% !TeX program = pdflatex %===================================================================== % Em01-F-v3.tex % % Standalone pdfLaTeX source; no external files or BibTeX are required. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error Em01-F-v3.tex % pdflatex -interaction=nonstopmode -halt-on-error Em01-F-v3.tex % % Scope: % A new application with constant gravitational and coordinate weights. % No identification with the inherited weighted observable is asserted. % No characteristic raw-limit decision is asserted. % % Changes from v2: % - explicit action--stress and coupling dictionary; % - separated coordinate and gravitational comparison weights; % - formula-level weighted comparison panel; % - explicit cubic density and action-to-source audit; % - full-index/ten-component normalization checks; % - exact internal regrouping; % - exact localized crossed-line and tensor-vertex simplification; % - preservation of the full eight-dimensional smoothing profile; % - additional exact cone equations retaining the residual variable; % - focused analytic continuation target; % - reconstruction-null identity and transposition audit. % % Execution record: % No TeX compiler, computer-algebra system, numerical integration program, % or proof assistant was executed during preparation of this revision. % In particular, the two suggested compilation passes have NOT been run. %===================================================================== \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=26mm]{geometry} \usepackage{microtype} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx} \usepackage{enumitem} \usepackage[hidelinks]{hyperref} \usepackage{tikz} \hypersetup{ pdftitle={Em01-F-v3: Constant-Weight Einstein--Scalar Substitution and Exact Regrouping}, pdfauthor={} } \allowdisplaybreaks \numberwithin{equation}{section} \setcounter{tocdepth}{1} \setlength{\emergencystretch}{3em} \setlist{leftmargin=*,itemsep=3pt,topsep=5pt} \renewcommand{\arraystretch}{1.13} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Nzero}{\mathbb N_0} \newcommand{\dd}{\,\mathrm d} \newcommand{\Test}{\mathcal D} \newcommand{\Dist}{\mathcal D'} \newcommand{\Tests}{\mathscr T} \newcommand{\PP}{\mathsf P} \newcommand{\Amap}{\mathsf A} \newcommand{\bmap}{\mathsf b} \newcommand{\Dmap}{\mathsf D} \newcommand{\Kret}{K_{\mathrm{ret}}} \newcommand{\Ksupp}{K_{\mathrm{supp}}} \newcommand{\Vertex}{\mathsf V} \newcommand{\deltaD}{\delta_{\mathrm D}} \newcommand{\pair}[2]{\langle #1,#2\rangle} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\Tr}{\operatorname{tr}_{\eta}} \newcommand{\Boxeta}{\Box_{\eta}} \newcommand{\fmap}{\mathfrak f} \newcommand{\Irr}{\mathsf I_{\mathrm{RR}}} \newcommand{\Iresp}{\mathsf I_{\mathrm{resp}}} \newcommand{\SeedOp}{\mathscr P} \newcommand{\Dz}{\mathscr Z} \newcommand{\Da}{\mathscr A} \newcommand{\Db}{\mathscr B} \newcommand{\Qbil}{\mathcal Q} \newcommand{\Glin}{\mathcal G_1} \newcommand{\Rlin}{\mathcal R_1} \newcommand{\Ddens}{D_{\mathrm{dens}}} \newcommand{\Bret}{\mathcal B_{f,g}} \newcommand{\App}{\mathrm{app}} \newcommand{\scw}{\sigma_{\mathrm c}} \newcommand{\sgw}{\sigma_{\mathrm{grav}}} \newcommand{\Cpair}{\mathcal C_{\eta}} \newcommand{\Geom}{\mathbf g} \newcommand{\Fiber}{\mathsf G} \newcommand{\tauI}{\vartheta_I} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\WF}{WF} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \newcommand{\scopebox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \title{\textbf{Em01-F-v3}\\[0.4em] Constant-Weight Einstein--Scalar Substitution\\ and Exact Positive-Scale Regrouping\\[0.4em] \large An explicitly normalized marked response insertion\\ with the characteristic limit left open} \author{} \date{} \begin{document} \maketitle \begin{abstract} We study a new gauge-fixed Einstein--scalar application with constant gravitational and coordinate weights. Its action, field normalization, free Hessians, contraction kernels, quadratic gravitational source, observation root, and density map are specified consistently. An explicit marked response insertion produces seven distributional rows. The revision separates these derived equalities from convention agreements and constant-weight specializations of supplied weighted formulas. A comparison panel displays the missing coordinate factors and density-weight derivative terms without adding an unproved weighted model to the main theorem. The cubic source is checked against an explicit Einstein--Hilbert density, and the action--stress signs, coupling powers, polarization factors, and symmetric-component conversion are recorded. We prove row-by-row substitution, exact transposition to cone measures, and a 42-entry positive-scale audit identity. On the strict time-ordering patch, an exact crossed-line reduction and trace-reversal identity compress this ledger into a combined scalar-profile formula. The full eight-dimensional convolution is preserved, including for nonfactorizing profiles. All cone tips and source incidences remain in the internal distributional construction. A conservative order-five estimate with scale power twenty-one is proved. This is not a leading asymptotic exponent. Neither the complete characteristic limit nor its failure is established. The application is not identified with an inherited weighted relational observable, and no quantum admission or gauge-invariant completion is asserted. \end{abstract} \scopebox{Object and scope}{% This paper proves an exact input-substitution and positive-scale reduction theorem for the marked insertion defined here. The gravitational and coordinate weights are identically one; the matter-interaction switch remains a separate function. \smallskip The comparison with supplied manuscripts is documentary and formula-level. It is not an independent verification of their parent sources and does not identify their observables with this one. The characteristic raw-limit question remains open.} \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) {Constant-weight insertion}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Positive-scale regrouping}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Characteristic raw limit}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {The regrouping is not the raw limit.}; \draw[->] (a) -- (b); \draw[->] (b) -- (c); \draw[->] (b) -- (s); \end{tikzpicture} \caption{A constant-weight substitution is not the characteristic limit. 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-Em01-F} \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) {Constant-weight insertion}; \node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Characteristic raw limit}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {The regrouping is not the raw limit.}; \draw[->] (a) -- (s); \draw[->] (b) -- (s); \end{tikzpicture} \caption{A constant-weight substitution is not the characteristic limit. 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-Em01-F} \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) {Constant-weight insertion}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Positive-scale regrouping}; \node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Characteristic raw limit}; \node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {The regrouping is not the raw limit.}; \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-Em01-F} \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) {Constant-weight insertion}; \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) {Characteristic raw limit}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Positive-scale regrouping. The regrouping is not the raw limit.}; \draw (a) -- (m); \draw (m) -- (b); \draw[->] (m) -- (s); \end{tikzpicture} \caption{Constant-weight insertion is not Characteristic raw limit. 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-Em01-F} \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: Constant-weight insertion}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Positive-scale regrouping}; \node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Characteristic raw limit}; \node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {The regrouping is not the raw limit.}; \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-Em01-F} \end{figure} Figure~\ref{fig:scope-Em01-F} records that separation. Figure~\ref{fig:split-Em01-F} records one view of the same separation. Figure~\ref{fig:chain-Em01-F} records one view of the same separation. Figure~\ref{fig:pair-Em01-F} records one view of the same separation. Figure~\ref{fig:gate-Em01-F} records one view of the same separation. The drawings add no hypothesis. \tableofcontents %===================================================================== \section{The theorem proved and the question not yet decided} \label{sec:scope} %===================================================================== The object of this paper is a new application, not a reconstruction of incompletely supplied predecessor data. The latter distinction is necessary because the supplied passages separate internal kernel construction from the still missing joint characteristic operation \cite{Packet}. The symbols below have different roles: \[ \begin{array}{ccl} f&=&\text{compact observation test},\\ g&=&\text{compact matter-interaction switch},\\ \scw&=&\text{coordinate weight, when discussing a comparison},\\ \sgw&=&\text{gravitational density weight, in that comparison}. \end{array} \] The proved application has \[ \scw=\sgw=1. \] Neither \(f\) nor \(g\) is a gravitational ramp. \begin{theorem}[Main constant-weight substitution and regrouping] \label{thm:main} Fix the action, contractions, observations, and marked insertion specified in Sections~\ref{sec:model}--\ref{sec:insertion}. Let \(U\Subset M^4\) be the separated time-ordering patch of Section~\ref{sec:geometry}. Let \(\rho\in C_c^\infty(\R^8;\R)\) satisfy \(\int\rho=1\), and let \(\chi\in C_c^\infty(\R^8)\) equal one near \(\fmap(\overline U)\). Every input in \begin{equation} t_\delta^{\App} = i\lambda^3\Vertex\left( \Kret[q^{\App},\Dmap^{\App},f,g,F], \rho_\delta*(\chi C^{\App}) \right)\big|_U, \qquad \rho_\delta(h)=\delta^{-8}\rho(h/\delta), \label{eq:apptarget} \end{equation} is determined by the displayed formulas of this paper. The seven kernels are exactly the coefficient kernels of the defined marked insertion. Their arbitrary compact joint-test actions have the cone-seed representation of Theorem~\ref{thm:seed}. The selected family has the exact 42-entry audit expansion of Theorem~\ref{thm:ledger}. For sufficiently small \(\delta>0\), the same complete family has the combined scalar-profile representation \eqref{eq:combined-seeds}; this representation agrees with the audit expansion on every compact joint test. For every \(L\Subset U\), there is \(C_L>0\), independent of \(\delta,\lambda,\hbar\), such that \begin{equation} |t_\delta^{\App}(\Phi)| \le C_L|\lambda|^3\hbar^2 \delta^{-21}\|\Phi\|_{5,L}, \qquad \supp\Phi\subset L,\quad 0<\delta\le1. \label{eq:mainbound} \end{equation} The remaining fixed data may enter \(C_L\). No raw-limit conclusion or leading scale exponent is asserted. \end{theorem} The overall factor \(i\) in \eqref{eq:apptarget} is part of the selected-family convention. There is no \(1/\hbar\) prefactor. Section~\ref{sec:normalization} explains its coupling and contraction multiplicity completely. The proof of Theorem~\ref{thm:main} is completed in Section~\ref{sec:estimate}. The distinction between ordinary contractions, time ordering, and their domains is retained throughout; these are separate operations in perturbative algebraic quantum field theory \cite[Sections~5--7]{FR}. %===================================================================== \section{Tensor conventions and separated characteristic geometry} \label{sec:geometry} %===================================================================== \subsection{Full-index and ten-component conventions} Let \[ M=\R^{1+3},\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Boxeta=\partial_t^2-\Delta_{\mathbf x}. \] All spacetime raising and lowering use \(\eta\). Distributional pairings are complex bilinear, with flat coordinate densities. Complex polarization is bilinear, not sesquilinear. Let \(S=\operatorname{Sym}^2((\R^4)^*)\). For \(I=(i,j)\), \(0\le i\le j\le3\), define \begin{equation} \begin{split} (e_{(i,j)})_{\mu\nu} &= \frac{\delta_{\mu i}\delta_{\nu j} +\delta_{\mu j}\delta_{\nu i}}{1+\delta_{ij}},\\ (e^{(i,j)})^{\mu\nu} &= \frac12\left( \delta^\mu_i\delta^\nu_j+\delta^\mu_j\delta^\nu_i \right). \end{split} \label{eq:basis} \end{equation} Thus \[ (e^I)^{\mu\nu}(e_J)_{\mu\nu}=\delta^I_J,\qquad h_I=(e^I)^{\mu\nu}h_{\mu\nu},\qquad h_{\mu\nu}=\sum_I(e_I)_{\mu\nu}h_I. \] An off-diagonal basis tensor has two entries equal to one; its algebraic dual has two entries equal to \(1/2\). The algebraic dual is not simply the raised basis tensor. Trace reversal is \[ (\PP h)_{\mu\nu} = h_{\mu\nu}-\frac12\eta_{\mu\nu}\Tr h, \qquad \PP^2=\operatorname{id}. \] Put \begin{equation} \Amap^{\mu\nu,I} = \frac12\eta^{\mu\nu}\Tr e_I-(e_I)^{\mu\nu}, \qquad \bmap^I=\frac12\Tr e_I, \qquad \PP^L{}_M=(e^L)^{\mu\nu}(\PP e_M)_{\mu\nu}. \label{eq:linear-maps} \end{equation} The subscripts in \(\Amap^{\mu\nu}{}_L\) and \(\bmap_L\) mean evaluation on \(e_L\); they do not prescribe an extra fiber-metric contraction. \begin{lemma}[Component conversion audit] \label{lem:components} In the basis \eqref{eq:basis}, define \[ \Fiber_{IJ}=(e_I)_{\mu\nu}(e_J)^{\mu\nu}. \] For the ordered labels \((00,01,02,03,11,12,13,22,23,33)\), \[ \Fiber=\operatorname{diag}(1,-2,-2,-2,1,2,2,1,2,1). \] If \(\mathcal E^{\mu\nu}\) is a contravariant Euler derivative, its coordinate Euler components and propagated covariant components are related by \begin{equation} E_I=(e_I)_{\mu\nu}\mathcal E^{\mu\nu}, \qquad J^L=(e^L)^{\mu\nu}\eta_{\mu\rho}\eta_{\nu\sigma} \mathcal E^{\rho\sigma} =(\Fiber^{-1})^{LI}E_I. \label{eq:sourceconversion} \end{equation} For arbitrary symmetric \(h,k\), \[ h_{\mu\nu}k^{\mu\nu} = \sum_{I,J}h_I\Fiber_{IJ}k_J. \] No additional diagonal/off-diagonal weights occur in a contraction of a coefficient dual to \(h_I\) with \(h_I\). \end{lemma} \begin{proof} The basis tensors have disjoint component supports. A diagonal contraction has one term; an off-diagonal contraction has two terms and the corresponding metric sign. This gives the displayed diagonal matrix. For \(I=00\), \(E_{00}=\mathcal E^{00}\) and \(J^{00}=\mathcal E^{00}\). For \(I=01\), \[ E_{01}=2\mathcal E^{01},\qquad J^{01}=-\mathcal E^{01}=-\frac12E_{01}. \] These agree with the corresponding entries of \(\Fiber^{-1}\). The general statement follows from the completeness identity \[ \sum_I(e_I)_{\mu\nu}(e^I)^{\rho\sigma} = \delta^\rho_{(\mu}\delta^\sigma_{\nu)}. \] It also proves the full-index pairing formula after substituting the component expansions. Therefore an extra off-diagonal factor in an already dualized coefficient pairing would count the same conversion a second time. \end{proof} \scopebox{Three locked conventions}{% \textbf{Inverse:} \(G_\gamma^{\mathrm R}=-\PP E_0^{\mathrm R}\). A coefficient-\(-4\) inverse belongs to a different normalization. \smallskip \textbf{Polarization:} for a homogeneous quadratic map \(J\), the half-polarization \(J(G,h)=\frac12(J[G+h]-J[G]-J[h])\) satisfies \(DJ[G]h=2J(G,h)\). The source and density maps below use the mixed derivative. \smallskip \textbf{Components:} the algebraic dual already handles symmetric components. No additional \(1,2\) diagonal/off-diagonal rule may be inserted without a derived change of convention.} \subsection{Tests} Write \(w=(a,b,z,y)\). The label \(J\) is dual to the marked tensor input of Section~\ref{sec:insertion}. For \(U\Subset M^4\), set \[ \Tests=\Test(U;S^*),\qquad \|\Phi\|_{m,L} = \max_{J,\ |\gamma|\le m}\sup_U|\partial^\gamma\Phi_J|, \quad \supp\Phi\subset L\Subset U. \] Strong distributional convergence is uniform convergence on bounded test sets. Such sets have common compact support and bounded derivative seminorms of every order. \subsection{Exact characteristic coordinates} Set \[ \ell_{\mathbf n}=(1,\mathbf n),\qquad k_{\mathbf n}=(1,-\mathbf n),\qquad |\mathbf n|=1, \] and choose a smooth orthonormal transverse frame \(E_A=(0,\mathbf e_A)\), \(A=1,2\), on a sphere subchart. Use \begin{equation} \begin{split} z&=a+r,\\ b&=a+\beta\ell_{\mathbf n}+u k_{\mathbf n},\\ y&=a+\alpha\ell_{\mathbf n} +v k_{\mathbf n}+\xi^AE_A. \end{split} \label{eq:coordinates} \end{equation} There are eight tangential coordinates \((a,\mathbf n,\alpha,\beta)\) and eight normal coordinates \((r,u,v,\xi)\). \begin{proposition}[Coordinates, cone equations, and density] \label{prop:geometry} Near \(r=u=v=\xi=0\), \(0<\alpha<\beta\), these are smooth coordinates, and \begin{align} q_R&=(b-a)^2=4\beta u, \label{eq:qR}\\ q_1&=(y-a)^2=4\alpha v-|\xi|^2, \label{eq:q1}\\ q_2&=(b-y)^2 =4(\beta-\alpha)(u-v)-|\xi|^2. \label{eq:q2} \end{align} The flat density is \begin{equation} \begin{split} \dd^4a\,\dd^4b\,\dd^4z\,\dd^4y ={}& 4(\beta-u)^2\, \dd^4a\,\dd\Omega\,\dd\alpha\,\dd\beta\\ &{}\times\dd^4r\,\dd u\,\dd v\,\dd^2\xi. \end{split} \label{eq:coordinate-density} \end{equation} The map and inverse have bounded derivatives on compact subcharts with positive lower bounds for \(\alpha,\beta-\alpha,\beta,\beta-u\). \end{proposition} \begin{proof} For \(s=b-a\), the inverse is \[ \mathbf n=\frac{\mathbf s}{|\mathbf s|},\qquad \beta=\frac{s^0+|\mathbf s|}{2},\qquad u=\frac{s^0-|\mathbf s|}{2}. \] For \(d=y-a\), \[ \alpha=\frac{d^0+\mathbf n\cdot\mathbf d}{2},\qquad v=\frac{d^0-\mathbf n\cdot\mathbf d}{2},\qquad \xi^A=\mathbf e_A\cdot\mathbf d,\qquad r=z-a. \] The null-frame identities give the cone equations. The absolute Jacobian from \((\beta,u)\) to \((s^0,|\mathbf s|)\) is two; its spatial radial factor is \((\beta-u)^2\). At fixed \(\mathbf n\), the map \((\alpha,v,\xi)\mapsto d\) has absolute determinant two. The full map is block triangular after choosing \(a,s\). This proves the density and the stated bounds. \end{proof} Let \[ N=\{r=u=v=\xi=0\},\qquad S_{\mathrm{resp}}=\{r=0,\ u=0\}. \] Their codimensions are eight and five. Along \(N\), \[ \dd q_R=4\beta\dd u,\qquad \dd q_1=4\alpha\dd v,\qquad \dd q_2=4(\beta-\alpha)(\dd u-\dd v). \] The three differentials have rank two. Their crossed restrictions to \(S_{\mathrm{resp}}\) have rank one along \(N\). Also, throughout the chart, \begin{equation} \frac{q_1}{\alpha}+\frac{q_2}{\beta-\alpha} = \frac{q_R}{\beta} -\frac{\beta}{\alpha(\beta-\alpha)}|\xi|^2. \label{eq:defect} \end{equation} Choose a small relatively compact patch about \[ a_*=z_*=(0,0,0,0),\qquad y_*=a_*+2\ell_{(0,0,1)},\qquad b_*=a_*+4\ell_{(0,0,1)}. \] We require throughout its closure \[ b^0>y^0>a^0,\qquad b^0>z^0, \] and all positive separation bounds above. In particular \(b\ne a\) and \(b\ne z\). Strict separation of \(a,y,b\) does not remove \(z=a\). \subsection{Residual cones and true smoothing displacements} Write \[ r=A_r\ell_{\mathbf n}+B_r k_{\mathbf n}+Z_r^AE_A. \] Then the additional internal cone equations are exactly \begin{equation} (b-z)^2 = 4(\beta-A_r)(u-B_r)-|Z_r|^2, \qquad r^2=4A_rB_r-|Z_r|^2. \label{eq:residualcones} \end{equation} The \(K_{5,+}\) tip line has argument \(-r\), and the \(K_{5,-}\) tip line has argument \(r\); their retarded time orientations must not be interchanged. If \(\beta-A_r>0\), the equation \((b-z)^2=0\) gives \begin{equation} B_r=u-\frac{|Z_r|^2}{4(\beta-A_r)}. \label{eq:residualconstraint} \end{equation} This is a geometric constraint, not a scaling rule for \(A_r\). For actual smoothing displacements \[ H_i=A_i\ell_{\mathbf n}+B_i k_{\mathbf n}+Z_i^AE_A, \qquad d=\beta-\alpha, \] one has \begin{align} (y-a-\delta H_1)^2 &= 4(\alpha-\delta A_1)(v-\delta B_1) -|\xi-\delta Z_1|^2, \label{eq:shift1}\\ (y-b-\delta H_2)^2 &= 4(d+\delta A_2)(u-v+\delta B_2) -|\xi-\delta Z_2|^2. \label{eq:shift2} \end{align} Thus \[ (y-b-\delta H_2)^2 = (b-y)^2+2\delta(b-y)\cdot H_2+\delta^2H_2^2. \] These equations retain the residual variable and its cone-tip transitions. The geometry does not assign a common rescaling weight to \(r\). %===================================================================== \section{One action and its normalization dictionary} \label{sec:model} %===================================================================== \subsection{Action, background, field, and gauge} Fix \(m\ge0\), \(\lambda\in\R\setminus\{0\}\), and \(\hbar>0\). Coefficient expansions are formal at \(\lambda=0\). The metric variable and background are \[ \Geom_{\mu\nu}=\eta_{\mu\nu}+\lambda\gamma_{\mu\nu}, \qquad (\Geom,\phi)_{\mathrm{background}}=(\eta,0). \] Use \[ \Gamma^\rho_{\mu\nu} = \frac12\Geom^{\rho\sigma} \left( \partial_\mu\Geom_{\sigma\nu} +\partial_\nu\Geom_{\sigma\mu} -\partial_\sigma\Geom_{\mu\nu} \right), \] \[ R_{\mu\nu} = \partial_\rho\Gamma^\rho_{\mu\nu} -\partial_\nu\Gamma^\rho_{\mu\rho} +\Gamma^\rho_{\rho\alpha}\Gamma^\alpha_{\mu\nu} -\Gamma^\rho_{\nu\alpha}\Gamma^\alpha_{\mu\rho}, \qquad R=\Geom^{\mu\nu}R_{\mu\nu}. \] Let \[ d_\nu(\gamma) = \partial^\mu\gamma_{\mu\nu} -\frac12\partial_\nu\Tr\gamma, \qquad S_{\mathrm{gf}}[\gamma] = \int d_\nu(\gamma)d^\nu(\gamma)\dd^4x. \] The gauge-fixing functional is explicitly quadratic; no nonlinear gauge-fixing interaction is implicit. With \(g\in C_c^\infty(M;\R)\), define \begin{equation} \begin{split} S_{\lambda,g}[\gamma,\phi] ={}& -\frac2{\lambda^2}\int \sqrt{-\det\Geom}\,R[\Geom]\dd^4x +S_{\mathrm{gf}}[\gamma]\\ &+\frac12\int \left(\partial^\mu\phi\,\partial_\mu\phi-m^2\phi^2\right) \dd^4x\\ &+\frac12\int g(x)\Big[ \sqrt{-\det\Geom} \left(\Geom^{\mu\nu}\partial_\mu\phi\partial_\nu\phi -m^2\phi^2\right)\\ &\hspace{9em} -\left(\partial^\mu\phi\,\partial_\mu\phi-m^2\phi^2\right) \Big]\dd^4x. \end{split} \label{eq:action} \end{equation} The action is evaluated on compact perturbations, or under compact local variations. Its linear gravitational term is a divergence. The gravitational action is unswitched. The coordinate prescription below also has constant weight. The matter interaction is switched by \(g\). The model is used off shell; no existence result for an invariant switched Einstein jet is assumed. \subsection{Action--stress sign} \label{sec:normalization} For a compact covariant metric variation, \begin{equation} \delta_{\Geom}S_{\mathrm{EH}} = \frac2{\lambda^2}\int \sqrt{-\det\Geom}\, G^{\mu\nu}[\Geom]\delta\Geom_{\mu\nu}\dd^4x. \label{eq:EHvariation} \end{equation} The curved matter stress in the invariant matter density is \[ T_{\mu\nu}[\Geom,\phi] = \partial_\mu\phi\,\partial_\nu\phi -\frac12\Geom_{\mu\nu} \left(\Geom^{\rho\sigma}\partial_\rho\phi\partial_\sigma\phi -m^2\phi^2\right). \] Its covariant metric variation satisfies \begin{equation} \delta_{\Geom}S_{\mathrm m} = -\frac12\int \sqrt{-\det\Geom}\, T^{\mu\nu}[\Geom,\phi]\delta\Geom_{\mu\nu}\dd^4x. \label{eq:mattervariation} \end{equation} Indeed, \[ \delta\sqrt{-\det\Geom} = \frac12\sqrt{-\det\Geom}\,\Geom^{\mu\nu}\delta\Geom_{\mu\nu}, \] \[ \delta\Geom^{\alpha\beta} = -\Geom^{\alpha\mu}\Geom^{\beta\nu}\delta\Geom_{\mu\nu}. \] These identities give \eqref{eq:mattervariation}. For \eqref{eq:EHvariation}, vary the Ricci tensor by \[ \delta R_{\mu\nu} = \nabla_\rho\delta\Gamma^\rho_{\mu\nu} -\nabla_\nu\delta\Gamma^\rho_{\mu\rho}. \] Metric compatibility puts the derivative contribution into a divergence; the remaining inverse-metric and determinant variations give the displayed Einstein tensor and sign. Where \(g=1\), the invariant part of the model therefore has metric equation \begin{equation} G_{\mu\nu}[\Geom]=\frac{\lambda^2}{4}T_{\mu\nu}[\Geom,\phi]. \label{eq:Einstein-normalization} \end{equation} This equation concerns the invariant part, not the additional gauge-fixing Euler equation. The comparison dictionary, when adopting the Newtonian normalization reported in the supplied XI/XII passages, is \begin{equation} \lambda=\kappa,\qquad \lambda^2=32\pi G_N, \qquad \frac{\lambda^2}{4}=8\pi G_N \quad(c=1). \label{eq:Newton-dictionary} \end{equation} It is a convention agreement with those passages \cite{Packet}, not an identification with every use of \(\kappa\) elsewhere. \subsection{Derived quadratic blocks} The covariant tensor pairing is \[ \langle h,k\rangle_\eta = \int h_{\mu\nu}k^{\mu\nu}\dd^4x. \] \begin{proposition}[Free Hessians and their inverses] \label{prop:hessians} At the zero-field background, \begin{equation} S^{(2)} = \frac12\langle\gamma,-\Boxeta\PP\gamma\rangle_\eta +\frac12\langle\phi,-(\Boxeta+m^2)\phi\rangle. \label{eq:quadratic} \end{equation} The mixed free block vanishes. For the equation-normalized Green operators of \(P_m=\Boxeta+m^2\), the action-Hessian inverses are \begin{equation} G_\gamma^{\mathrm R,\mathrm A} = -\PP E_0^{\mathrm R,\mathrm A}, \qquad G_\phi^{\mathrm R,\mathrm A} = -E_m^{\mathrm R,\mathrm A}. \label{eq:inverse} \end{equation} \end{proposition} \begin{proof} The linearized Einstein tensor is \[ \Glin(h)_{\mu\nu} = -\frac12\Boxeta(\PP h)_{\mu\nu} +\partial_{(\mu}d_{\nu)}(h) -\frac12\eta_{\mu\nu}\partial^\rho d_\rho(h). \] The gravitational Euler derivative in the \(\gamma\)-variable has linear covariant output \(2\Glin(\gamma)\), by \eqref{eq:EHvariation} and \(\delta\Geom=\lambda\delta\gamma\). The lowered gauge-fixing Euler derivative is \[ -2\partial_{(\mu}d_{\nu)}(\gamma) +\eta_{\mu\nu}\partial^\rho d_\rho(\gamma). \] It cancels the divergence terms, leaving \(-\Boxeta\PP\gamma\). The free scalar Euler derivative is \(-P_m\phi\). Every matter interaction correction contains at least one metric field and two scalar fields, so there is no free mixed block at \((\eta,0)\). Homogeneity gives the quadratic action. Finally \(\PP\) commutes with \(\Boxeta\) and squares to the identity, so direct composition proves \eqref{eq:inverse}. \end{proof} In particular, \[ (-\Boxeta\PP)(c\PP E_0^{\mathrm R})=-c\,\operatorname{id}. \] The coefficient is forced to be \(c=-1\). The reported X-track coefficient \(-4\) cannot be used with this Hessian \cite{Packet}. \subsection{Coupling, quantum power, and multiplicity} The three counts are distinct: \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.43\linewidth}Y@{}} \toprule Quantity & Convention in this paper\\ \midrule Cubic Einstein--Hilbert action coefficient & \(\lambda\int\ell_3\); one coupling power.\\ Marked insertion & \(\mathcal O_G^{\mathrm{ret}}=\lambda^2\Bret(\gamma,G,\phi)\); the prefactor is part of its definition.\\ Matter vertex & \(\lambda V_g\); one coupling power.\\ Two selected crossed lines & One graviton line and one scalar line; each contains \(\hbar\).\\ Selected numerical family & \(i\) times the contraction difference, with multiplicity one: overall degree \(\lambda^3\hbar^2\).\\ Response normalization & No \(1/\hbar\), \(1/(i\hbar)\), or other response prefactor is included.\\ Loop classification & Not assigned by the coupling count or by this numerical definition.\\ \bottomrule \end{tabularx} \end{center} For different species, the mixed term in \(\frac12(\Gamma_\gamma+\Gamma_\phi)^2\) is \(\Gamma_\gamma\Gamma_\phi\). The two species assignments cancel the factor \(1/2\). The mixed matter derivative already includes the choice of either scalar field in its quadratic stress. No further factor of two is added. Thus a later definition with an additional \(1/\hbar\) response normalization would have a different quantum degree. It would not be the family \eqref{eq:apptarget}. The distinction between contraction expansion and interacting response normalization is also explicit in the standard functional framework \cite[Sections~5--6]{FR}. %===================================================================== \section{Density and scalar vertex} \label{sec:density} %===================================================================== At flat background use \[ T_{\mu\nu}[\phi] = \partial_\mu\phi\,\partial_\nu\phi -\frac12\eta_{\mu\nu} \left(\partial^\rho\phi\,\partial_\rho\phi-m^2\phi^2\right). \] Then \[ \partial^\mu T_{\mu\nu}[\phi] = (\Boxeta+m^2)\phi\,\partial_\nu\phi. \] The mass sign therefore agrees with \(P_m\). Its polarization is \[ T_{\mu\nu}(u,v) = \partial_{(\mu}u\,\partial_{\nu)}v -\frac12\eta_{\mu\nu} \left(\partial^\rho u\,\partial_\rho v-m^2uv\right). \] Define \[ a_\lambda^{\mu\nu}(\gamma) = \sqrt{-\det(\eta+\lambda\gamma)} (\eta+\lambda\gamma)^{-1\,\mu\nu}, \qquad b_\lambda(\gamma)=\sqrt{-\det(\eta+\lambda\gamma)}. \] \begin{proposition}[Identified density derivative] \label{prop:density} The expansions are \[ a_\lambda(\gamma) = \eta^{-1}+\lambda a_1[\gamma]+\lambda^2a_2[\gamma] +O(\lambda^3), \qquad b_\lambda(\gamma)=1+\lambda b_1[\gamma]+O(\lambda^2), \] with \[ a_1^{\mu\nu}[h] = \frac12\eta^{\mu\nu}\Tr h-h^{\mu\nu}, \qquad b_1[h]=\frac12\Tr h, \] \[ a_2^{\mu\nu}[h] = \left(\frac18(\Tr h)^2-\frac14h_{\rho\sigma}h^{\rho\sigma}\right) \eta^{\mu\nu} -\frac12(\Tr h)h^{\mu\nu} +h^{\mu\rho}h_\rho{}^\nu. \] The mixed derivative is \begin{equation} \begin{split} \Ddens^{\mu\nu}(G,h) ={}& \left( \frac14(\Tr G)(\Tr h)-\frac12G_{\rho\sigma}h^{\rho\sigma} \right)\eta^{\mu\nu}\\ &-\frac12\left( (\Tr G)h^{\mu\nu}+(\Tr h)G^{\mu\nu} \right)\\ &+G^{\mu\rho}h_\rho{}^\nu+h^{\mu\rho}G_\rho{}^\nu. \end{split} \label{eq:Ddensity} \end{equation} For the defined insertion, \begin{equation} \Dmap^{\alpha\mu,I,J} = \Ddens^{\alpha\mu}(e_J,e_I). \label{eq:Didentified} \end{equation} The full mixed metric-field variation carries \(\lambda^2\Ddens(G,h)\). \end{proposition} \begin{proof} Expand the inverse matrix and square-root determinant: \[ (I+\lambda H)^{-1} = I-\lambda H+\lambda^2H^2+O(\lambda^3), \] \[ \sqrt{\det(I+\lambda H)} = 1+\frac\lambda2\operatorname{tr}H +\lambda^2\left[ \frac18(\operatorname{tr}H)^2-\frac14\operatorname{tr}(H^2) \right]+O(\lambda^3), \qquad H=\eta^{-1}h. \] Multiplication gives \(a_1,a_2,b_1\). Differentiation of \(a_2[G+th]\) at \(t=0\) gives \eqref{eq:Ddensity}. The insertion uses this derivative, not its half-polarization. \end{proof} For example, \[ \Ddens(e_{00},e_{00}) = \operatorname{diag}\left(\frac34,\frac14,\frac14,\frac14\right), \] \[ \Ddens(e_{01},e_{01}) = \operatorname{diag}(-1,1,-1,-1). \] These follow both from \eqref{eq:Ddensity} and from directly inverting the corresponding one-parameter metric matrices. For arbitrary \(h,G\), bilinearity and basis completeness give \[ \Ddens^{\mu\nu}(G,h) = \sum_{I,J}\Dmap^{\mu\nu,I,J}h_IG_J. \] Thus the full-index and ten-component density calculations agree. \begin{corollary}[Derived matter vertex] \label{cor:vertexnormalization} The first matter-interaction coefficient is \(\lambda V_g\), where \begin{equation} V_g[\gamma,\phi] = -\frac12\int g\,\gamma_{\mu\nu}T^{\mu\nu}[\phi]\dd^4y. \label{eq:mattervertex} \end{equation} Its mixed derivative is \[ V_g''[h,\psi] = -\int g\,h_{\mu\nu}T^{\mu\nu}(\phi,\psi)\dd^4y. \] \end{corollary} \begin{proof} The first density coefficient is \[ \frac12g\left( a_1^{\mu\nu}[\gamma]\partial_\mu\phi\partial_\nu\phi -m^2b_1[\gamma]\phi^2 \right) = -\frac12g\,\gamma_{\mu\nu}T^{\mu\nu}[\phi]. \] The scalar derivative doubles the symmetric polarization. \end{proof} %===================================================================== \section{Explicit cubic density and source audit} \label{sec:source} %===================================================================== \subsection{Unintegrated density representative} For a covariant symmetric field \(h\), let \[ S(h)_{\sigma\mu\nu} = \partial_\mu h_{\sigma\nu} +\partial_\nu h_{\sigma\mu} -\partial_\sigma h_{\mu\nu}, \] \[ u_0^{\mu\nu}=\eta^{\mu\nu},\qquad u_1^{\mu\nu}=-h^{\mu\nu},\qquad u_2^{\mu\nu}=h^{\mu\rho}h_\rho{}^\nu. \] For \(n=1,2,3\), set \begin{align} \Gamma_n^\rho{}_{\mu\nu} &=\frac12u_{n-1}^{\rho\sigma}S(h)_{\sigma\mu\nu},\\ R_{n,\mu\nu} &=\partial_\rho\Gamma_n^\rho{}_{\mu\nu} -\partial_\nu\Gamma_n^\rho{}_{\mu\rho} \notag\\ &\quad+ \sum_{\substack{a+b=n\\a,b\ge1}} \left( \Gamma_a^\rho{}_{\rho\alpha}\Gamma_b^\alpha{}_{\mu\nu} -\Gamma_a^\rho{}_{\nu\alpha}\Gamma_b^\alpha{}_{\mu\rho} \right),\\ r_n&=\sum_{i=0}^{n-1}u_i^{\mu\nu}R_{n-i,\mu\nu}. \end{align} With \[ b_1[h]=\frac12\Tr h,\qquad b_2[h]=\frac18(\Tr h)^2-\frac14h_{\rho\sigma}h^{\rho\sigma}, \] the declared cubic density is \begin{equation} \boxed{\ell_3[h]=-2\bigl(r_3+b_1[h]r_2+b_2[h]r_1\bigr).} \label{eq:ell3} \end{equation} It is the coefficient of \(\lambda\) in the unintegrated Einstein--Hilbert density of \eqref{eq:action}. The source is calculated from this density with unit weight. No density-weighted integration-by-parts equivalence is assumed. At zero scalar field, write the lowered metric Euler equation as \[ -\Boxeta\PP\gamma +\lambda\mathcal J_2[\gamma]+O(\lambda^2)=0. \] The source is homogeneous quadratic. The label \(J_3^{\App}\) may equivalently denote \(\mathcal J_2\), emphasizing that it comes from the cubic action coefficient, not from a cubic field-degree source. \subsection{First action-to-source calculation} Put \[ C(h)^\rho_{\mu\nu} = \frac12\eta^{\rho\sigma}S(h)_{\sigma\mu\nu}, \qquad C_2(h)^\rho_{\mu\nu} = -\frac12h^{\rho\sigma}S(h)_{\sigma\mu\nu}. \] Then \(C=\Gamma_1\), \(C_2=\Gamma_2\). Define \[ V_1(h)_{\mu\nu} = R_{1,\mu\nu}-\frac12\eta_{\mu\nu}r_1, \] \[ V_2(h)_{\mu\nu} = R_{2,\mu\nu}-\frac12\eta_{\mu\nu}r_2 -\frac12h_{\mu\nu}r_1. \] \begin{proposition}[Euler derivative of the cubic density] \label{prop:cubicEuler} For compact \(h,k\), \begin{equation} \delta\int\ell_3[h]\dd^4x = \int \mathcal J_2[h]_{\mu\nu}k^{\mu\nu}\dd^4x, \label{eq:cubicvariation} \end{equation} where \begin{equation} \begin{split} \mathcal J_2[h]_{\mu\nu} = 2\Big[ &V_2(h)_{\mu\nu}+b_1[h]V_1(h)_{\mu\nu}\\ &-h_\mu{}^\alpha V_1(h)_{\alpha\nu} -h_\nu{}^\alpha V_1(h)_{\mu\alpha} \Big]. \end{split} \label{eq:J2evaluated} \end{equation} \end{proposition} \begin{proof} Expand \eqref{eq:EHvariation} in the field variable \(\Geom=\eta+\lambda h\). Its lowered Euler output is \[ \frac2\lambda \eta_{\mu\rho}\eta_{\nu\sigma} \sqrt{-\det\Geom}\, \Geom^{\rho\alpha}\Geom^{\sigma\beta} G_{\alpha\beta}[\Geom]. \] The coefficient of \(\lambda^2\) inside the last density is \[ V_2+b_1V_1-h_\mu{}^\alpha V_{1,\alpha\nu} -h_\nu{}^\alpha V_{1,\mu\alpha}. \] Multiplication by \(2/\lambda\) gives \eqref{eq:J2evaluated} at order \(\lambda\). The coefficient of the action at that order is precisely \eqref{eq:ell3}. Formal coefficient extraction commutes with compact variation, proving \eqref{eq:cubicvariation}. The quadratic gauge-fixing term has no contribution at this order. \end{proof} For an independent density-level check, define \[ \mathcal H_{\mu\nu}(A,B) = A^\rho{}_{\mu\sigma}B^\sigma{}_{\nu\rho} -A^\rho{}_{\mu\nu}B^\sigma{}_{\rho\sigma}. \] The coordinate Ricci formula and metric compatibility give \[ -\frac2{\lambda^2}\sqrt{-\det\Geom}\,R[\Geom] = -\frac2{\lambda^2}a_\lambda^{\mu\nu} \mathcal H_{\mu\nu}(\Gamma,\Gamma) +\partial_\rho\mathcal V^\rho, \] where \[ \mathcal V^\rho = -\frac2{\lambda^2}a_\lambda^{\mu\nu}\Gamma^\rho_{\mu\nu} +\frac2{\lambda^2}a_\lambda^{\rho\mu}\Gamma^\nu_{\mu\nu}. \] Hence \(\ell_3\), modulo a displayed divergence, equals the first-jet cubic density \begin{equation} \ell_3^{(1)} = -2\left[ a_1^{\mu\nu}[h]\mathcal H_{\mu\nu}(C,C) +\eta^{\mu\nu} \bigl(\mathcal H_{\mu\nu}(C,C_2) +\mathcal H_{\mu\nu}(C_2,C)\bigr) \right]. \label{eq:firstjetdensity} \end{equation} Here the superscript \((1)\) refers to first-jet dependence, not coupling order. For a variation \(k\), \[ \delta C=C(k),\qquad \delta C_2 = -\frac12\left( k^{\rho\sigma}S(h)_{\sigma\mu\nu} +h^{\rho\sigma}S(k)_{\sigma\mu\nu} \right) =D(h,k)^\rho_{\mu\nu}. \] Thus its entire directional variation is \begin{equation} \begin{split} \delta\ell_3^{(1)}=-2\Big\{ &a_1^{\mu\nu}[k]\mathcal H_{\mu\nu}(C,C)\\ &+a_1^{\mu\nu}[h]\big[ \mathcal H_{\mu\nu}(C(k),C) +\mathcal H_{\mu\nu}(C,C(k))\big]\\ &+\eta^{\mu\nu}\big[ \mathcal H_{\mu\nu}(C(k),C_2) +\mathcal H_{\mu\nu}(C,D(h,k))\\ &\hspace{6em} +\mathcal H_{\mu\nu}(D(h,k),C) +\mathcal H_{\mu\nu}(C_2,C(k))\big]\Big\}. \end{split} \label{eq:firstjetvariation} \end{equation} The exact divergence identity proves that its compactly integrated variation is also \eqref{eq:cubicvariation}. This supplies a second density representation for checking the source. It is valid here because the weight is constant. Multiplication by a nonconstant gravitational weight would generate additional terms from the divergence and is not authorized by this check. \subsection{Mixed curvature calculation} The insertion uses \[ \Qbil(h,G)=D\mathcal J_2[G]h. \] The dynamical field \(h\) occupies the \(a\)-slot; the marked field \(G\) occupies the \(z\)-slot. Define \begin{equation} D(h,G)^\rho_{\mu\nu} = -\frac12\left( h^{\rho\sigma}S(G)_{\sigma\mu\nu} +G^{\rho\sigma}S(h)_{\sigma\mu\nu} \right), \label{eq:mixedconnection} \end{equation} \[ \Rlin(h)_{\mu\nu} = \partial_\rho C(h)^\rho_{\mu\nu} -\partial_\nu C(h)^\rho_{\mu\rho}, \qquad r_1(h)=\eta^{\mu\nu}\Rlin(h)_{\mu\nu}, \] \[ \Glin(h)_{\mu\nu} = \Rlin(h)_{\mu\nu}-\frac12\eta_{\mu\nu}r_1(h). \] Set \begin{equation} \begin{split} U(h,G)_{\mu\nu}={}& \partial_\rho D(h,G)^\rho_{\mu\nu} -\partial_\nu D(h,G)^\rho_{\mu\rho}\\ &+C(h)^\rho_{\rho\alpha}C(G)^\alpha_{\mu\nu} +C(G)^\rho_{\rho\alpha}C(h)^\alpha_{\mu\nu}\\ &-C(h)^\rho_{\nu\alpha}C(G)^\alpha_{\mu\rho} -C(G)^\rho_{\nu\alpha}C(h)^\alpha_{\mu\rho}, \end{split} \label{eq:mixedRicci} \end{equation} \[ s(h,G) = \eta^{\mu\nu}U(h,G)_{\mu\nu} -h^{\mu\nu}\Rlin(G)_{\mu\nu} -G^{\mu\nu}\Rlin(h)_{\mu\nu}, \] \begin{equation} E(h,G)_{\mu\nu} = U(h,G)_{\mu\nu} -\frac12\eta_{\mu\nu}s(h,G) -\frac12\bigl(h_{\mu\nu}r_1(G)+G_{\mu\nu}r_1(h)\bigr). \label{eq:mixedEinstein} \end{equation} \begin{theorem}[Evaluated mixed source and coefficient rule] \label{thm:source} The mixed source is \begin{equation} \begin{split} \Qbil(h,G)_{\mu\nu}=2\Big[ &E(h,G)_{\mu\nu} +\frac12(\Tr h)\Glin(G)_{\mu\nu} +\frac12(\Tr G)\Glin(h)_{\mu\nu}\\ &-h_\mu{}^\alpha\Glin(G)_{\alpha\nu} -h_\nu{}^\alpha\Glin(G)_{\mu\alpha}\\ &-G_\mu{}^\alpha\Glin(h)_{\alpha\nu} -G_\nu{}^\alpha\Glin(h)_{\mu\alpha} \Big]. \end{split} \label{eq:Qevaluated} \end{equation} It is symmetric and has total derivative order at most two. Every constant coefficient in \begin{equation} (\Qbil(h,G))^L = \sum_{\alpha,\beta,I,J} q_{\alpha\beta}^{L,I,J} (\partial^\alpha G_J)(\partial^\beta h_I) \label{eq:qnormalform} \end{equation} is determined by \begin{equation} q_{\alpha\beta}^{L,I,J} = [p^\alpha k^\beta]\, (e^L)^{\mu\nu}e^{-(k+p)\cdot x} \Qbil(e_Ie^{k\cdot x},e_Je^{p\cdot x})_{\mu\nu}. \label{eq:q-extraction} \end{equation} The dot in this formal generating formula is the coordinate covector pairing. The expression is independent of \(x\). The coefficients vanish for \(|\alpha|+|\beta|>2\). \end{theorem} \begin{proof} Expand the connection of \(\eta+s h+tG\). Its linear terms are \(sC(h)+tC(G)\), and its mixed coefficient is \(stD(h,G)\). Its mixed Ricci, scalar-curvature, and Einstein coefficients are exactly \eqref{eq:mixedRicci}--\eqref{eq:mixedEinstein}. The determinant supplies the two trace terms, and the two inverse metrics in the lowered Euler density supply the four index conversion terms in \eqref{eq:Qevaluated}. The Euler prefactor \(2/\lambda\) supplies the factor two. Alternatively, differentiate \eqref{eq:J2evaluated} at \(G\) in direction \(h\). Its \(V_2\) derivative is \(E(h,G)\); the derivatives of \(b_1V_1\) and of the index-conversion terms are the remaining six terms. The density calculation and the mixed curvature calculation therefore agree. All resulting terms are products of jets with total derivative order at most two. Exponential substitution converts them into the independent monomials \(p^\alpha k^\beta\), so coefficient extraction determines every coefficient uniquely. \end{proof} If \[ \mathcal J_2(G,h) = \frac12\bigl( \mathcal J_2[G+h]-\mathcal J_2[G]-\mathcal J_2[h]\bigr), \] then \[ \Qbil(h,G)=2\mathcal J_2(G,h). \] This is the source analogue of \(\Ddens(G,h)=2a_2(G,h)\). No factor is adjusted after contraction. \subsection{Exact algebraic checks} The source formula gives \[ \Qbil(h,G)=0 \quad\text{for constant }h,G, \] and \[ \Qbil(h,c\eta)=-2c\,\Glin(h) \quad\text{for constant }c. \] For \(h=e_{01}\) constant and \(G=v\eta\), \[ \Qbil(h,v\eta)_{\mu\nu} = h_\nu{}^\rho\partial_\mu\partial_\rho v +h_\mu{}^\rho\partial_\nu\partial_\rho v -\eta_{\mu\nu}h^{\rho\sigma}\partial_\rho\partial_\sigma v -h_{\mu\nu}\Boxeta v. \] For \(v=x^0x^2\), its only nonzero components are \[ \Qbil(e_{01},v\eta)_{12} = \Qbil(e_{01},v\eta)_{21}=1. \] For conformal directions, \begin{equation} \begin{split} \Qbil(u\eta,v\eta)_{\mu\nu}={}& 2u\,\partial_\mu\partial_\nu v +2v\,\partial_\mu\partial_\nu u\\ &+3\bigl(\partial_\mu u\,\partial_\nu v +\partial_\mu v\,\partial_\nu u\bigr)\\ &-2\eta_{\mu\nu}(u\Boxeta v+v\Boxeta u) -3\eta_{\mu\nu}\partial^\rho u\,\partial_\rho v. \end{split} \label{eq:conformalcheck} \end{equation} For \(u=v=x^0\), all four diagonal components equal \(3\). These evaluations follow by substitution into \eqref{eq:Qevaluated}. An independent conformal check uses \(\Geom=e^{2\omega}\eta\): \[ G_{\mu\nu}[\Geom] = -2\partial_\mu\partial_\nu\omega +2\eta_{\mu\nu}\Boxeta\omega +2\partial_\mu\omega\,\partial_\nu\omega +\eta_{\mu\nu}(\partial\omega)^2. \] Set \(e^{2\omega}=1+s u+t v\) and extract its mixed coefficient. The determinant and inverse-metric factors cancel in this four-dimensional conformal Euler density, giving \eqref{eq:conformalcheck}. Finally, completeness of the basis implies for generic symmetric \(h,G\) \[ \Qbil(h,G)_{\mu\nu} = \sum_L(e_L)_{\mu\nu} \sum_{\alpha,\beta,I,J} q_{\alpha\beta}^{L,I,J} (\partial^\alpha G_J)(\partial^\beta h_I). \] Together with Lemma~\ref{lem:components}, this checks the full-index and ten-component calculations without adding off-diagonal weights. %===================================================================== \section{The advanced root and observation domain} \label{sec:root} %===================================================================== Let \(P_m=\Boxeta+m^2\). Its advanced inverse is specified by \[ P_mE_m^{\mathrm A}f=f,\qquad \supp(E_m^{\mathrm A}f)\subset J^-(\supp f), \] and vanishing future data. This is the equation-normalized inverse of \(P_m\), not the inverse of the scalar action Hessian \(-P_m\). With \[ \widehat f(s,\mathbf p) = \int e^{-i\mathbf p\cdot\mathbf x}f(s,\mathbf x)\dd^3\mathbf x, \qquad \omega_{\mathbf p}=\sqrt{|\mathbf p|^2+m^2}, \] define \begin{equation} F(t,\mathbf x)=E_m^{\mathrm A}f(t,\mathbf x) = \int_{\R^3}\frac{\dd^3\mathbf p}{(2\pi)^3} e^{i\mathbf p\cdot\mathbf x} \int_t^\infty \frac{\sin(\omega_{\mathbf p}(s-t))}{\omega_{\mathbf p}} \widehat f(s,\mathbf p)\dd s. \label{eq:root} \end{equation} The massless zero-momentum quotient means \(s-t\). \begin{lemma}[Advanced root] \label{lem:root} For compact smooth \(f\), the root is smooth, satisfies \(P_mF=f\), and is the unique advanced solution. For fixed compact source support and every compact \(L\subset M\), each \(C^r(L)\)-seminorm of \(F\) is bounded by finitely many compact smooth seminorms of \(f\). \end{lemma} \begin{proof} The spatial Fourier transform of \(f\) decays rapidly and uniformly on its time support. After taking sufficiently many spatial derivatives of \(f\), this decay dominates all differentiated integrands. Time derivatives introduce the source boundary terms and polynomially bounded oscillatory factors. The Fourier equation is \[ (\partial_t^2+\omega_{\mathbf p}^2)\widehat F=\widehat f \] with zero future data. Finite propagation and uniqueness follow from the Klein--Gordon energy identity on a backward causal domain. For \(m=0\), a standard auxiliary local \(L^2\)-term can be added to the energy estimate; it does not alter the finite-propagation conclusion. These arguments also give the stated seminorm estimates. \end{proof} The advanced observation root and the retarded internal propagations below have independently specified source and support domains. Their coexistence is not a single interchangeable prescription. Lemma~\ref{lem:root} is a theorem for this application, not an identification of a predecessor root. Choose a compact set \(K\) containing neighborhoods of \(b_*,y_*\). Choose a real compact smooth \(g\) equal to one near \(K\). The admissible observations are real \(f\), compactly supported in the interior of \(K\), with \[ f(b_*)\ne0,\qquad \partial_tE_m^{\mathrm A}f(b_*)\ne0. \] This class is nonempty. Indeed, choose \(f_0=1\) near \(b_*\) and a compact smooth \(\theta\) equal to \(t-b_*^0\) there. Then \[ P_m\theta(b_*)=0,\qquad E_m^{\mathrm A}P_m\theta=\theta. \] For \(f=f_0+cP_m\theta\), \(f(b_*)=1\), and \[ \partial_tE_m^{\mathrm A}f(b_*) = \partial_tE_m^{\mathrm A}f_0(b_*)+c. \] Avoid the one cancelling value of \(c\). After this choice, \(F\) is still determined solely by \eqref{eq:root}, not independently prescribed. Shrinking \(U\) preserves these nonvanishing conditions near \(b_*\), and \(g=1\) near \(b_*,y_*\). This proves support activity, not survival of any complete asymptotic coefficient. %===================================================================== \section{Four contraction kernels in the actual relative slots} \label{sec:crossed} %===================================================================== \subsection{Scalar prescription and its equation normalization} Define distributionally \begin{equation} W_m(t,\mathbf x) = \int_{\R^3}\frac{\dd^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}} e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x}, \label{eq:Wm} \end{equation} \begin{equation} D_{T,m}(t,\mathbf x) = \int_{\R^3}\frac{\dd^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}} e^{-i\omega_{\mathbf p}|t|+i\mathbf p\cdot\mathbf x}. \label{eq:DTm} \end{equation} The second Fourier formula defines the extension across \(t=0\). The symbol \(T\) denotes chronological time ordering. For \(t>0\), \(D_{T,m}=W_m\), and for \(t<0\), \(D_{T,m}(x)=W_m(-x)\). Here \(W_m\) is the vacuum positive-frequency distribution, not a symmetric Hadamard kernel or a parametrix. Put \[ \Delta_m=E_m^{\mathrm R}-E_m^{\mathrm A}. \] The mode identities are \begin{equation} P_mW_m=0,\qquad P_mD_{T,m}=-i\deltaD^{(4)},\qquad W_m(x)-W_m(-x)=-i\Delta_m(x). \label{eq:scalar-kernel-identities} \end{equation} Indeed, the first derivative of \(e^{-i\omega|t|}/(2\omega)\) jumps by \(-i\) at \(t=0\), while the antisymmetric positive-frequency mode is \(-i\sin(\omega t)/\omega\). Thus \[ (-P_m)(\hbar D_{T,m})=i\hbar\deltaD^{(4)}, \] and the scalar antisymmetric contraction is \(i\hbar\) times the scalar action-Hessian causal inverse. \subsection{Graviton tensor factor and endpoint maps} Define \[ \Pi_{\alpha\beta\rho\sigma} = \frac12\left( \eta_{\alpha\rho}\eta_{\beta\sigma} +\eta_{\alpha\sigma}\eta_{\beta\rho} -\eta_{\alpha\beta}\eta_{\rho\sigma} \right). \] It is the inverse tensor pairing determined by \(-\Boxeta\PP\) in \eqref{eq:quadratic}. The endpoint maps raise the tensor at \(y\) and extract the component at \(a\): \[ \mathcal L^{\mu\nu,\alpha\beta} = \eta^{\mu\alpha}\eta^{\nu\beta}, \qquad \mathcal R_I^{\rho\sigma}=(e^I)^{\rho\sigma}. \] There are no endpoint derivatives or extra trace-reversal maps. Set \begin{equation} N^{\mu\nu}{}_I = \eta^{\mu\alpha}\eta^{\nu\beta} \Pi_{\alpha\beta\rho\sigma}(e^I)^{\rho\sigma}, \qquad \tauI=\eta_{\rho\sigma}(e^I)^{\rho\sigma}. \label{eq:Nfactor} \end{equation} In the relative variables \[ x_1=y-a,\qquad x_2=y-b, \] the four lines are \begin{equation} \boxed{ \begin{aligned} H_\gamma^{\mu\nu}{}_I(x_1) &=\hbar N^{\mu\nu}{}_I W_0(x_1),\\ H_{T,\gamma}^{\mu\nu}{}_I(x_1) &=\hbar N^{\mu\nu}{}_I D_{T,0}(x_1),\\ H_\phi(x_2)&=\hbar W_m(x_2),\\ H_{T,\phi}(x_2)&=\hbar D_{T,m}(x_2). \end{aligned}} \label{eq:Hdictionary} \end{equation} Each includes \(\hbar\), with no additional \(i\) or \(\lambda\). Composition with the tensor Hessian gives \(i\hbar\) times its identity kernel, and its antisymmetric part is \(i\hbar\) times \(-\PP(E_0^{\mathrm R}-E_0^{\mathrm A})\). The covariant graviton factor is indefinite on unrestricted metric components. It is a gauge-fixed quasifree contraction prescription, not a positive state on the whole metric-field algebra. No physical-state quotient is constructed. Endpoint derivatives satisfy \[ \partial_yH(y-a)=\partial_{x_1}H(x_1),\qquad \partial_aH(y-a)=-\partial_{x_1}H(x_1), \] and similarly for \(x_2=y-b\). Although \(D_{T,m}(-x)=D_{T,m}(x)\), the same equality is false for \(W_m\) in general. \subsection{The grouped array} The actual crossed distribution is \begin{equation} C_I^{\mu\nu}(x_1,x_2) = \hbar^2N^{\mu\nu}{}_I \left[ D_{T,0}(x_1)\otimes D_{T,m}(x_2) - W_0(x_1)\otimes W_m(x_2) \right]. \label{eq:actualC} \end{equation} Its factors are in independent variables. The linear map \[ \fmap(a,b,z,y)=(y-a,y-b) \] is a submersion, so the pullback of this array exists. This does not define its singular product with \(\Kret\). The role of a contraction prescription, and the additional extension issue for time-ordered products, are distinct even when their formal kernel expressions are available \cite[Sections~5--7]{FR}. %===================================================================== \section{The independently defined marked insertion} \label{sec:insertion} %===================================================================== \subsection{Retarded cone measures} Define \begin{equation} R(\psi) = \frac1{4\pi}\int_{\R^3} \frac{\psi(|\mathbf x|,\mathbf x)}{|\mathbf x|} \dd^3\mathbf x, \qquad B(t,\mathbf x) = \frac1{8\pi}\mathbf1_{\{t\ge|\mathbf x|\}}. \label{eq:RB} \end{equation} The measure defining \(R\) is locally finite at the tip. Away from it, \[ R(t,\mathbf x) = \frac{\Theta(t)}{2\pi}\deltaD(t^2-|\mathbf x|^2). \] Appendix~\ref{app:cone} proves \begin{equation} \Boxeta R=\deltaD^{(4)},\qquad R*R=B,\qquad \partial_{s^\alpha}B(s)=\frac{s_\alpha}{2}R(s) \quad(s^0>0,\ s\ne0). \label{eq:cone-identities} \end{equation} For compact smooth \(h\), put \[ X_h^\mu=-R*\partial_\alpha a_1^{\alpha\mu}[h]. \] Its kernel is \[ X_h^\mu(b) = \int L^{\mu,I}(b,a)h_I(a)\dd^4a,\qquad L^{\mu,I}(s,a) = \Amap^{\alpha\mu,I}\partial_{a^\alpha}R(s-a). \] It solves \[ \Boxeta X_h^\mu=-\partial_\alpha a_1^{\alpha\mu}[h] \] with retarded support. Define \begin{equation} r[h,G]=\PP R*\Qbil(h,G). \label{eq:rresponse} \end{equation} Then \[ (-\Boxeta\PP)r[h,G]=-\Qbil(h,G). \] This is the response to the negative nonlinear Euler source. Its coupling-normalized field correction carries a factor \(\lambda\). This forcing normalization is separate from the explicit prefactor assigned to the insertion. The retarded Green operators are used on compact sources and, for the nested response, on the resulting past-compact sources. All relevant intermediate integrations lie in causal diamonds over compact endpoint sets. \subsection{Marked observable} The tensor \(G\) is an external compact smooth probe, not a second quantum species. For compact smooth \(h,G,\psi\), define \begin{equation} \begin{split} \Bret(h,G,\psi)={}& \int \left[ f\,\partial_\nu X_G^\mu -\delta^\mu_\nu\partial_\rho(fX_G^\rho) \right]X_h^\nu\partial_\mu\psi\dd^4b\\ &+\int f(b) \left[ R*\partial_\alpha \left( a_1^{\alpha\beta}[h]\partial_\beta X_G^\mu +a_1^{\alpha\beta}[G]\partial_\beta X_h^\mu \right) \right](b)\partial_\mu\psi(b)\dd^4b\\ &+\int f(b) \left[R*\partial_\alpha\Ddens^{\alpha\mu}(G,h)\right](b) \partial_\mu\psi(b)\dd^4b\\ &-\int f(b)X_{r[h,G]}^\mu(b)\partial_\mu\psi(b)\dd^4b\\ &+\int g(b)\left[ a_1^{\mu\nu}[r[h,G]]\partial_\mu F\,\partial_\nu\psi -m^2b_1[r[h,G]]F\psi \right](b)\dd^4b. \end{split} \label{eq:insertion} \end{equation} Set \begin{equation} \mathcal O_G^{\mathrm{ret}}[\gamma,\phi] = \lambda^2\Bret(\gamma,G,\phi). \label{eq:observable} \end{equation} The ordering of \(h,G\) in the first line is part of the definition. The insertion is not assumed symmetric in these roles or identified with an unmarked metric Hessian. \begin{proposition}[Smooth-slot existence and regrouping] \label{prop:regroup-internal} The insertion is a continuous trilinear functional on fixed compact smooth input stages. Let \[ C_G^\mu{}_\nu = f\partial_\nu X_G^\mu -\delta^\mu_\nu\partial_\rho(fX_G^\rho), \] \begin{equation} \mathcal B^{\alpha\mu}[G;h] = a_1^{\alpha\beta}[h]\partial_\beta X_G^\mu +a_1^{\alpha\beta}[G]\partial_\beta X_h^\mu +\Ddens^{\alpha\mu}(G,h) +a_1^{\alpha\mu}[r[h,G]]. \label{eq:combinedB} \end{equation} Then, exactly, \begin{equation} \begin{split} \Bret(h,G,\psi) ={}& \int C_G^\mu{}_\nu X_h^\nu\partial_\mu\psi\dd^4b\\ &+\int f(b) [R*\partial_\alpha\mathcal B^{\alpha\mu}[G;h]](b) \partial_\mu\psi(b)\dd^4b\\ &+\int g(b)\left[ a_1^{\mu\nu}[r[h,G]]\partial_\mu F\,\partial_\nu\psi -m^2b_1[r[h,G]]F\psi \right](b)\dd^4b. \end{split} \label{eq:internalregroup} \end{equation} \end{proposition} \begin{proof} All compact-source retarded responses are smooth. The local products involving \(h,G\) are compact smooth sources, while the nested response is well defined by causal properness. Final integrations are compact because of \(f,g\). Finite-order differential formulas and compact-local Green estimates prove continuity. By definition, \[ -X_{r[h,G]}^\mu = R*\partial_\alpha a_1^{\alpha\mu}[r[h,G]]. \] Linearity of retarded propagation combines the second, third, and fourth lines of \eqref{eq:insertion}, giving \eqref{eq:internalregroup}. No crossed-line composition or asymptotic approximation is used. \end{proof} There are no undisplayed terms in this definition. This is not a statement that a different complete relational observable has no companion or contact terms. %===================================================================== \section{Substitution and the weighted comparison panel} \label{sec:substitution} %===================================================================== \subsection{Source incidence} Define \begin{equation} Q^{L,I,J}(v;a,z) = \sum_{|\alpha|+|\beta|\le2} q_{\alpha\beta}^{L,I,J} \partial_v^\alpha\deltaD^{(4)}(v-z) \partial_v^\beta\deltaD^{(4)}(v-a). \label{eq:Qincidence} \end{equation} Pairing against \(h_I(a)G_J(z)\) gives \(\Qbil(h,G)^L(v)\). Put \[ T^{L,I,J}(b;a,z) = \PP^L{}_M\int R(b-v)Q^{M,I,J}(v;a,z)\dd^4v. \] \begin{lemma}[Response reduction] \label{lem:response} For \(D_0=R,B\), \begin{equation} \begin{split} &\int D_0(b-v)Q^{L,I,J}(v;a,z)\dd^4v\\ &\quad= \sum_{\alpha,\beta}(-1)^{|\alpha|+|\beta|} \partial_z^\alpha\partial_a^\beta \left[ q_{\alpha\beta}^{L,I,J}D_0(b-a)\deltaD^{(4)}(a-z) \right]. \end{split} \label{eq:response} \end{equation} The incidence and response distributions exist, including at \(a=z\). \end{lemma} \begin{proof} The map \((v,a,z)\mapsto(v-z,v-a)\) is a submersion. Adding \(b-v\) provides an independent propagation variable. Thus the products are tensor products in independent coordinates. The incidence makes the integration locally proper on compact external test supports. Move each delta derivative to its external variable, producing \((-1)^{|\alpha|+|\beta|}\). The undifferentiated integration is \(q_{\alpha\beta}D_0(b-a)\deltaD^{(4)}(a-z)\). \end{proof} \subsection{Specialization, not reproduction} \label{sec:comparison} The seven-row display that follows belongs to the independently defined marked insertion with \[ \scw=\sgw=1. \] Its resemblance to supplied weighted rows is not an identification of the insertions. For comparison only, the supplied coordinate leg has the form \begin{equation} L_{\scw}^{\mu,I}(b,a) = \scw(a)\Amap^{\alpha\mu,I}\partial_{a^\alpha}R(b-a). \label{eq:weightedL} \end{equation} Corresponding displayed outer factors are \begin{align} K_{5,+}^{[\scw]} &= \partial_{b^\mu}\left[ f(b)\scw(a)\partial_{a^\alpha}R(b-a) \Amap^{\alpha\beta,I} \partial_{a^\beta}L_{\scw}^{\mu,J}(a,z)\right], \label{eq:weightedK5p}\\ K_{5,-}^{[\scw]} &= \partial_{b^\mu}\left[ f(b)\scw(z)\partial_{z^\alpha}R(b-z) \Amap^{\alpha\beta,J} \partial_{z^\beta}L_{\scw}^{\mu,I}(z,a)\right], \label{eq:weightedK5m}\\ K_6^{[\scw]} &= \partial_{b^\mu}\left[ f(b)\scw(a)\Dmap^{\alpha\mu,I,J} \partial_{a^\alpha}R(b-a)\deltaD^{(4)}(a-z)\right]. \label{eq:weightedK6} \end{align} These equations record factors in the supplied passages \cite{Packet}. Equality of the comparison density map with \eqref{eq:Didentified} is not inferred. The reported density-weighted cubic source has the form \begin{equation} \begin{split} J_{3,\sgw}^{\mu\nu} &=\sgw J_3^{\mu\nu}+\mathcal D_{\sgw}^{\mu\nu},\\ \mathcal D_{\sgw}^{\mu\nu} &= 2C_2^{\mu\nu\alpha\beta}\partial_\alpha\partial_\beta\sgw -2M_2^{\mu\nu\rho}\partial_\rho\sgw. \end{split} \label{eq:weightedsource} \end{equation} For fixed weights its mixed variation is \begin{equation} D J_{3,\sgw}[G]h = \sgw D J_3[G]h+D\mathcal D_{\sgw}[G]h. \label{eq:weightedsourcevariation} \end{equation} The tensors \(C_2,M_2\) belong to the reported unintegrated density representative. Their general components are not reconstructed here. The extra source incidence would generate additional contributions schematically denoted \[ K_{8,\mathrm{layer}},\qquad K_{9,\mathrm{der,layer}},\qquad K_{9,\mathrm{mass,layer}}. \] These names identify comparison obligations, not newly derived rows in the theorem of this paper. At constant \(\sgw=1\), the terms containing its derivatives vanish identically. A source weighting \(\sgw J_3\) and a density weighting are otherwise different prescriptions. \paragraph{Intermediate coordinate weight.} Even if a weighted source array were supplied, a nonconstant coordinate weight changes the intermediate composition. The relevant scalar kernel is of the form \begin{equation} \mathcal H_{\scw,\alpha}(b,v) = \int R(b-s)\partial_{s^\alpha} \bigl(\scw(s)R(s-v)\bigr)\dd^4s. \label{eq:weightedintermediate} \end{equation} The tensor product in \((b-s,s-v)\), its differentiated smooth weight, and causal-diamond properness define this kernel. At \(\scw=1\), \[ \mathcal H_{1,\alpha}(b,v) = \partial_{b^\alpha}B(b-v). \] For general \(\scw\), \eqref{eq:weightedintermediate} contains both \(\partial\scw\) terms and a weighted retarded integration. Replacing \(q\) by \(q_{\sgw}\) in the translation-invariant \(B\)-formula does not account for it. Thus a weighted \(K_8\) would require the intermediate kernel \(\mathcal H_{\scw,\alpha}\), as well as the weighted source incidence. Setting \(\scw=\sgw\) would be a further explicit model choice. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.27\linewidth}Y p{.23\linewidth}@{}} \toprule Item & Equality or comparison & Status\\ \midrule Free operator and inverse & \(-\Boxeta\PP\), \(-\PP E_0^{\mathrm R}\) & Derived here; convention agreement with supplied XI passages.\\[0.3em] Action--stress sign & \(G_{\mu\nu}=\lambda^2T_{\mu\nu}/4\) for the invariant part & Derived here; normalization agreement, not model identity.\\[0.3em] Coordinate leg & \(L_{\scw}|_{\scw=1}=L\) & Constant-weight specialization of a displayed formula.\\[0.3em] \(K_{5,\pm},K_6\) factors & The displayed weight factors disappear at \(\scw=1\) & Formula specialization; insertion identification not established.\\[0.3em] Cubic source & \(\mathcal J_2=\operatorname{EL}(\ell_3)\); \(\Qbil=D\mathcal J_2\) & Derived here. Agreement of all inherited representative/domain choices is not asserted.\\[0.3em] Source layers & \(D\mathcal D_{\sgw}[G]h=0\) at constant \(\sgw=1\) & Constant-weight specialization; general component recovery pending.\\[0.3em] \(K_8\) convolution & \(\mathcal H_{1,\alpha}=\partial_{b^\alpha}B\) & Derived here; nonconstant-weight reduction is a different operation.\\[0.3em] Advanced root & \(F=E_m^{\mathrm A}f\) with the stated operator and domain & Derived here; predecessor root identification not established.\\[0.3em] Complete marked insertion & The seven rows below are its coefficient kernel & Derived here; no inherited observable identity claimed.\\ \bottomrule \end{tabularx} \end{center} \subsection{Seven kernels of the constant-weight insertion} Define \[ \Kret=K_{47}+K_{5,+}+K_{5,-}+K_6+K_8 +K_{9,\mathrm{der}}+K_{9,\mathrm{mass}}, \] where \begin{align} K_{47}^{I,J} &=-\partial_{b^\mu}\Big[ \big( f(b)\partial_{b^\nu}L^{\mu,J}(b,z) -\delta^\mu_\nu\partial_{b^\rho}(f(b)L^{\rho,J}(b,z)) \big)L^{\nu,I}(b,a)\Big], \label{eq:K47}\\ K_{5,+}^{I,J} &=\partial_{b^\mu}\Big[ f(b)\partial_{a^\alpha}R(b-a) \Amap^{\alpha\beta,I} \partial_{a^\beta}L^{\mu,J}(a,z)\Big], \label{eq:K5p}\\ K_{5,-}^{I,J} &=\partial_{b^\mu}\Big[ f(b)\partial_{z^\alpha}R(b-z) \Amap^{\alpha\beta,J} \partial_{z^\beta}L^{\mu,I}(z,a)\Big], \label{eq:K5m}\\ K_6^{I,J} &=\partial_{b^\mu}\Big[ f(b)\Dmap^{\alpha\mu,I,J} \partial_{a^\alpha}R(b-a)\deltaD^{(4)}(a-z)\Big], \label{eq:K6}\\ K_8^{I,J} &=-\partial_{b^\mu}\Big[ f(b)\Amap^{\alpha\mu}{}_L\PP^L{}_M \partial_{b^\alpha} \int B(b-v)Q^{M,I,J}(v;a,z)\dd^4v\Big], \label{eq:K8}\\ K_{9,\mathrm{der}}^{I,J} &=-\partial_{b^\nu}\Big[ g(b)\partial_{b^\mu}F(b) \Amap^{\mu\nu}{}_LT^{L,I,J}(b;a,z)\Big], \label{eq:K9d}\\ K_{9,\mathrm{mass}}^{I,J} &=-m^2g(b)F(b)\bmap_LT^{L,I,J}(b;a,z). \label{eq:K9m} \end{align} At the \(K_8\) reduction, the assumptions are explicitly \[ b^0>a^0,\qquad b\ne a. \] After source-incidence reduction, they justify \[ \partial_{b^\alpha}B(b-a) = \frac{(b-a)_\alpha}{2}R(b-a). \] The tip is not used in this derivative identity. It is nevertheless retained in the measure defining \(R\), including the internal tip lines of \(K_{5,\pm}\). \begin{theorem}[Row-by-row substitution] \label{thm:substitution} The coefficient kernel of \eqref{eq:insertion} is exactly \eqref{eq:K47}--\eqref{eq:K9m}. For compact smooth inputs, \begin{equation} \Bret(h,G,\psi) = \sum_{I,J} \pair{\Kret^{I,J}}{h_I(a)G_J(z)\psi(b)}. \label{eq:insertionkernel} \end{equation} The \(y\)-variable is inactive internally; adjoining it means tensoring with the constant distribution one. The substituted source, density, root, and crossed array are respectively \eqref{eq:q-extraction}, \eqref{eq:Didentified}, \eqref{eq:root}, and \eqref{eq:actualC}. \end{theorem} \begin{proof} Insert the \(L\)-kernels in the first line of the insertion, then transfer its scalar derivative to \(b\). This gives \(K_{47}\). In the next two insertion lines, transferring the source divergence to the incoming \(R\)-kernel gives a minus sign; transferring the scalar derivative gives a second minus sign. The two tensor placements give \(K_{5,+},K_{5,-}\). The local mixed density gives \(K_6\). Its \(a\)-derivative is on the retarded line alone, not on the incidence delta. For the fourth line, \[ X_{r[h,G]}^\mu = -\Amap^{\alpha\mu}{}_L\PP^L{}_M \partial_\alpha(B*\Qbil(h,G)^M). \] The insertion sign and scalar transposition therefore give \(K_8\). On its supported incidence, the separated future assumptions above justify the derivative reduction for \(B\). The last insertion line gives the derivative and mass root rows, with their displayed signs. Causal properness justifies the intermediate integrations. The explicit distributional kernels and equality on compact product tests determine the equality on joint tests. \end{proof} \begin{proposition}[Existence and internal order] \label{prop:internal} All seven kernels exist. Their separated-patch order bounds are \[ 4,\quad4,\quad4,\quad2,\quad3,\quad3,\quad2. \] The complete kernel has local order at most four. \end{proposition} \begin{proof} The two-cone seeds are tensor-product pullbacks under the invertible maps \[ (a,b,z,y)\mapsto(b,b-a,b-z,y), \] \[ (a,b,z,y)\mapsto(a,b-a,a-z,y),\qquad (a,b,z,y)\mapsto(z,b-z,z-a,y). \] The response seed is a tensor product in \((a,b-a,z-a,y)\). All seeds have order zero. The first three rows have at most four derivatives, and \(K_6\) has two. The root rows have at most two source derivatives and one outer derivative, or no outer derivative in the mass row. For \(K_8\), commute the inner \(b\)-derivative through \eqref{eq:response} and use the separated \(B\)-derivative identity. It becomes a smooth multiple of an \(R\)-seed with two source derivatives and one outer derivative. This proves its order-three bound. \end{proof} %===================================================================== \section{Exact cone-seed transposition} \label{sec:seed} %===================================================================== For compact smooth \(V\), define \begin{equation} \Irr(V) = \frac1{16\pi^2} \int \frac{ V(c,(|\mathbf x|,\mathbf x),(|\mathbf x'|,\mathbf x'),y) }{|\mathbf x|\,|\mathbf x'|} \dd^4c\,\dd^4y\,\dd^3\mathbf x\,\dd^3\mathbf x', \label{eq:Irr} \end{equation} \begin{equation} \Iresp(V) = \frac1{4\pi} \int\frac{V(c,(|\mathbf x|,\mathbf x),0,y)}{|\mathbf x|} \dd^4c\,\dd^4y\,\dd^3\mathbf x. \label{eq:Iresp} \end{equation} Both are ordinary absolutely convergent integrals. The cone-tip weights are locally integrable. The determinant-one charts are \[ \begin{aligned} \kappa_{47}(c,s,t,y)&=(c-s,c,c-t,y),\\ \kappa_{5,+}(c,s,t,y)&=(c,c+s,c-t,y),\\ \kappa_{5,-}(c,s,t,y)&=(c-t,c+s,c,y),\\ \kappa_{\mathrm{resp}}(c,s,r,y)&=(c,c+s,c+r,y). \end{aligned} \] Write \(W_j=W\circ\kappa_j\). In the \(47\)-chart, \[ \Db_\mu=\partial_{c^\mu}+\partial_{s^\mu}+\partial_{t^\mu}. \] Set \begin{equation} \begin{split} \SeedOp_{47}^{I,J}W={}& -\Amap^{\rho\mu,J}\Amap^{\sigma\nu,I} \partial_{s^\sigma}\partial_{t^\nu}\partial_{t^\rho} (f(c)\Db_\mu W)\\ &-\delta^\mu_\nu\Amap^{\rho\alpha,J}\Amap^{\sigma\nu,I} \partial_{s^\sigma}\partial_{t^\rho} ((\partial_\alpha f)(c)\Db_\mu W)\\ &+\delta^\mu_\nu\Amap^{\rho\alpha,J}\Amap^{\sigma\nu,I} \partial_{s^\sigma}\partial_{t^\alpha}\partial_{t^\rho} (f(c)\Db_\mu W). \end{split} \label{eq:P47} \end{equation} For the other two-cone charts, \begin{align} \SeedOp_{5,+}^{I,J}W &= \Amap^{\alpha\beta,I}\Amap^{\rho\mu,J} \partial_{s^\alpha}\partial_{t^\beta}\partial_{t^\rho} (f(c+s)\partial_{s^\mu}W), \label{eq:P5p}\\ \SeedOp_{5,-}^{I,J}W &= \Amap^{\alpha\beta,J}\Amap^{\rho\mu,I} \partial_{s^\alpha}\partial_{t^\beta}\partial_{t^\rho} (f(c+s)\partial_{s^\mu}W). \label{eq:P5m} \end{align} In the response chart let \[ \Dz_\mu=\partial_{r^\mu},\qquad \Da_\mu=\partial_{c^\mu}-\partial_{s^\mu}-\partial_{r^\mu}. \] They represent original \(\partial_z,\partial_a\), respectively. Define \begin{equation} \SeedOp_6^{I,J}W = -\Dmap^{\alpha\mu,I,J} \partial_{s^\alpha}(f(c+s)\partial_{s^\mu}W), \label{eq:P6} \end{equation} \begin{equation} \SeedOp_8^{I,J}W = \sum_{\alpha,\beta,M,L} q_{\alpha\beta}^{M,I,J}\PP^L{}_M \Amap^{\rho\mu}{}_L\frac{s_\rho}{2} \Dz^\alpha\Da^\beta(f(c+s)\partial_{s^\mu}W), \label{eq:P8} \end{equation} \begin{equation} \begin{split} \SeedOp_{9,\mathrm{der}}^{I,J}W = \sum_{\alpha,\beta,M,L} &q_{\alpha\beta}^{M,I,J}\PP^L{}_M\Amap^{\mu\nu}{}_L\\ &{}\times \Dz^\alpha\Da^\beta (g(c+s)(\partial_\mu F)(c+s)\partial_{s^\nu}W), \end{split} \label{eq:P9d} \end{equation} \begin{equation} \begin{split} \SeedOp_{9,\mathrm{mass}}^{I,J}W = -m^2\sum_{\alpha,\beta,M,L} &q_{\alpha\beta}^{M,I,J}\PP^L{}_M\bmap_L\\ &{}\times\Dz^\alpha\Da^\beta(g(c+s)F(c+s)W). \end{split} \label{eq:P9m} \end{equation} Every derivative acts on the entire parenthesis. \begin{theorem}[Exact seed action] \label{thm:seed} For compact smooth scalar \(W\) supported in \(U\), \begin{equation} \pair{K_j^{I,J}}{W} = \begin{cases} \Irr(\SeedOp_j^{I,J}W_j), &j=47,5+,5-,\\[2pt] \Iresp(\SeedOp_j^{I,J}W_{\mathrm{resp}}), &j=6,8,9_{\mathrm{der}},9_{\mathrm{mass}}. \end{cases} \label{eq:seedformula} \end{equation} \end{theorem} \begin{proof} In the \(47\)-chart, \[ L^{\mu,J}(b,z) = -\Amap^{\rho\mu,J}\partial_{t^\rho}R(t),\qquad L^{\nu,I}(b,a) = -\Amap^{\sigma\nu,I}\partial_{s^\sigma}R(s). \] Expand the derivative of \(fL\) before transposing. The outer minus derivative supplies a plus test derivative; three seed derivatives supply a minus sign, and two supply a plus sign. The three resulting terms are \eqref{eq:P47}. For \(K_{5,+}\), \[ \partial_{a^\alpha}R(b-a)=-\partial_{s^\alpha}R(s), \qquad \partial_{a^\beta}L^{\mu,J}(a,z) = -\Amap^{\rho\mu,J} \partial_{t^\beta}\partial_{t^\rho}R(t). \] These line signs cancel. Transposition of the outer derivative and the three seed derivatives gives another cancellation, yielding \eqref{eq:P5p}. The second placement gives \eqref{eq:P5m}. For \(K_6\), the line-only derivative is \(-\partial_{s^\alpha}R(s)\). Transposing the outer derivative and this line derivative gives \eqref{eq:P6}. For response rows, \eqref{eq:response} becomes \[ \sum_{\alpha,\beta}(-1)^{|\alpha|+|\beta|} \Dz^\alpha\Da^\beta (q_{\alpha\beta}D_0(s)\deltaD^{(4)}(r)). \] Its sign cancels on transposition. In \(K_8\), the inner derivative gives \(\partial_{s^\rho}B=(s_\rho/2)R\) on the separated future support. The outer minus derivative then gives \eqref{eq:P8}. The root terms give \eqref{eq:P9d} and \eqref{eq:P9m}, with the original mass sign retained. Finally apply the measure \eqref{eq:RB} once or twice. The charts have unit absolute determinant, giving the ordinary-integral formulas. \end{proof} %===================================================================== \section{The exact 42-entry audit expansion} \label{sec:ledger} %===================================================================== Write \(C=C_+-C_-\), where \[ C_+^{\mu\nu}{}_I = \hbar^2N^{\mu\nu}{}_I(D_{T,0}\otimes D_{T,m}), \qquad C_-^{\mu\nu}{}_I = \hbar^2N^{\mu\nu}{}_I(W_0\otimes W_m). \] Set \[ H_\delta^\pm=\rho_\delta*(\chi C_\pm),\qquad C_\delta=H_\delta^+-H_\delta^-. \] For a smooth symmetric array \(H\), define \[ Z_I^\nu(H) = \fmap^*\left( \partial_{2,\mu}H_I^{\mu\nu} -\frac12\eta_{\mu\alpha}\partial_2^\nu H_I^{\mu\alpha} \right), \qquad M_I(H)=\fmap^*(\eta_{\mu\nu}H_I^{\mu\nu}), \] \begin{equation} \Vertex(K,H)^J = \partial_{y^\nu}\left[g(y)\sum_IK^{I,J}Z_I^\nu(H)\right] -\frac{m^2}{2}g(y)\sum_IK^{I,J}M_I(H). \label{eq:vertex} \end{equation} This is the coefficient from the mixed matter derivative after transferring its remaining scalar derivative. The three compact joint tests are \begin{align} W_{\mathrm{k}}^{I,J}(H,\Phi) &=-ig(y)\fmap^*(\partial_{2,\mu}H_I^{\mu\nu}) \partial_{y^\nu}\Phi_J, \label{eq:Wk}\\ W_{\mathrm{t}}^{I,J}(H,\Phi) &=\frac i2g(y)\eta_{\mu\alpha} \fmap^*(\partial_2^\nu H_I^{\mu\alpha}) \partial_{y^\nu}\Phi_J, \label{eq:Wt}\\ W_{\mathrm{m}}^{I,J}(H,\Phi) &=-\frac{im^2}{2}g(y) \fmap^*(\eta_{\mu\nu}H_I^{\mu\nu})\Phi_J. \label{eq:Wmass} \end{align} For \(\epsilon_+=1,\epsilon_-=-1\), put \begin{equation} \mathsf L_{j,v}^{\varepsilon}(\delta;\Phi) = \epsilon_\varepsilon\sum_{I,J} \pair{K_j^{I,J}}{W_v^{I,J}(H_\delta^\varepsilon,\Phi)}. \label{eq:ledgerentry} \end{equation} \begin{theorem}[Positive-scale audit identity] \label{thm:ledger} For every \(\delta>0\) and every compact joint test, \begin{equation} t_\delta^{\App}(\Phi) = \lambda^3 \sum_j\sum_{v\in\{\mathrm{k},\mathrm{t},\mathrm{m}\}} \sum_{\varepsilon\in\{+,-\}} \mathsf L_{j,v}^{\varepsilon}(\delta;\Phi). \label{eq:ledgeridentity} \end{equation} Each entry is exactly \begin{equation} \mathsf L_{j,v}^{\varepsilon}(\delta;\Phi) = \epsilon_\varepsilon\sum_{I,J} \mathsf I_j\left( \SeedOp_j^{I,J} [W_v^{I,J}(H_\delta^\varepsilon,\Phi)\circ\kappa_j] \right), \label{eq:ledgerseed} \end{equation} with \(\mathsf I_j=\Irr\) for the first three rows and \(\mathsf I_j=\Iresp\) for the four response rows. The family exists at positive scale, is strongly smooth on compact positive-scale intervals, and depends only on the actual reconstructed internal kernel. \end{theorem} \begin{proof} All smoothed crossed components are smooth. The mixed matter derivative and the insertion-kernel theorem therefore give legitimate internal test pairings. The vertex transposition gives \eqref{eq:Wk}--\eqref{eq:Wmass}, and linearity gives \eqref{eq:ledgeridentity}. The seed theorem gives \eqref{eq:ledgerseed}. The normalization is that of Section~\ref{sec:normalization}: one selected mixed species pair, the prefactors \(\lambda^2\) and \(\lambda\), and two \(\hbar\)-weighted lines. Null reconstructions pair to zero with every smooth test, so all reconstruction identities are respected simultaneously. On a compact positive-scale interval, the smoothing profiles and their scale derivatives have common compact support and bounded smooth seminorms. The finite-order kernel estimates then prove strong smoothness. \end{proof} The index set is \[ \{47,5+,5-,6,8,9d,9m\} \times\{\mathrm{k},\mathrm{t},\mathrm{m}\} \times\{+,-\}. \] Its 42 entries are an audit expansion before tensor-component summation, not 42 separately admitted singular products. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.16\linewidth}Y p{.15\linewidth}@{}} \toprule Row & Seed operator & Internal order\\ \midrule \(47\) & \(\Irr,\ \eqref{eq:P47}\) & \(4\)\\ \(5+\) & \(\Irr,\ \eqref{eq:P5p}\) & \(4\)\\ \(5-\) & \(\Irr,\ \eqref{eq:P5m}\) & \(4\)\\ \(6\) & \(\Iresp,\ \eqref{eq:P6}\) & \(2\)\\ \(8\) & \(\Iresp,\ \eqref{eq:P8}\) & \(3\)\\ \(9d\) & \(\Iresp,\ \eqref{eq:P9d}\) & \(3\)\\ \(9m\) & \(\Iresp,\ \eqref{eq:P9m}\) & \(2\)\\ \bottomrule \end{tabularx} \end{center} Every row includes all six vertex/crossed choices from \eqref{eq:ledgerentry}. A parameter-dependent zero, such as a mass term at \(m=0\), is not a deletion from the structural inventory. %===================================================================== \section{Exact localized crossed-line and tensor regrouping} \label{sec:localregroup} %===================================================================== \subsection{Crossed-line reduction on the actual time-ordering region} Let \[ \mathcal O=\{(x_1,x_2):x_1^0>0,\ x_2^0<0\}. \] The compact relative image \(\fmap(\overline U)\) lies inside \(\mathcal O\). \begin{lemma}[Localized scalar crossed distribution] \label{lem:localcross} On \(\mathcal O\), \begin{equation} C_I^{\mu\nu} = i\hbar^2N^{\mu\nu}{}_I(W_0\otimes\Delta_m). \label{eq:localC} \end{equation} Choose \(\zeta\in C_c^\infty(\mathcal O)\) equal one on a neighborhood of \(\fmap(\overline U)\), and define \begin{equation} d_\delta = \rho_\delta* \bigl(\zeta\chi(W_0\otimes\Delta_m)\bigr). \label{eq:dprofile} \end{equation} For sufficiently small \(\delta>0\), \begin{equation} C_{\delta,I}^{\mu\nu}(x) = i\hbar^2N^{\mu\nu}{}_I\,d_\delta(x) \label{eq:localized-smoothing} \end{equation} on a neighborhood of \(\fmap(\overline U)\), with equality of all relative derivatives. \end{lemma} \begin{proof} On \(\mathcal O\), \[ D_{T,0}(x_1)=W_0(x_1),\qquad D_{T,m}(x_2)=W_m(-x_2). \] By \eqref{eq:scalar-kernel-identities}, \[ W_m(-x_2)-W_m(x_2)=i\Delta_m(x_2). \] This proves \eqref{eq:localC} as an identity of distributions in independent relative variables. The compact relative image has a positive distance from the region where \(\zeta\ne1\) or the time signs change. Let \(R_\rho\) bound the Euclidean radius of \(\supp\rho\). For sufficiently small \(\delta\), every translated convolution test centered near this image has support in the region where the distributional identity holds and \(\zeta=1\). The difference convolves to zero there. The same support argument applies to every profile derivative, proving the jet equality. \end{proof} The scalar profile \eqref{eq:dprofile} is still an eight-dimensional convolution: \[ d_\delta(x_1,x_2) = \pair{\zeta\chi(W_0\otimes\Delta_m)}{ \delta^{-8}\rho\bigl((x_1-v_1)/\delta,(x_2-v_2)/\delta\bigr)}. \] A general \(\rho(H_1,H_2)\) need not factor. No replacement by two independently chosen four-dimensional mollifiers is made. The lemma is an exact positive-scale localization identity. It does not authorize multiplying \(W_0\otimes\Delta_m\) by an internal singular kernel at \(\delta=0\). \subsection{Trace-reversal simplification} \begin{lemma}[Complete tensor-vertex simplification] \label{lem:tensorsimplify} The endpoint factor satisfies \begin{equation} N^{\mu\nu}{}_I = (e^I)^{\mu\nu}-\frac12\eta^{\mu\nu}\tauI, \qquad \eta_{\mu\nu}N^{\mu\nu}{}_I=-\tauI. \label{eq:Ntrace} \end{equation} For every smooth scalar \(d\), \begin{equation} Z_I^\nu(Nd) = \fmap^*((e^I)^{\mu\nu}\partial_{2,\mu}d), \qquad M_I(Nd)=-\tauI\fmap^*d. \label{eq:verticesimplify} \end{equation} These identities retain the complete kinetic, trace-kinetic, and mass vertex. \end{lemma} \begin{proof} Contract the tensor numerator in \eqref{eq:Nfactor} with the endpoint maps. This gives the first identity in \eqref{eq:Ntrace}; taking its trace in dimension four gives the second. Substitute into \(Z\): \[ \partial_{2,\mu}N^{\mu\nu}{}_I d -\frac12\eta_{\mu\alpha}\partial_2^\nu N^{\mu\alpha}{}_I d = (e^I)^{\mu\nu}\partial_{2,\mu}d. \] The two trace contributions cancel exactly. The mass identity is the second trace formula. \end{proof} For \(I=00\), \(N^{\mu\nu}{}_{00} =\delta^\mu_0\delta^\nu_0-\eta^{\mu\nu}/2\), and the kinetic combination is \(\delta^\nu_0\partial_{2,0}d\). For \(I=01\), \(\tauI=0\) and \[ N^{\mu\nu}{}_{01} = \frac12(\delta^\mu_0\delta^\nu_1+\delta^\mu_1\delta^\nu_0). \] This checks both a diagonal and an off-diagonal endpoint. The generic result follows from the same exact numerator calculation, not a component truncation. \subsection{Combined expression for analysis} For compact \(\Phi\), define \begin{equation} \mathcal U_\delta^{I,J}[\Phi] = g(y)\left[ (e^I)^{\mu\nu}\fmap^*(\partial_{2,\mu}d_\delta) \partial_{y^\nu}\Phi_J -\frac{m^2}{2}\tauI\fmap^*d_\delta\,\Phi_J \right]. \label{eq:Ucombined} \end{equation} Let \[ \SeedOp_{\mathrm{supp}}^{I,J} = \SeedOp_6^{I,J} +\SeedOp_8^{I,J} +\SeedOp_{9,\mathrm{der}}^{I,J} +\SeedOp_{9,\mathrm{mass}}^{I,J}. \] \begin{theorem}[Exact combined positive-scale formula] \label{thm:combined} For all sufficiently small positive scales, \begin{equation} t_\delta^{\App}(\Phi) = \lambda^3\hbar^2\sum_{I,J} \pair{\Kret^{I,J}}{\mathcal U_\delta^{I,J}[\Phi]}. \label{eq:combinedpairing} \end{equation} Equivalently, \begin{equation} \begin{split} t_\delta^{\App}(\Phi) = \lambda^3\hbar^2\sum_{I,J}\Big\{ &\Irr\bigl( \SeedOp_{47}^{I,J} [\mathcal U_\delta^{I,J}[\Phi]\circ\kappa_{47}] \bigr)\\ &+\Irr\bigl( \SeedOp_{5,+}^{I,J} [\mathcal U_\delta^{I,J}[\Phi]\circ\kappa_{5,+}] \bigr)\\ &+\Irr\bigl( \SeedOp_{5,-}^{I,J} [\mathcal U_\delta^{I,J}[\Phi]\circ\kappa_{5,-}] \bigr)\\ &+\Iresp\bigl( \SeedOp_{\mathrm{supp}}^{I,J} [\mathcal U_\delta^{I,J}[\Phi]\circ\kappa_{\mathrm{resp}}] \bigr)\Big\}. \end{split} \label{eq:combined-seeds} \end{equation} This formula equals the original 42-entry ledger on every compact joint test. It has not removed any insertion row or vertex term. \end{theorem} \begin{proof} Use \eqref{eq:localized-smoothing} in the complete vertex and then apply Lemma~\ref{lem:tensorsimplify}. The factor \(i\) in the selected family multiplies the factor \(i\) in the localized crossed array. Consequently \[ t_\delta^{\App} = -\lambda^3\hbar^2\Vertex(\Kret,Nd_\delta). \] Transposing the output derivative gives the positive kinetic term and the negative mass term in \eqref{eq:Ucombined}. This proves \eqref{eq:combinedpairing}, including its overall sign. Apply Theorem~\ref{thm:seed} to each row and combine the four response seed actions by linearity. This gives \eqref{eq:combined-seeds}. All preceding steps are equalities with smooth crossed coefficients at positive scale. The ledger expansion is the same complete vertex before these algebraic regroupings, so the two expressions agree on arbitrary joint tests. \end{proof} The supported contribution is now one combined seed operator, but a cancellation verdict would still require analysis of the complete expression, including the other three seed sectors. %===================================================================== \section{The preliminary bound and the exact analytic target} \label{sec:estimate} %===================================================================== \subsection{A concrete finite-order estimate} \begin{lemma}[Crossed order bound] \label{lem:crossorder} On compact sets, \(W_m,D_{T,m}\) have order at most four. The compact arrays \(\chi C_\pm\) have order at most eight, with an overall factor \(\hbar^2\). \end{lemma} \begin{proof} For a compact smooth test \(\psi(t,\mathbf x)\), spatial integration by parts gives \[ |\widetilde\psi(t,\mathbf p)| \le C(1+|\mathbf p|^2)^{-2}\|\psi\|_4 \] uniformly on its bounded time support. Both oscillatory time factors have absolute value one. Hence their pairings are bounded by \[ C\|\psi\|_4 \int_{\R^3} \frac{(1+|\mathbf p|^2)^{-2}}{\omega_{\mathbf p}} \dd^3\mathbf p. \] The integral converges, including at massless zero momentum. For the tensor products, perform the integration by parts in each spatial variable separately. This uses at most eight total derivatives and gives an integrable double-momentum majorant. Fixed cutoff derivatives and finite tensor factors preserve the order bound. \end{proof} \begin{theorem}[Complete polynomial smoothing estimate] \label{thm:moderate} The bound \eqref{eq:mainbound} holds for every compact \(L\Subset U\). \end{theorem} \begin{proof} A compact distribution of order at most eight in dimension eight satisfies \[ \sup_x|\partial^\gamma(\rho_\delta*u)(x)| \le C_\gamma\delta^{-8-8-|\gamma|}. \] Apply its finite-order estimate to the translated differentiated profile. The complete internal kernel has order at most four. Four derivatives of a kinetic or trace-kinetic vertex test require at most five crossed derivatives and at most five joint-test derivatives. The mass test requires no more. The resulting conservative power is \(\delta^{-8-8-5}=\delta^{-21}\). The smooth observations, the fixed root, and finite sums enter only the compact-local constant. Restore \(|\lambda|^3\hbar^2\). \end{proof} \begin{proof}[Conclusion of Theorem~\ref{thm:main}] The action, source, density, root, and crossed formulas were proved or explicitly prescribed before substitution. Theorem~\ref{thm:substitution} identifies the insertion kernel; Theorems~\ref{thm:seed} and \ref{thm:ledger} give its complete positive-scale action. Theorem~\ref{thm:combined} proves the exact regrouping, and Theorem~\ref{thm:moderate} proves the bound. \end{proof} The number twenty-one is a sufficient upper-bound power. It is not a candidate leading exponent and does not decide between convergence and divergence. \subsection{One precise remaining problem} For the fixed data and fixed profile of this paper, the remaining problem is to decide the strong raw limit of \eqref{eq:combined-seeds} on arbitrary compact joint tests meeting the characteristic region. An affirmative quantitative result would give a distribution \(T\) such that for every \(L\Subset U\), \begin{equation} |t_\delta^{\App}(\Phi)-T(\Phi)| \le \varepsilon_L(\delta)\|\Phi\|_{m_L,L}, \qquad \varepsilon_L(\delta)\longrightarrow0. \label{eq:convergencetarget} \end{equation} A negative result would exhibit one fixed compact joint test \(\Phi_*\), independent of \(\delta\), for which the complete pairing is unbounded. If a finite singular expansion is actually established, it may have the form \begin{equation} t_\delta^{\App} = \sum_{\nu=1}^Na_\nu(\delta)A_\nu +A_{\mathrm{fin}}+R_\delta, \qquad |R_\delta(\Phi)| \le\varepsilon_L(\delta)\|\Phi\|_{m_L,L}, \quad \varepsilon_L(\delta)\to0. \label{eq:possibleexpansion} \end{equation} Equation \eqref{eq:possibleexpansion} is an analytic target, not an assumed expansion class or a theorem of this paper. The scale factors, complete coefficients, and remainder must be derived. No power--logarithm ansatz is imposed in advance. \subsection{Sector obligations and residual scaling} A calculation must cover, with overlap control: \begin{enumerate} \item the complete supported seed \(\SeedOp_{\mathrm{supp}}\), including both root terms; \item the \(K_{47}\) two-cone seed; \item the two \(K_5\) placements separately; \item their cone-tip transitions; \item complementary compact regions and noncompact rescaled tails. \end{enumerate} Equations \eqref{eq:residualcones}--\eqref{eq:residualconstraint} are exact input for this task. They do not show that \(A_r\), \(B_r\), and \(Z_r\) have common scale weights. The alignment equations in \((a,b,y)\) do not themselves impose \(r=0\). For example, \[ \{u=v=\xi=0\} \] contains points with nonzero \(r\); the \(K_{47}\) support then imposes its additional \(b-z\) cone. Therefore the codimension-eight marked locus \(N\) must not be declared an exhaustive singular set merely from the cone-rank calculation. The residual intersections also require analysis. For the supported rows, \(r=0\) is exact. For the other rows, fixed compact \(r\)-support can become unbounded after rescaling. Compact support of \(\rho\) alone does not estimate these tails. No exhaustive scale-sector decomposition or quantified characteristic remainder has been completed here. The exact regrouping is the starting expression for that calculation. \subsection{Complete sum before a verdict} At any candidate singular scale, the coefficient must be collected from all contributing rows, tensor components, root terms, and vertex terms before cancellation or divergence is declared. The specialization \(f=0,\ F(b)=b^0\) is not admissible here: the advanced root condition gives \(F=0\) when \(f=0\). It can only be a separately declared diagnostic model, not a detecting test for this application. A moving-frequency witness is also insufficient. For example, \[ A=\partial_x\deltaD,\qquad \Phi_\omega(x)=\psi(x)e^{i\omega x},\qquad \psi=1\text{ near }0, \] gives \(A(\Phi_\omega)=-i\omega\), although the constant family \(t_\delta=A\) converges strongly. The moving tests are not a bounded test family. %===================================================================== \section{Release boundary and later microlocal obligations} \label{sec:release} %===================================================================== The results proved for this application are: \begin{enumerate}[label=\textup{(\roman*)}] \item consistent action, stress, inverse, coupling, and component normalizations; \item explicit cubic density and evaluated mixed source; \item an identified density derivative and advanced root; \item explicit crossed kernels in the actual relative slots; \item the marked insertion and its seven distributional kernels; \item exact seed transposition and the positive-scale audit ledger; \item exact localized crossed-line and tensor regrouping; \item the conservative complete smoothing bound. \end{enumerate} The weighted comparison panel is not a weighted successor theorem. No inherited observable identity, characteristic raw decision, finite part, regulator-independence result, or quantum admission is released. A later ordinary-product agreement theorem would need its actual localized wavefront cones and nonopposition hypotheses. Multiplication is hypocontinuous in the normal topology under the hypotheses of \cite[Theorem~6.1]{BDH}. Proper pushforward has separate support and continuity requirements \cite[Theorem~6.3]{BDH}. Neither result is applied here to an unverified full product at the characteristic set. A weighted rescaling is not automatically the Euler normal action required by a submanifold extension framework. For instance, \[ u\partial_u+v\partial_v+\frac12\xi^A\partial_{\xi^A} \] has transverse \(\xi\)-weight \(1/2\), not one. An application of \cite[Section~1]{Dang} needs its own scaling and microlocal verification. Even a future strong finite-part limit would not alone give normal-topology convergence or a wavefront theorem. Uniform localized Fourier estimates remain necessary. Authorization, normalization, global overlap agreement, and quantum admission are later independent questions. %===================================================================== \appendix \section{Cone normalization and causal convolution} \label{app:cone} %===================================================================== \begin{lemma}[Retarded cone identities] \label{lem:cone} The measures \eqref{eq:RB} satisfy \eqref{eq:cone-identities}. Their causal convolution is support-proper over compact external sets. \end{lemma} \begin{proof} With spatial Fourier phase \(e^{-i\mathbf k\cdot\mathbf x}\), \[ \widehat R(t,\mathbf k) = \Theta(t)\frac{\sin(\varpi t)}{\varpi}, \qquad \varpi=|\mathbf k|. \] The limiting value is \(\Theta(t)t\). Its value at \(0+\) is zero and its derivative there is one, so \[ (\partial_t^2+\varpi^2)\widehat R=\deltaD(t). \] Fourier injectivity gives \(\Boxeta R=\deltaD^{(4)}\). If \(s,x-s\) both lie in the closed future cone, \(s\) lies in the compact causal diamond from \(0\) to \(x\). These diamonds are uniformly bounded over compact \(x\)-sets. The temporal convolution is \[ \int_0^t \frac{\sin(\varpi(t-\tau))}{\varpi} \frac{\sin(\varpi\tau)}{\varpi}\dd\tau = \frac{\sin(\varpi t)-\varpi t\cos(\varpi t)}{2\varpi^3}, \] with limit \(t^3/6\). The spatial transform of \(B\) gives the same expression, multiplied by \(\Theta(t)\). Hence \(R*R=B\). For \(s^0>0,s\ne0\), the regular cone equation gives \[ \partial_{s^\alpha}B(s) = \frac{2s_\alpha}{8\pi}\deltaD(s^2) = \frac{s_\alpha}{2}R(s). \] This separated derivative formula does not replace the cone-tip measure definition of \(R\). \end{proof} %===================================================================== \section{Translation identity and reconstruction-null test} \label{app:translation} %===================================================================== For compact \(\Phi\), define compact relative distributions \[ a_{I,\nu}[\Phi] = \fmap_*\left( \sum_Jg(y)\partial_{y^\nu}\Phi_JK^{I,J}\right), \qquad d_I[\Phi] = \fmap_*\left(\sum_Jg(y)\Phi_JK^{I,J}\right), \] \[ \begin{split} \Lambda_{I,\mu\nu}[\Phi]={}& \frac i2\left( \partial_{2,\mu}a_{I,\nu} +\partial_{2,\nu}a_{I,\mu}\right)\\ &-\frac i2\eta_{\mu\nu}\partial_2^\rho a_{I,\rho} -\frac{im^2}{2}\eta_{\mu\nu}d_I. \end{split} \] With \(\mathfrak d(x,v)=x-v\), put \[ \Omega_\Phi = \sum_{I,\mu,\nu} \mathfrak d_* \left(\Lambda_{I,\mu\nu}[\Phi]\otimes\chi C_I^{\mu\nu}\right). \] \begin{proposition}[Independent translation check] The compact distribution \(\Omega_\Phi\) exists, and \[ t_\delta^{\App}(\Phi)=\lambda^3\Omega_\Phi(\rho_\delta). \] \end{proposition} \begin{proof} Compact coefficient--test factors make the first pushforwards proper; both factors in the final tensor product are compact. Transpose the relative vertex derivatives, using symmetry in \(\mu,\nu\). The resulting pairing is against \(\rho_\delta(x-v)\), which is precisely the convolution pairing. \end{proof} This is not a factorwise pairing with every singular translated crossed array. It is a tensor-product distribution tested against a smooth profile. \begin{lemma}[A reconstruction-null identity] \label{lem:null} In one normal coordinate \(q\), \[ q\deltaD'(q)=-\deltaD(q). \] Consequently, for any distribution \(u\) in independent variables, \[ (q\deltaD'(q)+\deltaD(q))\otimes u=0. \] Every positive-scale pairing constructed here sends this null distribution to zero, regardless of a presentation's polynomial degree assignment. \end{lemma} \begin{proof} For a compact smooth \(\psi\), \[ \pair{q\deltaD'}{\psi} = -\partial_q(q\psi(q))\big|_{q=0} = -\psi(0). \] Tensor products preserve the identity. The positive-scale map depends only on the actual reconstructed distribution, proving the last assertion. \end{proof} %===================================================================== \section{Transposition and execution audit} \label{app:audit} %===================================================================== \subsection{Derivative and sign checks} The line-only derivative in \(K_6\), \[ \partial_{a^\alpha}R(b-a)\deltaD^{(4)}(a-z), \] is not \[ \partial_{a^\alpha} \bigl(R(b-a)\deltaD^{(4)}(a-z)\bigr). \] The response rows, by contrast, contain derivatives of the full incidence seed as proved in \eqref{eq:response}. The chart derivatives are \[ \begin{aligned} (c,s,t)=(b,b-a,b-z):\quad& \partial_b=\partial_c+\partial_s+\partial_t,\quad \partial_a=-\partial_s,\quad \partial_z=-\partial_t,\\ (c,s,r)=(a,b-a,z-a):\quad& \partial_b=\partial_s,\quad \partial_z=\partial_r,\quad \partial_a=\partial_c-\partial_s-\partial_r. \end{aligned} \] Their coefficients are constant and the charts have unit absolute determinant. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.19\linewidth}Y@{}} \toprule Row & Sign mechanism checked in the seed proof\\ \midrule \(47\) & Outer minus derivative gives a plus test derivative. Three- and two-derivative seed terms then produce their distinct signs in \eqref{eq:P47}.\\ \(5+\), \(5-\) & Two line signs cancel. The outer plus derivative and the three seed derivatives have cancelling transposition signs.\\ \(6\) & The line-only \(a\)-derivative has a minus sign; the resulting two transfers give the minus sign in \eqref{eq:P6}.\\ \(8\) & The response-incidence sign cancels on transfer; the outer minus derivative gives a plus test derivative.\\ \(9d\) & The same incidence cancellation and outer minus-derivative transfer give \eqref{eq:P9d}.\\ \(9m\) & There is no outer derivative. The original mass minus sign remains.\\ Localized vertex & The selected-family \(i\) times the local crossed-line \(i\) gives \(-1\); output transposition gives \eqref{eq:Ucombined}.\\ \bottomrule \end{tabularx} \end{center} \subsection{A remainder cannot be differentiated without another estimate} For fixed nonzero \(A\in\Dist\), \[ r_\delta=\delta\sin(\delta^{-2})A \] has a vanishing \(O(\delta)\) finite-order bound, but \[ \delta\partial_\delta r_\delta = \left( \delta\sin(\delta^{-2})-2\delta^{-1}\cos(\delta^{-2}) \right)A. \] Thus a future family-level remainder is not automatically a scale-integrability estimate. \subsection{Actual execution record} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.35\linewidth}Y@{}} \toprule Audit item & Recorded outcome\\ \midrule First pdfLaTeX pass & Not run; no compilation log is available.\\ Second pdfLaTeX pass & Not run; no generated PDF or cross-reference audit is available.\\ Computer-algebra checks & Not run.\\ Proof-assistant verification & Not run.\\ Written source/density check & Explicit density, compact variation, and mixed curvature calculations are included in Section~\ref{sec:source}.\\ Component checks & Diagonal, off-diagonal, generic-basis, density, and endpoint checks are included.\\ Distributional transposition & All seven rows and the combined vertex are checked by written arguments.\\ Reconstruction identity & Lemma~\ref{lem:null} proves the exact null identity.\\ Independent referee or submission acceptance & Not asserted.\\ \bottomrule \end{tabularx} \end{center} The source uses nonconflicting command names instead of redefining the existing \verb|\Pr| and \verb|\AA| commands. No successful compilation or independent certification is represented as an observed event. The suggested two-pass build is printed at the beginning of this file. Actual compilation and log inspection remain necessary before submission. The absence of an execution tool does not justify inventing that outcome. %===================================================================== \begin{thebibliography}{99} \bibitem{Packet} N.~Cheung, attributed manuscript excerpts supplied in the preparation exchange. Formula-level comparison sources include: XI-A, \texttt{lem:vertex}; XI-B, \texttt{thm:graviton}; XII-C, \texttt{eq:ell3} and \texttt{thm:actualsource}; Em01-C@3, \texttt{eq:basis-duality}, \texttt{eq:source-conversion}, and the seven-row display; XIV-H, \texttt{prop:K8exact} and \texttt{prop:K5exact}; XIV-I, \texttt{thm:I3}; XIV-J, \texttt{thm:reference}, \texttt{lem:null}, and \texttt{thm:J2}. These are supplied excerpts, not independently audited complete parent manuscripts. No cross-series identity or characteristic result is imported from them. \bibitem{FR} K.~Fredenhagen and K.~Rejzner, \emph{Perturbative algebraic quantum field theory}. arXiv:1208.1428v2 (2013). \url{https://arxiv.org/abs/1208.1428v2}. Sections~5--7: contraction products, time ordering, interacting normalizations, and extension problems. No product for the present nonlocal insertion is imported. \bibitem{BDH} C.~Brouder, N.~V.~Dang, and F.~H\'elein, ``Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker),'' \emph{Studia Mathematica} \textbf{232} (2016), 201--226. doi:10.4064/sm8316-3-2016. arXiv:1409.7662v2. \url{https://arxiv.org/abs/1409.7662v2}. Theorems~6.1 and 6.3 are cited only with their actual wavefront and proper-support hypotheses. \bibitem{Dang} N.~V.~Dang, \emph{The extension of distributions on manifolds, a microlocal approach}. arXiv:1412.2808v1 (2014). \url{https://arxiv.org/abs/1412.2808v1}. Section~1: Euler vector fields and their normal action. No characteristic extension theorem is applied here. \end{thebibliography} \end{document}