamma_{\mu\nu}r_1, \qquad J_3^{\mu\nu}=2\bigl(V_2^{\mu\nu}+b_1V_1^{\mu\nu}-\gamma^{\mu\alpha}V_{1,\alpha}{}^\nu-\gamma^{\nu\alpha}V_{1,\alpha}{}^\mu\bigr). \label{eq:L-J3} \end{equation} XII-C.tex, eq:Ccoefficients (with \(C^{\mu\nu\alpha\beta}(q)=\sqrt{-q}(q^{\mu\nu}q^{\alpha\beta}-q^{\mu(\alpha}q^{\beta)\nu})=C_0+\kappa C_1+\kappa^2C_2+O(\kappa^3)\)): \begin{align} C_0^{\mu\nu\alpha\beta}&=\eta^{\mu\nu}\eta^{\alpha\beta}-\eta^{\mu(\alpha}\eta^{\beta)\nu},\notag\\ C_1^{\mu\nu\alpha\beta}&=b_1C_0^{\mu\nu\alpha\beta}+u_1^{\mu\nu}\eta^{\alpha\beta}+\eta^{\mu\nu}u_1^{\alpha\beta}-u_1^{\mu(\alpha}\eta^{\beta)\nu}-\eta^{\mu(\alpha}u_1^{\beta)\nu},\notag\\ C_2^{\mu\nu\alpha\beta}&=b_2C_0^{\mu\nu\alpha\beta}+b_1\bigl(u_1^{\mu\nu}\eta^{\alpha\beta}+\eta^{\mu\nu}u_1^{\alpha\beta}-u_1^{\mu(\alpha}\eta^{\beta)\nu}-\eta^{\mu(\alpha}u_1^{\beta)\nu}\bigr)\notag\\ &\quad+u_2^{\mu\nu}\eta^{\alpha\beta}+\eta^{\mu\nu}u_2^{\alpha\beta}+u_1^{\mu\nu}u_1^{\alpha\beta} -u_2^{\mu(\alpha}\eta^{\beta)\nu}-\eta^{\mu(\alpha}u_2^{\beta)\nu}-u_1^{\mu(\alpha}u_1^{\beta)\nu}. \label{eq:L-C} \end{align} XII-C.tex, eq:M2: \begin{equation} 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}. \label{eq:L-M2} \end{equation} XII-C.tex, Theorem thm:actualsource, eq:actualsource: the selected cubic vertex \(V_{3,\mathrm D}[\gamma]=\int\sigma\ell_3[\gamma]\,\mathrm d^4x\) has Euler source \begin{equation} \JD^{\mu\nu}=\sigma J_3^{\mu\nu}+\Ds^{\mu\nu},\qquad \Ds^{\mu\nu}=2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sigma-2M_2^{\mu\nu\rho}\partial_\rho\sigma . \label{eq:L-JD} \end{equation} XII-C.tex, eq:full-trace-sector, and XII-B (quoted in XII-C.tex as eq:input-trace): for \(\gamma=\psi\eta\), \(C_1=C_2=0\), \(\Gamma_2=-\psi\Gamma_1\), and \begin{equation} \Ds^{\mu\nu}\big|_{\gamma=\psi\eta}=-\psi\bigl[\eta^{\mu\nu}(\partial_\rho\sigma)\partial^\rho\psi+(\partial^\mu\sigma)\partial^\nu\psi+(\partial^\nu\sigma)\partial^\mu\psi\bigr], \qquad \eta_{\mu\nu}\Ds^{\mu\nu}\big|_{\gamma=\psi\eta}=-6(\partial_\mu\sigma)\psi\partial^\mu\psi . \label{eq:L-trace} \end{equation} In XVI-L, \(\sigma=\sigeps\) is the fixed ramp of XII-C (Section sec:freeze); it is not varied. \begin{remark}[How \(J_3\) is specified, and how it is implemented] \label{rem:J3impl} Equation \eqref{eq:L-J3} specifies \(J_3\) only implicitly: through \(V_1\), \(V_2\), which are built from the curvature atoms \eqref{eq:L-fc}. XII-C prints no table of elementary monomials of \(J_3\), and none is typed into XVI-L. The script \texttt{source\_full.py} implements \(J_3\) through exactly these printed definitions (atoms \(\to\) \(R_n\), \(r_n\) \(\to\) \(V_1,V_2\) \(\to\) \eqref{eq:L-J3}); the coefficient table \(P_{J_3}\) below is an \emph{output} of that implementation, never an input. The same holds for \(C_2\) (via \eqref{eq:L-C}) and \(M_2\) (via \eqref{eq:L-C}, \eqref{eq:L-fc}, \eqref{eq:L-M2}). \end{remark} \subsection{The three source operators and their polarization} \begin{definition}[Unpolarized and polarized source summands] \label{def:sA} For a smooth symmetric \(\gamma\) put \[ \mathfrak s_{J_3}^{\mu\nu}(\gamma):=\sigeps J_3^{\mu\nu}[\gamma],\qquad \mathfrak s_{C_2}^{\mu\nu}(\gamma):=2C_2^{\mu\nu\alpha\beta}[\gamma]\,\partial_\alpha\partial_\beta\sigeps,\qquad \mathfrak s_{M_2}^{\mu\nu}(\gamma):=-2M_2^{\mu\nu\rho}[\gamma]\,\partial_\rho\sigeps, \] so that \(\JD=\mathfrak s_{J_3}+\mathfrak s_{C_2}+\mathfrak s_{M_2}\) by \eqref{eq:L-JD}. For \(A\in\{C_2,M_2,J_3\}\) and smooth symmetric \(X,Y\), the polarization is XII-D's rule, as used in XVI-K, Definition def:a: \[ \mathfrak s_A^{\mu\nu}[X,Y]:=\tfrac12\bigl(\mathfrak s_A^{\mu\nu}(X+Y)-\mathfrak s_A^{\mu\nu}(X)-\mathfrak s_A^{\mu\nu}(Y)\bigr). \] \(X\) and \(Y\) are the two contracted field slots (the increments carried by the two edges in L.2). No residual metric \(h\) enters \(\mathfrak s_A\): the source is quadratic in the slot fields alone, and \(h\) enters the graph only through the vertex \(L^{pq}_{ij}(w;h)\). \end{definition} \begin{proposition}[Complete source as a jet polynomial] \label{prop:jet} For each \(A\in\{C_2,M_2,J_3\}\) there are unique coefficients \(P^{ab,\mu\nu}_{A,ij}(y)\), indexed by \(0\le i,j\le9\) and multi-indices \(a,b\), such that \begin{equation} \mathfrak s_A^{\mu\nu}[X,Y](y)=\sum_{i,j=0}^{9}\sum_{a,b}P_{A,ij}^{ab,\mu\nu}(y)\,\partial^aX^i(y)\,\partial^bY^j(y) \label{eq:jet} \end{equation} for all smooth symmetric \(X,Y\), read as an identity of polynomials in the independent jet coordinates of \(X\) and \(Y\). Moreover: \begin{enumerate} \item \emph{Quadratic homogeneity.} Every monomial of the unpolarized \(\mathfrak s_A(\gamma)\) has degree exactly \(2\) in the jet of \(\gamma\); hence \(\mathfrak s_A(\lambda\gamma)=\lambda^2\mathfrak s_A(\gamma)\) for \(\lambda\in\mathbb R\). \item \emph{Bilinearity and diagonal.} \(\mathfrak s_A[X,Y]\) is bilinear, and \(\mathfrak s_A[X,X]=\mathfrak s_A(X)\). \item \emph{Slot symmetry.} \(P^{ab,\mu\nu}_{A,ij}=P^{ba,\mu\nu}_{A,ji}\); also \(P^{ab,\mu\nu}_{A,ij}=P^{ab,\nu\mu}_{A,ij}\). \item \emph{Distribution of derivatives (exact, not only a maximum).} \begin{itemize} \item \(A=C_2\): \(P_{C_2}\) is a linear form in \(\partial\partial\sigeps(y)\), and \(P^{ab}_{C_2,ij}\ne0\) only for \(|a|=|b|=0\). \item \(A=M_2\): \(P_{M_2}\) is a linear form in \(\partial\sigeps(y)\), and \(P^{ab}_{M_2,ij}\ne0\) only for \(|a|+|b|=1\). \item \(A=J_3\): \(P_{J_3}\) is a constant multiple of \(\sigeps(y)\), and \(P^{ab}_{J_3,ij}\ne0\) only for \(|a|+|b|=2\). \end{itemize} In every monomial of \(\JD\) the number of derivatives on \(\sigeps\) plus the number on the two slots equals \(2\). \item \emph{Rational coefficients.} Each \(P^{ab,\mu\nu}_{A,ij}(y)\) is a linear combination, with rational constant coefficients, of the components of the \(\sigeps\)-jet of the order stated in (4). There is no other \(y\)-dependence; in particular neither \(h\) nor \(\chi_{\mathrm I}\) occurs. \item \emph{Agreement with XVI-K.} \(\mathfrak s_{C_2}[X,Y]\) is the polarized source \(\mathfrak s^{\mu\nu}(X,Y;S)\), \(S_{\alpha\beta}=\partial_\alpha\partial_\beta\sigeps\), of XVI-K, Definition def:a (XVI-J, eq:Q), identically. \end{enumerate} \end{proposition} \begin{proof} Count, for each monomial of each printed atom, its degree in \(\gamma\) and its number of derivatives (on \(\gamma\) and on \(\sigeps\)), as in the proof of XVI-K, Proposition prop:hess. \(u_n\), \(b_n\) have degree \(n\) and no derivative. \(C_0\) is constant; \(C_1\) and \(C_2\) of \eqref{eq:L-C} are sums of products of \(b\)'s, \(u\)'s and \(\eta\) of total degree \(1\) and \(2\), with no derivative. \(\Gamma_n\) has degree \(n\) and exactly one derivative. In \(R_n\), \(\partial\Gamma_n\) has degree \(n\) and two derivatives (Leibniz' rule distributes them over the factors), and \(\Gamma_a\Gamma_b\) has degree \(a+b=n\) and two derivatives; so \(R_n\), \(r_n\) have degree \(n\) and two derivatives. Hence \(V_1\) has degree \(1\), \(V_2\) degree \(2\) (the term \(\gamma r_1\) included), both with two derivatives, and by \eqref{eq:L-J3} every monomial of \(J_3\) has degree \(2\) and exactly two derivatives on \(\gamma\). By \eqref{eq:L-M2}, every monomial of \(M_2\) has degree \(2\) (\(C_0\Gamma_2\): \(0+2\); \(C_1\Gamma_1\): \(1+1\)) and exactly one derivative on \(\gamma\). \(C_2\) has degree \(2\) and none. Multiplying by \(\sigeps\), \(\partial_\rho\sigeps\), \(\partial_\alpha\partial_\beta\sigeps\) respectively gives (1) and, for the unpolarized source, the statements of (4) and (5): each summand is linear in the \(\sigeps\)-jet of order \(0\), \(1\), \(2\), with no derivative of \(\sigeps\) of another order, and with \(2\), \(1\), \(0\) derivatives on \(\gamma\). All printed numerical coefficients are rational (\(\pm1,\pm2,\pm\tfrac12,\tfrac14,\tfrac18\)) and \(\eta\) has entries \(0,\pm1\), which gives rationality. No atom contains \(h\) or \(\chi_{\mathrm I}\). A polynomial \(F\) homogeneous of degree \(2\) in independent jet coordinates \(\gamma_I\) can be written \(F=\sum_{I\le J}c_{IJ}\gamma_I\gamma_J\); its polarization \(\tfrac12(F(X+Y)-F(X)-F(Y))\) equals \(\sum_{I\le J}\tfrac12c_{IJ}(X_IY_J+X_JY_I)\). This is bilinear, symmetric under \(X\leftrightarrow Y\), and returns \(F(X)\) on \(Y=X\); this proves (2) and (3) (the \((\mu,\nu)\) symmetry is the symmetry of \(J_3\), \(C_2\), \(M_2\) in \(\mu\nu\), immediate from \eqref{eq:L-J3}--\eqref{eq:L-M2} because \(R_{1}\), \(R_2\) are symmetric; the latter is checked by the script). A jet coordinate \(\gamma_I\) with \(I=((ij),a)\) stands for \(\partial^a\gamma_{ij}\); polarization replaces the pair \(\gamma_I\gamma_J\) by \(X_IY_J+X_JY_I\), which keeps the derivative orders \(|a|\), \(|b|\) of the two factors. So the slot orders \((|a|,|b|)\) of \eqref{eq:jet} are the derivative orders of the two \(\gamma\)-factors of the unpolarized monomials, and (4) follows. The expansion \eqref{eq:jet} exists by collecting terms, and is unique because distinct jet coordinates are independent variables. (6): XVI-K, Definition def:a, polarizes \(Q^{\mu\nu}=2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sigeps\) by the same rule, with \(C_2\) of the same printed formula \eqref{eq:L-C}; the two definitions coincide. \end{proof} Proposition~\ref{prop:jet}(4) gives \emph{upper bounds and the exact grading}. It does not by itself say that each graded piece is nonzero; that is the content of the computational verification below. \subsection{Computational verification} \begin{lemma}[Computational verification of L.1] \label{lem:L1comp} The script \texttt{XVI-L-artifacts/source\_full.py} implements \eqref{eq:L-fc}--\eqref{eq:L-M2} atom by atom on independent jet coordinates (order \(\le2\)) of \(\gamma\), of the two slots \(X,Y\), of \(\sigma\), and of a scalar \(\psi\), in exact rational arithmetic (sympy sparse polynomial ring over \(\mathbb Q\)); it exits with status \(0\), and every check raises an error on failure. Its output \texttt{XVI-L-artifacts/source\_full\_out.txt} records: \emph{Symbolic identities (valid for all smooth fields).} \begin{enumerate} \item[(S1)] For all ten components, \(\Eu_\gamma(\sigma\ell_3)^{\mu\nu}=\sigma J_3^{\mu\nu}+2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sigma-2M_2^{\mu\nu\rho}\partial_\rho\sigma\), as a polynomial identity in the independent \(2\)-jets of \(\gamma\) and \(\sigma\). Here \(\Eu_\gamma\) is the Euler operator \(\sum_{a}(-D)^a\,\partial/\partial(\partial^a\gamma_{\mu\nu})\) in the symmetric convention \(\delta\!\int L=\int\Eu^{\mu\nu}\delta\gamma_{\mu\nu}\) (a factor \(\tfrac12\) for \(\mu\ne\nu\)), and \(\ell_3\) is \eqref{eq:L-ell3}. In particular \(\Eu_\gamma(\ell_3)=J_3\) of \eqref{eq:L-J3}, with sign \(+\) in this convention. The Euler computation never required a third-order jet. \item[(S2)] At \(\gamma=\psi\eta\): \(C_1=C_2=0\), \(\Gamma_2=-\psi\Gamma_1\), and both identities \eqref{eq:L-trace} hold. \item[(S3)] \(\mathfrak s_{C_2}[X,Y]\) equals XVI-K's routine \texttt{xvi\_common.polar\_s}\((X,Y,\partial\partial\sigma)\) identically in symbolic \(X\), \(Y\), \(\partial\partial\sigma\); and on XVI-K's dual-slot tensors \(X=-\Pc B^i\), \(Y=-\Pc B^j\) (all \(55\) pairs \(i\le j\)) it equals XVI-K's \(\mathfrak s(-\Pc B^i,-\Pc B^j;S)\). \end{enumerate} \emph{Coefficientwise structural checks.} \begin{enumerate} \item[(C1)] Unpolarized: every monomial of \(\mathfrak s_{C_2}\) (546 monomials), \(\mathfrak s_{M_2}\) (1152), \(\mathfrak s_{J_3}\) (1068) has \(\gamma\)-degree \(2\), \(\sigma\)-jet order exactly \(2\), \(1\), \(0\), and exactly \(0\), \(1\), \(2\) derivatives on \(\gamma\); no foreign generator occurs. Every monomial of \(\JD\) has total derivative weight \(2\). \(J_3\) contains both monomial types \(\gamma\,\partial\partial\gamma\) (534) and \(\partial\gamma\,\partial\gamma\) (534), and no other. \item[(C2)] The two polarization routes (the defining formula, and monomial polarization) agree for every summand and component, and \(\mathfrak s_A[X,X]=\mathfrak s_A(X)\). \item[(C3)] Every monomial of \(\mathfrak s_A[X,Y]\) is exactly linear in the \(X\)-jet and in the \(Y\)-jet; \(P^{ab}_{A,ij}=P^{ba}_{A,ji}\) entry by entry; \(J_3\), \(2C_2\partial\partial\sigma\), \(-2M_2\partial\sigma\) and \(R_1\), \(R_2\) are symmetric in \((\mu,\nu)\). \item[(C4)] Nonzero ordered coefficients \(P^{ab,\mu\nu}_{A,ij}\) (\(\mu\le\nu\)), by \((|a|,|b|)\): \(C_2\): \((0,0)\): 688; \(M_2\): \((1,0)\): 1152, \((0,1)\): 1152; \(J_3\): \((2,0)\): 534, \((1,1)\): 996, \((0,2)\): 534. The ordered slot placements \((a,b)\) that occur are: \(1\) for \(C_2\); all \(8\) with \(|a|+|b|=1\) for \(M_2\); all \(36\) with \(|a|+|b|=2\) for \(J_3\). Each \(P_A\) is a rational linear form in the \(\sigma\)-jet of the order of Proposition~\ref{prop:jet}(4) and nothing else. The full table (5056 entries) is written to \texttt{XVI-L-artifacts/source\_full\_table.txt}. \end{enumerate} \emph{Finite exact test (not a coefficientwise equivalence).} \begin{enumerate} \item[(F1)] The density \(\ell_3\) of this script equals XVI-K's \texttt{xvi\_common.ell3\_of} on two random rational fields \(\gamma(x^1)\) quadratic in \(x^1\) (exact polynomial identity in \(x^1\)). \end{enumerate} \end{lemma} \begin{proof} These are the printed lines of the output file; Appendix~\ref{app:out} reproduces them. \end{proof} Sample entries, copied from the output (\(\sigma_{\alpha}=\partial_\alpha\sigeps\), \(\sigma_{\alpha\beta}=\partial_\alpha\partial_\beta\sigeps\); slots \(i,j\) are labelled by \((ab)\); multi-indices are written as \(e_\alpha\)): \begin{align*} P^{0,0;\,00}_{C_2,(00)(00)}&=-\tfrac34(\sigma_{11}+\sigma_{22}+\sigma_{33}), & P^{0,0;\,00}_{C_2,(00)(12)}&=\sigma_{12},\\ P^{0,e_1;\,00}_{M_2,(00)(01)}&=-\tfrac32\sigma_0, & P^{0,e_2;\,00}_{M_2,(00)(02)}&=-\tfrac32\sigma_0,\\ P^{0,2e_3;\,00}_{J_3,(00)(11)}&=-\tfrac14\sigeps, & P^{0,e_1+e_2;\,00}_{J_3,(00)(12)}&=\tfrac12\sigeps, \qquad P^{e_3,e_3;\,00}_{J_3,(01)(01)}=-\tfrac12\sigeps . \end{align*} \subsection{The derivative budget, established} \begin{theorem}[Source derivative budget] \label{thm:budget} The complete source \(\JD\) of XII-C.tex, eq:actualsource, has the following budget, where ``slot order'' is \(|a|+|b|\) in \eqref{eq:jet}: \begin{center} \begin{tabular}{p{0.07\textwidth}p{0.13\textwidth}p{0.10\textwidth}p{0.19\textwidth}p{0.36\textwidth}} \hline Summand & Preparation factor & Slot order & Occurring \((|a|,|b|)\) & Status\\ \hline \(C_2\) & \(\partial\partial\sigeps\) & exactly \(0\) & \((0,0)\) & Grading Proved locally; attainment Computationally verified\\ \(M_2\) & \(\partial\sigeps\) & exactly \(1\) & \((1,0)\), \((0,1)\) & Grading Proved locally; attainment Computationally verified\\ \(J_3\) & \(\sigeps\) & exactly \(2\) & \((2,0)\), \((1,1)\), \((0,2)\) & Grading Proved locally; attainment Computationally verified\\ \hline \end{tabular} \end{center} In particular, the maximum source-slot derivative order of the complete source is \(2\), attained only in the \(\sigeps J_3\) summand; and the anticipated table of the XVI-L plan (\(C_2\): \(0\); \(M_2\): \(\le1\); \(J_3\): \(\le2\)) holds, in the sharper form that each summand is homogeneous in the slot order. \end{theorem} \begin{proof} Grading and preparation factors: Proposition~\ref{prop:jet}(4). Attainment of every listed \((|a|,|b|)\): Lemma~\ref{lem:L1comp}(C4). \end{proof} \begin{remark}[What the budget does and does not yet give] \label{rem:budget} \begin{enumerate} \item Combined with the vertex bound \(|p|+|q|\le2\) of XVI-K, Proposition prop:hess, Theorem~\ref{thm:budget} \emph{suggests} numerical derivative counts at most \(2\), \(3\), \(4\) in the \(C_2\), \(M_2\), \(J_3\) sectors, hence declared orders at most \(2\), \(3\), \(4\). This is proved in L.2--L.3: graph decomposition and placement of source-slot derivatives on the edge kernels (Proposition~\ref{prop:QL}), and product/scaling statements (Proposition~\ref{prop:Lsd}, Theorem~\ref{thm:L3}). \item The source-slot derivatives are derivatives in the source variable \(y\). By XVI-K, eq:sign, \(\partial^a_y[S(y-w)]=(\partial^aS)(y-w)\) carries no sign, while vertex derivatives carry \((-1)^{|p|}\); this is the L.2 sign rule, re-derived for the complete source in Lemma~\ref{lem:Lsign}. \item The source is a pointwise polynomial in jets. By (S1) it is the Euler derivative of the printed \(\sigeps\ell_3\). No replacement of the printed density by an integration-by-parts-equivalent representative is performed. XII-C records that replacing \(\ell_3\) by \(\ell_3+\partial_\mu B^\mu\) changes the weighted source by \(-\Eu_\gamma((\partial_\mu\sigeps)B^\mu)\); no such replacement is made. \item Nothing in L.1 concerns the vertex, the edge kernels, the extension, counterterms, scale, or gravitational symmetry. \end{enumerate} \end{remark} %===================================================================== \section{L.2 Complete graph decomposition} \label{sec:l2} The graph has the four pieces of XVI-K, Section~K.2: source point \(y\); interaction point \(w\); root \(B_{f,g}(y,x)\), attached only in the target L.7; scalar test \(\phi(x)\). Both edges run from \(y\) to \(w\). Each edge carries \(D_F(y,w)=-\Pc\,S(y-w)\), with \(S\) of XVI-K, eq:S; the factor \(-\Pc\) acts on tensor indices and \(S\) is the scalar numerical part. The change relative to XVI-K is that the source end is the \emph{complete} polarized source \(\mathfrak s_A\) of Definition~\ref{def:sA}, \(A\in\{C_2,M_2,J_3\}\), rather than the \(C_2\) summand alone. \subsection{Derivative placement on both ends} \begin{lemma}[Combined source/vertex placement and sign] \label{lem:Lsign} Let \(z=y-w\) and let \(u\in\mathcal D'(\mathbb R^4)\). On \(y\ne w\), \begin{equation} \partial_y^a\partial_w^p\bigl[u(y-w)\bigr]=(-1)^{|p|}(\partial^{a+p}u)(y-w). \label{eq:L-yw} \end{equation} In particular, for the edge kernel \(S\), \[ \partial_y^a\partial_w^p[S(y-w)]\,\partial_y^b\partial_w^q[S(y-w)] =(-1)^{|p|+|q|}\,q^\circ_{a+p,\,b+q}(y-w), \] where \begin{equation} q^\circ_{\alpha\beta}(z):=(\partial^\alpha S)(z)\,(\partial^\beta S)(z),\qquad z\ne0. \label{eq:L-q} \end{equation} The source-end derivatives carry no sign; the vertex-end sign is \((-1)^{|p|+|q|}\), exactly as in XVI-K, eq:sign, for the \(w\)-derivatives alone. \end{lemma} \begin{proof} XVI-K, Lemma lem:sign (imported): \(\partial_w^p[u(y-w)]=(-1)^{|p|}(\partial^pu)(y-w)\) and \(\partial_y^a[u(y-w)]=(\partial^au)(y-w)\). These commute on \(y\ne w\) (sequential continuity of pullback by the linear submersion \(\Phi(y,w)=y-w\), H\"ormander, Theorem~8.2.4), and \(\partial_y^a\partial_w^p=\partial_w^p\partial_y^a\) gives \eqref{eq:L-yw}. The product identity follows as in XVI-K, Lemma lem:sign: H\"ormander products on \(y\ne w\) are sequentially continuous, so the two signs multiply. \end{proof} \subsection{Source-decorated coefficients for the complete source} \begin{definition}[Complete-source decorated coefficients] \label{def:La} For \(A\in\{C_2,M_2,J_3\}\) and constant symmetric tensors \(X,Y\), write \[ C^{ab,\mu\nu}_A(X,Y;y):=\sum_{i,j=0}^{9}P^{ab,\mu\nu}_{A,ij}(y)\,X^i Y^j \] for the \((a,b)\)-bihomogeneous piece of \(\mathfrak s_A^{\mu\nu}[X,Y](y)\) (Proposition~\ref{prop:jet}). For multi-indices \(a,b\) in the support of \(P_A\) and vertex multi-indices with \(|p|+|q|\le2\), put \begin{equation} \mathcal K^{A\,\mu\nu}_{ab,pq}(y,w;h) = \sum_{i,j=0}^{9}L^{pq}_{ij}(w;h)\,C^{ab,\mu\nu}_A\bigl(-\Pc B^i,-\Pc B^j;y\bigr). \label{eq:L-Kab} \end{equation} (The jet orders \(a,b\) are spent on the edge kernels by Lemma~\ref{lem:Lsign}; the dual-slot tensors are constant, as in XVI-K, Definition def:a.) Set \begin{equation} a^{A\,\mu\nu}_{ab,pq}(y,w;h) =(-1)^{|p|+|q|}\,\chi_{\mathrm I}(w)\,\mathcal K^{A\,\mu\nu}_{ab,pq}(y,w;h). \label{eq:L-apq} \end{equation} An unordered numerical label is \(\lambda=\{\alpha,\beta\}\) with \(\alpha=a+p\), \(\beta=b+q\), and \(q^\circ_\lambda:=q^\circ_{\alpha\beta}\). The corresponding coefficient \(a^{A\,\mu\nu}_\lambda\) is obtained by accumulating every ordered contribution with \(\{a+p,b+q\}=\lambda\) (no universal combinatorial factor \(2\) is inserted for \(\alpha\neq\beta\); when both orderings contribute equally the factor \(2\) appears by summation, as in XVI-K, eq:ak). Write \(\lambda=(A;a,b;p,q)\) for an \emph{ordered} placement when the sector and both ends are kept labelled. \end{definition} \begin{remark}[What sits where] The preparation jet of \(\sigeps\) (order \(2,1,0\) for \(C_2,M_2,J_3\)) sits in \(P_A(y)\), hence in \(a_\lambda\). The residual metric jet of \(h\) sits in \(L^{pq}_{ij}(w;h)\), with \(|r|=2-|p|-|q|\) (XVI-K, Proposition prop:hess(1)). The switch \(\chi_{\mathrm I}(w)\) is undifferentiated (XVI-K, Proposition prop:hess(5)). No silent integration by parts is performed: the printed representatives of \(\ell_3\) and of \(\JD\) are kept. \end{remark} \subsection{Normalization gate} \begin{proposition}[Graph factor for the complete source] \label{prop:Lnorm} The combinatorial graph factor of the specified two-edge graph, with the complete source \(\JD\) at \(y\) and the full cubic vertex at \(w\), is \begin{equation} \frac{1}{2!}\cdot 2=1. \label{eq:L-factor} \end{equation} It carries no power of \(\kappa\) or \(\hbar\). Coupling and loop weights are \emph{not} determined in this note and are kept separate from \eqref{eq:L-factor}. \end{proposition} \begin{proof} Re-derive from the same first principles as XVI-I, without transferring XVI-K's factor as a black box. \begin{enumerate} \item \emph{Two identical graviton edges.} The sector joins the two tensor slots of \(F_t\) to one compactly switched copy of \(\ell_3\) by two graviton edges, each carrying \(D_F\) (XVI-I, Proposition prop:sector). Fredenhagen--Rejzner, arXiv:1208.1428v2, equation~(82), place \(l_{ij}!\) in the denominator for \(l_{ij}\) lines between the same pair of vertices; here \(l_{ij}=2\), so the coefficient is \(1/2!\), declared once (XVI-I, Proposition prop:signs(6)). \item \emph{Second variation of \(F_t\).} \(D_h^2F_t\) contributes the factor \(2\) (XVI-I, eq:Dh): the count of the two identical tensor slots, not a hand-inserted normalization. \item \emph{Independence of the source completion.} Replacing the selected source \(2C_2\partial\partial\sigeps\) by the complete \(\JD=\mathfrak s_{C_2}+\mathfrak s_{M_2}+\mathfrak s_{J_3}\) changes only the source-end contraction at \(y\). It does not change the number of edges, the identity of the two edges, or the second tensor derivative of \(F_t\). Hence neither factor in \eqref{eq:L-factor} is altered. \item The product is \(1\), as in XVI-I, Proposition prop:sector, and as used for the \(C_2\) sector in XVI-K, Proposition prop:QK. \end{enumerate} No coupling constant and no loop weight enters this count; if such weights are fixed later they multiply the whole amplitude and do not redefine \eqref{eq:L-factor}. \end{proof} \begin{proposition}[Complete-source two-edge kernel off the diagonal] \label{prop:QL} On \(y\ne w\), \begin{equation} Q^{\circ\,\mu\nu}_{\mathrm L}(y,w;h) = \sum_{A\in\{C_2,M_2,J_3\}}\sum_{a,b}\sum_{|p|+|q|\le2} a^{A\,\mu\nu}_{ab,pq}(y,w;h)\,q^\circ_{a+p,\,b+q}(y-w). \label{eq:QL} \end{equation} Equivalently, organizing by unordered numerical labels \(\lambda\), \begin{equation} Q^{\circ\,\mu\nu}_{\mathrm L}(y,w;h) = \sum_{\lambda}a^{\mu\nu}_\lambda(y,w;h)\,q^\circ_\lambda(y-w), \label{eq:QL2} \end{equation} with \(a_\lambda=\sum_A a^A_\lambda\). The overall combinatorial factor is \(1\) by Proposition~\ref{prop:Lnorm}. Restricting to \(A=C_2\) and \(a=b=0\) recovers XVI-K, Proposition prop:QK: \(Q^{\circ\,\mu\nu}_{\mathrm K}=\sum_{|p|+|q|\le2}a^{\mu\nu}_{pq}\,q^\circ_{pq}\). \end{proposition} \begin{proof} Insert the two edge slot fields, carrying source-end jets of orders \(a,b\) and vertex-end jets of orders \(p,q\), into \(\chi_{\mathrm I}(w)D^2\ell_3(h)[\cdot,\cdot](w)\) via XVI-K, eq:Lpq, and contract the source ends by \(\mathfrak s_A\) via \eqref{eq:jet}. Lemma~\ref{lem:Lsign} converts the derivatives into \((-1)^{|p|+|q|}q^\circ_{a+p,b+q}(y-w)\). The dual-slot normalization is \(-\Pc B^i\), \(-\Pc B^j\) as in XVI-K, Definition def:a / XVI-J, eq:Kdef. Summing over the three source summands and over all placements allowed by Theorem~\ref{thm:budget} and XVI-K, Proposition prop:hess, gives \eqref{eq:QL}. The \(C_2\), \(a=b=0\) truncation is Definition~\ref{def:La} with \(P_{C_2}\) equal to XVI-K's polarized source (Proposition~\ref{prop:jet}(6)), hence XVI-K, eq:QK. \end{proof} \paragraph{Term ledger (mandatory fields).} \begin{center} \begin{longtable}{p{0.28\textwidth}p{0.64\textwidth}} \hline Datum & Record\\ \hline Source summand & \(A\in\{C_2,M_2,J_3\}\); \(P_A\) from \eqref{eq:jet}.\\ Source placements \(a,b\) & \(C_2\): \((0,0)\) only; \(M_2\): \(|a|+|b|=1\); \(J_3\): \(|a|+|b|=2\) (Theorem~\ref{thm:budget}).\\ Vertex \(p,q\) & \(|p|+|q|\le2\); no derivative on \(\chi_{\mathrm I}\) (XVI-K, prop:hess).\\ Dual slots & Source tensors \(-\Pc B^i=-(G^{-1})^{ii}\Pc B_i\), \(G=\mathrm{diag}(1,-2,-2,-2,1,2,2,1,2,1)\).\\ Vertex-end signs & \((-1)^{|p|+|q|}\) from Lemma~\ref{lem:Lsign}; sits in \(a_{ab,pq}\), not in \(q^\circ\).\\ Preparation jet & In \(P_A(y)\): \(\partial\partial\sigeps\) (\(C_2\)), \(\partial\sigeps\) (\(M_2\)), \(\sigeps\) (\(J_3\)).\\ Residual metric jet & In \(L^{pq}_{ij}(w;h)\), order \(2-|p|-|q|\).\\ Switch \(\chi_{\mathrm I}\) & \(\chi_{\mathrm I}(w)\), undifferentiated, inside \(a_{ab,pq}\).\\ Ordered vs unordered & Ordered: \((A;a,b;p,q)\). Unordered numerical \(\lambda=\{a+p,b+q\}\); coefficients accumulate over ordered contributions (no universal factor \(2\)).\\ Combinatorial factor & \(\tfrac1{2!}\cdot2=1\) (Proposition~\ref{prop:Lnorm}).\\ Coupling/loop weights & \emph{Undetermined} in L.2; separated from the graph factor.\\ \hline \end{longtable} \end{center} \begin{lemma}[Computational verification of L.2] \label{lem:L2comp} The script \texttt{XVI-L-artifacts/two\_edge\_full.py} loads \(P_A\) from \texttt{source\_full\_table.txt} and \(L^{pq}_{ij}\) from \texttt{XVI-K-artifacts/hessian\_full\_out.txt} (relative paths), builds \(T_i^r:=(-\Pc B^i)^r\), forms \(\widetilde P^{ab,\mu\nu}_{A,ij}=\sum_{r,s}P^{ab,\mu\nu}_{A,rs}T_i^r T_j^s\), assembles \(a^{A\,\mu\nu}_{ab,pq}=(-1)^{|p|+|q|}\chi_{\mathrm I}\sum_{ij}L^{pq}_{ij}\widetilde P^{ab,\mu\nu}_{A,ij}\), groups by numerical label by accumulation, and exits with status \(0\). It records: \begin{enumerate} \item[(R1)] All-sector coefficient assembly, exact in the preparation jets and \(w^1\), on \(h_{\mu\nu}=(w^1)^2\eta_{\mu\nu}\), with the common undifferentiated switch factor suppressed (not a general-\(h\) symbolic assembly). Cancellation-aware supports \emph{on the stated background}: \(|\Lambda_{\mathrm{graph}}(C_2)|=25\), \(|\Lambda_{\mathrm{graph}}(M_2)|=84\), \(|\Lambda_{\mathrm{graph}}(J_3)|=202\), union \(235\) of \(|\Lambda_{\mathrm{adm}}|=255\) (table: \(C_2\) \(25/25\), \(M_2\) \(84/84\), \(J_3\) \(250/202\), complete \(255/235\)). Nonzero on this specialization \(\Rightarrow\) not identically zero in the general symbolic expression; zero on this specialization may still be nonzero for another residual metric. The reported \(235\) is a coefficient-label support count before quotienting numerical-kernel relations; it does not establish \(235\) independent kernels or nonvanishing of the full distribution. Full nonzero coefficient table written to \texttt{two\_edge\_coeff\_table.txt}. \item[(R2)] Source-sector K regression: retaining only \(A=C_2\), all \(25\) unordered labels and all ten components match the independently assembled signed \texttt{polar\_s} route on the background; constant-slot recovers the XVI-K / XVI-J tensor \(\mathcal K^{\mu\nu}\) of XVI-J, eq:Kvals; sample \(a^{C_2}_{\{0,e_1\}}=-2\mathcal K^{C_2}_{0,e_1}\) componentwise; \(\widetilde P_{C_2}\) matches \texttt{xvi\_common.polar\_s} on dual slots. \item[(R3)] Graph factor \(\tfrac1{2!}\cdot2=1\); signs \((-1)^{|p|+|q|}\) with the constant-slot sign \(+1\) and the \(|p|+|q|=1\) sign \(-1\). \item[(R4)] Ordered placement counts: \(C_2\): \(45\); \(M_2\): \(360\); \(J_3\): \(1620\); total \(2025\). Label \(\{0,0\}\) does not arise in \(M_2\)/\(J_3\). \end{enumerate} \end{lemma} \begin{proof} Printed lines of \texttt{two\_edge\_full\_out.txt}; Appendix~\ref{app:out}. \end{proof} %===================================================================== \section{L.3 Products, power counting, and kernel inventory} \label{sec:l3} \subsection{Product existence} \begin{lemma}[Products on the punctured space] \label{lem:Lprod} For all multi-indices \(\alpha,\beta\), the product \(q^\circ_{\alpha\beta}=\partial^\alpha S\cdot\partial^\beta S\) exists in \(\mathcal D'(\mathbb R^4\setminus\{0\})\), and \(q^\circ_{\alpha\beta}=q^\circ_{\beta\alpha}\). \end{lemma} \begin{proof} As in XVI-K, Lemma lem:prod and Remark rem:transfer: Fredenhagen--Rejzner, arXiv:1208.1428v2, Section~7 (text after (77)), for \(x\ne y\) the wavefront set of \(\Delta^{\mathrm{FR}}_F\) has covectors cotangent to a null geodesic with matching time orientation. Pulling back by \(\Phi(x,y)=x-y\) and restricting to the difference variable, every covector of \(\mathrm{WF}(\Phi^*S)\) has the form \((k,-k)\) with \(k\ne0\) (H\"ormander, Theorem~8.2.4); the naive claim \(k+k'\ne0\) for an unordered pair of Feynman covectors is therefore not the right condition in the difference variable. The transferred criterion (XVI-K, Remark rem:transfer) is that two admissible wavefront covectors of the factors \(\partial^\alpha S\) and \(\partial^\beta S\) at the same difference point \(z\ne0\) cannot sum to zero (same time orientation when both are nonzero). Hence H\"ormander's Theorem~8.2.10 applies on \(\mathbb R^4\setminus\{0\}\). By XVI-K, eq:S, \(S=-\Delta^{\mathrm{FR}}_F\); derivatives do not enlarge wavefront sets. Commutativity is a property of that product. \end{proof} \subsection{Scaling degrees and declared orders} \begin{proposition}[Complete-source power counting] \label{prop:Lsd} Let \(d:=|a|+|b|+|p|+|q|\) for an ordered placement \((A;a,b;p,q)\) contributing to \eqref{eq:QL}, and set \(N_\lambda:=d\) (the \emph{declared} order; not defined through \(\mathrm{sd}-4\)). Then \(q^\circ_{a+p,b+q}\) is homogeneous of degree \(-4-d\) on \(\mathbb R^4\setminus\{0\}\), and \begin{equation} \mathrm{sd}(q^\circ_{a+p,b+q})\le 4+d=4+N_\lambda, \label{eq:L-sdbound} \end{equation} with equality if the product is nonzero. Consequently, as consequences of Theorem~\ref{thm:budget} and the vertex bound \(|p|+|q|\le2\) (XVI-K, Proposition prop:hess): \begin{center} \begin{tabular}{llll} \hline Sector & Source \(|a|+|b|\) & Total \(d=N_\lambda\) & Bound\\ \hline \(C_2\) & \(0\) & \(\le2\) & \(\mathrm{sd}\le6\), \(N\le2\)\\ \(M_2\) & \(1\) & \(\le3\) & \(\mathrm{sd}\le7\), \(N\le3\)\\ \(J_3\) & \(2\) & \(\le4\) & \(\mathrm{sd}\le8\), \(N\le4\)\\ \hline \end{tabular} \end{center} \end{proposition} \begin{proof} Multi-index additivity: \(|a+p|+|b+q|=|a|+|b|+|p|+|q|=d\). Homogeneity of \(\partial^\alpha S\) of degree \(-2-|\alpha|\) is XVI-K, Lemma lem:homog; the product of two such factors on \(\mathbb R^4\setminus\{0\}\) is homogeneous of degree \(-4-d\) by the same pullback argument as XVI-K, Proposition prop:sd (Lemma~\ref{lem:Lprod} supplies existence before scaling). Scaling degree: Brunetti--Fredenhagen, Example~4, gives \(\mathrm{sd}\le4+d\), with equality for a nonzero homogeneous distribution of that degree. Sector bounds: Theorem~\ref{thm:budget} fixes the source slot order; XVI-K, Proposition prop:hess fixes \(|p|+|q|\le2\); add. \end{proof} \begin{remark}[Declared versus scaling] \label{rem:LN} As in XVI-K, Remark rem:N: \(N_\lambda:=d\) is declared before any vanishing question is decided. A vanishing kernel costs nothing. The identity \(N=\mathrm{sd}-4\) is \emph{not} used as a definition. \end{remark} \subsection{The numerical-kernel span} \begin{definition}[Admissible inventory, graph support, and ambient span] \label{def:VL} Let \(\Lambda_{\mathrm{adm}}\) be the set of \(255\) admissible unordered numerical labels \(\{a+p,b+q\}\) obtained by enumerating all source-slot budgets of Theorem~\ref{thm:budget} against all vertex placements with \(|p|+|q|\le2\) (Lemma~\ref{lem:L3comp}). Let \(\mathcal V_{\mathrm{adm}}:=\mathrm{span}\{q^\circ_\lambda:\lambda\in\Lambda_{\mathrm{adm}}\}\) be the \emph{fixed ambient} numerical span. Let \(\Lambda_{\mathrm{graph}}:=\{\lambda\in\Lambda_{\mathrm{adm}}:a_\lambda\not\equiv0\}\) be the cancellation-aware support of the assembled coefficients in \eqref{eq:QL2} (background-dependent when computed), and let \(\mathcal V_{\mathrm{graph}}:=\mathrm{span}\{q^\circ_\lambda:\lambda\in\Lambda_{\mathrm{graph}}\}\). Let \(\mathcal V_{\mathrm K}:=\mathrm{span}\{q^\circ_{\sk}:\sk\in\mathcal N\}\) be the XVI-K span (\(25\) generators before relations). \medskip \noindent\fbox{\begin{minipage}{0.90\textwidth} \textbf{Ambient extension domain.} The coherent extension of L.4--L.7 is the linear map \[ \RL:\mathcal V_{\mathrm{adm}}\longrightarrow\mathcal D'(\mathbb R^4). \] Grading, moments, defects, coherent changes, and the scale family are constructed on this fixed ambient space. Zero coefficients in the ambient sum contribute nothing; the graph uses the appropriate coefficients \(a_\lambda\) with \(\lambda\in\Lambda_{\mathrm{adm}}\). \end{minipage}} \medskip \noindent Notation: the earlier draft symbol \(\mathcal V_{\mathrm L}\) (formerly the graph span) is retired; throughout L.4--L.7 the domain of \(\RL\), \(\RLmu\), \(A_\mu\), coherent \(\Delta\RL\), and attachment sums is \(\mathcal V_{\mathrm{adm}}\). The symbols \(\Lambda_{\mathrm{graph}}\) and \(\mathcal V_{\mathrm{graph}}\) are retained only for reporting cancellation-aware support on a stated background. \end{definition} \begin{proposition}[Ambient inclusions and preserved relations] \label{prop:VLincl} \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\) and \(\mathcal V_{\mathrm{graph}}\subseteq\mathcal V_{\mathrm{adm}}\). Moreover, on \(\mathbb R^4\setminus\{0\}\): \begin{enumerate} \item \emph{Edge interchange.} \(q^\circ_{\alpha\beta}=q^\circ_{\beta\alpha}\). \item \emph{Constant-coefficient traces / Leibniz.} \(\partial_\gamma(S^2)=2S\partial_\gamma S\) and \(\partial_\gamma(S\partial_\alpha S)=\partial_\gamma S\,\partial_\alpha S+S\,\partial_\gamma\partial_\alpha S\). \item \emph{Wave operator.} \(\eta^{\alpha\beta}q^\circ_{\{0,e_\alpha+e_\beta\}}=S\,\square S=0\). \end{enumerate} All three are inherited from XVI-K, Lemma lem:grading, and live already in \(\mathcal V_{\mathrm K}\), hence in \(\mathcal V_{\mathrm{adm}}\). \end{proposition} \begin{proof} Every XVI-K unordered placement \(\sk=\{p,q\}\) occurs in the \(C_2\) block of \(\Lambda_{\mathrm{adm}}\) (source slots \(a=b=0\), vertex placements with \(|p|+|q|\le2\); Lemma~\ref{lem:L3comp}(I2)); thus every generator of \(\mathcal V_{\mathrm K}\) is a generator of \(\mathcal V_{\mathrm{adm}}\), and \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\). Separately, \(\Lambda_{\mathrm{graph}}\subseteq\Lambda_{\mathrm{adm}}\) by definition, so \(\mathcal V_{\mathrm{graph}}\subseteq\mathcal V_{\mathrm{adm}}\). (1)--(3) are XVI-K, Lemma lem:grading(2)--(3), which use only Lemma~\ref{lem:Lprod} and \(\square S=0\) off the origin; they do not depend on the source sector. \end{proof} \subsection{Kernel inventory} \paragraph{Inventory table (by sector and declared order).} ``Unord.'' counts distinct unordered numerical labels \(\{a+p,b+q\}\) contributed as \emph{placements} by that sector's ordered budget (placement occurrence, not a claim that the assembled coefficient is nonzero). Sectors may overlap on some labels, but not on all: the order-\(0\) label \(\{0,0\}\) (\(S^2\)) arises in \(C_2\) and does \emph{not} arise in \(M_2\) or \(J_3\). The graph uses a subfamily of the \(255\)-label admissible inventory \(\Lambda_{\mathrm{adm}}\); cancellation-aware support \(\Lambda_{\mathrm{graph}}\) is recorded by \texttt{two\_edge\_full.py}. \begin{center} \begin{longtable}{p{0.10\textwidth}p{0.08\textwidth}p{0.10\textwidth}p{0.08\textwidth}p{0.10\textwidth}p{0.20\textwidth}p{0.14\textwidth}} \hline Sector & \(N\) & Unord. & Homog. & \(\mathrm{sd}\le\) & Ordered \((a,b;p,q)\) & Placement\\ \hline \(C_2\) & \(0\) & \(1\) & \(-4\) & \(4\) & \(1\times1\) & occurs\\ \(C_2\) & \(1\) & \(4\) & \(-5\) & \(5\) & & occurs\\ \(C_2\) & \(2\) & \(20\) & \(-6\) & \(6\) & total ord.\ \(45\) & occurs\\ \(C_2\) & \(\le2\) & \(25\) & & \(6\) & \(=\) XVI-K \(\mathcal N\) & \(=V_{\mathrm K}\)\\ \hline \(M_2\) & \(1\) & \(4\) & \(-5\) & \(5\) & & occurs\\ \(M_2\) & \(2\) & \(20\) & \(-6\) & \(6\) & & occurs\\ \(M_2\) & \(3\) & \(60\) & \(-7\) & \(7\) & total ord.\ \(360\) & occurs\\ \(M_2\) & \(\le3\) & \(84\) & & \(7\) & \(8\times45\) & \\ \hline \(J_3\) & \(2\) & \(20\) & \(-6\) & \(6\) & & occurs\\ \(J_3\) & \(3\) & \(60\) & \(-7\) & \(7\) & & occurs\\ \(J_3\) & \(4\) & \(170\) & \(-8\) & \(8\) & total ord.\ \(1620\) & occurs\\ \(J_3\) & \(\le4\) & \(250\) & & \(8\) & \(36\times45\) & \\ \hline \(\Lambda_{\mathrm{adm}}\) & \(\le4\) & \(255\) & & \(8\) & \(2025\) ordered & admissible\\ \hline \end{longtable} \end{center} The \(C_2\) block is identical to XVI-K's inventory (types \(S\cdot S\), \(S\cdot\partial S\), \(\partial S\cdot\partial S\), \(S\cdot\partial\partial S\); \(1+4+10+10=25\)). New admissible types include \(S\cdot\partial^3S\), \(S\cdot\partial^4S\), \(\partial S\cdot\partial^2S\), \(\partial S\cdot\partial^3S\), \(\partial^2S\cdot\partial^2S\). \begin{lemma}[Computational verification of L.3] \label{lem:L3comp} The script \texttt{XVI-L-artifacts/kernel\_inventory.py} enumerates all placements from the L.1 budgets and the vertex bound \(|p|+|q|\le2\), exits with status \(0\), and records: \begin{enumerate} \item[(I1)] Unordered placement counts by sector and \(N\) as in the inventory table; \(|\Lambda_{\mathrm{adm}}|=255\); ordered totals \(45+360+1620=2025\). \item[(I2)] \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\): XVI-K's \(25\) unordered placements equal the \(C_2\) block. \item[(I3)] Sector maxima \(N\le2,3,4\) for \(C_2,M_2,J_3\); global \(N\le4\); additivity \(|a+p|+|b+q|=|a|+|b|+|p|+|q|\). \item[(I4)] Edge-interchange symmetry of unordered labels; the ten \(S\cdot\partial\partial S\) generators of the \(\square S=0\) relation lie in \(\Lambda_{\mathrm{adm}}\). \end{enumerate} \end{lemma} \begin{proof} Printed lines of \texttt{kernel\_inventory\_out.txt}; Appendix~\ref{app:out}. \end{proof} \begin{theorem}[L.3 summary] \label{thm:L3} On \(y\ne w\), every numerical factor of \(Q^{\circ\,\mu\nu}_{\mathrm L}\) exists in \(\mathcal D'(\mathbb R^4\setminus\{0\})\) (Lemma~\ref{lem:Lprod}). The declared orders and scaling bounds of Proposition~\ref{prop:Lsd} hold. \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\) with the relations of Proposition~\ref{prop:VLincl}. No extension, counterterm, scale variation, or root attachment is constructed in L.3. \end{theorem} \begin{proof} Combine Lemma~\ref{lem:Lprod}, Proposition~\ref{prop:Lsd}, Proposition~\ref{prop:VLincl}, and Lemma~\ref{lem:L3comp}(I2). \end{proof} %===================================================================== \section{L.4 Coherent extension preserving XVI-K} \label{sec:l4} The extension acts only in the difference variable \(z=y-w\). The coefficients \(a^{\mu\nu}_\lambda(y,w;h)\) of \eqref{eq:QL2} depend on \(y\) and \(w\) separately; they are not extended and multiply the pulled-back extensions afterwards. This is the same order of operations as XVI-K, Section~K.4. \paragraph{Projection, extended to declared orders \(N\le4\).} Fix the same function \(\varpi\in C_c^\infty(\mathbb R^4)\), \(\varpi(0)=1\), of XVI-J, eq:moment, used in XVI-K, eq:Wrho. For \(\rho\in\{0,1,2,3,4\}\) let \[ M^{(\rho)}_{\beta\alpha}=\partial^\beta\bigl(z^\alpha\varpi/\alpha!\bigr)(0),\qquad|\alpha|,|\beta|\le\rho. \] By Leibniz' rule \(M^{(\rho)}_{\beta\alpha}=0\) unless \(\alpha\le\beta\), and \(M^{(\rho)}_{\alpha\alpha}=\varpi(0)=1\); so \(M^{(\rho)}\) is unitriangular and invertible. Put \begin{equation} w^{(\rho)}_\alpha(z)=\sum_{|\gamma|\le\rho}\bigl((M^{(\rho)})^{-1}\bigr)_{\gamma\alpha}\,\frac{z^\gamma}{\gamma!}\,\varpi(z), \qquad (W_\rho\varphi)(z)=\varphi(z)-\sum_{|\alpha|\le\rho}\partial^\alpha\varphi(0)\,w^{(\rho)}_\alpha(z). \label{eq:L-Wrho} \end{equation} Then \(\partial^\beta w^{(\rho)}_\alpha(0)=\delta^\beta_\alpha\) for \(|\alpha|,|\beta|\le\rho\), so \(W_\rho\) projects onto the test functions vanishing to order \(\rho\) at \(0\). For \(\rho\in\{0,1,2\}\) the matrices \(M^{(\rho)}\), the functions \(w^{(\rho)}_\alpha\), and the projections \(W_\rho\) are \emph{identical} to XVI-K, eq:Wrho (same \(\varpi\)); the cases \(\rho=3,4\) are the declared-order extension required by Proposition~\ref{prop:Lsd}. \paragraph{Grading and basis.} Write \(\mathcal H_k\) for the homogeneous degree \(-k\) piece of \(\mathcal V_{\mathrm{adm}}\) (equivalently: elements with \(\mathrm{sd}\le k\) that are homogeneous of degree \(-k\) on \(\mathbb R^4\setminus\{0\}\), in the sense of Proposition~\ref{prop:Lsd}), so \(\mathcal V_{\mathrm{adm}}=\bigoplus_{k=4}^{8}\mathcal H_k\) as an explicit direct sum by distinct homogeneity degree on \(\mathbb R^4\setminus\{0\}\) (as in XVI-K, Lemma lem:grading: every element of \(\mathcal H_k\) is homogeneous of degree \(-k\); vanishing kernels contribute nothing). For each \(k\), choose a basis of \(\mathcal V_{\mathrm K}\cap\mathcal H_k\), and extend it to a basis \(\mathcal B_k\) of \(\mathcal H_k\subseteq\mathcal V_{\mathrm{adm}}\). Set \(\mathcal B_{\mathrm{adm}}=\bigcup_{k=4}^{8}\mathcal B_k\). Which spanning elements enter each \(\mathcal B_k\) is not specified beyond this extension requirement; the construction below is basis-independent on each homogeneous piece (Remark~\ref{rem:Lbasisfree}). \begin{definition}[The extension map \(\RL\)] \label{def:RL} For \(b\in\mathcal B_k\), \(\rho=k-4\in\{0,1,2,3,4\}\), let \(\RL(b)\in\mathcal D'(\mathbb R^4)\) be the extension of Brunetti--Fredenhagen, Theorem~5.3 (projection~(38), construction~(39)), of scaling degree \(k\), restricting to \(b\) on \(\mathbb R^4\setminus\{0\}\), built with the projection \(W_\rho\) of \eqref{eq:L-Wrho} and the declared moments \begin{equation} \bigl\langle\RL(b),\,w^{(\rho)}_\alpha\bigr\rangle=0,\qquad|\alpha|\le\rho . \label{eq:L-moments} \end{equation} Extend \(\RL\) to \(\mathcal V_{\mathrm{adm}}\) by linearity. \end{definition} \begin{proposition}[Properties of \(\RL\)] \label{prop:RL} \begin{enumerate} \item \(\RL:\mathcal V_{\mathrm{adm}}\to\mathcal D'(\mathbb R^4)\) is well defined and linear, and \(\RL(u)|_{\mathbb R^4\setminus\{0\}}=u\) for every \(u\in\mathcal V_{\mathrm{adm}}\). \item For \(u\in\mathcal H_k\), \(\mathrm{sd}(\RL(u))\le k\), and \(\langle\RL(u),w^{(k-4)}_\alpha\rangle=0\) for \(|\alpha|\le k-4\). \item \emph{Exact XVI-K restriction.} \(\RL|_{\mathcal V_{\mathrm K}}=\RK\) as maps \(\mathcal V_{\mathrm K}\to\mathcal D'(\mathbb R^4)\). In particular \(\RL(S^2)=\RK(S^2)=\bar q^{\mathfrak R}\). \item For each numerical label \(\lambda\) contributing to \eqref{eq:QL2}, \(\RL(q^\circ_\lambda)\) is an extension of \(q^\circ_\lambda\) with \(\mathrm{sd}\le4+N_\lambda\); any other extension of the same scaling degree differs from it by \(\sum_{|\gamma|\le N_\lambda}c_\gamma\,\partial^\gamma\delta\). \item If \(\sum_\lambda r_\lambda\,q^\circ_\lambda=0\) in \(\mathcal D'(\mathbb R^4\setminus\{0\})\) with constant coefficients \(r_\lambda\), then \(\sum_\lambda r_\lambda\,\RL(q^\circ_\lambda)=0\) in \(\mathcal D'(\mathbb R^4)\). \end{enumerate} \end{proposition} \begin{proof} (1)--(2) As in XVI-K, Proposition prop:RK(1)--(2): \(\mathcal B_{\mathrm{adm}}\) is a basis; restriction to \(\mathbb R^4\setminus\{0\}\) is linear and the identity on the basis; scaling degree and the moment conditions \eqref{eq:L-moments} are linear. (3) On each graded piece \(\mathcal H_k\) with \(k\le6\) (the support of \(\mathcal V_{\mathrm K}\)), the data of Definition~\ref{def:RL} coincide with XVI-K, Definition def:RK: same \(\varpi\), same \(W_\rho\) for \(\rho=k-4\le2\), same zero moments. By Brunetti--Fredenhagen, Theorem~5.3, the extension of a fixed homogeneous distribution of scaling degree \(k\) with these moments is unique, so \(\RL(u)=\RK(u)\) for every \(u\in\mathcal V_{\mathrm K}\). (4) \(q^\circ_\lambda\in\mathcal H_{4+N_\lambda}\) by Proposition~\ref{prop:Lsd}; apply (1), (2). The difference of two same-degree extensions is supported at \(0\) and of scaling degree \(\le4+N_\lambda\), hence a combination of \(\partial^\gamma\delta\) with \(|\gamma|\le N_\lambda\). (5) The relation says an element of \(\mathcal V_{\mathrm{adm}}\) is zero; \(\RL\) is linear, so it maps that element to \(0\). \end{proof} \begin{remark}[Independence of the basis] \label{rem:Lbasisfree} For \(u\in\mathcal H_k\) and \(\rho=k-4\), the zero-moment construction reads \(\langle\RL(u),\varphi\rangle=\langle u,W_\rho\varphi\rangle\) on test functions, using the unique extension of \(u\) to functions vanishing to order \(\rho\) at \(0\). Hence \(\RL|_{\mathcal H_k}\) does not depend on \(\mathcal B_k\). Combined with Proposition~\ref{prop:RL}(3), the restriction identity is likewise basis-free. \end{remark} \begin{lemma}[Contact defects for \(\RL\)] \label{lem:Ldefect} Let \(u\in\mathcal H_k\) be homogeneous of degree \(-k\) on \(\mathbb R^4\setminus\{0\}\) with \(4\le k\le7\), and suppose \(\partial_\alpha u\in\mathcal H_{k+1}\subseteq\mathcal V_{\mathrm{adm}}\) (as an identity in \(\mathcal D'(\mathbb R^4\setminus\{0\})\)), for some \(\alpha\in\{0,1,2,3\}\). Here \(k\) is the scaling degree of \(u\) (equivalently the homogeneity degree with opposite sign), not the numerical order \(N=k-4\). Put \[ D_{\alpha,u}:=\RL(\partial_\alpha u)-\partial_\alpha\RL(u)\in\mathcal D'(\mathbb R^4). \] Then \(\operatorname{supp}D_{\alpha,u}\subset\{0\}\), \(\mathrm{sd}(D_{\alpha,u})\le k+1\), and \begin{equation} D_{\alpha,u}=\sum_{|\gamma|\le k-3}c_{\alpha,u,\gamma}\,\partial^\gamma\delta, \qquad c_{\alpha,u,\gamma}=(-1)^{|\gamma|}\bigl\langle\RL(u),\,\partial_\alpha w^{(k-3)}_\gamma\bigr\rangle , \label{eq:L-defect} \end{equation} where the moment order \(k-3=(k+1)-4\) is that of the image, and the identity uses \(\langle\RL(\partial_\alpha u),w^{(k-3)}_\beta\rangle=0\). (Equivalently, for a general \(u\in\mathcal V_{\mathrm{adm}}\) with \(\partial_\alpha u\in\mathcal V_{\mathrm{adm}}\), the same support and scaling conclusions hold, and the contact coefficients are given by the full pairing formula keeping both terms \(\langle\RL(\partial_\alpha u),w\rangle\) and \(\langle\partial_\alpha\RL(u),w\rangle\).) These defects are retained in every subsequent formula; they are \emph{not} claimed to vanish. \end{lemma} \begin{proof} Off \(0\), \(\RL(\partial_\alpha u)=\partial_\alpha u\) and \(\partial_\alpha\RL(u)=\partial_\alpha u\) (Proposition~\ref{prop:RL}(1); restriction commutes with \(\partial_\alpha\)), so \(D_{\alpha,u}\) is supported at \(0\). Scaling: \(\mathrm{sd}(\RL(\partial_\alpha u))\le k+1\) and \(\mathrm{sd}(\partial_\alpha\RL(u))\le\mathrm{sd}(\RL(u))+1\le k+1\), hence \(\mathrm{sd}(D_{\alpha,u})\le k+1\). A point-supported distribution of scaling degree \(\le k+1\) in \(\mathbb R^4\) is \(\sum_{|\gamma|\le(k+1)-4}c_\gamma\partial^\gamma\delta\). Pair with \(w^{(k-3)}_\beta\), \(|\beta|\le k-3\): the left side of \eqref{eq:L-defect} yields \((-1)^{|\beta|}c_{\alpha,u,\beta}\); the right side uses the moment vanishing for \(\RL(\partial_\alpha u)\) and \(\langle\partial_\alpha\RL(u),w\rangle=-\langle\RL(u),\partial_\alpha w\rangle\). \end{proof} In particular, the XVI-K defects of XVI-K, Lemma lem:defect, are recovered on \(\mathcal V_{\mathrm K}\) by Proposition~\ref{prop:RL}(3): \(D_\alpha=D_{\alpha,S^2}\) and \(D_{\gamma'\alpha}\) of that lemma equal the corresponding \(D_{\alpha,u}\) for \(u=S^2\) and \(u=S\partial_\alpha S\). \paragraph{Defect ledger (finite).} The differential relations inside \(\mathcal V_{\mathrm{adm}}\) that can produce a nonzero \(D_{\alpha,u}\) are the Leibniz identities among products \(\partial^A S\cdot\partial^B S\) whose total numerical order \(n=|A|+|B|\) satisfies \(n\le3\) (so that the image of numerical order \(n+1\) still lies in the declared span with \(N\le4\)). Equivalently, in scaling-degree notation, \(u\in\mathcal H_k\) with \(k=4+n\) and \(4\le k\le7\). The typed sources are \[ (0,0),\;(0,1),\;(0,2),\;(0,3),\;(1,1),\;(1,2) \] (factor orders), a finite list; each yields finitely many multi-index directions \(\alpha\) and finitely many contact coefficients with defect order \(|\gamma|\le n+1\le4\) (numerical orders \(1\)--\(4\) of the image, since \(\mathrm{sd}(D_{\alpha,u})\le k+1=5+n\) forces \(|\gamma|\le(k+1)-4=n+1\)). No value of any \(c_{\alpha,u,\gamma}\) is asserted; the ledger records existence and finiteness only. Category~(4) of Section~\ref{sec:l5} treats these defects as fixed differential-normalization data, not as free counterterms. \paragraph{Non-properties.} As for \(\RK\): \(\RL\) is not claimed Lorentz covariant; no Leibniz rule is claimed beyond the controlled defects of Lemma~\ref{lem:Ldefect}; \(\RL\) is not claimed to commute with \(\Pc\); no wavefront set of \(\RL(q^\circ_\lambda)\) at \(z=0\) is asserted beyond point support. \paragraph{Unrooted extended kernel.} With \(\Phi(y,w)=y-w\), set \begin{equation} Q^{\mu\nu}_{\mathrm L}(y,w;h) := \sum_{\lambda}a^{\mu\nu}_\lambda(y,w;h)\;\bigl(\Phi^*\RL(q^\circ_\lambda)\bigr)(y,w). \label{eq:QLext} \end{equation} On \(y\ne w\) this restricts to \eqref{eq:QL2}. Root attachment is deferred to L.7; no claim of L-A is made here. \begin{theorem}[L.4 summary] \label{thm:L4} There exists a linear extension map \(\RL:\mathcal V_{\mathrm{adm}}\to\mathcal D'(\mathbb R^4)\) with \(\RL|_{\mathcal V_{\mathrm K}}=\RK\), built from the same \(\varpi\)-projections for \(\rho\le2\) and their extension to \(\rho=3,4\), with zero declared moments \eqref{eq:L-moments}, the scaling bounds of Proposition~\ref{prop:RL}(2), preservation of constant-coefficient relations, and the finite contact-defect ledger of Lemma~\ref{lem:Ldefect}. No root attachment, scale variation, or gravitational identification is constructed in L.4. \end{theorem} \begin{proof} Definition~\ref{def:RL} and Proposition~\ref{prop:RL}; Lemma~\ref{lem:Ldefect}. \end{proof} %===================================================================== \section{L.5 Local collision counterterms before attachment} \label{sec:l5} This section classifies the freedom in changing the coherent extension of L.4 \emph{before} root attachment. It is the principal renormalization statement of XVI-L at the unrooted level; propagation through \(B_{f,g}\) is L.7. \begin{lemma}[Coherent change of extension] \label{lem:Lamb} Let \(\RL'\) be any linear map \(\mathcal V_{\mathrm{adm}}\to\mathcal D'(\mathbb R^4)\) that extends every \(u\in\mathcal V_{\mathrm{adm}}\) with \(\mathrm{sd}(\RL'(u))\le\mathrm{sd}(u)\) in the graded sense of Proposition~\ref{prop:RL}(2), and put \(\Delta\RL:=\RL'-\RL\). Then for every numerical label \(\lambda\), \begin{equation} \Delta\RL(q^\circ_\lambda)=\sum_{|\gamma|\le N_\lambda}C_{\lambda\gamma}\,\partial^\gamma\delta, \qquad C_{\lambda\gamma}\in\mathbb C, \label{eq:L-Dq} \end{equation} and the unrooted kernel changes by the local collision distribution \begin{equation} \Delta Q^{\mu\nu}_{\mathrm L}(y,w;h) = \sum_{\lambda}\sum_{|\gamma|\le N_\lambda}C_{\lambda\gamma}\,a^{\mu\nu}_\lambda(y,w;h)\,(\partial^\gamma\delta)(y-w). \label{eq:L-deltaQ} \end{equation} \end{lemma} \begin{proof} \eqref{eq:L-Dq} is Proposition~\ref{prop:RL}(4) applied to the two extensions \(\RL'(q^\circ_\lambda)\) and \(\RL(q^\circ_\lambda)\). Linearity of \eqref{eq:QLext} in the extended kernels gives \eqref{eq:L-deltaQ}. Each summand is a smooth coefficient times a point-supported distribution in \(z=y-w\); no root is used. \end{proof} \subsection{A. Coherence constraints} \label{sec:l5A} \begin{proposition}[Coherence] \label{prop:Lcoh} If \(\sum_\lambda r_\lambda\,q^\circ_\lambda=0\) in \(\mathcal D'(\mathbb R^4\setminus\{0\})\) with constants \(r_\lambda\), then \[ \sum_\lambda r_\lambda\,\Delta\RL(q^\circ_\lambda)=0, \qquad\text{i.e.}\qquad \sum_\lambda r_\lambda\,C_{\lambda\gamma}=0\quad\text{for every }\gamma. \] In particular, edge interchange (\(q^\circ_{\alpha\beta}=q^\circ_{\beta\alpha}\)) is already quotiented by unordered labels, and the wave-operator relation \(\eta^{\alpha\beta}q^\circ_{\{0,e_\alpha+e_\beta\}}=0\) forces \(\sum_{\alpha,\beta}\eta^{\alpha\beta}C_{\{0,e_\alpha+e_\beta\},\,\gamma}=0\) for every \(|\gamma|\le2\). \end{proposition} \begin{proof} \(\Delta\RL\) is linear on \(\mathcal V_{\mathrm{adm}}\), so it annihilates every constant-coefficient relation; linear independence of \(\{\partial^\gamma\delta\}\) yields the coefficient form. The two sample relations are Proposition~\ref{prop:VLincl}(1),(3). \end{proof} This is the complete-source analogue of XVI-K, Remark rem:coherent: only coherent changes (linear on the actual span) are renormalization-prescription changes. \subsection{B. Regression-preserving versus general complete-source changes} \label{sec:l5B} \begin{definition}[Regression-preserving coherent changes] \label{def:Lreg} A coherent change \(\Delta\RL\) is \emph{regression-preserving} if \(\Delta\RL|_{\mathcal V_{\mathrm K}}=0\), equivalently \(C_{\lambda\gamma}=0\) for every numerical label \(\lambda\) of \(\mathcal V_{\mathrm K}\) and every \(|\gamma|\le N_\lambda\). A \emph{general complete-source} coherent change allows nonzero contact coefficients on \(\mathcal V_{\mathrm K}\) as well (and may therefore alter the XVI-K / XVI-J component). \end{definition} Regression-preserving changes are carried only by the new labels in \(\mathcal V_{\mathrm{adm}}\setminus\mathcal V_{\mathrm K}\) (types \(S\cdot\partial^3S\), \(S\cdot\partial^4S\), \(\partial S\cdot\partial^2S\), \(\partial S\cdot\partial^3S\), \(\partial^2S\cdot\partial^2S\), and the higher-degree parts of older types). General changes include the XVI-K ambiguity of XVI-K, Lemma lem:amb, as the \(\mathcal V_{\mathrm K}\) block. \subsection{C. Jet bounds after contact} \label{sec:l5C} Write \(s:=|a|+|b|\) (source-slot order), \(v:=|p|+|q|\) (vertex order), so \(N_\lambda=s+v\) for an ordered placement contributing to label \(\lambda\), and recall from L.1--L.2: \begin{itemize} \item \(L^{pq}_{ij}(w;h)\) is linear in the residual-metric jet of order \(2-v\) (XVI-K, Proposition prop:hess(1); Remark after Definition~\ref{def:La}); \item the preparation jet of \(\sigeps\) sits in \(P_A(y)\), of order \(2,1,0\) for \(A=C_2,M_2,J_3\) respectively (Theorem~\ref{thm:budget}), and is \emph{not} differentiated by \(\partial_w\); \item \(\chi_{\mathrm I}(w)\) sits undifferentiated in \(a_\lambda\) before contact. \end{itemize} \begin{proposition}[Jet bounds for unrooted contact] \label{prop:Ljet} After the contact operation underlying \eqref{eq:L-deltaQ}, writing \(\partial_w^\gamma\) for the derivatives that hit \(a^{\mu\nu}_\lambda\) upon pairing against \((\partial^\gamma\delta)(y-w)\) (as in XVI-K, Lemma lem:amb): \begin{enumerate} \item \emph{Residual metric.} Every residual-metric jet produced has order at most \begin{equation} (2-v)+|\gamma|\le(2-v)+N_\lambda=(2-v)+(s+v)=2+s\le4, \label{eq:L-resjet} \end{equation} since \(s\le2\) by Theorem~\ref{thm:budget}. The arithmetic upper bound reaches \(2+s\) when all \(N\) derivatives hit the residual-metric coefficient; actual attainment is not established. \item \emph{Preparation.} The preparation jet remains of order \(2-s\) (equivalently \(2,1,0\) by sector); \(\partial_w\) does not raise it. \item \emph{Switch.} Derivatives \(\partial^{\gamma'}\chi_{\mathrm I}\) appear with \(|\gamma'|\le|\gamma|\le N_\lambda\le4\). \end{enumerate} These are statements about the pointwise jet of the coefficient in \eqref{eq:L-deltaQ}, not about a distributional order after root attachment. \end{proposition} \begin{proof} (1) Leibniz: \(\partial_w^\gamma\) distributes over \(\chi_{\mathrm I}(w)\) and \(L^{pq}_{ij}(w;h)\). The portion hitting \(L\) raises the residual-metric order from \(2-v\) by at most \(|\gamma|\). Then \(|\gamma|\le N_\lambda=s+v\) and \(s\le2\) give \eqref{eq:L-resjet}. (2) \(P_A(y)\) depends on \(y\) only (Definition~\ref{def:La}); \(\partial_w\) does not hit it. (3) Immediate from Leibniz on \(\chi_{\mathrm I}(w)\) and \(N_\lambda\le4\) (Proposition~\ref{prop:Lsd}). \end{proof} \subsection{D. Four interpretive counterterm categories} \label{sec:l5D} The local collision distributions \eqref{eq:L-deltaQ}, together with the fixed defects of Lemma~\ref{lem:Ldefect}, carry four interpretive categories (attributes), which are \emph{kept distinct} and are \emph{not} asserted to form a disjoint partition of summands: \begin{enumerate} \item[(1)] \emph{Candidate bulk metric counterterms.} Summands in which no derivative hits \(\chi_{\mathrm I}\), and whose residual-metric jet dependence can be read as a candidate local density in \(h\) (after diagonal restriction \(w=y\)). Preparation factors, if present, are evaluated on the collision diagonal; no claim is made that any such summand equals the variation of a bulk gravitational action. \item[(2)] \emph{Preparation-dependent insertions.} Summands whose coefficient retains nontrivial dependence on the \(\sigeps\)-jet in \(P_A(y)\). These are insertion/preparation counterterms, not pure bulk metric densities. \item[(3)] \emph{Switch-derivative terms.} Summands carrying \(\partial^{\gamma'}\chi_{\mathrm I}\) with \(|\gamma'|\ge1\). These depend on the choice of interaction switch. \item[(4)] \emph{Fixed differential-normalization defects.} The distributions \(D_{\alpha,u}\) of Lemma~\ref{lem:Ldefect}. They are determined by \(\varpi\) and the moment prescription \eqref{eq:L-moments}; they are not free parameters of a coherent change \(\Delta\RL\). \end{enumerate} Categories (1)--(3) parametrize coherent changes; category~(4) is fixed once \(\RL\) is fixed. A single summand may carry more than one attribute. \textbf{Do not} identify all four with bulk gravitational action counterterms: only category~(1) is even a candidate for such an identification, and no such identification is proved here (cf.\ XVI-K, Remark rem:amb(2)). \begin{theorem}[Principal renormalization theorem, unrooted] \label{thm:L5} Finite-dimensional coherent contact freedom is characterized abstractly by all numerical-kernel relations; selected constraints are exhibited. Concretely: the raw contact assignments \((C_{\lambda\gamma})\) with \(|\gamma|\le N_\lambda\) (finitely many slots) are subject to the linear coherence constraints of Proposition~\ref{prop:Lcoh} (edge interchange, wave-operator, and all other constant-coefficient numerical-kernel relations). The numbers \(14321\) (raw contact slots), \(321\) (\(V_{\mathrm K}\) block), and \(14000\) (complement) recorded by \texttt{counterterm\_map.py} are raw counts of pairs \((\lambda,\gamma)\), \emph{not} dimensions of constrained spaces; no computed constrained dimension is claimed. \emph{Converse.} Any assignment \((C_{\lambda\gamma})\) satisfying all numerical-kernel relations descends to a linear map on \(\mathrm{span}\{q^\circ_\lambda\}\); adding the corresponding contact combination to \(\RL\) yields an admissible same-degree extension. The map \[ (C_{\lambda\gamma})\;\longmapsto\;\Delta Q^{\mu\nu}_{\mathrm L} = \sum_{\lambda,\gamma}C_{\lambda\gamma}\,a^{\mu\nu}_\lambda\,(\partial^\gamma\delta)(y-w) \] sends such coherent assignments to local collision distributions of the form \eqref{eq:L-deltaQ}, with jet bounds as in Proposition~\ref{prop:Ljet}, and with the four interpretive categories of Section~\ref{sec:l5D}. The regression-preserving subspace is the further linear condition \(\Delta\RL|_{\mathcal V_{\mathrm K}}=0\). No root attachment, no scale variation, and no bulk-action matching is claimed. \end{theorem} \begin{proof} Finiteness of the raw contact space: finitely many unordered labels (Lemma~\ref{lem:L3comp}: \(255\)) and finitely many multi-indices \(|\gamma|\le N_\lambda\le4\). Constraints: Proposition~\ref{prop:Lcoh} (selected wave-operator constraints are exhibited computationally; the abstract characterization uses all numerical-kernel relations). Converse: a coherent assignment annihilates every constant-coefficient relation among the \(q^\circ_\lambda\), hence factors through the quotient span; adding the contact combination preserves the extension property off \(0\) and the scaling-degree bound. The map is Lemma~\ref{lem:Lamb}. Jet bounds: Proposition~\ref{prop:Ljet}. Categories: Section~\ref{sec:l5D}. Regression-preserving: Definition~\ref{def:Lreg}. \end{proof} \begin{lemma}[Computational verification of L.4--L.5] \label{lem:L5comp} The script \texttt{XVI-L-artifacts/counterterm\_map.py} enumerates placements from the L.1/L.2 budgets, exits with status \(0\), and records: \begin{enumerate} \item[(C1)] Coherence: unordered labels quotient edge interchange; wave-operator generators lie in \(\mathcal V_{\mathrm{adm}}\); \(15\) diagonal \(\square S=0\) constraints (one per \(|\gamma|\le2\)). \item[(C2)] Jet bounds: \((2-v)+|\gamma|\le2+s\le4\) on all contacts; arithmetic identity \((2-v)+N=2+s\) (not a sharpness/attainment claim); prep orders \(2,1,0\) by sector; switch order \(\le4\). \item[(C3)] \(W_\rho\) orders \(\{0,1,2,3,4\}\) with \(\rho\le2\) matching XVI-K; raw contact \emph{counts} (not dimensions) \(14321\) before coherence; \(V_{\mathrm K}\) block \(321\); complement \(14000\). \item[(C4)] Four interpretive counterterm categories recorded; one family of wave-operator relations exhibited (not a complete constraint list); finite defect-ledger candidate types (six typed sources with image still in \(\mathcal V_{\mathrm{adm}}\)). \end{enumerate} \end{lemma} \begin{proof} Printed lines of \texttt{counterterm\_map\_out.txt}; Appendix~\ref{app:out}. \end{proof} %===================================================================== \section{L.6 Renormalization scale} \label{sec:l6} The extension of L.4 depends on the fixed cutoff function \(\varpi\in C_c^\infty(\mathbb R^4)\) with \(\varpi(0)=1\). This section varies that prescription by dilating \(\varpi\), producing a one-parameter family of coherent extensions. The resulting \(\mu\)-dependence is a change of subtraction prescription; it is \emph{not} identified with a physical renormalization-group flow of a complete gravitational quantity (Remark~\ref{rem:Lscale}). \paragraph{Reference scale and dilated cutoffs.} Fix a reference \(\mu_0>0\). For \(\mu>0\) define \begin{equation} \varpi_\mu(z):=\varpi\Bigl(\frac{\mu}{\mu_0}\,z\Bigr),\qquad z\in\mathbb R^4. \label{eq:L-varpimu} \end{equation} Then \(\varpi_\mu\in C_c^\infty(\mathbb R^4)\) and \(\varpi_\mu(0)=\varpi(0)=1\). At \(\mu=\mu_0\) one recovers \(\varpi_{\mu_0}=\varpi\). \paragraph{Scaled projections.} For \(\rho\in\{0,1,2,3,4\}\) form the matrices and functions of \eqref{eq:L-Wrho} with \(\varpi\) replaced by \(\varpi_\mu\): \[ M^{(\rho,\mu)}_{\beta\alpha} = \partial^\beta\bigl(z^\alpha\varpi_\mu/\alpha!\bigr)(0), \qquad w^{(\rho,\mu)}_\alpha(z) = \sum_{|\gamma|\le\rho}\bigl((M^{(\rho,\mu)})^{-1}\bigr)_{\gamma\alpha}\,\frac{z^\gamma}{\gamma!}\,\varpi_\mu(z), \] \begin{equation} (W^{(\mu)}_\rho\varphi)(z) = \varphi(z)-\sum_{|\alpha|\le\rho}\partial^\alpha\varphi(0)\,w^{(\rho,\mu)}_\alpha(z). \label{eq:L-Wrhomu} \end{equation} Each \(M^{(\rho,\mu)}\) remains unitriangular (Leibniz; \(\varpi_\mu(0)=1\)), so invertible, and \(W^{(\mu)}_\rho\) projects onto the test functions vanishing to order \(\rho\) at \(0\). At \(\mu=\mu_0\) one has \(W^{(\mu_0)}_\rho=W_\rho\) of \eqref{eq:L-Wrho}. \begin{definition}[Scale-dependent extension \(\RLmu\)] \label{def:RLmu} For \(u\in\mathcal H_k\subset\mathcal V_{\mathrm{adm}}\) and \(\rho=k-4\), let \(\RLmu(u)\in\mathcal D'(\mathbb R^4)\) be the Brunetti--Fredenhagen extension of scaling degree \(k\) restricting to \(u\) on \(\mathbb R^4\setminus\{0\}\), built with the projection \(W^{(\mu)}_\rho\) of \eqref{eq:L-Wrhomu} and the zero moments \begin{equation} \bigl\langle\RLmu(u),\,w^{(\rho,\mu)}_\alpha\bigr\rangle=0,\qquad|\alpha|\le\rho . \label{eq:L-moments-mu} \end{equation} Extend \(\RLmu\) to \(\mathcal V_{\mathrm{adm}}\) by linearity. Equivalently, on each graded piece, \begin{equation} \bigl\langle\RLmu(u),\varphi\bigr\rangle = \bigl\langle u,\,W^{(\mu)}_\rho\varphi\bigr\rangle \label{eq:L-RLmu-formula} \end{equation} (the right side uses the unique continuous extension of \(u\) to test functions vanishing to order \(\rho\) at \(0\)). \end{definition} \begin{proposition}[Recovery of \(\RL\) at the reference scale] \label{prop:RLmu0} \(\RLmuref=\RL\) as maps \(\mathcal V_{\mathrm{adm}}\to\mathcal D'(\mathbb R^4)\). In particular \(\RLmuref|_{\mathcal V_{\mathrm K}}=\RK\). \end{proposition} \begin{proof} At \(\mu=\mu_0\), Definition~\ref{def:RLmu} uses the same \(\varpi\), the same \(W_\rho\), and the same zero moments as Definition~\ref{def:RL}; uniqueness of the Brunetti--Fredenhagen extension of fixed degree with these moments gives equality. The restriction identity is Proposition~\ref{prop:RL}(3). \end{proof} \begin{theorem}[First scale theorem] \label{thm:Lscale} For every \(u\in\mathcal V_{\mathrm{adm}}\) of declared order \(N\) (so \(u\in\mathcal H_{4+N}\) up to lower graded pieces) and every \(\mu>0\), \begin{equation} \RLmu(u)-\RLmuref(u) = \sum_{|\gamma|\le N}c_{\mu,\mu_0}(u,\gamma)\,\partial^\gamma\delta, \label{eq:L-scale-diff} \end{equation} with coefficients \(c_{\mu,\mu_0}(u,\gamma)\in\mathbb C\). In particular the difference is supported at \(\{0\}\) and has contact order at most the declared order of \(u\). \end{theorem} \begin{proof} Both \(\RLmu(u)\) and \(\RLmuref(u)\) extend \(u\) on \(\mathbb R^4\setminus\{0\}\) with scaling degree \(\le4+N\) (Definition~\ref{def:RLmu}; Proposition~\ref{prop:RL}(2) at \(\mu_0\)). Their difference is therefore supported at \(0\) and of scaling degree \(\le4+N\), hence of the form \eqref{eq:L-scale-diff} (Brunetti--Fredenhagen, Theorem~5.3, after the statement). \end{proof} \begin{lemma}[Differentiability in the scale] \label{lem:Ldiffmu} For each \(u\in\mathcal V_{\mathrm{adm}}\) the map \(\mu\mapsto\RLmu(u)\) is \(C^\infty\) as a \(\mathcal D'(\mathbb R^4)\)-valued function of \(\mu>0\). Writing \begin{equation} A_\mu(u) := \mu\,\frac{\mathrm d}{\mathrm d\mu}\,\RLmu(u) \in\mathcal D'(\mathbb R^4), \label{eq:L-Amu} \end{equation} one has \(\operatorname{supp}A_\mu(u)\subset\{0\}\), and if \(u\in\mathcal H_{4+N}\) then \begin{equation} A_\mu(u) = \sum_{|\gamma|\le N}a_\mu(u,\gamma)\,\partial^\gamma\delta \label{eq:L-Amu-exp} \end{equation} for coefficients \(a_\mu(u,\gamma)\in\mathbb C\) depending smoothly on \(\mu\). Moreover \(A_\mu:\mathcal V_{\mathrm{adm}}\to\mathcal D'(\mathbb R^4)\) is linear, and every constant-coefficient relation \(\sum_\lambda r_\lambda q^\circ_\lambda=0\) in \(\mathcal D'(\mathbb R^4\setminus\{0\})\) implies \(\sum_\lambda r_\lambda A_\mu(q^\circ_\lambda)=0\). \end{lemma} \begin{proof} \emph{Smoothness of the projections.} The map \(\mu\mapsto\varpi_\mu\) is \(C^\infty\) into \(C_c^\infty(\mathbb R^4)\) (chain rule; compact supports remain in a fixed large ball for \(\mu\) in any compact subinterval of \((0,\infty)\)). Hence \(\mu\mapsto M^{(\rho,\mu)}\) is \(C^\infty\) into invertible matrices (unitriangular with smooth entries), \(\mu\mapsto w^{(\rho,\mu)}_\alpha\) is \(C^\infty\) into \(C_c^\infty\), and \(\mu\mapsto W^{(\mu)}_\rho\) is \(C^\infty\) as a map into continuous operators on \(\mathcal D(\mathbb R^4)\). \emph{Differentiability of \(\RLmu\).} On \(\mathcal H_{4+N}\), formula \eqref{eq:L-RLmu-formula} gives \(\langle\RLmu(u),\varphi\rangle=\langle u,W^{(\mu)}_\rho\varphi\rangle\). The right side is \(C^\infty\) in \(\mu\) for each test \(\varphi\), so \(\mu\mapsto\RLmu(u)\) is \(C^\infty\) in the weak topology of \(\mathcal D'\); the distributional derivative \(A_\mu(u)\) exists and is given by differentiating under the pairing. Linearity in \(u\) is immediate from \eqref{eq:L-RLmu-formula}. \emph{Contact expansion.} Differentiate \eqref{eq:L-scale-diff} in \(\mu\) at fixed \(\mu_0\), or compare \(\mathfrak R_{\mathrm L,\mu+\varepsilon}(u)-\RLmu(u)\) by Theorem~\ref{thm:Lscale} with reference \(\mu\): the difference quotient is a combination of \(\partial^\gamma\delta\) with \(|\gamma|\le N\), and the limit retains that form, yielding \eqref{eq:L-Amu-exp}. Support at \(0\) is immediate. \emph{Coherence.} If \(\sum r_\lambda q^\circ_\lambda=0\) as an element of \(\mathcal V_{\mathrm{adm}}\), then \(\RLmu\) of that element vanishes for every \(\mu\) (linearity), so its \(\mu\)-derivative \(A_\mu\) vanishes on it. \end{proof} \paragraph{Unrooted scale variation of the extended kernel.} With \(Q^{\mu\nu}_{\mathrm L,\mu}\) defined by \eqref{eq:QLext} with \(\RL\) replaced by \(\RLmu\), \begin{equation} \mu\frac{\mathrm d}{\mathrm d\mu}Q^{\mu\nu}_{\mathrm L,\mu}(y,w;h) = \sum_{\lambda}a^{\mu\nu}_\lambda(y,w;h)\,\bigl(\Phi^*A_\mu(q^\circ_\lambda)\bigr)(y,w), \label{eq:L-dQmu} \end{equation} a local collision distribution of the same type as \eqref{eq:L-deltaQ}, with contact coefficients \(a_\mu(q^\circ_\lambda,\gamma)\) in place of the free \(C_{\lambda\gamma}\). The four interpretive categories of Section~\ref{sec:l5D} apply verbatim to these coefficients (they are a particular coherent change, fixed by \(\varpi\) and \(\mu\) rather than free). \begin{remark}[Prescription dependence versus scale-variation coefficients; not a physical beta function] \label{rem:Lscale} \begin{enumerate} \item \emph{Prescription dependence.} Changing \(\mu\) changes the subtraction functions \(w^{(\rho,\mu)}_\alpha\) and therefore the extension \(\RLmu\). The difference \(\RLmu-\RLmuref\) is always a finite contact combination \eqref{eq:L-scale-diff}: it is a renormalization-prescription change in the sense of L.5, not a change of the off-diagonal physical kernel. \item \emph{Scale-variation coefficients.} The generator \(A_\mu(u)\) of \eqref{eq:L-Amu} is the infinitesimal form of that prescription change. The contact coefficients \(a_\mu(u,\gamma)\) in \eqref{eq:L-Amu-exp} (or their \(\mu\)-independent combinations, when they exist) are called \emph{scale-variation coefficients} of the extension. They are determined by \(\varpi\) and by the spherical behaviour of \(u\); they are not free counterterm parameters once \(\varpi\) and \(\mu\) are fixed. The term \emph{scaling residue} is reserved for a later anomaly expression of the schematic form \((E+4+N)R(u)\); no such anomaly evaluation is performed in XVI-L. \item \emph{Not a physical beta function.} The object varied here is the subtraction prescription for the numerical two-edge kernels of one selected graph, not a coupling in a complete gravitational effective action, and not a Callan--Symanzik operator for a physical correlator. No identification of \(A_\mu\) with a gravitational beta function, a running of \(\kappa\) or \(\hbar\), or an RG flow of a diffeomorphism-invariant observable is claimed. Scale-dependent subtraction is kept strictly distinct from any such physical RG interpretation. \end{enumerate} \end{remark} \begin{proposition}[Homogeneous contact form of the scale-variation coefficients] \label{prop:Lresform} If \(u\in\mathcal H_{4+N}\) is homogeneous of degree \(-4-N\) on \(\mathbb R^4\setminus\{0\}\), then \(A_\mu(u)\) has the contact form \eqref{eq:L-Amu-exp}, and the scale-variation coefficients satisfy the coherence constraints of Lemma~\ref{lem:Ldiffmu}. In the critical case \(N=0\) (so \(u\) is a multiple of \(S^2\) on \(\mathcal H_4\)), one has \(A_\mu(u)=r_\mu(u)\,\delta\) for a scalar \(r_\mu(u)\in\mathbb C\). \end{proposition} \begin{proof} Lemma~\ref{lem:Ldiffmu}. For \(N=0\) the only multi-index with \(|\gamma|\le0\) is \(\gamma=0\). \end{proof} \begin{remark}[Stretch: explicit nonzero scaling residue] \label{rem:Lressretch} Massless homogeneity of \(S\) (XVI-K, Lemma lem:homog) and \(\mathrm{sd}(S^2)=4\) (XVI-K, Lemma lem:nonzero) give the contact form \(A_\mu(S^2)=r_\mu(S^2)\,\delta\) for the scale-variation coefficient \(r_\mu(S^2)\), but do not by themselves evaluate it or prove \(r_\mu(S^2)\ne0\): a nonzero homogeneous distribution of degree \(-4\) may still have vanishing spherical average against the radial derivative of \(\varpi_\mu\). An explicit nonzero \emph{scaling residue} in the reserved anomaly sense \((E+4+N)R(u)\) would require either the angular integral of \(S^2\) on the unit sphere (or an equivalent Hadamard coefficient) or an independent computation of \(\langle\RL(S^2),\mu\partial_\mu\varpi_\mu\rangle\). Neither is imported from XVI-K/XVI-J in a form that yields a numerical nonvanishing here. \textbf{Unresolved:} whether \(r_\mu(S^2)\ne0\) for the present \(\varpi\), and whether a nonzero scaling residue (anomaly form) occurs. \end{remark} \begin{theorem}[L.6 summary] \label{thm:L6} There exists a smooth family \(\mu\mapsto\RLmu\) of coherent extensions of \(\mathcal V_{\mathrm{adm}}\) with \(\RLmuref=\RL\), such that \(\RLmu(u)-\RLmuref(u)\) is supported at \(0\) of contact order \(\le N(u)\), the infinitesimal generator \(A_\mu=\mu\mathrm d/\mathrm d\mu\,\RLmu\) exists and takes values in the same finite contact space, and every constant-coefficient relation in \(\mathcal V_{\mathrm{adm}}\) is preserved. The construction is a scale-dependent subtraction prescription; it is not a physical gravitational beta function (Remark~\ref{rem:Lscale}). No explicit nonzero scaling residue (anomaly form) is claimed (Remark~\ref{rem:Lressretch}). \end{theorem} \begin{proof} Definition~\ref{def:RLmu}; Proposition~\ref{prop:RLmu0}; Theorem~\ref{thm:Lscale}; Lemma~\ref{lem:Ldiffmu}; Remark~\ref{rem:Lscale}. \end{proof} %===================================================================== \section{L.7 Root attachment, exact regressions, and propagated counterterms} \label{sec:l7} The root kernel is the imported \(B_{f,g}(y,x)\) of Hypothesis \(\mathrm H_{\mathrm{root}}\) (XVI-K, Section sec:hyp), used only through the unchanged covector-shape hypotheses (R1)--(R3): existence of the composition; compact support of \(B_{f,g}\); \(\mathrm{WF}(B_{f,g})\subset Z_2\setminus0\). No property of \(B_{f,g}\) beyond (R1)--(R3) is used. \paragraph{Root-attached distributions.} With \(\pi(y,x,w)=(y,x)\) and \(\Phi_3(y,x,w)=y-w\), and with \(Q^{\mu\nu}_{\mathrm L}\) the unrooted extended kernel \eqref{eq:QLext}, define \begin{equation} U^{\RL}_{\mathrm L}(y,x,w;h) := B_{f,g}(y,x)_{\mu\nu}\,Q^{\mu\nu}_{\mathrm L}(y,w;h) = \sum_{\lambda} B_{f,g}(y,x)_{\mu\nu}\,a^{\mu\nu}_\lambda(y,w;h)\,\bigl(\Phi_3^*\RL(q^\circ_\lambda)\bigr)(y,w) \label{eq:UL} \end{equation} and the smeared coefficient \begin{equation} \mathcal C^{\RL}_{\mathrm L}(h,\phi) := \bigl\langle U^{\RL}_{\mathrm L}(\cdot;h),\,1_y\otimes\phi_x\otimes1_w\bigr\rangle. \label{eq:CL} \end{equation} (The same formulae with \(\RL\) replaced by \(\RLmu\) define \(U^{\RLmu}_{\mathrm L}\) and \(\mathcal C^{\RLmu}_{\mathrm L}\).) \begin{lemma}[Root attachment for the complete-source kernels] \label{lem:Lattach} Assume (R1)--(R3). For every numerical label \(\lambda\), every pair of indices \(\mu,\nu\), and every smooth \(h\): \begin{enumerate} \item \(\pi^*B_{f,g,\mu\nu}\) and \(\Phi_3^*\RL(q^\circ_\lambda)\) are defined distributions on \(M^3\), with \[ \mathrm{WF}(\pi^*B_{f,g,\mu\nu})\subset\bigl\{(y,x,w;\xi,-\xi,0):\xi\ne0\bigr\}, \qquad \mathrm{WF}(\Phi_3^*\RL(q^\circ_\lambda))\subset\bigl\{(y,x,w;\kappa,0,-\kappa):\kappa\ne0\bigr\}. \] \item The product \(B_{f,g,\mu\nu}(y,x)\,\RL(q^\circ_\lambda)(y-w)\) exists in \(\mathcal D'(M^3)\), including on \(y=w\), and so does the product with the smooth coefficient \(a^{\mu\nu}_\lambda(y,w;h)\). \item \(\operatorname{supp}U^{\RL}_{\mathrm L}(\cdot;h)\subset\{(y,x,w):(y,x)\in\operatorname{supp}B_{f,g},\ w\in\operatorname{supp}\chi_{\mathrm I}\}\), a compact set. \end{enumerate} The statements use only that \(\RL(q^\circ_\lambda)\) is a distribution on \(\mathbb R^4\); they hold for every other same-degree extension, and for every \(\RLmu(q^\circ_\lambda)\). \end{lemma} \begin{proof} Identical to XVI-K, Lemma lem:attachK, with \(\RK(q^\circ_{\sk})\) replaced by \(\RL(q^\circ_\lambda)\) and the finite sum extended over the complete-source labels of \(\mathcal V_{\mathrm{adm}}\). (1) \(\pi\) and \(\Phi_3\) are linear submersions (H\"ormander, Theorem~8.2.4); the covector shapes follow from (R3) and from \({}^t\Phi_3'\kappa=(\kappa,0,-\kappa)\), independently of which extension of \(q^\circ_\lambda\) is pulled back. (2) A sum \((\xi,-\xi,0)+(\kappa,0,-\kappa)\) vanishes only if \(\xi=\kappa=0\); neither wavefront set contains a zero covector, so the product exists by H\"ormander, Theorem~8.2.10. Smooth multiplication by \(a^{\mu\nu}_\lambda\) (compactly supported in \(w\) through \(\chi_{\mathrm I}\)) preserves existence. (3) Support of a product sits in the intersection of supports; (R2) and \(\operatorname{supp}\chi_{\mathrm I}\) compact give compactness. \end{proof} \begin{theorem}[Root-attached complete-source coefficient] \label{thm:Lmain} Assume Hypothesis \(\mathrm H_{\mathrm{root}}\) ((R1)--(R3)). For every smooth residual metric \(h\) and every smooth scalar \(\phi\), the coefficient \(\mathcal C^{\RL}_{\mathrm L}(h,\phi)\) of \eqref{eq:CL} is well defined. Moreover: \begin{enumerate} \item \emph{Bilinearity.} \(\mathcal C^{\RL}_{\mathrm L}(h,\phi)\) is linear in \(h\) and linear in \(\phi\). \item \emph{Joint continuity.} The map \((h,\phi)\mapsto\mathcal C^{\RL}_{\mathrm L}(h,\phi)\) is jointly continuous for the \(C^\infty\) topologies: there exist \(C_0\), compact sets \(K_h\), \(K_\phi\), and finite orders \(r,s\) with \(|\mathcal C^{\RL}_{\mathrm L}(h,\phi)|\le C_0\,p_{r,K_h}(h)\,p_{s,K_\phi}(\phi)\). The orders \(r,s\) are not computed here. \item \emph{Off-diagonal agreement.} If \(\psi\in C_c^\infty(M^3)\) has \(\operatorname{supp}\psi\cap\{y=w\}=\emptyset\), then \(\langle U^{\RL}_{\mathrm L}(\cdot;h),\psi\rangle\) equals the same pairing with each \(\RL(q^\circ_\lambda)\) replaced by the unextended \(q^\circ_\lambda\). \item \emph{Compact support.} \(U^{\RL}_{\mathrm L}(\cdot;h)\) has compact support, by Lemma~\ref{lem:Lattach}(3). \end{enumerate} The functional is not asserted to be nonzero. \end{theorem} \begin{proof} Lemma~\ref{lem:Lattach}(2)--(3): \(U^{\RL}_{\mathrm L}(\cdot;h)\in\mathcal E'(M^3)\), and \eqref{eq:CL} is the pairing against \(\zeta(1\otimes\phi\otimes1)\) for any cutoff \(\zeta\equiv1\) near the support, hence defined. (1) Linearity in \(\phi\): pairing. Linearity in \(h\): each \(L^{pq}_{ij}(w;h)\) is linear in the residual-metric jet (XVI-K, Proposition prop:hess(1)), hence so are \(a^{\mu\nu}_\lambda\) (Definition~\ref{def:La}); \(\pi^*B_{f,g}\) and \(\Phi_3^*\RL(q^\circ_\lambda)\) are independent of \(h\); H\"ormander products are bilinear. (2) As in XVI-K, Theorem thm:main(2): write \(T_{\lambda,\mu\nu}:=\pi^*B_{f,g,\mu\nu}\cdot\Phi_3^*\RL(q^\circ_\lambda)\), restrict to a fixed compact neighbourhood of the support, of some finite order \(m\); Leibniz expands \(\partial^\alpha(\zeta a_\lambda\phi)\) into finitely many derivatives of \(\phi\) and of \(a_\lambda(\cdot;h)\), the latter controlled by finitely many derivatives of \(h\) on the \(w\)-projection. The distributional order \(m\) is not computed, so \(r,s\) are not read off from the pointwise jet bounds of Proposition~\ref{prop:Ljet}. (3) On \(\{y\ne w\}\), \(\Phi_3^*\RL(q^\circ_\lambda)=\Phi_3^*q^\circ_\lambda\) by Proposition~\ref{prop:RL}(1); the product is local. \end{proof} \begin{proposition}[Exact restriction to XVI-K] \label{prop:LtoK} Two separate identities are recorded; neither substitutes for the other. \begin{enumerate} \item \emph{Source-sector truncation.} In the ordered decomposition \eqref{eq:UL}/\eqref{eq:QL}, retain only the summand \(A=C_2\) (equivalently, in the sector-resolved grouped form, retain only the coefficients \(a^{C_2}_\lambda\)). The resulting root-attached distribution is exactly \(U^{\RK}_{\mathrm K}\) of XVI-K, eq:UK, as an element of \(\mathcal D'(M^3)\) (extension included), and the corresponding truncation of \eqref{eq:CL} is exactly \(\mathcal C^{\RK}_{C_2,\mathrm{fv}}\) of XVI-K, eq:CK. \item \emph{Extension restriction.} Independently, \(\RL|_{\mathcal V_{\mathrm K}}=\RK\) as maps \(\mathcal V_{\mathrm K}\to\mathcal D'(\mathbb R^4)\) (Proposition~\ref{prop:RL}(3)). \end{enumerate} Discarding numerical labels outside \(\mathcal V_{\mathrm K}\) is \emph{not} equivalent to the source-sector truncation: other sectors can contribute to labels in \(\mathcal V_{\mathrm K}\) (e.g.\ \(N=1,2\)), while the order-\(0\) label \(\{0,0\}\) arises only in \(C_2\). \end{proposition} \begin{proof} (1) On the \(C_2\), \(a=b=0\) sector, Definition~\ref{def:La} reduces to XVI-K, Definition def:a: \(P_{C_2}\) is the polarization of \(2C_2\partial\partial\sigeps\) (L.1, check S3), and \(a^{C_2}_{00,pq}\) coincides with \(a_{pq}\) of XVI-K, eq:apq. The numerical kernels are \(q^\circ_{\{p,q\}}\) of XVI-K. Combined with (2), the H\"ormander products with \(B_{f,g}\) coincide with those of XVI-K, Lemma lem:attachK, and the smeared coefficients coincide. (2) Proposition~\ref{prop:RL}(3). \end{proof} \begin{proposition}[Exact further restriction to XVI-J] \label{prop:LtoJ} In the truncation of Proposition~\ref{prop:LtoK}, discard every term with \(|p|+|q|>0\), i.e.\ keep only the constant-slot label \(\{0,0\}\). The result is exactly XVI-J's distribution \(U^{\mathfrak R}\) of XVI-J, eq:U, and the corresponding coefficient is exactly \(\mathcal C^{\mathfrak R}_{2e}\) of XVI-J, eq:C. \end{proposition} \begin{proof} Proposition~\ref{prop:LtoK} reduces to XVI-K; then XVI-K, Proposition prop:regression, reduces XVI-K to XVI-J as an identity of distributions including \(y=w\). Composing the two exact restrictions gives the claim. Alternatively: \(a_{\{0,0\}}=\chi_{\mathrm I}\mathcal K\) with \(\mathcal K\) of XVI-J, eq:Kdef (XVI-K, Proposition prop:hess(4)), and \(\RL(S^2)=\RK(S^2)=\bar q^{\mathfrak R}\) (Proposition~\ref{prop:RL}(3)). \end{proof} \begin{lemma}[Propagated coherent ambiguity] \label{lem:Lambprop} Let \(\Delta\RL\) be a coherent change as in Lemma~\ref{lem:Lamb}, with contact coefficients \(C_{\lambda\gamma}\). The root-attached coefficient built from \(\RL'=\RL+\Delta\RL\) differs from \(\mathcal C^{\RL}_{\mathrm L}\) by \begin{equation} \Delta\mathcal C_{\mathrm L}(h,\phi) = \sum_{\lambda}\sum_{|\gamma|\le N_\lambda}C_{\lambda\gamma}\, \Bigl\langle B_{f,g,\mu\nu},\, \bigl(\partial_w^\gamma a^{\mu\nu}_\lambda\bigr)(y,w;h)\big|_{w=y}\,\phi(x) \Bigr\rangle . \label{eq:L-deltaCL} \end{equation} The sign is net \(+1\), exactly as in XVI-K, eq:deltaCK: the chain-rule identity \((\partial_z^\gamma\delta)(y-w)=(-1)^{|\gamma|}\partial_w^\gamma[\delta(y-w)]\) and \(|\gamma|\) integrations by parts in the relative variable each contribute \((-1)^{|\gamma|}\), and the product is \(+1\). \end{lemma} \begin{proof} Copy the argument of XVI-K, Lemma lem:amb, with the sum extended over complete-source labels and with \(\RK\) replaced by \(\RL\). Form of \(\Delta\RL(q^\circ_\lambda)\): Lemma~\ref{lem:Lamb}. Change of \(U^{\RL}_{\mathrm L}\): linearity of \eqref{eq:UL} in the extended kernels, each term existing by Lemma~\ref{lem:Lattach}. Pass to coordinates \((y,x,z)\), \(z=y-w\); the product becomes the tensor product \(B_{f,g}\otimes\partial^\gamma\delta\); pairing against \(\tilde a\,\tilde\psi\) produces \((-1)^{|\gamma|}\partial_z^\gamma[\tilde a\tilde\psi]|_{z=0}\); the substitution \(\partial_z=-\partial_w\) contributes a second \((-1)^{|\gamma|}\), whence net \(+1\) and \eqref{eq:L-deltaCL}. \end{proof} \begin{lemma}[Propagated scale variation] \label{lem:Lscaleprop} Under the same root hypotheses, the scale derivative of the root-attached coefficient is \begin{equation} \mu\frac{\mathrm d}{\mathrm d\mu}\mathcal C^{\RLmu}_{\mathrm L}(h,\phi) = \sum_{\lambda}\sum_{|\gamma|\le N_\lambda}a_\mu(q^\circ_\lambda,\gamma)\, \Bigl\langle B_{f,g,\mu\nu},\, \bigl(\partial_w^\gamma a^{\mu\nu}_\lambda\bigr)(y,w;h)\big|_{w=y}\,\phi(x) \Bigr\rangle , \label{eq:L-dCLmu} \end{equation} i.e.\ \eqref{eq:L-deltaCL} with the free coefficients \(C_{\lambda\gamma}\) replaced by the scale-variation coefficients \(a_\mu(q^\circ_\lambda,\gamma)\) of \eqref{eq:L-Amu-exp}. \end{lemma} \begin{proof} Differentiate \(\mathcal C^{\RLmu}_{\mathrm L}\) under the pairing (justified by Lemma~\ref{lem:Ldiffmu} and compact support, Lemma~\ref{lem:Lattach}(3)): the \(\mu\)-derivative hits only \(\RLmu(q^\circ_\lambda)\), producing \(A_\mu(q^\circ_\lambda)\). Apply Lemma~\ref{lem:Lambprop} to the coherent infinitesimal change \(\varepsilon A_\mu\) and take \(\varepsilon\to0\), or repeat the relative-variable computation with \(\partial^\gamma\delta\) coefficients \(a_\mu(q^\circ_\lambda,\gamma)\). \end{proof} \begin{remark}[Locality before versus after propagation] \label{rem:Lloc} The unrooted changes are collision-local; their propagated images are given by the root pairing. Locality or nonlocality of those functionals is not established here. \begin{enumerate} \item \emph{Before propagation.} The unrooted collision distributions \(\Delta Q_{\mathrm L}\) of \eqref{eq:L-deltaQ} and \(\mu\partial_\mu Q_{\mathrm L,\mu}\) of \eqref{eq:L-dQmu} are collision-local in the difference variable \(z=y-w\): they are finite sums of smooth coefficients of \((y,w)\) times \(\partial^\gamma\delta(y-w)\). \item \emph{After propagation.} The functionals \(\Delta\mathcal C_{\mathrm L}\) and \(\mu\partial_\mu\mathcal C^{\RLmu}_{\mathrm L}\) are pairings of these local collisions against \(B_{f,g}\) after the diagonal restriction \(w=y\). Locality or nonlocality of those propagated functionals on the base manifold is \emph{not} established here; no replacement of \eqref{eq:L-deltaCL} by a bulk local density is proved (cf.\ XVI-K, Remark rem:amb(2)--(3)). The two levels are kept separate. \end{enumerate} \end{remark} \begin{theorem}[L.7 summary] \label{thm:L7} Assume (R1)--(R3). The root-attached coefficient \(\mathcal C^{\RL}_{\mathrm L}\) exists, has compactly supported integrand, is bilinear and jointly continuous, agrees off the diagonal with the unextended graph, restricts exactly to XVI-K's \(\mathcal C^{\RK}_{C_2,\mathrm{fv}}\) on the \(C_2\)-only sector and exactly to XVI-J's \(\mathcal C^{\mathfrak R}_{2e}\) on the further constant-slot truncation, and has propagated coherent ambiguity \eqref{eq:L-deltaCL} and propagated scale variation \eqref{eq:L-dCLmu} with net sign \(+1\). The unrooted changes are collision-local; their propagated images are given by the root pairing. Locality or nonlocality of those functionals is not established here. \end{theorem} \begin{proof} Lemmas~\ref{lem:Lattach}, \ref{lem:Lambprop}, \ref{lem:Lscaleprop}; Theorems~\ref{thm:Lmain}; Propositions~\ref{prop:LtoK}, \ref{prop:LtoJ}; Remark~\ref{rem:Lloc}. \end{proof} \begin{lemma}[Computational verification of L.7 regressions] \label{lem:L7comp} The script \texttt{XVI-L-artifacts/regression\_L.py} exits with status \(0\) and records: \begin{enumerate} \item[(R1)] \(C_2\)-only truncation: numerical labels equal \(\mathcal V_{\mathrm K}\) (25 unordered); ordered count \(45\); max \(N=2\). \item[(R2)] Constant-slot truncation of that sector: single label \(\{0,0\}\); recovers the XVI-J slot pattern \(a=b=p=q=0\). \item[(R3)] Combinatorial prerequisites for the exact-restriction identities \(\RL|_{\mathcal V_{\mathrm K}}=\RK\) and \(\RK(S^2)=\bar q^{\mathfrak R}\) (written analytic proofs; the script does not claim distributional scale proofs); net sign \(+1\) of the relative-variable computation recorded. \item[(R4)] Scale-family combinatorial bookkeeping / illustrative recovery: \(\varpi_\mu(z)=\varpi((\mu/\mu_0)z)\) recovers \(\varpi\) at \(\mu=\mu_0\); contact-order bound \(N\) for scale differences recorded. \end{enumerate} \end{lemma} \begin{proof} Printed lines of \texttt{regression\_L\_out.txt}; Appendix~\ref{app:out}. \end{proof} %===================================================================== \section{L.8 Bridge to gravitational EFT (matching questions only)} \label{sec:l8} This section records the interpretive context in which the collision counterterms of L.5--L.7 may later be compared with gravitational effective-field-theory (EFT) operators. It does \emph{not} claim that the bridge is crossed, and it does not claim any instance of release level L-D. \subsection{Short matching context} A gravitational EFT organizes ultraviolet-sensitive corrections in terms of higher-curvature local operators (curvature-squared densities, and higher), while infrared-sensitive quantum corrections typically produce long-distance nonanalytic contributions that cannot be absorbed into a local metric Lagrangian of finite derivative order. The unrooted collision distributions \(\Delta Q_{\mathrm L}\) of \eqref{eq:L-deltaQ} are collision-local: they are finite sums of smooth coefficients times \(\partial^\gamma\delta(y-w)\). Their root-propagated images \(\Delta\mathcal C_{\mathrm L}\) of \eqref{eq:L-deltaCL} are given by the root pairing; locality or nonlocality of those functionals is not established here. The unrooted contact sector is the appropriate context in which to interpret the unrooted changes as candidates for local counterterm freedom. It is \textbf{not} evidence that the specified two-edge graph determines gravitational Wilson coefficients. A Wilson coefficient in a gravitational EFT is a coefficient in a diffeomorphism-covariant local effective action (or in a complete set of composite insertions) after all graphs, gauge-fixing data, and symmetry constraints of a chosen truncation have been assembled. XVI-L constructs one complete-source contribution for one two-edge graph, with its coherent contact freedom and scale variation; it does not assemble that larger package, and no numerical or structural output of L.1--L.7 is promoted to a Wilson coefficient here. \begin{remark}[Four-derivative jet bound is not a curvature-squared identity] \label{rem:L4der} Proposition~\ref{prop:Ljet} gives the residual-metric jet bound \((2-v)+|\gamma|\le2+s\le4\) after unrooted contact. That bound enumerates the differential order available in the collision coefficients; it does \emph{not} identify any summand of \(\Delta Q_{\mathrm L}\) with a curvature-squared density \(R^2\), \(R_{\mu\nu}R^{\mu\nu}\), \(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\), or with the variation of any such density. Equating the four-derivative jet bound with a proved curvature-squared counterterm identity would be a category error between a jet-counting statement and an action-matching statement; L-D remains unclaimed. \end{remark} \subsection{Matching questions for each candidate collision term} \label{sec:l8Q} Write \(\Delta Q_{\mathrm L}\) in the four interpretive categories of Section~\ref{sec:l5D}, and let \(T\) be any individual summand (or any coherent linear combination that one proposes to match). Before any claim that \(T\) is a gravitational or insertion counterterm, the following questions are to be answered affirmatively or negatively with a written argument; none is answered here. \begin{enumerate} \item[(Q1)] \emph{Local metric functional?} Can \(T\) arise as the metric variation (or as a contribution to the metric variation) of a local functional of the residual metric \(h\), after the collision diagonal \(w=y\) is taken and after preparation and switch factors are specialized? Only class~(1) of Section~\ref{sec:l5D} is even a candidate; classes~(2)--(4) are not bulk metric densities by construction. \item[(Q2)] \emph{Composite-insertion renormalization?} Is \(T\) instead a renormalization of the composite insertion carried by the complete source \(\JD\) (preparation-dependent counterterms of class~(2)), rather than a bulk gravitational action counterterm? \item[(Q3)] \emph{Essential \(\sigeps\) dependence?} Does the coefficient of \(T\) depend essentially on the preparation \(\sigeps\) (or on its jet), in the sense that it fails to factor through a \(\sigeps\)-independent local density of \(h\) after the fixed preparation of the Scope box is restored? Essential \(\sigeps\) dependence places \(T\) in class~(2), not in class~(1). \item[(Q4)] \emph{\(\chi_{\mathrm I}\) derivatives?} Does \(T\) involve derivatives \(\partial^{\gamma'}\chi_{\mathrm I}\) with \(|\gamma'|\ge1\) (class~(3))? Such terms depend on the choice of interaction switch and are not bulk metric counterterms. \item[(Q5)] \emph{Removable only by IBP / field-equation reduction?} Is \(T\) removable only by an integration-by-parts identity or by a field-equation reduction that the present off-shell setup forbids? XVI-L works with an off-shell smooth residual metric \(h\) and with the printed density representative of the cubic vertex (no IBP; cf.\ XVI-K, Proposition prop:hess, and the artifact boundary of Appendix~\ref{app:script}). A term that vanishes only after IBP or after imposing the linearized Einstein equation is not identified with zero in this note, and is not thereby matched to a bulk on-shell counterterm. \item[(Q6)] \emph{Additional graphs and ghosts for a symmetry test?} What additional graphs, gauge-fixing data, and ghost (or antifield) contributions would be required before a gravitational symmetry test --- diffeomorphism Ward identity, BRST cohomology statement, or BV master-equation compatibility --- could be stated for the sector containing this graph? Answering (Q6) is a prerequisite for any claim that a matched counterterm is symmetry-compatible; it is not supplied by extending numerical distributions. \end{enumerate} \begin{proposition}[Matching ledger; no match claimed] \label{prop:Lmatch} The questions (Q1)--(Q6) are well-posed for every coherent collision summand classified in Theorem~\ref{thm:L5} and for every propagated image under Lemma~\ref{lem:Lambprop}. No answer to (Q1)--(Q6) is proved in this note. In particular, no instance of release level L-D is claimed: no particular counterterm is matched to a local gravitational functional or to a composite-insertion functional. \end{proposition} \begin{proof} Well-posedness: the four interpretive categories of Section~\ref{sec:l5D} and the jet bounds of Proposition~\ref{prop:Ljet} supply the data against which (Q1)--(Q5) are checked; (Q6) is a scope question about the ambient theory, not about the numerical extension. Absence of a match: no identity equating a summand of \(\Delta Q_{\mathrm L}\) or \(\Delta\mathcal C_{\mathrm L}\) to the variation of a named local gravitational or insertion functional is stated or proved above; Remark~\ref{rem:L4der} blocks the jet-bound shortcut. \end{proof} \subsection{BV boundary} \label{sec:l8BV} Renormalized perturbative gravity is organized, in the locally covariant framework, by a Batalin--Vilkovisky (BV) treatment of gauge symmetry: classical master equation, ghost and antifield content, and renormalized interacting products compatible with that structure. XVI-L prepares objects that a later symmetry analysis may use --- the complete-source jet, the coherent extension \(\RL\), the unrooted collision space, the scale family \(\RLmu\), and the root-attached coefficient \(\mathcal C^{\RL}_{\mathrm L}\) under (R1)--(R3) --- but it does \textbf{not} assert BRST closure of the completed graph, does not write a BV antibracket or master action for this sector, and does not claim a Ward identity. \begin{remark}[Compatibility is separate from extending distributions] \label{rem:LBV} Extending the numerical kernels \(q^\circ_\lambda\in\mathcal V_{\mathrm{adm}}\) to \(\RL(q^\circ_\lambda)\in\mathcal D'(\mathbb R^4)\) (L.4), classifying coherent contact changes (L.5), introducing a scale family (L.6), and attaching the imported root (L.7) are distributional constructions. Compatibility of those constructions with a BV gauge-fixed gravitational action --- anomaly analysis, symmetry-restoring counterterms, ghost contributions --- is a separate task. No result of L.1--L.7 is a BRST or BV theorem, and none is transferred from the background orientation named in Section~\ref{sec:l0}. \end{remark} \begin{theorem}[L.8 summary] \label{thm:L8} The collision counterterms of XVI-L are interpreted in the gravitational-EFT distinction between local higher-curvature operators and long-distance nonanalytic contributions; matching questions (Q1)--(Q6) are recorded for every candidate; the four-derivative jet bound is not a curvature-squared identity (Remark~\ref{rem:L4der}); objects are prepared for later symmetry analysis without any assertion of BRST closure (Remark~\ref{rem:LBV}). Release level L-D remains \textbf{not claimed}. \end{theorem} \begin{proof} The interpretive paragraph of this section; Proposition~\ref{prop:Lmatch}; Remarks~\ref{rem:L4der}, \ref{rem:LBV}. \end{proof} %===================================================================== \section{Status} \label{sec:status} \subsection{Constructed} \begin{enumerate} \item L.0: frozen XVI-K dependency ledger; coherent versus placementwise changes; declared orders; artifact reference; verification boundary; target ledger and release ladder L-A--L-D. \emph{Recorded.} \item L.1: complete source \(\JD\) as polarized jet polynomial \eqref{eq:jet}; derivative budget (Proposition~\ref{prop:jet}, Theorem~\ref{thm:budget}, Lemma~\ref{lem:L1comp}). \item L.2: complete graph decomposition \(Q^{\circ\,\mu\nu}_{\mathrm L}\) \eqref{eq:QL}; term ledger; normalization gate with graph factor \(1\) re-derived for the complete source (Proposition~\ref{prop:Lnorm}, Proposition~\ref{prop:QL}, Lemma~\ref{lem:L2comp}). \item L.3: product existence; power counting \(N\le2,3,4\) and \(\mathrm{sd}\le6,7,8\) by sector; ambient span \(\mathcal V_{\mathrm{adm}}\) with \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\) and \(\mathcal V_{\mathrm{graph}}\subseteq\mathcal V_{\mathrm{adm}}\); kernel inventory (Proposition~\ref{prop:Lsd}, Proposition~\ref{prop:VLincl}, Theorem~\ref{thm:L3}, Lemma~\ref{lem:L3comp}). \item L.4: coherent extension \(\RL\) with \(\RL|_{\mathcal V_{\mathrm K}}=\RK\); same \(\varpi\)-projections for \(\rho\le2\), extended to \(N=3,4\); defect ledger (Definition~\ref{def:RL}, Proposition~\ref{prop:RL}, Lemma~\ref{lem:Ldefect}, Theorem~\ref{thm:L4}). \item L.5: coherent local collision counterterms before attachment; abstract finite-dimensional characterization by numerical-kernel relations; selected constraints exhibited; jet bounds \((2-v)+|\gamma|\le2+s\le4\); four interpretive categories (Lemma~\ref{lem:Lamb}, Propositions~\ref{prop:Lcoh}, \ref{prop:Ljet}, Theorem~\ref{thm:L5}, Lemma~\ref{lem:L5comp}). \item L.6: renormalization-scale family \(\RLmu\) with \(\RLmuref=\RL\); first scale theorem (contact order \(\le N\)); infinitesimal generator \(A_\mu\); coherence; prescription/residue distinction (Definition~\ref{def:RLmu}, Theorem~\ref{thm:Lscale}, Lemma~\ref{lem:Ldiffmu}, Theorem~\ref{thm:L6}). Explicit nonzero residue remains Unresolved (Remark~\ref{rem:Lressretch}). \item L.7: root-attached coefficient \(\mathcal C^{\RL}_{\mathrm L}\) under (R1)--(R3); bilinearity; joint continuity; compact support; off-diagonal agreement; exact restriction to XVI-K; exact further restriction to XVI-J; propagated \(\Delta\mathcal C_{\mathrm L}\) and scale variation through \(B_{f,g}\) with net sign \(+1\); locality before/after kept separate (Theorem~\ref{thm:Lmain}, Propositions~\ref{prop:LtoK}, \ref{prop:LtoJ}, Lemmas~\ref{lem:Lambprop}, \ref{lem:Lscaleprop}, Theorem~\ref{thm:L7}, Lemma~\ref{lem:L7comp}). \item L.8: matching context and questions (Q1)--(Q6); BV boundary without BRST assertion; L-D explicitly not claimed (Proposition~\ref{prop:Lmatch}, Theorem~\ref{thm:L8}, Remarks~\ref{rem:L4der}, \ref{rem:LBV}). \emph{Recorded as matching questions only.} \end{enumerate} Release ladder: L-A and L-C are \textbf{Constructed}, conditional on (R1)--(R3); L-B is an \textbf{abstract finite-dimensional characterization}; L-D is \textbf{not claimed}. \subsection{Proved obstructed} None new. XVI-L proves no obstruction theorem; no statement of this note is an obstruction result. \subsection{Unresolved} \begin{enumerate} \item Explicit nonzero scaling residue (anomaly form) of \(A_\mu(S^2)\) (Remark~\ref{rem:Lressretch}). \item Answers to the matching questions (Q1)--(Q6) of Section~\ref{sec:l8Q}; any instance of L-D. \item Nonvanishing of \(\mathcal C^{\RL}_{\mathrm L}\) after the switch \(\chi_{\mathrm I}\) and the smearing. \item The numerical value of \(\mathcal C^{\RL}_{\mathrm L}\) on explicit data. \item Locality of the propagated ambiguity \(\Delta\mathcal C_{\mathrm L}\) after propagation through \(B_{f,g}\) (Remark~\ref{rem:Lloc}). \item Whether the contact defects of Lemma~\ref{lem:Ldefect} are nonzero. \item Lorentz covariance and the Leibniz (derivative-compatibility) property of \(\RL\). \item The seminorm orders of the joint continuity statement in Theorem~\ref{thm:Lmain}. \item BRST/BV compatibility of the completed graph (Remark~\ref{rem:LBV}). \end{enumerate} \subsection{Non-claims} \begin{enumerate} \item L-D is not claimed; no bulk gravitational or insertion matching is proved (Proposition~\ref{prop:Lmatch}). \item The object constructed is the complete-source contribution for the specified two-edge graph; it is not called, and is not, the complete one-loop gravitational correction. \item Collision counterterms are not evidence that this graph determines gravitational Wilson coefficients (Section~\ref{sec:l8}). \item The four-derivative jet bound is not a proved curvature-squared counterterm identity (Remark~\ref{rem:L4der}). \item Scale-dependent subtraction is not a physical beta function of a complete gravitational quantity (Remark~\ref{rem:Lscale}). \item The four interpretive counterterm categories of Section~\ref{sec:l5D} are kept distinct as attributes (not a disjoint partition); they are not identified with bulk gravitational action counterterms. \item The unrooted changes are collision-local; their propagated images are given by the root pairing. Locality or nonlocality of those functionals is not established here (Remark~\ref{rem:Lloc}). \item Coupling and loop weights are undetermined; only the combinatorial graph factor \(1\) is fixed (Proposition~\ref{prop:Lnorm}). \item No BRST, BV, or Ward identity, no all-orders renormalization, no gravitational-symmetry compatibility is claimed (Remark~\ref{rem:LBV}). \item No limit \(\varepsilon\to0\); no state; no ultraviolet completion; no physical observable; nonvanishing of \(\mathcal C^{\RL}_{\mathrm L}\) is not claimed. \item XVI-K's statements are cited as stated; none is upgraded, and XVI-A--XVI-K are not modified. \item The imported root hypotheses (R1)--(R3) are used, not re-proved. \item Declared orders \(N_\lambda:=|a|+|b|+|p|+|q|\) are not defined through \(\mathrm{sd}-4\). \item Computational regressions support the stated identities; they do not replace the written proofs. \end{enumerate} \scopebox{Release.}{% \textbf{XVI-L constructs the complete-source contribution for the specified two-edge graph, conditional on the imported root hypotheses, using a coherent extension on the fixed admissible numerical span. It characterizes coherent collision freedom abstractly and constructs a smooth subtraction-scale family. All-sector contraction is computationally verified as reported on the stated residual-metric background. L-D is not claimed.} L-A, L-C: \textbf{Constructed}, conditional on (R1)--(R3). L-B: \textbf{abstract finite-dimensional characterization}. L-D: \textbf{not claimed}. Explicit nonzero scaling residue (anomaly form): Unresolved. L.0 recorded; L.1--L.7 constructed-as-stated; L.8 recorded as matching questions only (bridge not crossed). } %===================================================================== \appendix \section{Computational artifacts} \label{app:script} The folder \texttt{XVI-L-artifacts/} contains: \begin{itemize} \item \texttt{xvi\_common.py} (byte-identical copy of \texttt{XVI-K-artifacts/xvi\_common.py}; scripts compare SHA-256 when the original is present); \item \texttt{source\_full.py}, \texttt{source\_full\_out.txt}, \texttt{source\_full\_table.txt} (L.1); \item \texttt{two\_edge\_full.py}, \texttt{two\_edge\_full\_out.txt}, \texttt{two\_edge\_support.txt}, \texttt{two\_edge\_coeff\_table.txt} (L.2); \item \texttt{kernel\_inventory.py}, \texttt{kernel\_inventory\_out.txt} (L.3); \item \texttt{counterterm\_map.py}, \texttt{counterterm\_map\_out.txt} (L.4--L.5); \item \texttt{regression\_L.py}, \texttt{regression\_L\_out.txt} (L.6--L.7); \item \texttt{README.txt} (run commands and evidence boundary). \end{itemize} Outputs listed above accompany the scripts; \texttt{\_\_pycache\_\_/} is incidental and may be ignored. \paragraph{How to run.} From any working directory, with sympy available (the archive used \texttt{/workspace/tmp/venv/bin/python}): \begin{lstlisting} /workspace/tmp/venv/bin/python XVI-L-artifacts/source_full.py > XVI-L-artifacts/source_full_out.txt /workspace/tmp/venv/bin/python XVI-L-artifacts/two_edge_full.py > XVI-L-artifacts/two_edge_full_out.txt /workspace/tmp/venv/bin/python XVI-L-artifacts/kernel_inventory.py > XVI-L-artifacts/kernel_inventory_out.txt /workspace/tmp/venv/bin/python XVI-L-artifacts/counterterm_map.py > XVI-L-artifacts/counterterm_map_out.txt /workspace/tmp/venv/bin/python XVI-L-artifacts/regression_L.py > XVI-L-artifacts/regression_L_out.txt \end{lstlisting} All scripts locate companions relative to their own path. \texttt{two\_edge\_full.py} additionally reads \texttt{../XVI-K-artifacts/hessian\_full\_out.txt} (XVI-K is a frozen dependency; that folder is not modified by XVI-L). Every check raises \texttt{RuntimeError} on failure (fail-on-error); a run that exits with status \(0\) and prints \texttt{ALL CHECKS PASSED} has passed all of its checks. The runs used for this note exited with status \(0\). \paragraph{Script roles.} \texttt{source\_full.py}: implements XII-C eq:finite-curvature, eq:ell3, eq:J3, eq:Ccoefficients, eq:M2 atom by atom on independent jet coordinates (sympy sparse ring over \(\mathbb Q\)); refuses a third-order jet; no IBP; no typed monomial table of \(J_3\). \texttt{two\_edge\_full.py}: loads \(P_A\) and \(L^{pq}_{ij}\); builds \(T_i^r:=(-\Pc B^i)^r\), \(\widetilde P\), and \(a^{A}_{ab,pq}\); groups by numerical label by accumulation; all-sector coefficient assembly exact in the preparation jets and \(w^1\) on \(h_{\mu\nu}=(w^1)^2\eta_{\mu\nu}\) (not a general-\(h\) assembly); cancellation-aware \(\Lambda_{\mathrm{graph}}\) support on the stated background with explicit set checks against \(\Lambda_{\mathrm{adm}}\) and the XVI-K labels; full source-sector K regression (all \(25\) labels \(\times\) ten components vs signed \texttt{polar\_s}); exports nonzero coefficients to \texttt{two\_edge\_coeff\_table.txt}; writes \texttt{two\_edge\_support.txt}; fail-on-error. \texttt{kernel\_inventory.py}: enumerates placements from L.1 budgets and \(|p|+|q|\le2\); asserts \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\) with \(|\Lambda_{\mathrm{adm}}|=255\); records that \(\{0,0\}\) arises in \(C_2\) only; prints the inventory table. \texttt{counterterm\_map.py}: coherence constraints on contact coefficients (selected wave-operator relations exhibited); jet-bound checks \((2-v)+|\gamma|\le2+s\le4\); arithmetic identity \((2-v)+N=2+s\) (not attainment); \(W_\rho\) orders through \(4\); raw contact counts (not dimensions); four overlapping interpretive attributes; fail-on-error. \texttt{regression\_L.py}: exact \(C_2\)-only truncation to XVI-K and constant-slot truncation to XVI-J; scale-family illustrative floating-point recovery at \(\mu=\mu_0\) (not a compactly supported bump proxy; not a distributional scale proof); net-sign check for all \(|\gamma|\le4\); fail-on-error. \paragraph{Evidence kinds (symbolic/structural versus finite exact).} As in Section~\ref{sec:l0}(d) and XVI-K, Appendix app:script, three kinds of evidence are distinguished: \begin{itemize} \item \emph{Symbolic constructions}: exact polynomial identities in independent jet coordinates, valid for all smooth fields (e.g.\ the polarized source expansion and the Euler identity S1 in \texttt{source\_full.py}). \item \emph{Coefficientwise structural checks}: properties verified monomial by monomial on such a symbolic polynomial (homogeneity, slot symmetry, derivative budgets, \(\mathcal V_{\mathrm K}\subseteq\mathcal V_{\mathrm{adm}}\), jet bounds, coherence constraints). \item \emph{Finite exact tests}: exact rational evaluation on specified test data (e.g.\ agreement of \texttt{ell\_3} with XVI-K's \texttt{ell3\_of} on two random \(s\)-dependent fields). Agreement on those cases is \textbf{not} a coefficientwise equivalence proof of independent implementations, and the symbolic constructions do not rest on it. \end{itemize} A statement is called ``Proved locally'' only if it has a written argument in this note; a statement supported only by the script is called ``Computationally verified'' and its kind of evidence is named. \section{Script output excerpt} \label{app:out} Lines of \texttt{source\_full\_out.txt} (output of \texttt{grep -n}, line numbers refer to the output file). The \(60\) lines 35--94 (``polarization \dots equals monomial polarization'' and ``diagonal recovery'' for each of the three summands and ten components) all end in \texttt{True} and are omitted. \begin{lstlisting} 1:# source_full.py output (python 3.13.5, sympy 1.14.0, exact QQ arithmetic) 3:CHECK xvi_common.py is byte-identical to XVI-K-artifacts/xvi_common.py: True 5:CHECK R_{1,mu nu} and R_{2,mu nu} (eq:finite-curvature) are symmetric: True 6:CHECK symmetry in (mu,nu) of J_3: True 7:CHECK symmetry in (mu,nu) of 2C_2 dd sigma: True 8:CHECK symmetry in (mu,nu) of -2M_2 d sigma: True 9:# number of monomials of ell_3 (printed representative, eq:ell3): 1398 10:CHECK S1: Euler_gamma(sigma ell_3) = sigma J_3 + 2 C_2 dd sigma - 2 M_2 d sigma for all 10 components (XII-C thm:actualsource, symbolic in the 2-jets of gamma and sigma): True 11:CHECK S1': Euler_gamma(ell_3) = J_3 of eq:J3 (sign: +, convention dS = int E^{mu nu} d gamma_{mu nu}): True 12:CHECK finite exact test: ell_3 (this script) equals XVI-K's xvi_common.ell3_of on two random s-dependent fields gamma(x^1): True 13:# C_2 summand, unpolarized: (gamma-degree, derivatives on gamma, sigma-jet orders, foreign) -> #monomials: {(2, 0, (2,), False): 546} 14:CHECK quadratic homogeneity of the unpolarized C_2 summand: True 15:CHECK C_2 summand is linear in the sigma-jet of order exactly 2: True 16:CHECK C_2 summand: every monomial carries exactly 0 derivative(s) on gamma: True 17:CHECK C_2 summand contains no foreign generator (no residual metric, no psi): True 18:CHECK C_2 summand is not identically zero: True 19:# M_2 summand, unpolarized: (gamma-degree, derivatives on gamma, sigma-jet orders, foreign) -> #monomials: {(2, 1, (1,), False): 1152} 20:CHECK quadratic homogeneity of the unpolarized M_2 summand: True 21:CHECK M_2 summand is linear in the sigma-jet of order exactly 1: True 22:CHECK M_2 summand: every monomial carries exactly 1 derivative(s) on gamma: True 23:CHECK M_2 summand contains no foreign generator (no residual metric, no psi): True 24:CHECK M_2 summand is not identically zero: True 25:# J_3 summand, unpolarized: (gamma-degree, derivatives on gamma, sigma-jet orders, foreign) -> #monomials: {(2, 2, (0,), False): 1068} 26:CHECK quadratic homogeneity of the unpolarized J_3 summand: True 27:CHECK J_3 summand is linear in the sigma-jet of order exactly 0: True 28:CHECK J_3 summand: every monomial carries exactly 2 derivative(s) on gamma: True 29:CHECK J_3 summand contains no foreign generator (no residual metric, no psi): True 30:CHECK J_3 summand is not identically zero: True 31:CHECK rational coefficients (ring over QQ): True 32:CHECK every monomial of J_D carries total derivative weight 2 (on sigma and gamma together): True 33:# J_3 monomial types by derivative orders of the two gamma factors: {(0, 2): 534, (1, 1): 534} 34:CHECK J_3 contains both gamma*dd gamma and d gamma*d gamma monomials, and nothing else: True 96:CHECK bilinearity after polarization: every monomial of s_C_2[X,Y] is exactly linear in the X-jet and in the Y-jet, times a sigma-jet: True 97:CHECK slot symmetry P_{C_2,ij}^{ab,mu nu} = P_{C_2,ji}^{ba,mu nu}: True 98:# nonzero ordered coefficients P_{C_2,ij}^{ab,mu nu} (mu<=nu) by (|a|,|b|): {(0, 0): 688} total 688 99:CHECK bilinearity after polarization: every monomial of s_M_2[X,Y] is exactly linear in the X-jet and in the Y-jet, times a sigma-jet: True 100:CHECK slot symmetry P_{M_2,ij}^{ab,mu nu} = P_{M_2,ji}^{ba,mu nu}: True 101:# nonzero ordered coefficients P_{M_2,ij}^{ab,mu nu} (mu<=nu) by (|a|,|b|): {(0, 1): 1152, (1, 0): 1152} total 2304 102:CHECK bilinearity after polarization: every monomial of s_J_3[X,Y] is exactly linear in the X-jet and in the Y-jet, times a sigma-jet: True 103:CHECK slot symmetry P_{J_3,ij}^{ab,mu nu} = P_{J_3,ji}^{ba,mu nu}: True 104:# nonzero ordered coefficients P_{J_3,ij}^{ab,mu nu} (mu<=nu) by (|a|,|b|): {(0, 2): 534, (1, 1): 996, (2, 0): 534} total 2064 105:CHECK C_2: source-slot derivative orders (|a|,|b|) = (0,0) only: True 106:CHECK M_2: source-slot derivative orders (|a|,|b|) in {(1,0),(0,1)}, both occur: True 107:CHECK J_3: source-slot derivative orders (|a|,|b|) in {(2,0),(0,2),(1,1)}, all occur: True 108:CHECK every P_{C_2} is a rational linear form in the sigma-jet of order 2 (no other y-dependence): True 109:CHECK every P_{M_2} is a rational linear form in the sigma-jet of order 1 (no other y-dependence): True 110:CHECK every P_{J_3} is a rational linear form in the sigma-jet of order 0 (no other y-dependence): True 111:# C_2: ordered source-slot placements (a,b) occurring: 1 112:# M_2: ordered source-slot placements (a,b) occurring: 8 113:# J_3: ordered source-slot placements (a,b) occurring: 36 114:CHECK S3: C_2-only polarized source equals XVI-K's xvi_common.polar_s(X,Y,dd sigma) identically (symbolic): True 116:CHECK S3': on the XVI-K dual-slot tensors X=-P B^i, Y=-P B^j (all 55 pairs) the C_2 summand equals XVI-K's s(-P B^i,-P B^j; dd sigma): True 117:CHECK S2a: C_1 = C_2 = 0 at gamma = psi eta: True 118:CHECK S2b: Gamma_2 = -psi Gamma_1 at gamma = psi eta: True 119:CHECK S2c: D_sigma|_{gamma=psi eta} = -psi[eta^{mu nu} d_rho sigma d^rho psi + d^mu sigma d^nu psi + d^nu sigma d^mu psi] (XII-C eq:full-trace-sector): True 120:CHECK S2d: eta_{mu nu} D_sigma^{mu nu}|_{gamma=psi eta} = -6 (d_mu sigma) psi d^mu psi (XII-B eq:input-trace): True 134:# total runtime 53.3s 135:ALL CHECKS PASSED \end{lstlisting} The full script, its complete output and the coefficient table accompany this note. \paragraph{L.2 output excerpt (\texttt{two\_edge\_full\_out.txt}).} \begin{lstlisting} # cancellation-aware support |Lambda_graph| by sector: C_2: 25 M_2: 84 J_3: 202 # |Lambda_graph| (union after sector sum + cancel): 235 CHECK C_2 background support equals K labels: True CHECK complete background support contains K labels: True CHECK complete background support lies in admissible inventory: True # NOTE: support counts are on h=(w^1)^2 eta; NOT an established general-h support count # NOTE: 235 is coefficient-label support before quotienting numerical-kernel relations # NOTE: nonzero on this specialization => not identically zero in general; zero may still be nonzero elsewhere CHECK SOURCE-SECTOR REGRESSION: C_2 const-slot recovers XVI-K/XVI-J K tensor: True CHECK FULL C_2 K regression: all 25 labels x 10 mn match signed polar_s route: True # full C_2 comparisons checked: 250 CHECK a^{C_2}_{{0,e_1}} = -2 K^{C_2}_{0,e_1} componentwise on background: True # graph factor = (1/2!)*2 = 1 CHECK graph factor (1/2!)*2 = 1: True # coupling/loop weights: UNDETERMINED in L.2 (separated from graph factor 1) # All-sector coefficient assembly, exact in the preparation jets and w^1, # NOT a general-h symbolic assembly. Support counts ON THE STATED BACKGROUND. # TOTAL |Lambda_graph|=235 on the stated background (of |Lambda_adm|=255) # support table (adm/graph on stated background): C2 25/25, M2 84/84, J3 250/202, complete 255/235 # wrote two_edge_support.txt # wrote two_edge_coeff_table.txt with 1065 nonzero entries ALL CHECKS PASSED \end{lstlisting} \paragraph{L.3 output excerpt (\texttt{kernel\_inventory\_out.txt}).} \begin{lstlisting} # unordered numerical kernels in Lambda_adm: 255 CHECK V_K subset V_adm (XVI-K's 25 unordered among admissible): True CHECK C_2 sector numerical kernels equal V_K exactly: True CHECK degree histogram 1,4,20,60,170: True CHECK C_2: N <= 2: True CHECK M_2: N <= 3: True CHECK J_3: N <= 4: True ALL CHECKS PASSED \end{lstlisting} \paragraph{L.4--L.5 output excerpt (\texttt{counterterm\_map\_out.txt}).} \begin{lstlisting} CHECK residual metric jet (2-v)+|gamma| <= 2+s for all contacts: True CHECK global residual jet after contact <= 4: True CHECK switch jet |gamma| <= 4: True # C_2: residual jet bound 2+s = 2; prep = 2; N_max = 2 # M_2: residual jet bound 2+s = 3; prep = 1; N_max = 3 # J_3: residual jet bound 2+s = 4; prep = 0; N_max = 4 CHECK 15 wave-operator coherence constraints: True # raw V_K contact slots (regression-preserving sets these to 0): 321 # raw new (outside V_K) contact slots: 14000 === COUNTERTERM ATTRIBUTE LEDGER (four overlapping interpretive attributes) === CHECK four overlapping interpretive attributes recorded: True # attributes overlap: not asserted to form a disjoint partition of summands CHECK arithmetic identity (2-v)+N = 2+s on all 2025 placements: True # NOTE: arithmetic upper bound reaches 2+s when all N derivatives hit the residual-metric coefficient; # actual attainment is not established ALL CHECKS PASSED \end{lstlisting} \paragraph{L.6--L.7 output excerpt (\texttt{regression\_L\_out.txt}).} \begin{lstlisting} CHECK C_2-only unordered labels equal V_K (25): True CHECK C_2-only ordered placements = 45: True CHECK C_2-only max N = 2: True CHECK constant-slot truncation is exactly {0,0}: True CHECK XVI-J label sits inside V_adm: True # truncation chain V_adm --C2--> V_K --const--> {0,0} recorded CHECK net sign (+1)*(+1)=+1 for relative-variable propagation at |gamma|=0: True CHECK varpi_mu recovers varpi at mu=mu0: True CHECK varpi_mu(z)=varpi((mu/mu0)*z) dilation identity: True # scale family: reference mu0>0; illustrative floating-point recovery at mu=mu0 # NOTE: 1/(1+r^2) is NOT a compactly supported bump proxy for varpi CHECK scale difference contact order <= N for all labels: True CHECK on V_K block, scale contact order <= 2: True CHECK V_adm has 255 unordered placements (cross-check L.3): True ALL CHECKS PASSED \end{lstlisting} \end{document}