% !TEX program = pdflatex %===================================================================== % III.tex % % Self-contained manuscript source for pdfLaTeX. % % Build: % pdflatex -interaction=nonstopmode -halt-on-error III.tex % pdflatex -interaction=nonstopmode -halt-on-error III.tex % Repeat if cross-reference warnings remain. % % No external graphics, bibliography database, or shell escape. % % Revision and mathematical audit: % - The four opening rows are equation-space questions; the second % does not define a quotient for general compact gauge parameters. % - Failure of descent is proved for a fixed exhibited symbol, not % for every scalar functional. % - The massive free-shell tangency theorem is kept distinct from % the unchanged-product obstruction. % - Poisson and retarded compensation signs are fixed explicitly. % - The C1 data and solution topologies are defined by corresponding % support stages; the Cauchy isomorphism is proved stagewise. % - Newton reconstruction in the closed-range argument is a membership % test, not admission of noncompact gauge descriptors. % - Compact adjoint-exact Ward repairs are distinguished from the % explicit nonzero compact coupled Ward pair. % - The scalar symplectic result fixes the metric. % - The ADM kinetic normalization is declared. % - A gauge-fixed response need not solve the invariant equations; % it is not asserted that it can never do so. % - Gauge transformations, inverse operators, boundary prescriptions, % and reduced functionals are not identified. % - The sourced successor is split into invariant equations, % gauge-fixed response, and quotient descent. % - Companion results are documentary comparisons, not proof premises. % - H2/I3/J2 verdicts are attributed to their specified source versions. % % Verification status: % The source has not been compiled in the present interface. % No machine-checked proof, verified page count, or execution % certificate is claimed. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1.05in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage{xcolor} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,calc} \usepackage[font=small,labelfont=bf]{caption} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={Descent of first-order relational scalars and the free-product obstruction}, pdfauthor={Ngo Cheung}, pdfsubject={Classical equation-space quotients and coupled Ward observables} } \allowdisplaybreaks \numberwithin{equation}{section} \setlength{\emergencystretch}{3em} \setlength{\parskip}{2pt} \setlist{itemsep=3pt,topsep=5pt} \setcounter{secnumdepth}{2} \setcounter{tocdepth}{1} \urlstyle{same} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Dual}{\mathbb D} \newcommand{\dd}{\,\mathrm d} \newcommand{\bx}{\mathbf x} \newcommand{\by}{\mathbf y} \newcommand{\bk}{\mathbf k} \newcommand{\id}{\mathrm{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\tr}{\operatorname{tr}_{\eta}} \newcommand{\Glin}{G^{(1)}} \newcommand{\Boxeta}{\Box_{\eta}} \newcommand{\Dop}{\mathsf D} \newcommand{\Rev}{\mathsf r} \newcommand{\Om}{\Omega_m} \newcommand{\Sch}{\mathscr S} \newcommand{\Eplus}{\mathscr E_+} \newcommand{\GaugeR}{\mathscr G_{\mathrm R}} \newcommand{\MetricR}{\mathscr M_{\mathrm R}} \newcommand{\Jet}{\mathscr J} \newcommand{\Slice}{\mathscr Y} \newcommand{\SolC}{\mathscr S_{\mathrm C}} \newcommand{\DataC}{\mathscr X_{\mathrm C}} \newcommand{\GaugeC}{\mathscr G_{\mathrm C}} \newcommand{\Solphi}{\mathscr S_\phi} \newcommand{\Solg}{\mathscr S_g} \newcommand{\Ocal}{\mathcal O} \newcommand{\wP}{w_{\mathrm P}} \newcommand{\wR}{w_{\mathrm R}} \newcommand{\Fret}{\mathcal F} \newcommand{\Fpois}{F^{\mathrm P}} \newcommand{\Rbox}{R_{\Box}} \newcommand{\Norm}[1]{\left\lVert #1\right\rVert} \newcommand{\Abs}[1]{\left\lvert #1\right\rvert} \newcommand{\Lie}{\mathcal L} \newcommand{\SourceLabel}[1]{\texttt{\detokenize{#1}}} \newcommand{\doi}[1]{% \href{https://doi.org/#1}{\nolinkurl{#1}}} \newcommand{\smallbox}[2]{% \par\medskip\noindent \fbox{\begin{minipage}{% \dimexpr\linewidth-2\fboxsep-2\fboxrule\relax} \textbf{#1}\par\smallskip #2 \end{minipage}} \par\medskip } \tikzset{ proofbox/.style={ draw,rounded corners,align=center,font=\small, minimum height=1.0cm,inner sep=6pt }, proofarrow/.style={-{Stealth[length=2mm]},thick} } \title{\textbf{Descent of first-order relational scalars\\ and the free-product obstruction}\\[3pt] \large Classical equation spaces, retarded sections, and compact Ward observables} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We identify the classical equation spaces and gauge actions on which specified first-order relational scalar functionals descend. An exhibited Poisson-dressed symbol is not constant on the unchanged free product of linearized gravity with a massive scalar; the same two-representative gap survives spatial Fourier regularization. Moving the scalar does not repair that product: \(X_\xi\phi\) preserves every compact-data massive free solution only when \(\xi\) is Killing. Retaining the forced correction \(P\phi_1=-L_h\phi_0\) changes the equation space and supplies a genuine coefficient gauge action. A past-trivial retarded harmonic section then makes the complete first-order functional a quotient observable. For a triangular system with a fixed free scalar background, compact constrained de Donder Cauchy data modulo residual wave gauges form a Hausdorff, well-posed quotient. We construct a compact coupled Ward pair with strictly positive on-shell value, although the immediate adjoint-exact compact repair vanishes on shell. Poisson compensation, compensator choice, and canonical normalization are treated separately. Finally, the sourced successor is divided into invariant equations, a gauge-fixed forced response, and descent on a named quotient. None of those sourced constructions is completed here. \end{abstract} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.65\linewidth}X@{}} \toprule Equation space and identification being tested & Result\\ \midrule Unchanged product: \(q_0(h,\phi)=([h],\phi)\) & No descent for the exhibited symbol.\\ \addlinespace Unchanged massive free scalar shell, with proposed variation \(X_\xi\phi\) & No tangent action for general compact gauges.\\ \addlinespace Coupled jet: \((h,\phi_0,\phi_1)\), \(P\phi_1=-L_h\phi_0\) & Descent through first order for the declared retarded group.\\ \addlinespace C1 triangular system: \((h,\varphi)\), fixed \(\Phi\) & Hausdorff Cauchy quotient; nonzero compact Ward observables.\\ \bottomrule \end{tabularx} \end{center} \smallbox{Scope}{% This is a classical quotient paper. It proves no physical Hilbert-space correspondence, measurement instrument, quantum record, Born theorem, contact-span separation, or interacting quantum coefficient. It does not solve the next matter-sourced gravitational reduction, construct a compact gravitational apparatus, or establish an asymptotic scattering theory. Companion manuscripts appear only as explicitly scoped comparisons.} \tableofcontents %===================================================================== \section{Introduction: the equation space precedes descent} \label{sec:introduction} %===================================================================== A proposed relational expression must define a function on the equivalence classes actually declared. This question precedes any subsequent representation or measurement construction: \[ \text{classical functional} \longrightarrow \text{represented observable} \longrightarrow \text{instrument} \longrightarrow \text{record}. \] Only the first object and its classical descent are studied here. Two failures must be distinguished. An expression can assign different values to equivalent representatives. Alternatively, its proposed transformation can fail to preserve the equation space, so that the intended quotient action is not defined. An off-shell cancellation excludes neither failure. Relational and metric-dependent coordinate constructions have a substantial literature; see, for example, Brunetti, Fredenhagen, and Rejzner~\cite{BFR}. Explicit gravitational dressings and their nonlocal kernels are developed by Donnelly and Giddings~\cite{DG}. Our question is narrower: for the displayed first-order functionals, which equations, identifications, and support prescriptions make descent valid? \subsection{The four results} The first negative result fixes a free scalar and varies only its metric representative. A compact gauge change of a Poisson compensator produces \[ -\kappa\int f\,\xi^\mu\partial_\mu\phi. \] We choose \(f,\phi,\xi\) once so that this integral is nonzero. No function of the unchanged label \(([h],\phi)\) can then agree with the symbol on both representatives. The same representatives yield a quantitative regulated obstruction. The second negative result allows the scalar to move but retains its free equation. The required condition is \[ P(X_\xi\phi)=0 \quad\text{for every compact-data solution of }P\phi=0. \] A solution-jet argument proves that this holds exactly for Minkowski Killing fields. In particular, no nonzero compactly supported gauge vector satisfies it. The first positive result changes the equation space: \[ P\phi_0=0,\qquad P\phi_1=-L_h\phi_0. \] The coefficient transformation is an action on these equations. For a named past-trivial gauge group, a retarded harmonic compensator defines a unique orbit representative. The complete first-order scalar then descends. The second positive result fixes a free background \(\Phi\) and uses the linear triangular system \[ \Glin(h)=0,\qquad P\varphi=-L_h\Phi. \] Compact constrained harmonic data modulo residual wave gauges form a Hausdorff Cauchy quotient. A compact coupled Ward pair has a strictly positive value on an explicit solution. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[proofbox,text width=5.2cm] (product) at (0,0) {Unchanged product\\ Scalar fixed; metric quotiented\\ Exhibited symbol does not descend}; \node[proofbox,text width=5.2cm] (free) at (6.8,0) {Unchanged massive shell\\ Scalar variation proposed\\ General compact gauges are not tangent}; \node[proofbox,text width=5.2cm] (jet) at (0,-2.2) {Coupled first-order jet\\ Forced scalar coefficient retained\\ Retarded orbit section exists}; \node[proofbox,text width=5.2cm] (c1) at (6.8,-2.2) {C1 triangular system\\ Background scalar fixed\\ Residual-wave Cauchy quotient exists}; \end{tikzpicture} \caption{The four rows are different equation-space questions. The second row does not define a quotient by general compact gauges. No arrow identifying the rows is presumed.} \label{fig:descent-ledger} \end{figure} \subsection{What is not identified} The retarded group and the C1 residual-wave group differ. Their intersection is zero when the retarded support requirement is imposed. The section proof for one group cannot be transplanted to the other. Nor is C1 a linearization of the full Einstein--scalar system about a nonlinear solution \((\eta,\Phi)\). Its metric equation is vacuum linearized gravity; the stress of \(\Phi\) is not inserted into that equation. For the genuinely linearized Einstein--Klein--Gordon system on on-shell backgrounds, see Hack~\cite{Hack}. That system is not an additional premise here. \subsection{Analytical and bibliographic inputs} The differential identities, obstructions, orbit section, closed-range test, and compact Ward construction are proved below. The analytical background is smooth wave and Klein--Gordon Cauchy theory with finite propagation~\cite{BGP}; we include the estimates needed for its use. The Coulomb comparison concerns precisely the kernels of the cited version of Donnelly and Giddings~\cite{DG}. Cheung's companion manuscripts are cited as versioned unpublished source records~\cite{Registry}. Their comparison role is collected in Section~\ref{sec:nontransfer}. They are not hypotheses of the five principal theorems: Theorems~\ref{thm:product}, \ref{thm:Killing}, \ref{thm:jet-descent}, \ref{thm:reduced-Cauchy}, and \ref{thm:compact-pair}. %===================================================================== \section{Conventions, signs, and support} \label{sec:conventions} %===================================================================== \subsection{Geometry and operators} On \(M=\R^{1+3}\), use \[ \eta=\diag(-1,1,1,1),\qquad \Boxeta=-\partial_t^2+\Delta,\qquad P=\Boxeta-m^2,\qquad m>0, \] where \(\Delta=\partial_i\partial_i\). All fields and gauge parameters are real unless Fourier transforms or complex-valued comparison functions are explicitly used. Indices are raised with \(\eta\), and symmetrization includes the factor \(1/2\). For a covector \(\xi_\mu\), set \[ (K\xi)_{\mu\nu} =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu, \qquad X_\xi=\xi^\mu\partial_\mu. \] The same letter denotes its raised vector when used in \(X_\xi\). For a symmetric covariant tensor \(h\), define \[ (\Rev h)_{\mu\nu} =h_{\mu\nu}-\frac12\eta_{\mu\nu}\tr h, \] \[ (\Dop h)_\nu =\partial^\mu h_{\mu\nu}-\frac12\partial_\nu\tr h =\partial^\mu(\Rev h)_{\mu\nu}, \] and \begin{equation} L_hu =-h^{\mu\nu}\partial_\mu\partial_\nu u -(\Dop h)^\nu\partial_\nu u. \label{eq:Lh} \end{equation} In four dimensions, \(\Rev^2=\id\). Our linearized Einstein normalization is \begin{equation} \Glin(h)_{\mu\nu} =-\frac12\Boxeta(\Rev h)_{\mu\nu} +\partial_{(\mu}(\Dop h)_{\nu)} -\frac12\eta_{\mu\nu}\partial^\alpha(\Dop h)_\alpha. \label{eq:Glin} \end{equation} \begin{lemma}[Sign and differential identities] \label{lem:signs} For smooth \(\xi\) and \(u\), \[ \Dop K\xi=\Boxeta\xi,\qquad \Glin(K\xi)=0,\qquad \partial^\mu\Glin(h)_{\mu\nu}=0. \] Moreover, \begin{equation} [P,X_\xi]u =(\Boxeta\xi^\nu)\partial_\nu u +2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu u, \label{eq:commutator} \end{equation} and therefore \begin{equation} \boxed{L_{K\xi}=-[P,X_\xi].} \label{eq:sign-lock} \end{equation} \end{lemma} \begin{proof} In \(\Dop K\xi\), the derivatives of \(\partial_\mu\xi^\mu\) cancel, leaving \(\Boxeta\xi\). Insert this and \[ (\Rev K\xi)_{\mu\nu} =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu -\eta_{\mu\nu}\partial_\alpha\xi^\alpha \] into \eqref{eq:Glin} to obtain \(\Glin K\xi=0\). Taking the divergence of \eqref{eq:Glin} gives \[ -\frac12\Boxeta(\Dop h)_\nu +\frac12\Boxeta(\Dop h)_\nu +\frac12\partial_\nu\partial^\alpha(\Dop h)_\alpha -\frac12\partial_\nu\partial^\alpha(\Dop h)_\alpha=0. \] Apply the product rule twice to \(\Boxeta(\xi^\nu\partial_\nu u)\). The constant mass term commutes with \(X_\xi\), proving \eqref{eq:commutator}. Substituting \(K\xi\) in \eqref{eq:Lh} gives its negative. \end{proof} The consistent triangular signs are consequently \begin{equation} \boxed{ P\varphi=-L_h\Phi,\qquad (h,\varphi)\longmapsto(h+K\xi,\varphi+X_\xi\Phi). } \label{eq:triangular-signs} \end{equation} \subsection{Formal order} For first-order coefficients, use \[ \Dual=\R[\kappa]/(\kappa^2). \] Equality in \(\Dual\) is coefficient equality, not a numerical remainder estimate. In the two-representative estimates, \(\kappa\) can instead be a real parameter in an exact degree-one polynomial. We indicate that use explicitly. A real representation or cutoff parameter in a different model is not automatically this nilpotent formal parameter. \subsection{Support classes} Fix a reference time \(t_0\). For this paper, a smooth field is called spacelike compact if, for some \(R_0\), \[ \supp u\subset \{(t,\bx):|\bx|\le R_0+|t-t_0|\}. \] This concrete cone condition is sufficient for every use below. It follows from compact Cauchy data for homogeneous wave and Klein--Gordon solutions. Let \(\Eplus\) be the smooth fields supported in \(J_\eta^+(K_0)\) for some compact spacetime set \(K_0\). Use this notation componentwise for vectors and tensors. Such fields have zero data on a sufficiently early Cauchy slice and bounded spatial support on each finite time interval. Compact spacetime support, compact Cauchy data, and future support from a compact set are different conditions. \subsection{Smooth forced Cauchy evolution} \begin{lemma}[Smooth forced evolution] \label{lem:PDE} For \(\mu\ge0\), the equation \[ (\partial_t^2-\Delta+\mu^2)u=F \] has a unique global smooth solution for smooth compact Cauchy data and smooth forcing with bounded spatial support on every finite time interval. On such intervals, the solution depends continuously on data and forcing in their smooth topologies. Finite propagation gives the corresponding cone-support bounds when the data and source satisfy those bounds. \end{lemma} \begin{proof} Let \(\Omega_\mu=\sqrt{-\Delta+\mu^2}\). Spatial Fourier transformation gives \[ \begin{aligned} u(t)={}& \cos((t-t_0)\Omega_\mu)u_0 +\frac{\sin((t-t_0)\Omega_\mu)}{\Omega_\mu}u_1\\ &+\int_{t_0}^t \frac{\sin((t-s)\Omega_\mu)}{\Omega_\mu}F(s)\dd s. \end{aligned} \] For \(\mu=0\), the sine quotient has the continuous value \(t-s\) at zero frequency. On a finite interval, the source has uniformly compact spatial support and bounded smooth seminorms, so the required Sobolev integrals are defined. Differentiation verifies the equation and data. For an integer \(s\ge0\), use \[ E_s(t)=\frac12\sum_{|\alpha|\le s} \left( \|\partial_t\partial^\alpha u\|_2^2 +\|\nabla\partial^\alpha u\|_2^2 +(\mu^2+1)\|\partial^\alpha u\|_2^2 \right). \] Integration by parts and Cauchy--Schwarz give \[ E_s'(t)\le C_\mu E_s(t)+C\|F(t)\|_{H^s}^2. \] The backwards estimate and Gronwall's inequality give uniqueness and continuous dependence. Increasing \(s\), applying Sobolev embedding, and using the equation for time derivatives give smooth estimates. For finite propagation, integrate the local energy identity outside an expanding ball. The outward characteristic flux has the required nonnegative sign. Where initial data and forcing vanish, the exterior energy remains zero. The same argument after a time translation proves the retarded support statement. These are the flat-space estimates underlying the standard general Cauchy theory~\cite[Chapter~3]{BGP}. \end{proof} %===================================================================== \section{The unchanged free-product obstruction} \label{sec:product} %===================================================================== \subsection{The quotient being tested} Let \[ \Solg =\{h:\Glin(h)=0,\ h\text{ smooth and spacelike compact}\}, \] and \[ \Solphi =\{\phi:P\phi=0,\ \phi\text{ has compact smooth Cauchy data}\}. \] Choose any linear subspace \(\mathscr G_0\subset\Solg\) containing \[ \{K\xi:\xi\in C_c^\infty(M;T^*M)\}. \] Define \[ \mathscr P_g=\Solg/\mathscr G_0,\qquad \mathscr P_0=\mathscr P_g\times\Solphi, \qquad q_0(h,\phi)=([h],\phi). \] Only the metric is quotiented. \begin{definition}[Descent] A function \(F\) on \(X\) descends through \(q:X\to Y\) if there is a function \(\overline F\) on \(Y\) with \[ F=\overline F\circ q. \] Equivalently, \(F\) is constant on every fiber of \(q\). Failure of this set-theoretic condition excludes continuous or smooth descent as well. \end{definition} \subsection{Spatial Poisson compensation} For compact smooth or Schwartz spatial sources, define \begin{equation} (Ns)(\bx) =-\frac1{4\pi}\int_{\R^3} \frac{s(\by)}{|\bx-\by|}\dd^3y. \label{eq:Newton} \end{equation} The fundamental-solution identity gives \[ \Delta Ns=s,\qquad N\Delta s=s. \] For the second equality, the difference is smooth harmonic and tends to zero at infinity, hence vanishes. For smooth spacelike-compact \(h\), apply \(N\) at each time: \begin{align} A_i[h]&=\partial_jh_{ij}-\frac12\partial_i h_{jj}, & B_{\mathrm P}[h]&=\partial_i h_{0i} -\frac12\partial_t h_{ii}, \notag\\ \wP^i[h]&=-NA_i[h], & \wP^0[h]&=NB_{\mathrm P}[h]. \label{eq:Poisson} \end{align} These potentials are smooth but need not be spatially compact. Their sources have uniformly bounded support on finite time intervals, so time differentiation is legitimate. \begin{proposition}[Poisson compensation] \label{prop:Poisson} For compact smooth \(\xi\), \[ \wP^\mu(h+K\xi)=\wP^\mu(h)-\xi^\mu. \] \end{proposition} \begin{proof} Directly, \[ A_i[K\xi]=\Delta\xi_i,\qquad B_{\mathrm P}[K\xi]=\Delta\xi_0. \] Therefore \[ \wP^i[K\xi]=-\xi_i=-\xi^i, \qquad \wP^0[K\xi]=\xi_0=-\xi^0. \] Use linearity. \end{proof} For real \(f\in C_c^\infty(M)\), put \begin{align} \Fpois_0(f;h,\phi)&=\int_M f\phi\dd^4x, \notag\\ \Fpois_1(f;h,\phi) &=\int_M f\,\wP^\mu(h)\partial_\mu\phi\dd^4x, \notag\\ \Fpois_\kappa(f;h,\phi) &=\Fpois_0(f;h,\phi)+\kappa\Fpois_1(f;h,\phi). \label{eq:Poisson-functional} \end{align} The displayed expression is an exact degree-one polynomial, not an all-orders invariant. \subsection{A fixed nonzero gap} \begin{theorem}[Free-product obstruction] \label{thm:product} There exist fixed real \[ f\in C_c^\infty(M),\qquad \phi\in\Solphi,\qquad \xi\in C_c^\infty(M;T^*M) \] such that \[ J:=\int_M f\,\xi^\mu\partial_\mu\phi\dd^4x>0 \] and \begin{equation} \Fpois_\kappa(f;K\xi,\phi) -\Fpois_\kappa(f;0,\phi) =-\kappa J. \label{eq:product-gap} \end{equation} For every real \(\kappa\ne0\), this fixed symbol does not descend through \(q_0\). Its first-order coefficient does not descend either. \end{theorem} \begin{proof} Choose compact scalar data at \(t=0\): \[ \phi(0,\bx)=x^1\chi(\bx),\qquad \partial_t\phi(0,\bx)=0, \] where \(\chi=1\) near the origin. The resulting free solution has \(\partial_1\phi(0,0)=1\), so this derivative is positive on some relatively compact spacetime neighborhood \(U\). Take nonnegative nonzero \(f\in C_c^\infty(U)\). Choose a compact gauge vector equal to \(\partial_1\) near \(\supp f\). Then \(J=\int f\partial_1\phi>0\). Both \(0\) and \(K\xi\) solve the vacuum metric equation, and they have the same quotient class. Their scalar labels agree. Since \(\wP(K\xi)=-\xi\), equation \eqref{eq:product-gap} follows. A descended function cannot distinguish these representatives. \end{proof} The theorem exhibits one fixed symbol. It does not assert that every scalar functional fails to descend; metric-independent scalar functionals already descend through the metric quotient. The argument uses only \(\wP(K\xi)=-\xi\). Adding a linear compensator term annihilating \(K\xi\) does not remove this particular gap. \begin{lemma}[Two-representative estimate] \label{lem:two-point} If \(q(x_0)=q(x_1)\), then for \(H:X\to\C\) and \(G:Y\to\C\), \[ \max_{j=0,1}|G(q(x_j))-H(x_j)| \ge\frac12|H(x_1)-H(x_0)|. \] \end{lemma} \begin{proof} Insert the common value \(G(q(x_0))=G(q(x_1))\) between the two values of \(H\), and use the triangle inequality. \end{proof} \begin{corollary}[Nonvanishing approximation error] \label{cor:product-error} For the fixed data above and any \(G:\mathscr P_0\to\C\), \[ \begin{split} \max\bigl\{& |G([0],\phi)-\Fpois_\kappa(f;0,\phi)|,\\ & |G([0],\phi)-\Fpois_\kappa(f;K\xi,\phi)| \bigr\} \ge\frac{|\kappa|J}{2}. \end{split} \] Agreement to \(o(|\kappa|)\) on both representatives is impossible. \end{corollary} \begin{proof} Apply Lemma~\ref{lem:two-point}. \end{proof} \subsection{Spatial Fourier regularization} Use the unitary transform \[ \widehat s(\bk) =(2\pi)^{-3/2}\int_{\R^3} e^{-i\bk\cdot\bx}s(\bx)\dd^3x. \] Choose smooth \(a,b:[0,\infty)\to[0,1]\) with \[ a=0\text{ on }[0,1],\quad a=1\text{ on }[2,\infty), \] \[ b=1\text{ on }[0,1],\quad b=0\text{ on }[2,\infty). \] For \(0<2\mu<\Lambda\), let \[ \chi_{\mu,\Lambda}(\bk) =a(|\bk|/\mu)b(|\bk|/\Lambda), \qquad \rho=(\mu,\Lambda), \] and \[ \widehat{N_\rho s}(\bk) =-\frac{\chi_\rho(\bk)}{|\bk|^2}\widehat s(\bk). \] Replacing \(N\) in \eqref{eq:Poisson} gives \(\wP_\rho\) and the symbol \(\Fpois_{\kappa,\rho}\). \begin{lemma}[Regulated compensation] \label{lem:regulated} For compact smooth \(\xi\), \[ \wP_\rho^\mu(K\xi)=-\chi_\rho(D)\xi^\mu. \] For the fixed witness, \[ J_\rho =\int_M f\,(\chi_\rho(D)\xi^\mu)\partial_\mu\phi\dd^4x \longrightarrow J \] as \(\mu\downarrow0\), \(\Lambda\uparrow\infty\). \end{lemma} \begin{proof} The multiplier identity \(N_\rho\Delta=\chi_\rho(D)\) proves the first claim. On a compact time interval containing \(\supp f\), the spatial Fourier transforms of \(\xi\) have a uniform Schwartz bound. Thus \[ \sup_{\bx}|(\chi_\rho(D)-1)\xi^\mu(t,\bx)| \le (2\pi)^{-3/2}\int |1-\chi_\rho(\bk)|\,|\widehat\xi^\mu(t,\bk)|\dd^3k \] tends to zero uniformly on that interval. One can use a fixed integrable Schwartz envelope and dominated convergence. Integration against the fixed compact test factors gives the result. \end{proof} \begin{theorem}[Regulator-stable failure of descent] \label{thm:regulated-gap} For every family \(G_\rho:\mathscr P_0\to\C\), \[ \begin{split} \max\bigl\{& |G_\rho([0],\phi)-\Fpois_{\kappa,\rho}(f;0,\phi)|,\\ & |G_\rho([0],\phi)-\Fpois_{\kappa,\rho}(f;K\xi,\phi)| \bigr\} \ge\frac{|\kappa||J_\rho|}{2}. \end{split} \] For fixed real \(\kappa\ne0\), the bound tends to \(|\kappa|J/2>0\). \end{theorem} \begin{proof} The symbol difference is \(-\kappa J_\rho\), while the quotient labels remain identical. Apply Lemma~\ref{lem:two-point} and the convergence lemma. \end{proof} This regularizes a classical symbol only. It does not concern an operator or measurement. It also does not obstruct a different quotient that changes the scalar identification along with the metric. %===================================================================== \section{Moving the scalar: the massive free-shell obstruction} \label{sec:tangency} %===================================================================== \smallbox{Different premises}{% The product theorem fixes \(\phi\) and varies \(h\). The present theorem moves \(\phi\) by \(X_\xi\phi\), while retaining the massive free equation. Neither statement is a reformulation of the other.} \subsection{Realization of solution jets} \begin{lemma}[Second jets of compact-data free solutions] \label{lem:jets} At any \(p\in M\), let \(a\in\R\), \(b_\mu\in\R^4\), and symmetric \(H_{\mu\nu}\) satisfy \[ \eta^{\mu\nu}H_{\mu\nu}=m^2a. \] There is \(\phi\in\Solphi\) with \[ \phi(p)=a,\qquad \partial_\mu\phi(p)=b_\mu,\qquad \partial_\mu\partial_\nu\phi(p)=H_{\mu\nu}. \] \end{lemma} \begin{proof} Translate \(p\) to \((0,0)\). Choose compact smooth data \(q,\pi\) with \[ q(0)=a,\quad \partial_iq(0)=b_i,\quad \partial_i\partial_jq(0)=H_{ij}, \] \[ \pi(0)=b_0,\qquad \partial_i\pi(0)=H_{0i}. \] Finite polynomials multiplied by a bump equal to one near the origin realize these data jets. The free equation supplies \[ \partial_t^2\phi(0) =\sum_iH_{ii}-m^2a=H_{00}, \] which is the remaining required component. \end{proof} \begin{theorem}[Free-shell tangency is exactly Killing symmetry] \label{thm:Killing} For smooth \(\xi\), the following are equivalent: \begin{enumerate}[label=(\roman*)] \item \(P(X_\xi\phi)=0\) for every \(\phi\in\Solphi\); \item \(K\xi=0\); \item \[ \xi^\mu(x)=a^\mu+B^\mu{}_\nu x^\nu \] for constants with \(B_{\mu\nu}=-B_{\nu\mu}\). \end{enumerate} A field satisfying these conditions and vanishing on a nonempty open set is identically zero. \end{theorem} \begin{proof} Assume (i). At a fixed point, \eqref{eq:commutator} and the jet lemma imply \[ (\Boxeta\xi^\nu)b_\nu+A^{\mu\nu}H_{\mu\nu}=0 \] for every allowed jet, where \[ A^{\mu\nu} =\partial^\mu\xi^\nu+\partial^\nu\xi^\mu. \] Take \(a=0,H=0\) and arbitrary \(b\), obtaining \(\Boxeta\xi=0\). Take \(a=0,b=0\) and arbitrary trace-free symmetric \(H\). The annihilator of that codimension-one space is the span of the trace functional, so \[ A^{\mu\nu}=c\,\eta^{\mu\nu} \] at the point. Finally take \(a=1,b=0\) and a symmetric \(H\) of trace \(m^2\). Then \(cm^2=0\), hence \(c=0\). The point was arbitrary, so \(K\xi=0\). Differentiating the Killing equation gives \[ 2\partial_\mu\partial_\nu\xi_\rho = \partial_\mu(K\xi)_{\nu\rho} +\partial_\nu(K\xi)_{\mu\rho} -\partial_\rho(K\xi)_{\mu\nu}=0. \] Thus \(\xi\) is affine with antisymmetric linear part. Conversely, an affine Killing field has \(\Boxeta\xi=0\) and vanishing symmetric first derivative. The commutator identity gives (i). An affine field vanishing on an open set is zero. \end{proof} \begin{corollary}[No general compact gauge action on the free shell] \label{cor:free-shell} The prescription \[ \delta_\xi\phi=\kappa X_\xi\phi \] does not define a tangent action on the unchanged massive free solution space for all compact smooth \(\xi\). This conclusion does not depend on a proposed metric transformation. \end{corollary} \begin{proof} The tangent space of the linear solution space is itself. Tangency at every solution is condition (i) above. A compactly supported Killing field vanishes on an open set and is therefore zero. \end{proof} The assumption \(m>0\) is used in eliminating \(c\). No massless version of this classification is asserted. \subsection{A positive smeared defect} \begin{proposition}[Fixed tangency witness] \label{prop:tangency-witness} There exist real compact-data \(\phi\), compact smooth \(\xi\), and nonnegative \(f\in C_c^\infty(M)\), \(\int f=1\), such that \[ \int_M fP(X_\xi\phi)\dd^4x\ge\frac12. \] For \(\phi_\kappa=\phi+\kappa X_\xi\phi\), \[ \left|\int_M fP\phi_\kappa\dd^4x\right| \ge\frac{|\kappa|}{2}. \] \end{proposition} \begin{proof} Take \(\beta\in C_c^\infty(\R^3)\) equal to one near the origin and scalar data \[ \phi(0,\bx)=x^1\beta(\bx),\qquad \partial_t\phi(0,\bx)=0. \] For \(a\in C_c^\infty(\R)\) equal to one near zero, set \[ \xi^1=-\frac12t^2a(t)\beta(\bx),\qquad \xi^0=\xi^2=\xi^3=0. \] At the origin, \[ \partial_\mu\xi^\nu=0,\qquad \Boxeta\xi^1=1,\qquad \partial_1\phi=1. \] Thus \(P(X_\xi\phi)(0)=1\). It exceeds \(1/2\) on a neighborhood. A normalized nonnegative test supported there proves the first claim; \(P\phi=0\) proves the second. \end{proof} An additional regular formal coefficient of order \(\kappa^2\) cannot remove this order-\(\kappa\) equation defect. Numerical remainder claims would require additional estimates. %===================================================================== \section{The coupled first-order jet and retarded harmonic section} \label{sec:jet} %===================================================================== \subsection{The changed scalar equations} For \(g=\eta+\kappa h\), \[ g^{\mu\nu}=\eta^{\mu\nu}-\kappa h^{\mu\nu}\pmod{\kappa^2}, \] \[ \sqrt{|\det g|} =1+\frac{\kappa}{2}\tr h\pmod{\kappa^2}. \] Substitution into \[ \Box_g u =|\det g|^{-1/2} \partial_\mu\!\left( |\det g|^{1/2}g^{\mu\nu}\partial_\nu u \right) \] gives \begin{equation} \Box_g-m^2=P+\kappa L_h\pmod{\kappa^2}. \label{eq:operator-variation} \end{equation} Consequently \(\phi=\phi_0+\kappa\phi_1\) obeys the coefficient equations \begin{equation} P\phi_0=0,\qquad P\phi_1+L_h\phi_0=0. \label{eq:jet-scalar} \end{equation} The second scalar coefficient is forced, not independently free. \begin{proposition}[Coefficient gauge action] \label{prop:jet-action} For field-independent \(\xi\), define \[ T_\xi(h,\phi_0,\phi_1) =(h+K\xi,\phi_0,\phi_1+X_\xi\phi_0). \] This preserves \eqref{eq:jet-scalar} and \(\Glin(h)=0\), and \[ T_\xi T_\zeta=T_{\xi+\zeta}, \qquad T_\xi^{-1}=T_{-\xi}. \] \end{proposition} \begin{proof} The transformed forced equation has left-hand side \[ \begin{aligned} P(\phi_1+X_\xi\phi_0)+L_{h+K\xi}\phi_0 &=P\phi_1+L_h\phi_0 +PX_\xi\phi_0-[P,X_\xi]\phi_0\\ &=P\phi_1+L_h\phi_0+X_\xi P\phi_0=0. \end{aligned} \] The metric equation is preserved because \(\Glin K=0\). The group identities are immediate. \end{proof} This is the additive action on retained coefficients. Commutators of diffeomorphisms generated by \(\kappa\xi\) and \(\kappa\zeta\) begin at order \(\kappa^2\), which is not retained. \subsection{Past-trivial gauges} Let \[ \GaugeR=\Eplus(T^*M). \] For \(F\in\Eplus\), define \(\Rbox F\) to be the solution of \[ \Boxeta\Rbox F=F \] with zero data on a slice preceding \(\supp F\). The forced Cauchy lemma constructs compatible solutions on all finite intervals, and finite propagation gives \(\Rbox F\in\Eplus\). The prescription is independent of the earlier slice. We have \begin{equation} \Boxeta\Rbox F=F,\qquad \Rbox\Boxeta a=a\quad(a\in\Eplus). \label{eq:retarded-inverse} \end{equation} The second equality follows because the difference is homogeneous with zero past data. Define \[ \MetricR =\{h\in C^\infty(M;S^2T^*M): (\Dop h)^\sharp\in\Eplus(TM)\}, \] and \begin{equation} \wR(h)=\Rbox(\Dop h)^\sharp,\qquad \Pi_{\mathrm R}h=h-K\wR(h). \label{eq:retarded-compensator} \end{equation} In \(K\wR(h)\), lower the index first. \begin{lemma}[Retarded section identities] \label{lem:retarded-section} For \(\xi\in\GaugeR\), \[ \wR(K\xi)=\xi,\qquad \wR(h+K\xi)=\wR(h)+\xi. \] Also, \[ \Dop\Pi_{\mathrm R}h=0,\qquad \wR(\Pi_{\mathrm R}h)=0,\qquad \Pi_{\mathrm R}^2h=\Pi_{\mathrm R}h, \] and \[ \Glin(\Pi_{\mathrm R}h)=\Glin(h). \] \end{lemma} \begin{proof} Apply \(\Dop K\xi=\Boxeta\xi\) and \eqref{eq:retarded-inverse}. Then \[ \Dop\Pi_{\mathrm R}h =\Dop h-\Boxeta\Rbox\Dop h=0. \] The other projector identities follow, and \(\Glin K=0\) proves the final equality. \end{proof} \begin{lemma}[Support-dependent uniqueness] \label{lem:gauge-intersection} A homogeneous vector wave in \(\GaugeR\) is zero. Thus \[ \GaugeR\cap \{\xi:\Boxeta\xi=0,\ \xi\text{ has compact Cauchy data}\} =\{0\}. \] \end{lemma} \begin{proof} A field in \(\GaugeR\) has zero data on a sufficiently early slice. Homogeneous wave uniqueness makes it zero. \end{proof} \begin{figure}[htbp] \centering \begin{tikzpicture} \node[proofbox,text width=5.1cm] (ret) at (0,0) {Retarded gauge class \(\GaugeR\)\\ Future support from a compact set\\ Zero sufficiently early data}; \node[proofbox,text width=5.1cm] (res) at (6.7,0) {C1 residual-wave class\\ \(\Boxeta\xi=0\)\\ Arbitrary compact Cauchy data}; \node[proofbox,text width=5.8cm] (zero) at (3.35,-2.1) {Intersection: \(\xi=0\)\\ By homogeneous Cauchy uniqueness}; \draw[proofarrow] (ret.south)--(zero.north west); \draw[proofarrow] (res.south)--(zero.north east); \end{tikzpicture} \caption{The support requirement is part of the gauge group. The retarded section removes no nonzero C1 residual wave by the identity \(\Rbox\Boxeta\xi=\xi\), because that identity has a different domain.} \label{fig:gauge-support-classes} \end{figure} \subsection{The coupled jet} \begin{definition}[Coupled first-order jet] \label{def:jet} Let \(\Jet\) consist of triples satisfying \[ h\in\MetricR,\qquad \Glin(h)=0,\qquad \phi_0\in\Solphi,\qquad P\phi_1=-L_h\phi_0, \] with \(\phi_1\) smooth and spacelike compact. Let \(\Slice\subset\Jet\) have \(\Dop h=0\). The gauge group is \(\GaugeR\), acting by \(T_\xi\). \end{definition} There is no spacelike-compact requirement on \(h\) in this definition. Its harmonic part may be noncompact. The retarded prescription needs the condition on \(\Dop h\). The scalar forcing remains spacelike compact because its support is contained in that of \(\phi_0\). Set \begin{equation} \mathcal S_{\mathrm R}(h,\phi_0,\phi_1) = \left( \Pi_{\mathrm R}h,\, \phi_0,\, \phi_1-X_{\wR(h)}\phi_0 \right). \label{eq:jet-section} \end{equation} \begin{theorem}[Unique retarded harmonic section] \label{thm:retarded-section} The map \(\mathcal S_{\mathrm R}:\Jet\to\Slice\) obeys: \begin{enumerate}[label=(\roman*)] \item \(\mathcal S_{\mathrm R}x=T_{-\wR(h)}x\); \item \(\mathcal S_{\mathrm R}T_\xi x=\mathcal S_{\mathrm R}x\); \item \(\mathcal S_{\mathrm R}|_{\Slice}=\id\); \item every \(\GaugeR\)-orbit meets \(\Slice\) exactly once. \end{enumerate} It induces a bijection \[ \Jet/\GaugeR\longrightarrow\Slice. \] This is an orbit-space assertion, not an infinite-dimensional quotient-manifold theorem. \end{theorem} \begin{proof} Because \(\wR(h)\in\GaugeR\), part (i) follows from the action. The equations are preserved by Proposition~\ref{prop:jet-action}. The scalar correction remains spacelike compact because its support is contained in that of \(\phi_0\). The retarded section identities supply \(\Dop\Pi_{\mathrm R}h=0\). Under \(T_\xi\), the scalar correction is \[ \phi_1+X_\xi\phi_0-X_{\wR(h)+\xi}\phi_0 =\phi_1-X_{\wR(h)}\phi_0. \] The metric projection is unchanged as well, proving (ii). On the slice, \(\wR(h)=0\), proving (iii). Part (i) gives existence. If \(h\) and \(h+K\xi\) are harmonic with \(\xi\in\GaugeR\), then \(\Boxeta\xi=0\), hence \(\xi=0\) by Lemma~\ref{lem:gauge-intersection}. The scalar correction is consequently unchanged. \end{proof} \begin{figure}[htbp] \centering \begin{tikzpicture} \node[proofbox,text width=4.8cm] (x) at (0,0) {\(x=(h,\phi_0,\phi_1)\)}; \node[proofbox,text width=4.8cm] (tx) at (6.5,0) {\(T_\xi x=(h+K\xi,\phi_0,\phi_1+X_\xi\phi_0)\)}; \node[proofbox,text width=7.1cm] (slice) at (3.25,-2.2) {One harmonic representative\\ \((\Pi_{\mathrm R}h,\phi_0, \phi_1-X_{\wR(h)}\phi_0)\)}; \draw[proofarrow] (x)--node[above,font=\small]{\(T_\xi\)}(tx); \draw[proofarrow] (x.south)--node[left,font=\small] {\(\mathcal S_{\mathrm R}\)}(slice.north west); \draw[proofarrow] (tx.south)--node[right,font=\small] {\(\mathcal S_{\mathrm R}\)}(slice.north east); \end{tikzpicture} \caption{The complete scalar coefficient and the metric representative transform together. The commuting diagram uses the past-trivial retarded group, not the residual-wave group.} \label{fig:retarded-jet-section} \end{figure} \begin{lemma}[Nonempty scalar fibers] \label{lem:fibers} For smooth \(k\) satisfying \(\Dop k=0\), \(\Glin(k)=0\), and \(\phi_0\in\Solphi\), every pair of compact smooth scalar data determines a unique smooth spacelike-compact solution \[ P\psi_1=-L_k\phi_0. \] \end{lemma} \begin{proof} The source is smooth and supported in the cone of \(\phi_0\). The forced Cauchy lemma gives the solution. Propagation from both compact initial data and the source cone gives a cone of slope one with possibly enlarged initial radius. \end{proof} \subsection{The complete quotient functional} For real compact \(f\), define \begin{equation} \Fret_f(h,\phi_0,\phi_1) = \int_M f \left[\phi_0+\kappa\left( \phi_1-X_{\wR(h)}\phi_0 \right)\right]\dd^4x \quad\in\Dual. \label{eq:retarded-functional} \end{equation} \begin{theorem}[Descent on the coupled jet] \label{thm:jet-descent} The functional \(\Fret_f\) descends to \(\Jet/\GaugeR\). If \[ (k,\phi_0,\psi_1)=\mathcal S_{\mathrm R}(h,\phi_0,\phi_1), \] then \[ \Fret_f(h,\phi_0,\phi_1) =\int_M f(\phi_0+\kappa\psi_1)\dd^4x, \qquad P\psi_1=-L_k\phi_0. \] \end{theorem} \begin{proof} Both coefficients are invariant by the section calculation, so the functional factors through the orbit map. On the slice, \(\wR(k)=0\). Writing \(w=\wR(h)\), \[ \begin{aligned} P(\phi_1-X_w\phi_0)+L_{h-Kw}\phi_0 &=P\phi_1-PX_w\phi_0+L_h\phi_0+[P,X_w]\phi_0\\ &=P\phi_1+L_h\phi_0-X_wP\phi_0=0. \end{aligned} \] \end{proof} The \(\phi_1\) term cannot be omitted from the declared coupled equation space. It provides both the forced equation and the transformation needed for cancellation. \begin{proposition}[The reduced correction need not be free] \label{prop:nonfree} There are slice data with \(P\psi_1\ne0\). \end{proposition} \begin{proof} Let \[ k_{11}=\cos(t-x^3),\qquad k_{22}=-\cos(t-x^3), \] and all other components vanish. This tensor is trace free, divergence free, and a componentwise wave, so it is harmonic vacuum data. Use the jet lemma to choose \(\phi_0\) whose value and gradient vanish at the origin, with Hessian \[ \partial_1^2\phi_0=1,\qquad \partial_2^2\phi_0=-1 \] and all other components zero there. Then \(L_k\phi_0(0)=-2\). Every corresponding correction satisfies \(P\psi_1(0)=2\). \end{proof} \subsection{A scalar symplectic consequence with metric fixed} Fix the reduced metric \(k\); do not vary it. Put \[ c_k^{\mu\nu} =\frac12(\tr k)\eta^{\mu\nu}-k^{\mu\nu}, \quad a_k^{\mu\nu}=\eta^{\mu\nu}+\kappa c_k^{\mu\nu}, \quad b_k=1+\frac{\kappa}{2}\tr k, \] and \[ \mathcal P_k A =\partial_\mu(a_k^{\mu\nu}\partial_\nu A)-m^2b_kA. \] Expansion gives \[ \mathcal P_k =P+\kappa\left(L_k+\frac12(\tr k)P\right). \] Thus \(A=A_0+\kappa A_1\) solves \(\mathcal P_kA=0\) exactly when \[ PA_0=0,\qquad PA_1=-L_kA_0. \] \begin{corollary}[Prescribed-metric scalar symplectic form] \label{cor:symplectic} For coefficientwise spacelike-compact solutions, define \[ J_k^\mu(A,B) =a_k^{\mu\nu}(A\partial_\nu B-B\partial_\nu A), \] \[ \sigma_{k,t}(A,B) =-\int_{\{t\}\times\R^3}J_k^0(A,B)\dd^3x. \] The form is independent of \(t\), antisymmetric, \(\Dual\)-bilinear, and weakly nondegenerate. \end{corollary} \begin{proof} Symmetry of \(a_k^{\mu\nu}\) gives \[ \partial_\mu J_k^\mu(A,B) =A\mathcal P_kB-B\mathcal P_kA=0. \] Support is bounded on finite slabs, so integration has no spatial boundary term. At a fixed time set \[ q_A=A|_{t_0},\qquad p_A=-a_k^{0\nu}\partial_\nu A|_{t_0}. \] The leading coefficient of \(a_k^{00}\) is \(-1\), hence it is invertible in \(\Dual\). Arbitrary compact formal \((q_A,p_A)\) determine the coefficient Cauchy data. Solve first for \(A_0\), then for the forced \(A_1\). The pairing is \[ \sigma_k(A,B)=\int(q_Ap_B-q_Bp_A)\dd^3x. \] If it vanishes for every \(B\), take \(q_B=0\) and arbitrary real compact \(p_B\). Both coefficients of \(q_A\) vanish distributionally. Then take \(p_B=0\) and arbitrary real compact \(q_B\), obtaining \(p_A=0\). Cauchy uniqueness gives \(A=0\). \end{proof} This is not a dynamical gravity--matter symplectic reduction. The derivative of a field-dependent section has not been used to construct a full reduced symplectic form. %===================================================================== \section{Compensator choice and classical kernel comparison} \label{sec:choices} %===================================================================== \subsection{Different right inverses need not give the same observable} \begin{proposition}[Difference of compensators] \label{prop:choices} Let \(C,C'\) be linear maps on a common metric domain, valued in smooth vector fields, with \[ CK=C'K=\id \] on a common gauge class. For \(A=C'-C\), one has \(AK=0\), and \[ \Fret_f^{C'}-\Fret_f^C =-\kappa\int_M fX_{Ah}\phi_0\dd^4x. \] The difference is invariant under the common coefficient gauge action, but need not vanish. \end{proposition} \begin{proof} Subtract the right-inverse identities. Under \(h\mapsto h+K\xi\), \(Ah\) is unchanged. Subtract the complete coefficient functionals. \end{proof} An actual section also requires \(C(h)\) to belong to the declared gauge group and its transformed configuration to remain admissible. The algebraic condition \(CK=\id\) alone does not establish either fact. Two right inverses with different boundary, initial-data, or support prescriptions need not define the same observable even when they invert the same formal differential operator. On a common domain, \(-\wP\) has the right-inverse sign of \(\wR\), but may have tails or past behavior incompatible with \(\GaugeR\). This does not identify the two sections. A coupled Poisson coefficient would be \[ \phi_1+X_{\wP(h)}\phi_0. \] The unchanged-product expression tested earlier is different because it does not carry these complete coefficient data. \subsection{Poisson and Coulomb kernels} At fixed time let \[ \mathbf d=\by-\bx,\qquad d=|\mathbf d|, \qquad \widehat{\mathbf d}=\mathbf d/d. \] For spatial Schwartz fields define the Coulomb kernels and their deformations as in the version-specific formulas of Donnelly and Giddings~\cite[Eq.~(52), Eqs.~(73)--(74)]{DG}: \begin{align} V_C^0(x) &=-\frac{\kappa}{4\pi}\int_{\R^3} \left( \frac{\widehat d^i\widehat d^j\dot h_{ij}}{2d} +\frac{\widehat d^ih_{0i}}{d^2} \right)\dd^3y, \label{eq:VC0}\\ V_C^i(x) &=\frac{3\kappa}{8\pi}\int_{\R^3} \frac{\widehat d^i\widehat d^j\widehat d^k}{d^2} h_{jk}\dd^3y, \label{eq:VCi}\\ D_C^0(x) &=\frac{\kappa}{8\pi}\int_{\R^3} \frac{\widehat d^i\widehat d^j+\delta^{ij}}d \dot h_{ij}\dd^3y, \label{eq:DC0}\\ D_C^i(x) &=\frac{\kappa}{8\pi}\int_{\R^3} \frac{ 3\widehat d^i\widehat d^j\widehat d^k +\widehat d^i\delta^{jk} -2\delta^{ij}\widehat d^k}{d^2} h_{jk}\dd^3y. \label{eq:DCi} \end{align} Fields in the integrals are evaluated at \((t,\by)\). Our active convention is \(\delta h=K\xi\). The gauge parameter in the corresponding convention of \cite{DG} has the opposite sign. The displayed kernel definitions themselves are unchanged. \begin{proposition}[Classical kernel identity] \label{prop:DG} For \(V_P^\mu=\kappa\wP^\mu\), \[ V_P^0=V_C^0+D_C^0,\qquad V_P^i=V_C^i-D_C^i. \] \end{proposition} \begin{proof} Integrating by parts in the Poisson formula gives \[ V_P^0 =-\frac{\kappa}{4\pi}\int \left( \frac{\widehat d^ih_{0i}}{d^2} -\frac{\dot h_{ii}}{2d} \right)\dd^3y, \] \[ V_P^i =\frac{\kappa}{8\pi}\int \frac{2\widehat d^jh_{ij}-\widehat d^ih_{jj}}{d^2} \dd^3y. \] The kernels are locally integrable. Boundary terms from a shrinking inner sphere vanish; Schwartz decay removes the outer terms. Adding and subtracting the displayed Coulomb deformation kernels gives exactly these expressions. \end{proof} The deformations annihilate compact pure gauges. Their distributional divergence identities, including the shrinking-sphere flux checks, are proved in Appendix~\ref{app:audit}. This comparison identifies written kernels on a stated classical domain. It identifies neither retarded and instantaneous prescriptions nor all gauge-invariant additions. %===================================================================== \section{The C1 triangular Cauchy quotient} \label{sec:C1} %===================================================================== \smallbox{The C1 background is prescribed}{% The metric background is \(\eta\), and \(\Phi\) is a fixed free scalar. The metric perturbation is vacuum linearized, while the scalar perturbation is forced. This is not the fully coupled Einstein--scalar linearization about an on-shell nonlinear background. No stress source is inserted into the metric equation.} \subsection{A fixed compact-data background} Take \(A_\star>0\), \(R>0\), and real \(\chi\in C_c^\infty(\R^3)\) with \(\chi=1\) on \(|\bx|\le R\). Let \(\tau=t-t_0\) and define \begin{equation} P\Phi=0,\qquad \Phi(t_0,\bx)=A_\star\chi(\bx),\qquad \partial_t\Phi(t_0,\bx)=0. \label{eq:background} \end{equation} Then \[ \Phi(t)=A_\star\cos(\tau\Om)\chi, \qquad \Om=\sqrt{-\Delta+m^2}. \] Finite propagation and local uniqueness give \begin{equation} \Phi(t,\bx)=A_\star\cos(m\tau) \quad\text{if }|\bx|+|\tau|0. } \label{eq:positive-observable} \end{equation} Moreover, \[ c\notin PC_c^\infty(M). \] \end{theorem} \begin{proof} On the coefficient support, \[ \partial_i\Phi=0,\qquad \partial^0\Phi=A_\star m\sin(m\tau). \] For \(\nu=0\), the Ward expression is \[ -2a'b+ \frac{2a'b}{A_\star m\sin(m\tau)} A_\star m\sin(m\tau)=0. \] For spatial \(\nu\), both terms vanish. The identity also holds outside the compact support. The test solution has \(h=0\) and a free scalar with compact data. Its value is \[ \begin{aligned} \int_Mc\frac{\Phi}{A_\star}\dd^4x &=\frac{2}{A_\star m} \left(\int a'(\tau)\cot(m\tau)\dd\tau\right) \left(\int b\dd^3x\right)\\ &=\frac{2}{A_\star} \left(\int a(\tau)\csc^2(m\tau)\dd\tau\right) \left(\int b\dd^3x\right). \end{aligned} \] Integration by parts has no endpoint term. Both factors are positive. If \(c=P\psi\) for compact smooth \(\psi\), then \[ \int_Mc\frac{\Phi}{A_\star}\dd^4x =\int_M\psi P\left(\frac{\Phi}{A_\star}\right)\dd^4x=0, \] contradicting positivity. \end{proof} The pair is genuinely coupled: \[ \partial_\mu J^{\mu0}=a'(\tau)b(\bx)\not\equiv0. \] Neither term is separately invariant under unrestricted compact gauge variations. Its scalar coefficient cannot lie in the adjoint-exact family. The test solution has nonzero homogeneous scalar data. It is not in a prescription setting \(\varphi\)'s initial data to zero; that prescription is not preserved by all residual gauges. \begin{figure}[htbp] \centering \begin{tikzpicture} \node[proofbox,text width=5.0cm] (exact) at (0,0) {Adjoint-exact family\\ \(c=P\psi,\ J=J_\psi\)\\ \(\Ocal_{J,c}=0\) on shell}; \node[proofbox,text width=5.0cm] (nonzero) at (6.6,0) {Displayed compact coupled pair\\ \(c\notin PC_c^\infty(M)\)\\ Positive on an explicit solution}; \node[proofbox,text width=7.4cm] (ward) at (3.3,-2.1) {Both satisfy the same coupled Ward identity\\ Therefore compactness and Ward invariance alone do not imply on-shell vanishing}; \draw[proofarrow] (exact.south)--(ward.north west); \draw[proofarrow] (nonzero.south)--(ward.north east); \end{tikzpicture} \caption{The zero repair theorem is a statement about an exact family, not a classification of compact coupled observables.} \label{fig:compact-Ward} \end{figure} \subsection{Homogeneous data belong to the quotient tuple} Every C1 solution decomposes uniquely as \[ \varphi=S_{t_0}(h)\Phi+\mathcal H_{t_0}(u,v). \] Under a residual gauge, \[ \begin{aligned} \varphi+X_\xi\Phi-S_{t_0}(h+K\xi)\Phi = \varphi-S_{t_0}(h)\Phi+H_{t_0}(X_\xi\Phi). \end{aligned} \] The homogeneous scalar data change by \eqref{eq:scalar-gauge-data}. They therefore belong in the full quotient datum. The conclusion is existence: \[ \text{compact coupled Ward pair} \ \not\Longrightarrow\ \text{zero on shell}. \] No complete compact-current classification is claimed. %===================================================================== \section{What instantaneous Poisson compensation establishes} \label{sec:Poisson-scope} %===================================================================== \subsection{Formal cancellation is not unchanged-product descent} Consider the active formal variation \[ \delta_\xi h=K\xi+\kappa\Lie_\xi h,\qquad \delta_\xi\phi=\kappa X_\xi\phi. \] On smooth configurations in the compensator domain, \[ \delta_\xi\Fpois_\kappa=O(\kappa^2) \] formally, because \[ \delta_\xi\int f\phi=\kappa\int fX_\xi\phi \] and the leading variation of the correction is \[ \kappa\int fX_{\wP(K\xi)}\phi =-\kappa\int fX_\xi\phi. \] This varies the scalar. The map \(q_0\) does not. The varied scalar also need not remain free. Neither negative theorem is contradicted. In coefficient notation, the cancellation is \[ \delta_0F_1+\delta_1F_0=0, \] not \(\delta_0F_1=0\). \subsection{Canonical temporal normalization} Write \[ \gamma_{ij}=\delta_{ij}+\kappa h_{ij},\qquad \beta_i=\kappa h_{0i}+O(\kappa^2),\qquad N_{\mathrm{lapse}}=1+O(\kappa), \] and choose \[ K_{ij}^{\mathrm{ADM}} =\frac1{2N_{\mathrm{lapse}}} (\dot\gamma_{ij}-D_i\beta_j-D_j\beta_i). \] With \[ v_{ij}=\dot h_{ij}-\partial_i h_{0j}-\partial_j h_{0i}, \] declare the quadratic kinetic density \[ \mathcal L_{\mathrm{kin}}^{(2)} =\frac12\left(v_{ij}v_{ij}-(v_{ii})^2\right). \] This follows from the ADM kinetic term with Einstein--Hilbert coefficient \(2/\kappa^2\). \begin{proposition}[Temporal dressing in canonical variables] \label{prop:ADM} For this normalization, \[ \pi^{ij}=v_{ij}-\delta^{ij}v_{kk}, \qquad \Pi_{\mathrm{tr}}:=\pi^{ii} =-2\dot h_{ii}+4\partial_i h_{0i}. \] Thus \[ \Pi_{\mathrm{tr}}=4B_{\mathrm P}[h], \qquad \wP^0=\frac14N\Pi_{\mathrm{tr}}. \] \end{proposition} \begin{proof} Differentiate the kinetic density with the symmetric-tensor convention, and take its spatial trace. Then \(\pi^{ii}=-2v_{ii}\). Use the definition of \(B_{\mathrm P}\). \end{proof} The momentum agrees with the displayed ADM convention in \cite[Eq.~(70)]{DG}. It is not the momentum of an unspecified alternative gauge-fixed Lagrangian. \subsection{Leading instantaneous slice inversion} Use \[ \{h_{ij}(\bx),\pi^{kl}(\by)\} =\delta_i^{(k}\delta_j^{l)}\delta^3(\bx-\by), \] and \[ \mathcal R_{\mathrm{lin}}[h] =\partial_i\partial_jh_{ij}-\Delta h_{ii}. \] The generator \[ G_0[\xi] =\int_{\R^3} \left( -2\xi^0\mathcal R_{\mathrm{lin}}[h] -2\xi^i\partial_j\pi^{ij} \right)\dd^3x \] gives \[ \delta h_{ij}=\partial_i\xi_j+\partial_j\xi_i, \] \[ \delta\pi^{ij} =2(\partial_i\partial_j-\delta_{ij}\Delta)\xi^0. \] For \(\chi_0=\Pi_{\mathrm{tr}}\), \(\chi_i=A_i\), \[ \delta\chi=\mathcal M\xi,\qquad \mathcal M= \begin{pmatrix} -4\Delta&0\\ 0&\Delta\delta_{ij} \end{pmatrix}. \] \begin{proposition}[Decaying leading inverse] \label{prop:instantaneous-inverse} Let \(\mathscr X=N\Sch(\R^3)\). For Schwartz \(\Pi_{\mathrm{tr}},A_i\), the unique descriptor in \(\mathscr X^4\) sending the leading conditions to zero is \[ \xi^0=\frac14N\Pi_{\mathrm{tr}},\qquad \xi_i=-NA_i. \] Equivalently, \[ \mathcal M^{-1} =\begin{pmatrix} -\frac14N&0\\ 0&N\delta_{ij} \end{pmatrix}. \] \end{proposition} \begin{proof} The required descriptor is \(-\mathcal M^{-1}\chi\). Newton inversion gives existence. The difference of two descriptors is smooth harmonic and decays, hence vanishes. \end{proof} This decaying class differs from the compact gauge class. The inversion does not prove preservation of the slice by evolution. The conditions are not mutually commuting: \[ \{\Pi_{\mathrm{tr}}(\bx),A_i(\by)\} =\frac12\partial_{y^i}\delta^3(\bx-\by). \] A full Dirac-bracket calculation needs more than \(\mathcal M\). \subsection{No exact compact linear compensator} \begin{theorem}[Compact-kernel obstruction] \label{thm:no-compact} For fixed \(x\), no linear distributional functional \(w_x^\mu[h]\) of compact spacetime support satisfies \[ w_x^\mu[K\xi]=-\xi^\mu(x) \] for every compact smooth \(\xi\). The same holds for a compact spatially supported time-slice functional using only finitely many time derivatives of \(h\). \end{theorem} \begin{proof} Choose a compact set containing the distributional support and \(x\). Take a compact vector field constant on its neighborhood, with a nonzero \(\mu\)-component. Then \(K\xi\) vanishes near the support, so its pairing is zero, while the required value is nonzero. For a time-slice functional, make the field constant in time as well near the relevant support. All finitely many tested derivatives vanish there. \end{proof} This does not exclude compact coupled Ward pairs, which use different variables and a different cancellation. \subsection{A separate source-support fact} \begin{proposition}[Nonzero-total-source tails] \label{prop:source-support} If \(s\in C_c^\infty(\R^3)\) has \(M_s=\int s\ne0\), no smooth compact symmetric tensor solves \[ \mathcal R_{\mathrm{lin}}[h]=s. \] A decaying solution is \[ h_{ij}=\delta_{ij}H,\qquad H=-\frac12Ns, \] with \[ H(\bx)=\frac{M_s}{8\pi|\bx|}+O(|\bx|^{-2}). \] The expression \(\mathcal R_{\mathrm{lin}}\) vanishes on spatial TT tensors and is unchanged by spatial pure gauges. \end{proposition} \begin{proof} Integrating second derivatives of compact \(h\) gives zero. For \(h_{ij}=\delta_{ij}H\), \(\mathcal R_{\mathrm{lin}}[h]=-2\Delta H=s\). The tail is the leading Newton multipole. The TT statement is immediate. For a spatial gauge addition, \[ \partial_i\partial_j(\partial_i\xi_j+\partial_j\xi_i) -\Delta(2\partial_i\xi_i)=0. \] \end{proof} This is a prescribed elliptic source problem, not either quotient constructed above. It explains why compact-support assumptions cannot be transferred to a sourced metric problem by analogy. %===================================================================== \section{Constructions that do not discharge the sourced successor} \label{sec:nontransfer} %===================================================================== A later construction does not change the domain or equivalence relation of an earlier theorem. The four opening rows remain different questions. \subsection{Three successor objects} Before posing a sourced problem, specify an action and source normalization, the coefficient equation space, its gauge group, support classes, and initial or boundary conditions. An evaluation set is not a quotient unless an action preserving that set and the resulting equivalence relation have been supplied. For definiteness, one possible future gauge specification is the initial-surface-fixing subgroup \[ \mathscr G_{+,t_0} = \bigcup_{K_0\Subset\{t>t_0\}} \{\xi\in C^\infty(M;T^*M): \supp\xi\subset J_\eta^+(K_0)\}. \] It is contained in \(\GaugeR\), and its fields vanish on a neighborhood of the initial surface. A formal second-order prescription could take both coefficient descriptors in this group. This is not the C1 residual-wave group. For a sourced successor, allow smooth metric coefficients unless a separate asymptotic class is proved appropriate. Do not impose compact metric data merely because the scalar data are compact. Scalar coefficients may be required to be spacelike compact. Invariant initial metric data must satisfy their sourced constraints; arbitrary zero metric data are not automatically admissible. \begin{definition}[Three successor objects] \label{prop:three-successor} For a selected sourced Einstein--scalar model, distinguish: \begin{enumerate}[label=(\roman*)] \item \textbf{Invariant coupled jet.} The coefficients solve the expanded invariant equations, including the stress compatibility and Cauchy constraints implied by the chosen action and normalization. The allowed gauge transformations must preserve that equation space. \item \textbf{Gauge-fixed forced response.} A named reduced operator is inverted with a declared gauge-fixing and support or initial-data prescription. For example, one may prescribe de Donder-reduced wave equations and zero response data at \(t_0\). Such a reduced response must be checked separately against the invariant constraints. \item \textbf{Reduced functional.} A complete coefficient functional is shown to be constant on a named quotient of (i), or on a separately justified quotient of (ii). Its correction includes both the forced scalar coefficient and the coordinate compensator. \end{enumerate} The full sourced successor comprising these objects is not constructed in this paper. Existence of (ii) alone does not imply existence of (i), and it does not establish descent in (iii). \end{definition} A gauge transformation is not itself a right inverse. A plateau switch, a ramp, or a different initial-velocity prescription can define a different response problem or inverse operator. Its relation to the retarded section must be proved, not inferred from terminology. \begin{proposition}[A reduced wave inverse need not solve invariant equations] \label{prop:forced-not-invariant} There are smooth compact symmetric sources \(Q_{\mu\nu}\) for which the past-zero reduced equation \[ -\frac12\Boxeta(\Rev j)_{\mu\nu}=Q_{\mu\nu} \] has a smooth retarded solution, while \[ \Glin(j)=Q \] has no smooth solution. This example is not an Einstein--scalar source model. \end{proposition} \begin{proof} Choose compact smooth \(s\) with \(\partial_ts\not\equiv0\), and let \(Q_{00}=s\), all other components zero. Then \(\partial^\mu Q_{\mu0}=-\partial_ts\not\equiv0\). The reduced equation has solution \[ j=-2\Rev\Rbox Q \] by the retarded wave construction and \(\Rev^2=\id\). If the invariant equation had a solution, the linearized Bianchi identity would imply \(\partial^\mu Q_{\mu\nu}=0\), a contradiction. \end{proof} The proposition establishes the logical nonimplication. For a conserved source and compatible data, a reduced solution may also solve the invariant equations; that possibility is not excluded. \subsection{Higher coefficients do not alter lower-order obstructions} \begin{proposition}[Persistence on the unchanged identification] \label{prop:lower-order-persistence} Keep the map \(q_0\) and the fixed witness of Theorem~\ref{thm:product}. Adding regular formal terms beginning at order \(\kappa^2\) cannot make \[ \Fpois_0+\kappa\Fpois_1+\sum_{r\ge2}\kappa^rF_r \] constant on that witness fiber. Likewise, adding terms beginning at order \(\kappa^2\) to \(\phi+\kappa X_\xi\phi\) cannot remove the nonzero order-\(\kappa\) free-shell defect of Proposition~\ref{prop:tangency-witness}. \end{proposition} \begin{proof} The first difference has coefficient \(-J\ne0\) at order \(\kappa\). The second equation defect has coefficient \(P(X_\xi\phi)\ne0\). Formal equality with nonnegative powers requires each coefficient to vanish separately. \end{proof} Retaining \(P\phi_1=-L_h\phi_0\) changes the equation space and identification. It does not cancel the earlier theorems. A second metric coefficient likewise does not make the old Poisson symbol constant on \(q_0\), or make a general compact Lie derivative tangent to the unchanged massive shell. \subsection{Scoped companion comparisons} The following citations refer to source records, not new inputs. Where only a summary or excerpt is available, no independent verification of the complete companion proof is claimed. \paragraph{Certificate strength and selected representations.} Landmark II's finite certificate has a different object and domain from a quotient identity~\cite[statement \SourceLabel{thm:certificate}]{CheungII}. A finite fit is not automatically a global functional identity. No catalogue entry or target coefficient from II is a counterterm in this paper. Landmark IV describes selected represented models, not a representation of every observable on the present quotients~\cite[statement \SourceLabel{thm:model1}]{CheungIV}. Gauge invariance is not prescription independence. Its representation parameter is not identified with the formal nilpotent \(\kappa\). \paragraph{Fixed-background currents, response, and higher jets.} Landmark V uses the same sign identity and fixed-time response mechanism, but studies its own fixed-background metric functionals, compact representatives, and initial-surface terms \cite[statements \SourceLabel{lem:gaugeidentity}, \SourceLabel{thm:defect}, and \SourceLabel{thm:rawclassification}]{CheungV}. It is not a replacement for the coupled quotient. The nonzero Ward pair here is neither the vanished exact repair nor a residual witness imported from V. Landmark VI continues a coupled-coordinate construction to higher coefficients on a finite-time formal domain \cite[statements \SourceLabel{thm:descent} and \SourceLabel{prop:gauge-action}]{CheungVI}. Its transformations supported away from the initial surface are not identified with the whole group \(\GaugeR\). Neither result is a premise of the other, and neither constructs the sourced successor posed here. VII-A's plateau-supported intertwining and larger-support defect belong to its switched model \cite[statement \SourceLabel{thm:plateau} and corollary \SourceLabel{cor:nohigherrepair}]{CheungVIIA}. The lower-order persistence principle is shared; the defect is not. That defect is not a failure of Theorem~\ref{thm:retarded-section}. X-A's coordinate intertwiner retains equations, data, and switches \cite[statement \SourceLabel{thm:coordinate}]{CheungXA}. Its complete first relational coefficient includes the switched scalar solution minus the coordinate dressing. The dressing alone is not a forced scalar correction, and X-A is not a repair of the Fourier regulator in this paper. \paragraph{Invariant compatibility versus gauge-fixed solvability.} XI-A records an empty invariant jet for a specified input and switched model, while XI-B explicitly retains gauge-fixed constructions without repairing that jet~\cite{CheungXIA,CheungXIB}. XII-C recomputes compatibility for a different selected action; XII-D constructs its corresponding forced response \cite[statement \SourceLabel{thm:empty}]{CheungXIIC}% \cite[statement \SourceLabel{thm:response}]{CheungXIID}. These records illustrate the distinction in Definition~\ref{prop:three-successor}. Their switches, actions, inputs, and stress conditions are not those of an unspecified unswitched Einstein--scalar successor. They prove no emptiness theorem for the proposed sourced problem of Section~\ref{sec:conclusion} and repair none of the current obstructions. \paragraph{Shell tangency and relative evaluation.} XIII-D's causal zero-past variations preserve a separately named homogeneous free shell, forcing its induced variations to vanish \cite[statement \SourceLabel{thm:triviality}]{CheungXIIID}. This is a comparison with the principle ``declare the shell first'', not the massive Killing classification above. XIV-A's relative master-equation identity and XIV-B's free-shell ghost intersection concern their own relative spaces \cite[statement \SourceLabel{thm:relative}]{CheungXIVA}% \cite[statement \SourceLabel{thm:ghost}]{CheungXIVB}. An evaluation identity does not itself construct a quotient. Their relative ghost conditions are not the gauge group of \(\Jet\), and no embedding of their relative differential into the current coupled jet is supplied here. \paragraph{Other backgrounds and observational certificates.} Em02-E constructs a classical relational lapse on a prescribed power-law Einstein--scalar background through second order \cite[statements \SourceLabel{thm:lapse} and \SourceLabel{thm:release}]{CheungEm02E}. It is not the Minkowski jet used here, and its incomplete quantum audit is not a theorem of III. En01-B's sphere contrast and En01-C's statistical certificate concern their stated diagnostic and record models \cite[statement \SourceLabel{thm:contrast}]{CheungEn01B}% \cite[statement \SourceLabel{thm:certificate}]{CheungEn01C}. Neither derives the present quotient or Einstein dynamics. No claim that the entire I, Em, En, Psy, Hawk, or XVI series has been exhaustively inspected is made. No descended scalar on the four present domains is imported from those tracks. \subsection{Separate analytic stops} The supplied XIV-H version records its internal-amplitude stop at H2~\cite[statement \SourceLabel{thm:H2}]{CheungXIVH}. The subsequent XIV-I version supplies a named I2 internal module, but leaves the joint characteristic map unresolved at I3 \cite[statements \SourceLabel{thm:I2} and \SourceLabel{thm:I3}]{CheungXIVI}. XIV-J supplies a partial supported-sector reference on its off-characteristic domain and leaves the full map unresolved at J2 \cite[statements \SourceLabel{thm:J2}, \SourceLabel{thm:release}, and \SourceLabel{cor:noadmission}]{CheungXIVJ}. These are version-specific documentary statements. An internal reconstruction, a presentation-kernel descent condition, and field-equation quotient descent are different constructions. None supplies an interacting quantum coefficient here. This manuscript closes neither O.1 nor the external characteristic programme, admits no quantum coefficient, opens no XV-B admission exit, and claims no compilation. In particular, the stated O.1 status \SourceLabel{NOT_CLOSED} is not changed. A missing construction is not a nonexistence theorem. %===================================================================== \section{Conclusion and the sourced successor} \label{sec:conclusion} %===================================================================== The proved results concern their actual domains: \begin{enumerate} \item The exhibited Poisson symbol is not constant on unchanged free-product fibers, with a regulator-stable gap. \item General compact scalar Lie derivatives are not tangent to the unchanged massive free shell. The preserving vector fields are exactly Killing fields. \item The forced scalar coefficient changes the equation space and gives a genuine coefficient action. A past-trivial retarded harmonic section makes the complete first-order functional descend. \item The C1 constrained harmonic data modulo residual waves have a Hausdorff, well-posed Cauchy quotient. The explicit compact coupled Ward pair is nonzero. \item Poisson compensation, canonical slice inversion, and compensator choice remain separate from descent. \end{enumerate} None requires a represented observable or measurement construction. \subsection{An unswitched sourced model as a proposed next problem} For a concrete source convention, take the minimally coupled scalar action together with Einstein--Hilbert coefficient \(2/\kappa^2\). Write \[ T_{\mu\nu}(g,\phi)=\frac14T^{\mathrm{std}}_{\mu\nu}(g,\phi), \] where \[ T^{\mathrm{std}}_{\mu\nu}(g,\phi) = \partial_\mu\phi\,\partial_\nu\phi -\frac12g_{\mu\nu} \left(g^{\alpha\beta}\partial_\alpha\phi\partial_\beta\phi +m^2\phi^2\right). \] Then the Einstein equation is \[ G(g)=\kappa^2T(g,\phi). \] This normalization is compatible with the ADM subsection. It has no role in the preceding quotient proofs. Set \[ g=\eta+\kappa h+\kappa^2j,\qquad \phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2. \] The invariant order-\(\kappa^2\) metric equation is \begin{equation} \Glin(j)+G^{(2)}[h,h]=T(\eta,\phi_0), \label{eq:sourced-successor} \end{equation} where \(G^{(2)}[h,h]\) is defined as the coefficient of \(\kappa^2\) in \(G(\eta+\kappa h)\). The complete successor also includes the scalar equation through that order, its gauge action, and its constraints. By the linearized Bianchi identity, a necessary compatibility condition for \eqref{eq:sourced-successor} is \[ \partial^\mu \left(T(\eta,\phi_0)-G^{(2)}[h,h]\right)_{\mu\nu}=0. \] This must be checked for the actual model. For the unswitched minimal scalar, \[ \partial^\mu T_{\mu\nu}(\eta,\phi_0) =\frac14(P\phi_0)\partial_\nu\phi_0=0. \] This elementary conservation fact is not a sourced Cauchy-existence, constraint-solving, or quotient theorem. Switching the action or source changes the compatibility audit. \smallbox{Three sourced successor problems}{% Use a named gauge group and support class, for example the initial-surface-fixing specification \(\mathscr G_{+,t_0}\) of Section~\ref{sec:nontransfer}, rather than silently importing the C1 residual group. \medskip \textbf{S1. Invariant coupled jet.} Define and solve the full Einstein--scalar coefficient equations, including stress compatibility and sourced Cauchy constraints. Specify permitted metric asymptotics. The coupled first-order scalar result alone does not solve S1. \medskip \textbf{S2. Gauge-fixed response.} Choose the reduced operator, gauge-fixing, and initial-data or retarded prescription. Construct its forced responses and check which satisfy the invariant constraints. Existence of a reduced inverse alone does not solve S1. \medskip \textbf{S3. Quotient functional.} On a named quotient of S1, or a separately justified quotient associated with S2, construct the complete reduced functional or prove a new descent obstruction. A coordinate dressing or one forced response alone does not establish S3.} A later result must state its object, equation space, equivalence relation, gauge class, and inverse prescription. A higher coefficient, a different background, a finite certificate, or a measurement interface cannot substitute for those declarations. %===================================================================== \appendix \clearpage \section{Sign, inverse, boundary, and kernel audit} \label{app:audit} %===================================================================== \subsection{Five distinct maps} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.21\linewidth}p{.31\linewidth}X@{}} \toprule Map & Identity used & Domain or prescription\\ \midrule Newton \(N\) & \(\Delta Ns=s,\ N\Delta u=u\) & Compact or Schwartz sources; decaying inverse branch.\\ \addlinespace Finite-radius \(N_R\) & \(N_R\Delta u=u+\mathcal B_R[u]\) & Ball integral; explicit boundary traces.\\ \addlinespace Fourier \(N_\rho\) & \(N_\rho\Delta=\chi_\rho(D)\) & Smooth infrared/ultraviolet multiplier; not an exact inverse.\\ \addlinespace Retarded \(\Rbox\) & \(\Boxeta\Rbox F=F,\ \Rbox\Boxeta a=a\) & Future support from a compact set.\\ \addlinespace Canonical \(\mathcal M^{-1}\) & Inverse of \(\diag(-4\Delta,\Delta I)\) & Leading phase-space gauge conditions; decaying descriptors.\\ \bottomrule \end{tabularx} \end{center} The canonical inverse uses \(N\), but is a map between different phase-space quantities with a different normalization. None of the five maps is identified merely because their formulas contain an inverse Laplacian or wave operator. The compensation signs are \[ \wP^i[K\xi]=-\xi^i,\qquad \wP^0[K\xi]=\xi_0=-\xi^0, \] whereas \[ \wR(K\xi)=\Rbox\Boxeta\xi=\xi \] on \(\GaugeR\). A homogeneous wave is zero in the past-trivial class but need not be zero in the compact-Cauchy-data class. This is why the retarded section proof does not transplant to C1. \subsection{Exact finite-radius boundary error} For \(\bx\in\mathbb B_R\), put \[ G(\bx,\by)=-\frac1{4\pi|\bx-\by|}, \qquad N_Rf(\bx)=\int_{\mathbb B_R}G(\bx,\by)f(\by)\dd^3y, \] and \[ \mathcal B_R[u](\bx) =\int_{\partial\mathbb B_R} \left(G\,\partial_nu-u\,\partial_nG\right)\dd S. \] \begin{proposition}[Boundary error] \label{prop:boundary-error} For smooth \(u\), \[ N_R\Delta u=u+\mathcal B_R[u]. \] Replacing \(N\) by \(N_R\) in the Poisson compensator gives \[ \delta_0\wP_R^\mu=-\xi^\mu-\mathcal B_R[\xi^\mu]. \] Thus formally \[ \delta_\xi\left( \phi+\kappa\wP_R^\mu\partial_\mu\phi \right) = -\kappa\mathcal B_R[\xi^\mu]\partial_\mu\phi +O(\kappa^2). \] \end{proposition} \begin{proof} Green's second identity and \(\Delta_yG(\bx,\by)=\delta_{\bx}(\by)\) give \[ \int_{\mathbb B_R}G\Delta u-u(\bx)=\mathcal B_R[u]. \] Apply this to the variations of \(A_i,B_{\mathrm P}\), with \(\xi_0=-\xi^0\). The scalar leading variation cancels the \(-\xi^\mu\) term. \end{proof} For a constant \(u\), the left-hand side is zero, so \(\mathcal B_R[u]=-u\). Truncating the spatial integral does not preserve exact compensation without a boundary check. For gauges supported strictly inside the ball, both boundary traces vanish. That special case does not justify dropping them for a larger class. \subsection{Fourier compensation error} The regulated inverse obeys \[ N_\rho\Delta=\chi_\rho(D), \] so \[ \delta_0\wP_\rho^\mu=-\chi_\rho(D)\xi^\mu \] and \[ \delta_\xi\left( \phi+\kappa\wP_\rho^\mu\partial_\mu\phi \right) = \kappa((1-\chi_\rho(D))\xi^\mu)\partial_\mu\phi +O(\kappa^2). \] The multiplier need not be a projection. Its error is not the finite-radius boundary error. For fixed compact smooth test data, this formal compensation error tends to zero. It does not repair the unchanged-product gap, where the scalar is held fixed and the limiting difference is \(-\kappa J\). \subsection{Distributional kernel checks} For the temporal deformation, let \[ T_{ij}(\mathbf d) =\frac{d_id_j}{d^3}+\frac{\delta_{ij}}d. \] Away from zero, \[ \partial_i\left(\frac{d_id_j}{d^3}\right) =\frac{d_j}{d^3}, \qquad \partial_i\left(\frac{\delta_{ij}}d\right) =-\frac{d_j}{d^3}. \] The shrinking-sphere flux of this degree-\(-1\) kernel tends to zero. Thus \(\partial_iT_{ij}=0\) distributionally. Only the \(j,k\)-symmetric part of the spatial kernel contributes: \[ S_{ijk}(\mathbf d) = \frac{3d_id_jd_k}{d^5} +\frac{d_i\delta_{jk}}{d^3} -\frac{\delta_{ij}d_k+\delta_{ik}d_j}{d^3}. \] It satisfies \(\partial_jS_{ijk}=0\) away from zero. Its flux is \[ \begin{aligned} \int_{|\mathbf d|=\varepsilon}S_{ijk}n_j\dd S &=\int_{\mathbb S^2} (3n_in_k-\delta_{ik})\dd\Omega\\ &=3\frac{4\pi}{3}\delta_{ik}-4\pi\delta_{ik}=0. \end{aligned} \] Replacing a smooth test on the sphere by its value at zero leaves an error tending to zero. There is no delta contribution, and the divergence identity holds distributionally. Integration by parts against compact pure gauges therefore gives \[ D_C^0[K\xi]=0,\qquad D_C^i[K\xi]=0. \] Together with the kernel comparison and Poisson compensation, this also checks the Coulomb transformation sign in our convention. The general kernel comparison requires spatial Schwartz behavior. No boundary identity for arbitrary \(1/r\) metric tails is asserted. \subsection{Reconstruction is not gauge admission} In the closed-range proof, \(\mathfrak R(A,B)\) is defined for all compact data. It is not a gauge-fixing prescription on every datum. The equality \[ x=\mathfrak g(\mathfrak R(A,B)) \] forces the Newton tails to disappear. The canonical inverse instead explicitly permits decaying descriptors in \(N\Sch\). It does not prove compactness. Neither is the retarded spacetime inverse. \subsection{Fixed parameters and field-dependent sections} The action \(T_\xi\) is first proved for a field-independent parameter. At each configuration, the section evaluates that action at \(\xi=-\wR(h)\). This proves an orbit identity. Differentiating the section with respect to fields is another operation and introduces derivatives of \(\wR\). Those terms cannot be discarded in a dynamical symplectic reduction. The scalar symplectic corollary fixes \(k\) and does not perform that reduction. \subsection{Compact adjoint repair and initial surfaces} Both identities \[ \int hJ_\psi=\int\psi L_h\Phi, \qquad \int(P\psi)\varphi=\int\psi P\varphi \] use compact smooth spacetime \(\psi\). Replacing \(\psi\) by a time-truncated response kernel can introduce initial-surface distributions and changes the argument. The zero-repair theorem does not make that replacement. The positive Ward pair avoids it differently: its coefficient is not \(P\psi\) for any compact smooth \(\psi\), as shown by its positive pairing with a free homogeneous solution. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Nature of the work.} This is a theoretical study of classical differential equations, gauge actions, and quotient functionals. No empirical data are used. \paragraph{Materials and provenance.} The definitions and proofs required for the stated classical results are included. Standard Cauchy theory and the provenance of the compared Coulomb kernels are cited. Companion manuscripts are identified by registry identifier, file, version, and supplied statement locators. No publication date, DOI, or journal status is invented for them. \paragraph{Relation to Landmark III.} This manuscript is a revised self-contained presentation of the supplied Landmark III. The companion-comparison section adds equation-space and source discipline, not premises or sourced successor theorems. \paragraph{Verification status.} No machine-checked proof, successful compilation, or verified page count is claimed. The figures are newly drawn explanations of the displayed arguments; they are not claimed to reproduce an unavailable registered layout. No physical representation, quantum product, or measurement result is included in the theorem family. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle. \newblock \emph{Wave Equations on Lorentzian Manifolds and Quantization}. \newblock ESI Lectures in Mathematics and Physics. European Mathematical Society, 2007. \newblock doi:\ \doi{10.4171/037}. \newblock arXiv:\ \nolinkurl{0806.1036}. \bibitem{DG} W.~Donnelly and S.~B.~Giddings. \newblock Diffeomorphism-invariant observables and their nonlocal algebra. \newblock \emph{Physical Review D} \textbf{93}, 024030 (2016); erratum, \textbf{94}, 029903 (2016). \newblock doi:\ \doi{10.1103/PhysRevD.93.024030}. \newblock Version-specific formulas: arXiv:\ \nolinkurl{1507.07921v3}, Eqs.~(52), (70), (73), and (74). \bibitem{BFR} R.~Brunetti, K.~Fredenhagen, and K.~Rejzner. \newblock Quantum gravity from the point of view of locally covariant quantum field theory. \newblock \emph{Communications in Mathematical Physics} \textbf{345}, 741--779 (2016). \newblock doi:\ \doi{10.1007/s00220-016-2676-x}. \newblock arXiv:\ \nolinkurl{1306.1058}. \bibitem{Hack} T.-P.~Hack. \newblock Quantization of the linearised Einstein--Klein--Gordon system on arbitrary backgrounds and the special case of perturbations in Inflation. \newblock \emph{Classical and Quantum Gravity} \textbf{31}, 215004 (2014). \newblock doi:\ \doi{10.1088/0264-9381/31/21/215004}. \newblock arXiv:\ \nolinkurl{1403.3957}. \bibitem{Registry} N.~Cheung. \newblock \emph{Enter Calculus Companion}. \newblock Registered-manuscript index. \newblock \url{https://enter.grok.me/}. \newblock The following items are undated, versioned companion manuscripts, not independently verified journal publications. Their identifiers and supplied statement locators are used for documentary comparison only. \bibitem{CheungII} N.~Cheung. \newblock \emph{II}, \SourceLabel{II.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Supplied comparison locator: \SourceLabel{thm:certificate}, finite contact-span certificate. \newblock Enter Calculus Companion. \bibitem{CheungIV} N.~Cheung. \newblock \emph{IV}, \SourceLabel{IV.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Supplied comparison locator: \SourceLabel{thm:model1}, selected represented model. \newblock Enter Calculus Companion. \bibitem{CheungV} N.~Cheung. \newblock \emph{V}, \SourceLabel{V.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Supplied comparison locators: \SourceLabel{lem:gaugeidentity}, \SourceLabel{thm:defect}, and \SourceLabel{thm:rawclassification}. \newblock Enter Calculus Companion. \bibitem{CheungVI} N.~Cheung. \newblock \emph{VI}, \SourceLabel{VI.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locators: \SourceLabel{thm:descent} and \SourceLabel{prop:gauge-action}. \newblock Enter Calculus Companion. \bibitem{CheungVIIA} N.~Cheung. \newblock \emph{VII-A}, \SourceLabel{VII-A.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locators: \SourceLabel{thm:plateau} and \SourceLabel{cor:nohigherrepair}. \newblock Enter Calculus Companion. \bibitem{CheungXA} N.~Cheung. \newblock \emph{X-A}, \SourceLabel{X-A.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:coordinate}, coordinate intertwining with the declared equations and data. \newblock Enter Calculus Companion. \bibitem{CheungXIA} N.~Cheung. \newblock \emph{XI-A}, \SourceLabel{XI-A.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:empty}, fixed-input invariant-jet obstruction. \newblock Enter Calculus Companion. \bibitem{CheungXIB} N.~Cheung. \newblock \emph{XI-B}, \SourceLabel{XI-B.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: front-matter release distinguishing the gauge-fixed construction from repair of the invariant jet. \newblock Enter Calculus Companion. \bibitem{CheungXIIC} N.~Cheung. \newblock \emph{XII-C}, \SourceLabel{XII-C.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:empty}, recomputed compatibility for the selected action. \newblock Enter Calculus Companion. \bibitem{CheungXIID} N.~Cheung. \newblock \emph{XII-D}, \SourceLabel{XII-D.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:response}, local response construction and cutoff independence. \newblock Enter Calculus Companion. \bibitem{CheungXIIID} N.~Cheung. \newblock \emph{XIII-D}, \SourceLabel{XIII-D.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:triviality}, causal tangency on its declared shell. \newblock Enter Calculus Companion. \bibitem{CheungXIVA} N.~Cheung. \newblock \emph{XIV-A}, \SourceLabel{XIV-A.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:relative}, relative classical master equation. \newblock Enter Calculus Companion. \bibitem{CheungXIVB} N.~Cheung. \newblock \emph{XIV-B}, \SourceLabel{XIV-B.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:ghost}, relative/free-shell ghost intersection. \newblock Enter Calculus Companion. \bibitem{CheungEm02E} N.~Cheung. \newblock \emph{Em02-E}, \SourceLabel{Em02-E-v1.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locators: \SourceLabel{thm:lapse}, \SourceLabel{thm:release}, and \SourceLabel{prop:background}. \newblock Enter Calculus Companion. \bibitem{CheungEn01B} N.~Cheung. \newblock \emph{En01-B}, \SourceLabel{En01-B-v2.tex}, version~2. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:contrast}, nuisance cancellation and robust margin. \newblock Enter Calculus Companion. \bibitem{CheungEn01C} N.~Cheung. \newblock \emph{En01-C}, \SourceLabel{En01-C.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locator: \SourceLabel{thm:certificate}, complete-record statistical certificate. \newblock Enter Calculus Companion. \bibitem{CheungXIVH} N.~Cheung. \newblock \emph{XIV-H}, \SourceLabel{XIV-H.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locators: \SourceLabel{thm:H2} and \SourceLabel{thm:release}. \newblock Enter Calculus Companion. \bibitem{CheungXIVI} N.~Cheung. \newblock \emph{XIV-I}, \SourceLabel{XIV-I.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locators: \SourceLabel{thm:I2}, \SourceLabel{thm:I3}, and \SourceLabel{thm:release}. \newblock Enter Calculus Companion. \bibitem{CheungXIVJ} N.~Cheung. \newblock \emph{XIV-J}, \SourceLabel{XIV-J.tex}, version~1. \newblock Unpublished companion manuscript. \newblock Source locators: \SourceLabel{prop:descent}, \SourceLabel{thm:reference}, \SourceLabel{thm:J2}, \SourceLabel{thm:release}, and \SourceLabel{cor:noadmission}. \newblock Enter Calculus Companion. \end{thebibliography} \end{document}