% !TEX program = pdflatex %===================================================================== % landmark_V.tex % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error landmark_V.tex % pdflatex -interaction=nonstopmode -halt-on-error landmark_V.tex % Repeat if cross-reference warnings remain. % % Single-file source. No external bibliography, figures, or data files. % % Citation policy: % No supplied companion manuscript is cited. % The Enter calculus is not used as a premise and is not cited. % Standard external foundational references are identified explicitly. % % Mathematical repairs: % 1. The two supplied response conventions give the SAME response: % L_plus = -L, P S = L_plus = -L. % There is no additional sign change in the reconstructed current. % 2. For R_{1,1}=i(T_2-V phi), u_f=-G_A f, the correct compact test is % q_{g,h,f}=Q_{gh}u_f, % not -Q_{gh}u_f. % 3. Advanced-minus-retarded propagation is denoted by E_r; % the scalar commutator uses Delta_m=G_R-G_A. % 4. The compact-current smoothing argument is proved directly by % Green identities, without radical exhaustion or a presymplectic % nondegeneracy assumption. % % Verification disclosure: % No successful pdfLaTeX compilation or verified page count is claimed. % No numerical computation or software execution is used in the proofs. %===================================================================== \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{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,fit,backgrounds,matrix} \definecolor{mblue}{HTML}{1D4E89} \definecolor{sgreen}{HTML}{1F7A4D} \definecolor{rpurple}{HTML}{5B4B8A} \definecolor{corange}{HTML}{C05621} \definecolor{sgray}{HTML}{6B6560} \tikzset{ mbox/.style={draw=mblue,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=mblue!8}, sbox/.style={draw=sgreen,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=sgreen!10}, pbox/.style={draw=rpurple,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=rpurple!8}, cbox/.style={draw=corange,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=corange!10}, gbox/.style={draw=sgray,dashed,text=black,rounded corners=2pt,align=center, font=\footnotesize,inner sep=3.5pt,fill=sgray!8}, ibox/.style={draw,rounded corners=2pt,align=center,font=\footnotesize,inner sep=3.5pt}, arr/.style={-{Stealth[length=2mm]},semithick}, arrg/.style={-{Stealth[length=2mm]},semithick,draw=sgray}, arro/.style={-{Stealth[length=2mm]},semithick,draw=corange} } \usepackage[hidelinks]{hyperref} \allowdisplaybreaks \setlength{\emergencystretch}{3em} \linespread{1.06} \setlist{itemsep=3pt,topsep=5pt} \numberwithin{equation}{section} \setcounter{tocdepth}{1} \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{\Sch}{\mathcal S} \newcommand{\Cc}{C_c^\infty} \newcommand{\Csc}{C_{\mathrm{sc}}^\infty} \newcommand{\Sol}{\mathscr S} \newcommand{\SoldD}{\mathscr S_{\mathrm{dD}}} \newcommand{\Solret}{\mathscr S_{\mathrm{ret}}} \newcommand{\Dret}{\mathscr D_{\mathrm{ret}}} \newcommand{\Test}{\mathscr E} \newcommand{\Vfin}{\mathcal V_{\mathrm{fin}}} \newcommand{\GSch}{\mathscr G_{\Sch}} \newcommand{\LE}{L_{\mathrm E}} \newcommand{\Jraw}{J_{\mathrm{raw}}} \newcommand{\Jsec}{J_{\mathrm{sec}}} \newcommand{\Jadm}{J_{\mathrm{adm}}} \newcommand{\bfH}{\mathbf H} \newcommand{\sret}{\mathfrak s_{\mathrm{ret}}} \newcommand{\sini}{\mathfrak s_0} \newcommand{\E}{\mathsf E} \newcommand{\Qop}{\mathsf Q} \newcommand{\Kmap}{\mathcal K} \newcommand{\El}{\mathcal E_{\mathrm{el}}} \newcommand{\Dphi}{\mathscr D_{\phi}} \newcommand{\Dmix}{\mathscr D_{\mathrm{mix}}} \newcommand{\Hil}{\mathcal H} \newcommand{\Fock}{\Gamma_s} \newcommand{\vac}{\Omega} \newcommand{\one}{I} \newcommand{\WF}{W_m^{\mathrm F}} \newcommand{\W}{W_m} \newcommand{\Om}{\Omega_m} \newcommand{\Sw}{S_{\mathrm w}} \newcommand{\Cw}{C_{\mathrm w}} \newcommand{\form}{\mathrm{form}} \newcommand{\TT}{\mathrm{TT}} \newcommand{\dd}{\,\mathrm d} \newcommand{\bp}{\mathbf p} \newcommand{\bq}{\mathbf q} \newcommand{\bx}{\mathbf x} \newcommand{\bz}{\mathbf z} \newcommand{\bw}{\mathbf w} \newcommand{\cC}{\mathcal C} \newcommand{\cT}{\mathcal T} \newcommand{\cR}{\mathcal R} \newcommand{\init}{\mathrm{init}} \newcommand{\src}{\mathrm{src}} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\br}[1]{\langle #1\rangle} \newcommand{\wick}[1]{:\!#1\!:} \newcommand{\sharpT}[1]{\left(#1\right)^{\sharp\sharp}} \newcommand{\flatT}[1]{\left(#1\right)^{\flat\flat}} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\Dom}{Dom} \DeclareMathOperator{\sd}{sd} \DeclareMathOperator{\Data}{Data} \newcommand{\statementbox}[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{Admissible currents and first retarded corrections\\ for a fixed-time scalar response}} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} A fixed-time Klein--Gordon response to a linearized vacuum metric, with the scalar held fixed, is reconstructed as a compact tensor distribution with an explicit initial-surface term. On the spacelike-compact vacuum solution space it has a smooth compact conserved representative if and only if its gauge defect vanishes. After a retarded or zero-Cauchy harmonic prescription, the criterion is vanishing of the propagated defect. A compact-Cauchy residual wave proves failure on an open dense family of smearings in a specified Fr\'echet test space. The same pairing, restricted to any declared Fourier-gap two-dimensional spatial-TT plane, nevertheless has an explicit compact conserved representative, constructed from curvature currents dual to that plane. Separately, the first retarded product of the compactly switched minimal Einstein--scalar vertex with one scalar insertion is computed for an external metric and for a one-oscillator metric. The external coefficient is a linearly smeared free field; its contraction coefficients have scaling degree at most three and unique minimal extensions. The oscillator coefficient reconstructs the switched bilinear and defines an essentially self-adjoint real polynomial on a common core. The Newton--wave composition of stress, inverse, and scalar response is treated as a distinct cubic Wick form. Dense-TT representability, the adiabatic limit, and higher causal products remain open. \end{abstract} \begin{center} \small \begin{tabular}{@{}p{.30\linewidth}p{.30\linewidth}p{.32\linewidth}@{}} \toprule Question & Closed answer & Not answered\\ \midrule Raw on \(\Sol\) & \(\bfH=0\) & Minkowski radical\\ Harmonic on declared domains & \(\E_1\bfH=0\) & Smaller TT class automatically\\ Fourier-gap 2-plane & \(\exists J_c\in\Test\) & Dense TT\\ External \(R_{1,1}\) & Linear field, \(\sd\le3\) & \(g\to1,\ T_2(V,V)\)\\ Finite-mode \(R_{1,1}\) & Switched bilinear & Local metric QFT\\ \(\cC_f^{(3)}\) & Wick form & \(R_{1,1}\)\\ \bottomrule \end{tabular} \end{center} \noindent A compact current on a 2-plane is not a counterexample to generic failure on \(\Sol\). A linear \(R_{1,1}\) is not a cubic Newton--wave form. \newpage \tableofcontents %===================================================================== \section{Two questions and five distinct objects} \label{sec:introduction} %===================================================================== The first question is classical. Fix a free scalar solution \(\phi\), a Cauchy surface, and a scalar smearing \(f\). The metric-to-scalar response is linear in the metric perturbation. When does that functional equal a pairing with a smooth compact conserved tensor on a declared space of vacuum metric solutions? The second question is quantum and perturbative. Start from a specified compactly switched vertex, not from a presumed observable. What is its first retarded product with one scalar insertion, and on which domain is the resulting expression represented? The two questions have different inputs and different conclusions: \[ \text{fixed classical pairing} \longrightarrow \text{compact conserved representative or gauge obstruction}, \] \[ \text{specified switched vertex} \longrightarrow R_{1,1} \longrightarrow \text{represented form or polynomial operator}. \] A result along one arrow is not a premise completing the other. \subsection{Object ledger} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.23\linewidth}X@{}} \toprule Object & Statement established here\\ \midrule Full-space classical response & Exact criteria for raw and harmonic-prescription representability on explicitly named solution spaces.\\ \addlinespace Finite-plane restriction & An independent compact-current construction for the actual response on each fixed two-dimensional TT plane.\\ \addlinespace External-metric insertion & The first retarded scalar coefficient is a linearly smeared free field at compact switch.\\ \addlinespace One-oscillator insertion & The coefficient is a mixed quadratic polynomial, equal to the switched bilinear; its real polynomial realization is essentially self-adjoint on the stated core.\\ \addlinespace Newton--wave contrast & A different composed classical observable has a fully Wick-ordered cubic form, with a separate kernel estimate and cutoff limit.\\ \bottomrule \end{tabularx} \end{center} Figure~\ref{fig:object-roadmap} separates the two questions and the five objects treated in this manuscript. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[font=\small\bfseries, align=center] (title) {Different inputs, degrees, and conclusions.}; \node[gbox, text width=4.15cm, below left=8mm and 18mm of title] (c0) {fixed $\phi$, $f$, $t_0$}; \node[sbox, text width=4.15cm, below=of c0] (c1) {$B_f(h,\phi)$}; \node[cbox, text width=4.15cm, below=of c1] (c2) {compact conserved representative\\ or gauge obstruction}; \node[mbox, text width=2.0cm, below left=2mm and -1mm of c2] (c3) {full-space\\ classification}; \node[mbox, text width=2.0cm, below right=2mm and -1mm of c2] (c4) {restricted\\ TT plane}; \draw[arr] (c0)--(c1); \draw[arr] (c1)--(c2); \draw[arr] (c2.south)--(c3.north); \draw[arr] (c2.south)--(c4.north); \node[font=\scriptsize, sgray, above=0mm of c0] {classical}; \node[gbox, text width=4.15cm, right=7mm of c0] (q0) {specified compactly\\ switched vertex}; \node[pbox, text width=4.15cm, below=of q0] (q1) {$R_{1,1}$}; \node[pbox, text width=4.15cm, below=of q1] (q2) {represented coefficient}; \node[pbox, text width=2.0cm, below left=2mm and -1mm of q2] (q3) {external metric:\\ linear scalar}; \node[pbox, text width=2.0cm, below right=2mm and -1mm of q2] (q4) {one oscillator:\\ mixed quadratic}; \draw[arr] (q0)--(q1); \draw[arr] (q1)--(q2); \draw[arr] (q2.south)--(q3.north); \draw[arr] (q2.south)--(q4.north); \node[font=\scriptsize, sgray, above=0mm of q0] {quantum}; \node[sbox, text width=4.15cm, right=7mm of q0] (u0) {$T(\phi)$}; \node[mbox, text width=4.15cm, below=of u0] (u1) {$\mathcal R^A T(\phi)$}; \node[sbox, text width=4.15cm, below=of u1] (u2) {$B_f(\mathcal R^A T(\phi),\phi)$}; \node[sbox, text width=4.15cm, below=of u2] (u3) {cubic Wick form}; \draw[arr] (u0)--(u1); \draw[arr] (u1)--(u2); \draw[arr] (u2)--(u3); \node[font=\scriptsize, sgray, above=0mm of u0] {separate cubic lane}; \draw[arro] (q2.east)--(u2.west) node[cbox, font=\scriptsize, midway, fill=white] {not an equality}; \end{tikzpicture} \caption{Two questions and five distinct objects. The classical problem asks for compact conserved representatives on declared solution spaces. The perturbative problem starts from a compactly switched vertex and constructs its first retarded scalar coefficient. The Newton--wave composition is a separate cubic observable, not either retarded coefficient.} \label{fig:object-roadmap} \end{figure} Full-space nonrepresentability and finite-plane representability are compatible. The former requires equality on a space containing residual pure gauges. The latter asks for two interpolation conditions on a plane with no nonzero Schwartz pure gauge. Its annihilator is correspondingly much larger. Likewise, field degree depends on what is quantized. With an external metric, a scalar-quadratic vertex has a linear first retarded scalar coefficient. With one metric oscillator, the coefficient has one oscillator factor and one scalar factor. Substituting a scalar stress into a Newton--wave inverse before forming the classical response is another operation, with a different degree and singularity structure. \subsection{Scope and references} We call a smooth compact conserved tensor an \emph{FH test tensor}, following the observable class of Fewster and Hunt \cite{FH}. We prove the flat-space representability statements directly. In particular, we do not use radical exhaustion, separation by observables on the full Minkowski quotient, or a compact-Cauchy nondegeneracy theorem. For the quantum calculation we use an explicitly specified two-insertion Wick normalization. Scaling degree is used in the sense of the extension theory of \cite{BF}; the required coefficients and their uniqueness are also checked directly. The off-shell input convention and retarded subtraction are those of the retarded-product framework \cite{DF}. Only one insertion pair is constructed. The essential self-adjointness conclusion uses the analytic-vector criterion of Nelson \cite{Nelson}, after explicit number estimates. It does not assert essential self-adjointness of the cubic vertex or existence of an interacting Hamiltonian. No companion manuscript is a theorem premise. No result about the Enter calculus, a measurement instrument, or named record statistics is needed. There is consequently no Enter-calculus citation. \statementbox{Non-goals}{% There is no dense-TT representability theorem, adiabatic limit, construction of \(T_2(V,V)\), stress--stress product, graviton-loop calculation, instrument, Ramsey record, or contact-separation result. A finite oscillator-valued wave is not treated as a local relativistic metric field.} \subsection{Two sign corrections} Two potential transcription errors are resolved before the proofs. First, the convention \[ L_h^+=h^{\mu\nu}\partial_\mu\partial_\nu+D(h)^\nu\partial_\nu, \qquad PS=L_h^+\phi \] and our convention \(L_h=-L_h^+\), \(PS=-L_h\phi\), define the same response, not opposite responses. Their raw currents therefore agree. Second, with the retarded convention \[ R_{1,1}=i(T_2-V\phi),\qquad u_f=-G_Af, \] the represented test function is \[ q_{g,h,f}=\Qop_{gh}u_f. \] An additional minus sign in this formula would contradict both the retarded contraction and the Duhamel equation. Appendix~\ref{app:signs} gives the full audit. Figure~\ref{fig:sign-audit} summarizes the response and retarded sign conventions fixed throughout the paper. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[ibox, text width=4.35cm] (a) {% \textbf{(a)} identical response\\[1mm] $L_h^+=-L_h$\\ $PS=L_h^+\phi=-L_h\phi$\\[1mm] same Duhamel response\\ same raw current}; \node[cbox, below=of a, font=\scriptsize] (ax) {additional current sign reversal}; \draw[corange, thick] (ax.north west)--(ax.south east); \draw[corange, thick] (ax.north east)--(ax.south west); \node[ibox, text width=4.35cm, right=4mm of a] (b) {% \textbf{(b)} opposite causal orientations\\[1mm] classification\\ $\mathsf E_r=\mathsf G_{r,A}-\mathsf G_{r,R}$\\[1.5mm] scalar commutator\\ $\Delta_m=G_R-G_A$\\[1mm] symbols kept distinct}; \node[ibox, text width=4.55cm, right=4mm of b] (c) {% \textbf{(c)} retarded sign chain\\[1mm] $W_m^{\mathrm F}-W_m=iG_R$\\ $i(iG_R)=-G_R$\\ $\phi^{(1)}_{g,h}=-G_R\mathsf Q_{gh}\phi$\\ $u_f=-G_Af$\\ $q_{g,h,f}=+\mathsf Q_{gh}u_f$\\[1mm] TT minus belongs to $\mathsf Q$}; \node[gbox, below=2mm of b, text width=9.4cm] (n) {No additional convention beyond the displayed chain.}; \end{tikzpicture} \caption{Locked sign conventions. The two displayed notations for the fixed-time response are identical. Classification uses advanced-minus-retarded propagation, whereas the scalar commutator uses retarded-minus-advanced propagation. In the retarded coefficient, the sign of \(-G_R\) is already absorbed when \(u_f=-G_Af\), giving \(q_{g,h,f}=\mathsf Q_{gh}u_f\).} \label{fig:sign-audit} \end{figure} %===================================================================== \section{Locked signs, solution spaces, and Green operators} \label{sec:conventions} %===================================================================== \subsection{Differential operators} Work on \(M=\R^{1+3}\) with \[ \eta=\operatorname{diag}(-1,1,1,1),\qquad \Box=-\partial_t^2+\Delta,\qquad P=\Box-m^2=-(\partial_t^2+\Om^2), \qquad \Om=\sqrt{-\Delta+m^2}. \] The quantum and Newton--wave parts assume \(m>0\). The classical classification remains valid for \(m=0\), with the sine propagator interpreted continuously at zero. For a vector field \(\xi\), put \[ (K\xi)_{\mu\nu} =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu, \qquad X_\xi=\xi^\mu\partial_\mu. \] Trace reversal is \[ \bar h_{\mu\nu} =h_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta h, \qquad D(h)^\nu=\partial_\mu\bar h^{\mu\nu}. \] In four dimensions it is an involution. Direct differentiation gives \begin{equation} D(K\xi)=\Box\xi. \label{eq:DK} \end{equation} The linearized Einstein operator is \begin{align} (\LE h)_{\mu\nu} ={}&-\frac12\Box h_{\mu\nu} -\frac12\partial_\mu\partial_\nu\tr_\eta h +\partial^\alpha\partial_{(\mu}h_{\nu)\alpha} \nonumber\\ &-\frac12\eta_{\mu\nu} \left(\partial^\alpha\partial^\beta h_{\alpha\beta} -\Box\tr_\eta h\right). \label{eq:LE} \end{align} Its identities are \begin{equation} \LE K=0,\qquad \partial^\mu(\LE h)_{\mu\nu}=0,\qquad 2\overline{\LE h}=-\Box h+KD(h). \label{eq:LEidentities} \end{equation} They follow by expanding derivatives in \eqref{eq:LE}. The scalar-response operator is \begin{equation} L_h=-h^{\mu\nu}\partial_\mu\partial_\nu -D(h)^\nu\partial_\nu. \label{eq:L} \end{equation} For the response with zero Cauchy data at \(t_0\), \begin{equation} PS_{t_0}(h)\phi=-L_h\phi,\qquad S_{t_0}(h)\phi(t) =\int_{t_0}^t \frac{\sin((t-s)\Om)}{\Om}L_h\phi(s)\dd s. \label{eq:Duhamel} \end{equation} \subsection{The classical solution spaces} A smooth field is spacelike compact if its support is contained in \(J(C)\) for some compact \(C\subset M\). Define \[ \Sol=\{h\in\Csc(S^2T^*M):\LE h=0\}, \qquad \SoldD=\{h\in\Sol:D(h)=0\}. \] The retarded-prescription domain is \[ \Dret=\left\{ h\in\Csc(S^2T^*M): D(h)=0\text{ for all sufficiently negative times} \right\}, \qquad \Solret=\Dret\cap\Sol. \] These spaces are not identified with the Schwartz TT plane used later. The test-tensor space is \begin{equation} \Test= \left\{ J\in\Cc(S^2TM;\R): \partial_\mu J^{\mu\nu}=0 \right\}, \qquad F_J(h)=\int h_{\mu\nu}J^{\mu\nu}\dd^4x. \label{eq:testspace} \end{equation} \begin{definition}[Representability on a declared space] A real linear functional \(B\) on \(\mathscr V\subseteq\Sol\) is FH-representable on \(\mathscr V\) if a single \(J\in\Test\) satisfies \(B(h)=F_J(h)\) for every \(h\in\mathscr V\). For the non-spacelike-compact TT plane below, the same compact pairing defines a restricted extension, without identifying that plane with the phase space of an FH algebra. \end{definition} For every smooth \(\xi\) and every \(J\in\Test\), \begin{equation} F_J(K\xi) =-2\int \xi_\nu\partial_\mu J^{\mu\nu}\dd^4x=0. \label{eq:gaugeannihilation} \end{equation} This elementary identity is the necessary condition used in every negative result. The formal adjoints, with covectors paired against vectors, are \begin{align} (K^*J)_\nu&=-2\eta_{\nu\beta}\partial_\mu J^{\mu\beta}, \label{eq:Kstar}\\ (D^*a)^{\mu\nu} &=-\partial^{(\mu}a^{\nu)} +\frac12\eta^{\mu\nu}\partial_\alpha a^\alpha, \qquad K^*D^*a=\Box a. \label{eq:Dstar} \end{align} \subsection{Green operators and opposite causal conventions} For the componentwise massless wave equation on rank-\(r\) tensors, write \[ \mathsf G_{r,R},\qquad \mathsf G_{r,A},\qquad \E_r=\mathsf G_{r,A}-\mathsf G_{r,R}. \] Thus \(\E_r\) is advanced minus retarded. If \[ \Sw(t)=\frac{\sin(t|D|)}{|D|},\qquad \Cw(t)=\cos(t|D|), \] then the time kernel of \(\E_r\) is \(\Sw(t)\). For the massive scalar, use \(G_R,G_A\), and set \[ \Delta_m=G_R-G_A. \] The spatial Fourier kernels are \begin{equation} G_R(t)=-\Theta(t)\frac{\sin(t\Om)}{\Om}, \qquad G_A(t)=\Theta(-t)\frac{\sin(t\Om)}{\Om}. \label{eq:green} \end{equation} The scalar commutator below uses \(\Delta_m\), not \(\E_r\). Compact distributions are allowed as Green-operator sources, using transposition. In particular, \[ \pair{\mathsf G_{r,A}C}{u} =\pair{C}{\mathsf G_{r,R}u} \] for compact smooth tests \(u\). For smooth spacelike-compact sources bounded below in time, the retarded solution is defined by zero data sufficiently far in the past. All support statements used below follow from finite propagation and these kernels. We use the following compact-source consequence repeatedly: \begin{equation} \E_r C=0 \quad\Longleftrightarrow\quad \mathsf G_{r,A}C=\mathsf G_{r,R}C=:A \text{ has compact support}, \qquad \Box A=C. \label{eq:compactprimitive} \end{equation} In the forward direction the common support lies in \(J^+(\supp C)\cap J^-(\supp C)\). In the reverse direction the Green identities on compact distributions give \(\E_r\Box A=0\). %===================================================================== \section{Raw current and complete gauge defect} \label{sec:raw} %===================================================================== For this section and the classification, translate the initial surface to \(t_0=0\). Fix a real free scalar \[ P\phi=0 \] with compact smooth Cauchy data, and a real \[ f\in\Cc((0,T)\times\R^3). \] The scalar is fixed as a classical parameter. Define \[ B_{\mathrm{raw}}(h)=\int f\,S_0(h)\phi\dd^4x. \] Let \begin{equation} u(s)=\int_s^T \frac{\sin((t-s)\Om)}{\Om}f(t)\dd t=-G_Af(s). \label{eq:u} \end{equation} It has zero terminal data and \(Pu=-f\). It is smooth, including across a time later than the support of \(f\), where it vanishes. Self-adjointness of the spatial sine operator gives \begin{equation} B_{\mathrm{raw}}(h) =\int_0^T\int_{\R^3}u\,L_h\phi\dd^3x\dd t. \label{eq:rawadjoint} \end{equation} Put \(V=\Theta(t)u\). Its support is contained in the compact set \[ K_f=J^-(\supp f)\cap\{0\le t\le T\}. \] \subsection{Distributional reconstruction} \begin{theorem}[Raw current and surface measure] \label{thm:rawcurrent} For every smooth \(h\), \begin{equation} B_{\mathrm{raw}}(h)=\pair{\Jraw}{h}, \qquad \Jraw^{\mu\nu} =\partial^{(\mu}V\,\partial^{\nu)}\phi -\frac12\eta^{\mu\nu}\partial_\alpha(V\partial^\alpha\phi). \label{eq:Jraw} \end{equation} The current is a compact distribution of order at most zero. It decomposes as \[ \Jraw=\Theta(t)J_{\mathrm{bulk}}+\delta(t)j_\Sigma, \] where \(J_{\mathrm{bulk}}\) is \eqref{eq:Jraw} with \(V=u\), and \begin{equation} j_\Sigma^{\mu\nu} =-\delta_0^{(\mu}u_0\partial^{\nu)}\phi|_0 +\frac12\eta^{\mu\nu}u_0\dot\phi|_0. \label{eq:surface} \end{equation} Here \(u_0=u(0)\). In components, \[ j_\Sigma^{00}=\frac12u_0\dot\phi|_0,\qquad j_\Sigma^{0i}=-\frac12u_0\partial_i\phi|_0,\qquad j_\Sigma^{ij}=\frac12\delta^{ij}u_0\dot\phi|_0. \] \end{theorem} \begin{proof} Using \eqref{eq:L}, integrate the derivative of \(h\) in \(\pair{V}{L_h\phi}\) by parts: \begin{align*} \pair{V}{L_h\phi} ={}&-\pair{Vh^{\mu\nu}}{\partial_\mu\partial_\nu\phi} +\pair{h^{\mu\nu}}{\partial_\mu(V\partial_\nu\phi)}\\ &-\frac12\pair{\tr_\eta h} {\partial_\alpha(V\partial^\alpha\phi)}. \end{align*} The Hessian terms cancel. This proves the current formula first for compact \(h\), and then for arbitrary smooth \(h\) by inserting a cutoff equal to one near \(K_f\). The derivative of \(\Theta u\) is \[ \partial^\mu(\Theta u) =\Theta\partial^\mu u-\delta(t)\delta_0^\mu u_0. \] Substitution gives \eqref{eq:surface}. Only a bounded bulk density and a slice measure occur. Their support is compact by finite propagation. \end{proof} Figure~\ref{fig:raw-current} shows the truncated causal support and the initial-surface term in the raw-current reconstruction. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[font=\scriptsize, sgray] (sch) {schematic}; \draw[sgray] (0,-1.35) -- (4.6,-1.35) node[right, font=\scriptsize] {$t=0$}; \draw[sgray, ->] (0.25,-1.35) -- (0.25,1.95) node[above, font=\scriptsize] {$t$}; \draw[mblue, fill=mblue!8] (1.15,-1.35) -- (1.55,0.65) -- (3.35,0.65) -- (3.75,-1.35) -- cycle; \node[font=\scriptsize, mblue] at (2.45,-0.35) {$K_f$}; \node[sbox, font=\scriptsize] at (2.45,1.25) {$\operatorname{supp} f$}; \node[sbox, font=\scriptsize] (trunc) at (6.7,0.15) {$u=-G_Af$\\ $V=\Theta(t)u$}; \node[sbox, text width=5.6cm, below left=8mm and 0mm of trunc] (rec) {% $B_{\mathrm{raw}}(h)=\int_0^T u L_h\phi=\langle J_{\mathrm{raw}},h\rangle$\\[1mm] $J_{\mathrm{raw}}^{\mu\nu}=\partial^{(\mu}V\,\partial^{\nu)}\phi -\tfrac12\eta^{\mu\nu}\partial_\alpha(V\partial^\alpha\phi)$}; \node[cbox, text width=5.3cm, right=3mm of rec] (dist) {% $J_{\mathrm{raw}}=\Theta(t)J_{\mathrm{bulk}}+\delta(t)j_\Sigma$\\[1mm] $j_\Sigma^{00}=\tfrac12 u_0\dot\phi|_0$\\ $j_\Sigma^{0i}=-\tfrac12 u_0\partial_i\phi|_0$\\ $j_\Sigma^{ij}=\tfrac12\delta^{ij}u_0\dot\phi|_0$}; \node[gbox, text width=11.2cm, below=2mm of dist] {% compact order-zero distribution; not yet a smooth conserved test tensor}; \end{tikzpicture} \caption{Raw-current reconstruction. Truncation of the advanced scalar solution at the initial surface produces a compact bulk density and a slice measure. The latter is part of the fixed-time response and cannot be omitted. Compactness and order zero alone do not establish conservation or smooth FH representability.} \label{fig:raw-current} \end{figure} \subsection{The defect covector} \begin{theorem}[Complete Ward defect] \label{thm:defect} Distributionally, \begin{equation} \partial_\mu\Jraw^{\mu\nu} =\frac12(PV)\partial^\nu\phi,\qquad PV=-f-\delta(t)\dot u_0-\delta'(t)u_0. \label{eq:divraw} \end{equation} Consequently \begin{equation} \boxed{ \bfH:=K^*\Jraw =\gamma+\delta(t)a+\delta'(t)b, } \label{eq:H} \end{equation} where the covector-valued coefficients are \begin{align} \gamma_\nu&=f\,\partial_\nu\phi, \nonumber\\ a_\nu&=\dot u_0\,\partial_\nu\phi|_0 -u_0\,\partial_t\partial_\nu\phi|_0, \nonumber\\ b_\nu&=u_0\,\partial_\nu\phi|_0. \label{eq:Hab} \end{align} The density \(\gamma\) is smooth and compact in \(00\), there are compact smooth scalar Cauchy data, a nonnegative \(f\in\Cc((0,T)\times\R^3)\), and a nonzero residual pure gauge \(K\xi\), with compact smooth vector-wave Cauchy data, such that \[ B_{\mathrm{raw}}(K\xi)=B_R(K\xi)=B_0(K\xi)>0. \] \end{theorem} \begin{proof} Choose real \(F\in\Cc(\R^3)\) with \(\partial_1F\not\equiv0\), and put \(\beta=\partial_1F\). Let \[ \phi(0)=0,\quad \dot\phi(0)=F,\quad P\phi=0, \] and let the only nonzero component of the vector wave satisfy \[ \xi^1(0)=0,\quad \dot\xi^1(0)=\beta,\quad \Box\xi^1=0. \] Finite propagation gives spacelike compactness, and \((K\xi)_{01}|_0=\beta\), so \(K\xi\ne0\). Set \(w=\xi^1\partial_1\phi\). Its value and first time derivative vanish at zero. Equation~\eqref{eq:boundarygauge} therefore gives \(S_0(K\xi)\phi=w\). Locally uniformly in space, \[ w(t,x)=t^2\beta(x)\partial_1F(x)+O(t^4). \] It is positive on some open set compactly contained in \(00\). Both harmonic prescriptions fix \(K\xi\), by \eqref{eq:residualsurvives}. \end{proof} \begin{corollary}[Open dense failure in a specified test space] \label{cor:generic} Fix the scalar and vector wave of Theorem~\ref{thm:residualwitness}. Let \(Q\Subset(0,T)\times\R^3\) have interior meeting an open region where \(w=\xi^1\partial_1\phi>0\). In the Fr\'echet space \[ \mathscr D_Q=\{f\in C^\infty(M;\R):\supp f\subset Q\}, \] the failures of raw representability on \(\Sol\), and of harmonic representability on the domains of Theorem~\ref{thm:harmonic}, each form an open dense set. Their common failure set contains \[ \{f\in\mathscr D_Q:\ \int fw\ne0\}, \] which is itself open and dense. \end{corollary} \begin{proof} The last set is the complement of the kernel of a nonzero continuous linear functional. Such a kernel is closed with empty interior. For the exact failure sets, the map \(f\mapsto\bfH\) is continuous and linear into compact distributions with common support, and the map from \(f\) to the two data in \eqref{eq:zetadata} is continuous and linear into compact smooth data with common support. Their zero sets are closed linear subspaces. The witness makes each proper. The classification theorems identify their complements with the claimed failure sets. \end{proof} The scalar in this statement is fixed and specially constructed. No assertion of generic failure for every scalar background is made; the zero scalar, for example, gives the zero functional. Figure~\ref{fig:residual-generic-failure} displays the compact-data residual witness and the specified open dense failure set. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[font=\scriptsize, sgray] (sch) {schematic}; \node[sbox, text width=4.0cm, below=2mm of sch] (d) {% $\phi(0)=0$,\ $\dot\phi(0)=F$\\ $\xi^1(0)=0$,\ $\dot\xi^1(0)=\partial_1 F$\\ $\Box\xi^1=0$; other components $0$}; \node[sbox, text width=4.0cm, right=5mm of d] (w) {% $w=\xi^1\partial_1\phi$\\ $=t^2(\partial_1 F)^2+O(t^4)$}; \draw[sgreen!30, fill=sgreen!12] (5.55,-1.7) rectangle (7.15,-0.7); \node[font=\scriptsize, sgreen!70!black] at (6.35,-1.2) {$w>0$}; \node[font=\scriptsize] at (6.35,-2.05) {$f\ge 0$}; \node[cbox, text width=4.2cm, right=5mm of w] (pos) {% $B_{\mathrm{raw}}(K\xi)=B_R(K\xi)$\\ $=B_0(K\xi)=\int fw>0$}; \node[gbox, text width=12.4cm, below=8mm of d] (gen) {% $\mathscr D_Q$: kernel of $f\mapsto\int fw$ is a closed proper subspace\\ its complement is open dense}; \node[gbox, font=\scriptsize, below=of gen] (fix) {fixed specially constructed scalar; not a theorem for every scalar background}; \end{tikzpicture} \caption{Residual witness and generic failure. Compact smooth Cauchy data give a residual pure gauge with positive response for a suitable future smearing. For the fixed scalar and vector wave, the nonzero continuous functional \(f\mapsto\int fw\) supplies an open dense family of obstructed smearings in the stated Fr\'echet test space. No genericity claim is made for every scalar background.} \label{fig:residual-generic-failure} \end{figure} %===================================================================== \section{What the full-space classification does not settle} \label{sec:limitsclassification} %===================================================================== \subsection{Restriction is a new extension problem} For a smaller declared solution space \(\mathscr V\), the question is \begin{equation} \text{find }J_c\in\Test \quad\text{such that}\quad \Jraw-J_c\in \Ann_{\mathcal E'(S^2TM)}(\mathscr V). \label{eq:restrictedproblem} \end{equation} Here \(\mathcal E'\) denotes compactly supported distributions. Every pairing is therefore defined against smooth tensors, whether or not those tensors are spacelike compact. A residual-gauge obstruction proves failure on \(\mathscr V\) only if its witness belongs to \(\mathscr V\). The witness of Theorem~\ref{thm:residualwitness} has \((K\xi)_{01}|_0\ne0\), so it is not a spatial-TT wave. Its full-space conclusion does not answer \eqref{eq:restrictedproblem} on a smaller TT class. Noncompactness of one displayed conserved distribution is also not a general obstruction to an equivalent compact current. The negative proofs above used nonzero values on pure gauges. The compactness equivalence of Lemma~\ref{lem:compactconditions} was a separate fact specific to its advanced-potential construction. \subsection{A coupled Ward identity and an on-shell-zero candidate} For comparison, consider the triangular classical system \[ \LE h=0,\qquad P\varphi=-L_h\phi. \] It is preserved by \[ (h,\varphi)\mapsto(h+K\xi,\varphi+X_\xi\phi), \] but arbitrary such transformations do not preserve zero response data at the initial surface. For compact smooth \(J,c\), the joint pairing \[ \mathcal O_{J,c}(h,\varphi) =\int h_{\mu\nu}J^{\mu\nu}+\int c\varphi \] is invariant under all compact gauge parameters if and only if \begin{equation} -2\partial_\mu J^{\mu\nu} +c\,\partial^\nu\phi=0. \label{eq:jointWard} \end{equation} This follows directly by integration by parts. For any \(\psi\in\Cc(M)\), set \[ J_\psi^{\mu\nu} =\partial^{(\mu}\psi\,\partial^{\nu)}\phi -\frac12\eta^{\mu\nu} \partial_\alpha(\psi\partial^\alpha\phi), \qquad c_\psi=P\psi. \] The divergence identity proves \eqref{eq:jointWard}, but \[ \int h_{\mu\nu}J_\psi^{\mu\nu} +\int(P\psi)\varphi =\int\psi(L_h\phi+P\varphi)=0 \] on every solution of the triangular system. Thus this immediate compact joint construction is on-shell zero. A nontrivial coupled observable would require a separately specified solution space and quotient; the Ward identity alone does not supply one. \subsection{Support classes} The classification witnesses have literal compact smooth Cauchy data. The next section uses Schwartz data with a Fourier gap. These conditions must not be identified. A compactly supported smooth datum has an entire Fourier transform. If that transform vanishes on a nonempty real open set, real-analytic uniqueness forces the datum to vanish. Thus a nonzero Fourier-gap Schwartz wave cannot also have compact Cauchy data. Nor does compact spacetime support force an ungauged vacuum Einstein perturbation to vanish: compact pure gauges provide counterexamples. The corresponding uniqueness statement is valid for a homogeneous normally hyperbolic wave equation with zero data before the support. Figure~\ref{fig:support-and-restriction} separates the support classes and the restricted extension problem from the full-space classification. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[mbox, text width=5.3cm] (full) {% \textbf{full-space classification}\\[1mm] spacelike-compact vacuum solutions\\ witnesses: compact smooth Cauchy data\\ residual pure gauges included}; \node[pbox, text width=5.3cm, right=8mm of full] (tt) {% \textbf{restricted TT plane}\\[1mm] Fourier-gap Schwartz Cauchy data\\ spatial TT conditions\\ generally not spacelike compact\\ no nonzero Schwartz pure gauge}; \draw[arro] (full.east)--(tt.west) node[cbox, font=\scriptsize, midway, fill=white, align=center] (x) {nonzero datum\\ cannot be both}; \node[gbox, text width=11.6cm, below=7mm of x] (why) {% compact spatial support $\Rightarrow$ entire Fourier transform\\ an open Fourier gap forces that datum to vanish}; \node[cbox, text width=11.6cm, below=of why] (prob) {% $J_{\mathrm{raw}}-J_c\in\operatorname{Ann}(\mathcal V_{\mathrm{fin}})$\\[1mm] the full-space witness has a nonzero $0i$ component and is not spatial TT\\ failure on the larger space does not decide a smaller TT restriction}; \end{tikzpicture} \caption{Support classes and restriction. Compact-Cauchy residual witnesses and Fourier-gap Schwartz TT waves belong to different declared classes. A full-space obstruction transfers to a restriction only when its witness belongs to that restriction. On a fixed TT plane, representability is instead an extension problem modulo the plane's annihilator.} \label{fig:support-and-restriction} \end{figure} %===================================================================== \section{Restricted compact representatives on a two-dimensional TT plane} \label{sec:plane} %===================================================================== This construction is independent of Sections~\ref{sec:raw}--\ref{sec:limitsclassification}. It uses curvature separation on a finite plane, not the propagated-defect criterion. \subsection{The declared plane and its gauge intersection} Fix \(t_00\). Let \[ \mathfrak h_\phi=L^2(\R^3,\dd^3p),\qquad \Hil_\phi=\Fock(\mathfrak h_\phi), \] and define the dense domain \begin{equation} \Dphi= \bigoplus_{n=0}^{\mathrm{alg}} \Sch_{\mathrm{sym}}(\R^{3n}). \label{eq:scalar-domain} \end{equation} There is no fixed common particle ceiling. The scalar field is \[ \phi(x)= \int\frac{\dd^3p}{(2\pi)^{3/2}\sqrt{2\omega_{\bp}}} \left( a(\bp)e^{-i\omega_{\bp}t+i\bp\cdot\bx} +a^\dagger(\bp)e^{i\omega_{\bp}t-i\bp\cdot\bx} \right), \quad \omega_{\bp}=\sqrt{|\bp|^2+m^2}. \] For real compact smooth \(q\), \begin{equation} (\Kmap q)(\bp)= \frac{1}{(2\pi)^{3/2}\sqrt{2\omega_{\bp}}} \int q(t,\bx)e^{i\omega_{\bp}t-i\bp\cdot\bx}\dd^4x, \qquad \phi(q)=\Phi(\Kmap q), \label{eq:massshell} \end{equation} where \(\Phi(v)=a(v)+a^\dagger(v)\) and \(a(v)\) is antilinear in \(v\). The map \(\Kmap\) takes compact smooth tests continuously to Schwartz vectors. Take an external real \(h\in\Vfin\) and real \(g,f\in\Cc(M)\). The switch is always compact. The minimal stress is \[ T_{\mu\nu}(\phi) =\partial_\mu\phi\,\partial_\nu\phi -\frac12\eta_{\mu\nu} \left(\partial_\alpha\phi\,\partial^\alpha\phi+m^2\phi^2\right). \] Vacuum normal ordering is fixed once and for all. The vertex, without the perturbative parameter, is \begin{equation} V(g;h)=\frac12\int gh^{\mu\nu} \wick{T_{\mu\nu}(\phi)}\dd^4x. \label{eq:vertex} \end{equation} The interaction is \(+\kappa V(g;h)\). For the switched tensor \(H_{\mu\nu}=gh_{\mu\nu}\), define \begin{equation} \Qop_H =-\partial_\mu(\bar H^{\mu\nu}\partial_\nu) -\frac12m^2\tr_\eta H. \label{eq:Q} \end{equation} It is formally self-adjoint and is the first variation of the scalar-quadratic vertex. Expanding derivatives gives \begin{equation} \Qop_H=L_H+\frac12(\tr_\eta H)P. \label{eq:QL} \end{equation} On a free scalar solution, \begin{equation} \Qop_{gh}\phi =gL_h\phi-(\partial_\mu g)\bar h^{\mu\nu}\partial_\nu\phi. \label{eq:switch} \end{equation} For spatial TT \(h\), \[ \Qop_{gh}v=-\partial_i(gh^{ij}\partial_jv). \] The letter \(H_{\mu\nu}\) here is a switched metric coefficient. It is unrelated to the defect covector \(\bfH\). \subsection{Off-shell input and normalization} The time-ordered inputs retain their off-shell local-functional labels until the product is formed. Representation on the free on-shell Fock space is performed afterward. In particular, \(\phi(P\rho)=0\) as a represented free field does not require the time-ordered insertion with that label to vanish. Our first retarded coefficient is \begin{equation} R_{1,1}(V(g;h),\phi(f)) =i\left(T_2(V(g;h),\phi(f))-V(g;h)\phi(f)\right). \label{eq:Rdef} \end{equation} This specifies the ordering and sign convention. For this pair only, the normalization requires: \begin{enumerate}[label=(N\arabic*)] \item the fixed vacuum Wick prescription at one insertion; \item chronological operator ordering on ordered separated supports and symmetry of the time-ordered product; \item the independent-field Wick expansion displayed below; \item extension of each numerical coefficient without increasing its actual fixed-mass relative scaling degree; \item distributional differentiation of normalized contractions, including their field-equation contacts. \end{enumerate} These conditions do not define all higher causal products. \subsection{Complete contraction list} Use \begin{align} W_m(z) &=\int\frac{\dd^3p}{(2\pi)^3\,2\omega_{\bp}} e^{-i\omega_{\bp}z^0+i\bp\cdot\mathbf z}, \nonumber\\ W_m^{\mathrm F}(z) &=\Theta(z^0)W_m(z)+\Theta(-z^0)W_m(-z). \label{eq:W} \end{align} With the stated Green convention, \begin{equation} W_m(z)-W_m(-z)=i\Delta_m(z),\qquad PW_m^{\mathrm F}=i\delta,\qquad W_m^{\mathrm F}(y-x)-W_m(y-x)=iG_R(x-y). \label{eq:Widentities} \end{equation} For a numerical distribution \(D(y-x)\), put \begin{align} \mathcal C_{\mu\nu}[D](y,x) ={}& \partial_\mu D(y-x)\partial_\nu\phi(y) +\partial_\nu D(y-x)\partial_\mu\phi(y) \nonumber\\ &-\eta_{\mu\nu} \left( \partial_\alpha D(y-x)\partial^\alpha\phi(y) +m^2D(y-x)\phi(y) \right). \label{eq:contraction} \end{align} Derivatives of \(D\) are in \(y\). If \(\mathcal N_{\mu\nu}(y,x)\) is the completely normal ordered expression formed from \(T_{\mu\nu}(\phi(y))\phi(x)\), then \begin{align} T_2(\wick{T_{\mu\nu}(y)},\phi(x)) &=\mathcal N_{\mu\nu}(y,x) +\mathcal C_{\mu\nu}[W_m^{\mathrm F}](y,x), \nonumber\\ \wick{T_{\mu\nu}(y)}\,\phi(x) &=\mathcal N_{\mu\nu}(y,x) +\mathcal C_{\mu\nu}[W_m](y,x). \label{eq:Wickexpansion} \end{align} There is at most one cross-contraction: the second insertion contains only one scalar field. Contracting either differentiated factor in the stress gives the first two terms of \eqref{eq:contraction}; the trace and mass terms give the rest. There is no double-contraction vacuum block. The uncontracted normal block has scalar degree three and will cancel in \eqref{eq:Rdef}. \subsection{Scaling degree and unique minimal extension} \begin{lemma}[Actual relative scaling degrees] \label{lem:scaling} In four relative coordinates, \[ \sd(W_m)=\sd(W_m^{\mathrm F})=2,\qquad \sd(\partial_\alpha W_m), \sd(\partial_\alpha W_m^{\mathrm F})\le3. \] The same upper bounds hold for \(G_R\) and its first derivatives. Every nonzero numerical coefficient in \eqref{eq:Wickexpansion} therefore has degree below four. \end{lemma} \begin{proof} Changing spatial momentum variables gives \[ W_m(\lambda z)=\lambda^{-2}W_{\lambda m}(z). \] The distributions \(W_\mu\) converge to \(W_0\) as \(\mu\downarrow0\). After pairing with a spacetime test, its spatial Fourier transform decreases rapidly in momentum, uniformly for time in a fixed compact interval. At small momentum, \[ (|\bp|^2+\mu^2)^{-1/2}\le|\bp|^{-1}, \] which is locally integrable in three dimensions. Dominated convergence applies. Splitting the time integration into positive and negative parts gives the same convergence for \(W_\mu^{\mathrm F}\). Positive dilations preserve those two time regions, so \[ W_m^{\mathrm F}(\lambda z) =\lambda^{-2}W_{\lambda m}^{\mathrm F}(z). \] The nonzero massless limits give degree two; differentiation adds at most one inverse power of \(\lambda\). Equation~\eqref{eq:Widentities} gives the retarded bounds. The mass \(m\) is fixed in this scaling operation. \end{proof} \begin{proposition}[Extension for this insertion pair] \label{prop:extension} Under the stated normalization, the numerical coefficients have unique extensions across \(x=y\). They are the distributions in \eqref{eq:W} and their first derivatives. No independent diagonal parameter occurs for this pair. \end{proposition} \begin{proof} Existence is supplied by the displayed full-space distributions. For uniqueness, a difference of extensions is supported at the relative origin. Such a distribution is a finite sum of derivatives of a delta distribution. Every nonzero term has scaling degree at least four, and cannot be added to a coefficient of degree at most three. This is the subcritical case of the extension criterion in \cite{BF}. \end{proof} The uniqueness is coefficientwise and normalization-specific. A weaker total engineering-dimension condition may admit additional terms. A change of the Wick stress, such as an improvement term, changes the named vertex rather than this extension problem. The contact \(PW_m^{\mathrm F}=i\delta\) appears after applying a field equation; it is fixed by \eqref{eq:Widentities}, not an independent coefficient. Figure~\ref{fig:subcritical-extension} illustrates the coefficientwise subcritical extension argument of \cite{BF}. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \draw[->] (0,0) -- (9.2,0) node[right, font=\scriptsize] {relative scaling degree}; \draw (2.1,0.08)--(2.1,-0.08) node[below, font=\scriptsize, align=center] {$2$\\ $W_m,\ W_m^{\mathrm F}$}; \draw (4.3,0.08)--(4.3,-0.08) node[below, font=\scriptsize, align=center] {$\le 3$\\ $\partial W$}; \draw[corange, thick] (6.4,-0.35)--(6.4,0.85); \node[cbox, font=\scriptsize, above] at (6.4,0.85) {threshold $4$}; \node[cbox, font=\scriptsize, anchor=west] at (6.7,0.35) {$\operatorname{sd}(\delta)=4$\\ $\operatorname{sd}(\partial^\alpha\delta)=4+|\alpha|$}; \node[sbox, text width=8.6cm, anchor=north west] at (0,-1.7) {% no nonzero origin-supported difference preserving coefficientwise actual scaling degree}; \node[cbox, text width=8.6cm, anchor=north west] at (0,-2.85) {% $PW_m^{\mathrm F}=i\delta$: fixed differentiated field-equation contact;\\ not an independent extension parameter}; \node[gbox, text width=8.6cm, anchor=north west] at (0,-4.15) {% this insertion pair only; fixed mass; specified coefficientwise normalization;\\ no conclusion for $T_2(V,V)$}; \end{tikzpicture} \caption{Subcritical extension for the specified insertion pair. Numerical coefficients have relative scaling degree at most three in four dimensions, whereas every nonzero origin-supported counterterm has degree at least four. Thus the stated minimal normalization has no independent diagonal parameter. Differentiated field-equation contacts are fixed by the normalized contractions \cite{BF}.} \label{fig:subcritical-extension} \end{figure} \subsection{Representation and the external coefficient} \begin{lemma}[Domain for the required products] \label{lem:vertexdomain} The compactly smeared quadratic Wick vertex preserves \(\Dphi\). The time-ordered and ordinary products in \eqref{eq:Rdef} are continuous forms in the finite-sector Schwartz topology. \end{lemma} \begin{proof} Creation-pair kernels of a compactly smeared quadratic Wick polynomial contain the Fourier transform of a compact smooth coefficient at energy \(\omega_{\bp}+\omega_{\bq}\). That energy controls \(|\bp|+|\bq|\), so these kernels, including their derivative factors, are Schwartz. The number-preserving kernel has polynomial growth along \(\bp=\bq\) and rapid decrease in \(\bp-\bq\), with the same property after differentiation. The inequality \[ 1+|\bp|\le(1+|\bp-\bq|)(1+|\bq|) \] shows that it maps Schwartz functions to Schwartz functions. These facts prove invariance of the algebraic finite-sector domain. Completely normal ordered matrix elements between vectors of this domain are smooth in spacetime, with all derivatives bounded on compact sets by finite Schwartz seminorms. For contraction terms, first smear \(D(y-x)\) in \(x\) against \(f\). The result is smooth in \(y\). Multiplication by the compact coefficient \(gh\), followed by integration by parts, leaves a linearly smeared scalar field. This proves the asserted form bounds. \end{proof} \begin{theorem}[External first retarded coefficient] \label{thm:external} For real compact \(g,f\), and real external \(h\in\Vfin\), \begin{equation} \boxed{ R_{1,1}(V(g;h),\phi(f)) =\phi(q_{g,h,f}),\qquad q_{g,h,f}=\Qop_{gh}u_f,\qquad u_f=-G_Af. } \label{eq:external} \end{equation} The test \(q_{g,h,f}\) is real and compact smooth. The unsmeared coefficient is \begin{equation} \phi^{(1)}_{g,h}=-G_R\Qop_{gh}\phi,\qquad P\phi^{(1)}_{g,h}=-\Qop_{gh}\phi=-L_{gh}\phi. \label{eq:YangFeldman} \end{equation} On the TT plane, \[ q_{g,h,f}=-\partial_i(gh^{ij}\partial_j u_f). \] \end{theorem} \begin{proof} In the subtraction \eqref{eq:Rdef}, the normal three-scalar block cancels exactly. The contraction difference is \(iG_R(x-y)\); the additional factor \(i\) in the definition produces \(-G_R(x-y)\). Contracting with \(gh^{\mu\nu}/2\) and integrating by parts in the compact vertex variable gives \(-G_R\Qop_{gh}\phi\). Smearing with \(f\) gives \[ -\int (G_Af)\Qop_{gh}\phi =\int u_f\Qop_{gh}\phi =\phi(\Qop_{gh}u_f), \] using formal self-adjointness and compact coefficients. This proves the sign and compact-test formula. Applying \(P\) gives the equation in \eqref{eq:YangFeldman}; \eqref{eq:QL} uses \(P\phi=0\). \end{proof} Figure~\ref{fig:external-contraction} organizes the Wick subtraction yielding the external linear retarded coefficient. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[sbox, text width=5.2cm] (v) {$V(g;h)=\tfrac12\int gh^{\mu\nu}:T_{\mu\nu}:$}; \node[sbox, right=8mm of v] (phi) {$\phi(f)$}; \node[pbox, text width=4.6cm, below=of v] (t2n) {$T_2$: normal three-scalar block}; \node[pbox, text width=4.6cm, below=of t2n] (t2c) {$T_2$: one cross-contraction $W_m^{\mathrm F}$}; \node[pbox, text width=4.6cm, below=of phi] (vn) {$V\phi$: same normal block}; \node[pbox, text width=4.6cm, below=of vn] (vc) {$V\phi$: one cross-contraction $W_m$}; \draw[arr] (v)--(t2n); \draw[arr] (phi)--(vn); \draw[arr] (t2n)--(t2c); \draw[arr] (vn)--(vc); \node[cbox, below=4mm of t2c, xshift=2.6cm] (sub) {$R_{1,1}=i(T_2-V\phi)$}; \node[gbox, font=\scriptsize, below=of sub] (can) {normal blocks cancel}; \node[pbox, text width=9.4cm, below=of can] (out) {% $\phi^{(1)}_{g,h}=-G_R\mathsf Q_{gh}\phi$,\quad $R_{1,1}=\phi(\mathsf Q_{gh}u_f)$\\ at most one cross-contraction; no double-contraction vacuum block\\ off-shell labels retained before free-field representation}; \draw[arr] (t2c.south)--(sub.west); \draw[arr] (vc.south)--(sub.east); \draw[arr] (sub)--(can); \draw[arr] (can)--(out); \end{tikzpicture} \caption{External first retarded coefficient. The two products have the same completely normal-ordered three-scalar block, which cancels in the retarded subtraction. Their single cross-contractions differ by \(iG_R\), leaving a linearly smeared free field with compact test \(\mathsf Q_{gh}(-G_Af)\). The off-shell insertion label is retained until the product is constructed.} \label{fig:external-contraction} \end{figure} The off-shell retarded-product framework is described in \cite{DF}; Figure~\ref{fig:external-contraction} depicts the single insertion pair calculated here. If \(\Pi_{\le M}\) projects onto scalar sectors at most \(M\), \begin{equation} \norm{\phi(q_{g,h,f})\Pi_{\le M}} \le(\sqrt M+\sqrt{M+1})\norm{\Kmap q_{g,h,f}}, \label{eq:scalarbound} \end{equation} and \[ |\pair\Psi{\phi(q_{g,h,f})\Chi}| \le2\norm{\Kmap q_{g,h,f}}\, \norm\Psi\,\norm{(N_\phi+1)^{1/2}\Chi}. \] These follow from the creation and annihilation estimates. For temporally ordered supports, \(T_2=\phi(f)V(g;h)\) when the vertex is earlier, so \(R_{1,1}=i[\phi(f),V(g;h)]\). For the opposite ordering \(T_2=V(g;h)\phi(f)\), so the retarded coefficient is zero. It also vanishes at spacelike separation. It is \(T_2\), not the retarded coefficient without its subtraction, that equals an ordered product. \subsection{Nonzero vacuum--one-particle witness} \begin{theorem}[External witness with zero free term] \label{thm:externalwitness} For any nonzero real TT wave \(h\) in the plane, there are real compact \(f,g\) and a normalized scalar \(v\in\Cc(\R^3)\) such that \[ \phi(f)=0,\qquad \pair\vac{ [\phi(f)+\kappa R_{1,1}(V(g;h),\phi(f))] a^\dagger(v)\vac} =\kappa c,\qquad c\ne0. \] \end{theorem} \begin{proof} Choose \(x_*=(t_*,\bx_*)\) and \(\bp_*\) such that \(h^{ij}(x_*)p_{*i}p_{*j}\ne0\). Choose a nonnegative nonzero smooth momentum bump \(\beta\) near \(\bp_*\) on which that quadratic expression has one sign, and set \[ v(\bp)=\|\beta\|^{-1} e^{i\omega_{\bp}t_*-i\bp\cdot\bx_*}\beta(\bp). \] For \[ w_v(x)=\pair\vac{\phi(x)a^\dagger(v)\vac}, \] the number \[ h^{ij}(x_*)\partial_i\partial_jw_v(x_*) =-\|\beta\|^{-1} \int\frac{h^{ij}(x_*)p_ip_j\beta(\bp)} {(2\pi)^{3/2}\sqrt{2\omega_{\bp}}}\dd^3p \] is real and nonzero. Its real part retains that sign on a small neighborhood. Take a nonnegative nonzero real test \(\rho\) supported there, set \(f=P\rho\), and choose compact real \(g=1\) near \(\supp\rho\). Then \(\phi(f)=0\), \(u_f=-\rho\), and \(q_{g,h,f}=-\Qop_h\rho\). Hence \[ \pair\vac{\phi(q_{g,h,f})a^\dagger(v)\vac} =-\int\rho\,\Qop_hw_v =\int\rho h^{ij}\partial_i\partial_jw_v\ne0. \] This also proves that the mass-shell vector of \(q_{g,h,f}\) is nonzero. \end{proof} The witness uses the off-shell insertion label \(P\rho\). It is consistent with \(\phi(P\rho)=0\), because the time-ordered contraction satisfies \(PW_m^{\mathrm F}=i\delta\). Scalar parity excludes vacuum--two-particle and one--one witnesses for this linear coefficient. \subsection{Switching and the external gauge defect} The same coefficient construction applies to any smooth external symmetric tensor, because its product with \(g\) is compact. For a smooth gauge parameter, \[ L_{K\xi}=-[P,X_\xi]. \] Using \eqref{eq:switch} and \(P\phi=0\), \[ \Qop_{gK\xi}\phi =-gP(X_\xi\phi) -(\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi. \] Therefore \begin{equation} \boxed{ \delta_\xi\phi^{(1)}_g =gX_\xi\phi+ G_R\left[ (\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi -[P,g]X_\xi\phi \right], } \label{eq:gaugedefect} \end{equation} where \[ [P,g]a=(\Box g)a+2(\partial^\mu g)\partial_\mu a. \] The proof uses \(gP(X_\xi\phi)=P(gX_\xi\phi)-[P,g]X_\xi\phi\) and \(G_RP(gX_\xi\phi)=gX_\xi\phi\), valid on compactly supported domain matrix elements. Smearing gives \begin{align} \delta_\xi R_{1,1} ={}&\int fgX_\xi\phi +\int u_f[P,g]X_\xi\phi \nonumber\\ &-\int u_f(\partial_\mu g) \overline{K\xi}^{\mu\nu}\partial_\nu\phi. \label{eq:gaugesmeared} \end{align} Each term is a linearly smeared scalar field after integration by parts. It is generally nonzero even if \(g=1\) near \(\supp f\): switching terms in the causal past may remain, and the scalar coordinate variation is not a gauge-invariant observable by itself. For fixed compact supports, \((g,f)\mapsto q_{g,h,f}\) is continuous into compact smooth tests, hence into Schwartz mass-shell vectors. This implies continuity in \eqref{eq:scalarbound}. If \(g_1-g_2\) vanishes on a neighborhood of \(J^-(\supp f)\cap\supp h\), the coefficients coincide. That set need not be compact for a radiative \(h\). No exhausting-switch limit follows. Figure~\ref{fig:external-witness-switch} distinguishes the free-null insertion witness from the causal dependence on the interaction switch. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[font=\scriptsize\bfseries, sgray] at (-2.6,0.35) {(a) schematic}; \node[pbox, text width=5.4cm] (a) {% $f=P\rho$,\quad $\phi(f)=0$,\quad $u_f=-\rho$\\ $q_{g,h,f}=-\mathsf Q_{gh}\rho$\\[1mm] $\langle\Omega,\phi(q_{g,h,f})a^\dagger(v)\Omega\rangle\neq 0$\\ $g=1$ near $\operatorname{supp}\rho$\\[1mm] off-shell label before representation;\\ fixed contact retained}; \node[font=\scriptsize\bfseries, sgray, anchor=west] at (3.6,0.35) {(b) schematic}; \draw[sgray] (4.0,-1.6) -- (8.3,-1.6); \draw[sgreen!40, fill=sgreen!12] (6.3,-0.55) ellipse (0.55 and 0.32); \node[font=\scriptsize, sgreen!70!black] at (6.3,-0.55) {$\operatorname{supp} f$}; \draw[sgray, dashed] (5.4,-1.6) -- (5.75,-0.7) -- (6.85,-0.7) -- (7.2,-1.6); \node[font=\scriptsize, sgray] at (6.3,-1.15) {causal past}; \node[cbox, font=\scriptsize] at (7.7,-0.35) {$\partial g\neq 0$}; \node[sbox, text width=5.2cm, anchor=north] at (6.15,-1.9) {% $\mathsf Q_{gh}\phi=gL_h\phi-(\partial_\mu g)\bar h^{\mu\nu}\partial_\nu\phi$\\[1mm] $g=1$ near $\operatorname{supp} f$ does not remove all past-switching terms\\ compact-support continuity is not $g\to 1$}; \end{tikzpicture} \caption{External witness and switching. A free-null insertion label \(P\rho\) can have a nonzero first retarded coefficient because time ordering is formed before free-field representation. Separately, derivatives of the compact interaction switch contribute through the causal past; constancy near the final smearing does not generally eliminate them or establish an adiabatic limit.} \label{fig:external-witness-switch} \end{figure} %===================================================================== \section{The one-oscillator mixed insertion} \label{sec:mixed} %===================================================================== \subsection{One real symplectic plane and one oscillator} For this section assume the declared real TT plane is symplectic for \[ \sigma_{\TT}(e,r) =\int_{\R^3} (e_{ij}\dot r^{ij}-\dot e_{ij}r^{ij})\dd^3x, \] and choose its basis with \(\sigma_{\TT}(e,r)=1\). An arbitrary independent plane need not be symplectic; the restricted-current theorem did not need this extra assumption. The example with data \((a,0)\) and \((0,a)\), normalized by \(\int a_{ij}a^{ij}=1\), is symplectic. A real Darboux plane has one canonical oscillator. Let \[ Q=\frac{b+b^\dagger}{\sqrt2},\qquad \Pi=\frac{b-b^\dagger}{i\sqrt2},\qquad [Q,\Pi]=i\one, \] and use \[ \Hil=\Fock(\C)\otimes\Fock(\mathfrak h_\phi) \simeq L^2(\R)\otimes\Fock(\mathfrak h_\phi). \] The mixed core is \begin{equation} \Dmix= \bigoplus_{j,n\ge0}^{\mathrm{alg}} \C|j\rangle\otimes\Sch_{\mathrm{sym}}(\R^{3n}). \label{eq:mixedcore} \end{equation} Let \(N=N_g+N_\phi\). The oscillator-valued metric is \[ \widehat h_{\mu\nu}(x)=Qe_{\mu\nu}(x)+\Pi r_{\mu\nu}(x). \] It commutes with the scalar algebra, but \begin{equation} [\widehat h_{\mu\nu}(x),\widehat h_{\alpha\beta}(y)] =i\left(e_{\mu\nu}(x)r_{\alpha\beta}(y) -r_{\mu\nu}(x)e_{\alpha\beta}(y)\right)\one. \label{eq:nonlocalcomm} \end{equation} This smooth commutator is not generally supported on causally related pairs. The model is not a local relativistic metric field and does not support an unrestricted local-field causal induction. The product constructed here has only one metric factor. Its commutator with the scalar insertion inherits causal support from the scalar commutator, so \eqref{eq:nonlocalcomm} does not enter that calculation. \subsection{Mixed Wick product and retarded coefficient} Product-vacuum normal ordering gives \[ \wick{\widehat h^{\mu\nu}T_{\mu\nu}(\phi)} =\widehat h^{\mu\nu}\wick{T_{\mu\nu}(\phi)}. \] There are no oscillator--scalar contractions and no pair of metric factors at this insertion. For real compact \(g\), \[ V(g) =Q\otimes V_e(g)+\Pi\otimes V_r(g), \qquad V_a(g)=\frac12\int ga^{\mu\nu} \wick{T_{\mu\nu}(\phi)}. \] The vertex preserves \(\Dmix\), by Lemma~\ref{lem:vertexdomain} and finite oscillator number shifts. It is symmetric there. No essential self-adjointness claim is made for this mixed cubic vertex. In its product with one scalar insertion the normal uncontracted block has one oscillator and three scalar factors. It cancels in the retarded subtraction. The surviving block has one oscillator and one scalar. All numerical coefficients are the same ones as in Section~\ref{sec:external}; multiplication by matrix elements of \(e,r\) is smooth and does not raise relative scaling degree. \begin{theorem}[Mixed first retarded coefficient] \label{thm:mixed} For real compact \(g,f\), put \[ q_a=\Qop_{ga}u_f,\qquad v_a=\Kmap q_a,\qquad a=e,r, \quad u_f=-G_Af. \] Then on \(\Dmix\), \begin{equation} \boxed{ \mathcal R_{g,f}:=R_{1,1}(V(g),\phi(f)) =Q\otimes\Phi(v_e)+\Pi\otimes\Phi(v_r). } \label{eq:mixed} \end{equation} The unsmeared coefficient is \[ \phi_g^{(1)} =-Q\otimes G_R\Qop_{ge}\phi -\Pi\otimes G_R\Qop_{gr}\phi. \] The normalization has no extra diagonal parameter for this insertion pair. \end{theorem} \begin{proof} Apply the contraction calculation of Theorem~\ref{thm:external} separately to \(e\) and \(r\), retaining the commuting oscillator factors. The same normal-block cancellation and the same unique scalar contraction extensions give the result. \end{proof} For clarity about the particle blocks, define \[ w_+=\frac{v_e+iv_r}{\sqrt2},\qquad w_-=\frac{v_e-iv_r}{\sqrt2}. \] Antilinearity of annihilation gives \[ \mathcal R_{g,f} =b\,a(w_+)+b\,a^\dagger(w_-) +b^\dagger a(w_-)+b^\dagger a^\dagger(w_+). \] These are mixed pair annihilation, two exchange terms, and mixed pair creation. They are finite-rank cross kernels, not estimates for a dense gravitational one-particle space. Figure~\ref{fig:mixed-blocks-locality} displays the mixed particle blocks and the finite-mode locality limitation. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[mbox, text width=7.4cm] (h) {$\widehat h(x)=Qe(x)+\Pi r(x)$,\quad $[Q,\Pi]=iI$\\ plane symplectic in this section}; \node[pbox, text width=7.4cm, below=of h] (r) {$\mathcal R_{g,f}=Q\otimes\Phi(v_e)+\Pi\otimes\Phi(v_r)$}; \draw[arr] (h)--(r); \node[pbox, text width=2.5cm, below left=6mm and 2mm of r] (b1) {$ba(w_+)$\\ pair annihilation}; \node[pbox, text width=2.5cm, right=2mm of b1] (b2) {$ba^\dagger(w_-)$\\ exchange}; \node[pbox, text width=2.5cm, right=2mm of b2] (b3) {$b^\dagger a(w_-)$\\ exchange}; \node[pbox, text width=2.5cm, right=2mm of b3] (b4) {$b^\dagger a^\dagger(w_+)$\\ pair creation}; \node[gbox, font=\scriptsize, below=2mm of b2] (w) {$w_\pm=(v_e\pm iv_r)/\sqrt{2}$}; \node[gbox, font=\scriptsize, below=of w] (fin) {finite-rank cross kernels; no dense graviton space}; \node[cbox, text width=10.8cm, below=of fin] (loc) {% metric commutator of \eqref{eq:nonlocalcomm} is not generally causal-support restricted\\ this scalar insertion uses only one metric factor: no metric--metric contraction}; \end{tikzpicture} \caption{One-oscillator retarded coefficient. The scalar contraction calculation leaves one oscillator factor and one scalar factor, giving four mixed particle blocks. These are finite-rank cross kernels. The oscillator-valued metric is not a local relativistic metric field, although its commutator with the scalar insertion has the causal support required for this single calculation.} \label{fig:mixed-blocks-locality} \end{figure} \subsection{Exact switched bilinear and unswitched remainder} Choose \(t_0\) earlier than the temporal supports of both \(g\) and \(f\). For a classical \(h=qe+\pi r\), define \[ B_f^{g,t_0}(h,\varphi) =\int f\,S_{t_0}(gh)\varphi. \] Quantize this bilinear by replacing \(q,\pi\) by \(Q,\Pi\) after evaluating its scalar linear functionals. \begin{theorem}[Switched identity] \label{thm:switched} For compact real \(g,f\), \begin{equation} \widehat B_f^{g,t_0}=\mathcal R_{g,f} \quad\text{on }\Dmix. \label{eq:switchedidentity} \end{equation} This equality uses \(L_{ga}\), including derivatives of the switch. \end{theorem} \begin{proof} On free scalar matrix elements, \(\Qop_{ga}\phi=L_{ga}\phi\). Since \(ga\) is supported after \(t_0\), \[ -G_R\Qop_{ga}\phi=S_{t_0}(ga)\phi. \] Smearing and using Theorem~\ref{thm:mixed} proves the identity. The retarded product was constructed before this Duhamel comparison. \end{proof} For the unswitched fixed-time pairing choose \(T\) later than \(\supp f\), and set \begin{equation} q_a^0(t,\bx) =-\one_{[t_0,T]}(t) \partial_i(a^{ij}\partial_j u_f), \qquad v_a^0=\Kmap q_a^0. \label{eq:unswitchedvectors} \end{equation} Although \(q_a^0\) may jump at \(t_0\), it has compact spacetime support and is uniformly spatially smooth on the finite time slab. Its spatial Fourier transform decreases rapidly with all spatial derivatives. Momentum derivatives of the mass-shell exponential introduce bounded-time and polynomially controlled factors, so \(v_a^0\in\Sch(\R^3)\). This defines \[ \widehat B_f^{0,t_0} =Q\otimes\Phi(v_e^0)+\Pi\otimes\Phi(v_r^0). \] It is a mode-integral definition for these specific nonsmooth tests, not an extension of the free field to arbitrary distributions. The exact remainder is \begin{equation} \boxed{ \mathcal R_{g,f}-\widehat B_f^{0,t_0} =Q\otimes\Phi(v_e-v_e^0)+ \Pi\otimes\Phi(v_r-v_r^0), } \label{eq:remainder} \end{equation} where \[ v_a-v_a^0 =\Kmap\left( \one_{[t_0,T]}\Qop_{(g-1)a}u_f \right). \] Equivalently it is the quantization of \(\int fS_{t_0}((g-1)h)\varphi\). This is a switching and initial-time remainder, not a renormalization ambiguity. If one incorrectly substitutes \(gL_a\) for \(L_{ga}\), the missing contribution is exactly \begin{equation} -\sum_{a=e,r}O_a\otimes \int u_f(\partial_i g)a^{ij}\partial_j\phi, \qquad O_e=Q,\quad O_r=\Pi. \label{eq:missingswitch} \end{equation} Constancy of \(g\) merely near \(\supp f\) does not generally make this past-sampling term vanish. \subsection{Number graph and essential self-adjointness} \begin{theorem}[Mixed bound and real polynomial closure] \label{thm:ESA} For compact real \(g,f\), let \[ C_{g,f}=\sqrt2(\norm{v_e}+\norm{v_r}). \] Then \(\mathcal R_{g,f}\) is symmetric and preserves \(\Dmix\), and \begin{equation} \norm{\mathcal R_{g,f}\Psi} \le C_{g,f}\norm{(N+2)\Psi}. \label{eq:mixedbound} \end{equation} It extends continuously from \(\Dom(N)\), with its graph norm, to \(\Hil\), and is essentially self-adjoint on \(\Dmix\). For every real \(\lambda\), the real polynomial \begin{equation} A_f^g(\lambda) =\one\otimes\phi(f)+\lambda\mathcal R_{g,f} \label{eq:polynomial} \end{equation} is also essentially self-adjoint on \(\Dmix\). This is a statement about the displayed polynomial, not an all-orders interacting field. \end{theorem} \begin{proof} The oscillator estimates and scalar estimates are \[ \norm{Q\chi},\norm{\Pi\chi} \le\sqrt2\norm{(N_g+1)^{1/2}\chi}, \qquad \norm{\Phi(v)\psi} \le2\norm v\,\norm{(N_\phi+1)^{1/2}\psi}. \] They act on commuting factors. Using \[ \sqrt{(N_g+1)(N_\phi+1)}\le\frac{N+2}{2} \] proves \eqref{eq:mixedbound}. Schwartz \(v_e,v_r\) and finite oscillator shifts prove invariance. Reality gives symmetry. The domain is a core for \(N\): truncate particle number and approximate each remaining scalar wavefunction in Schwartz space. Completion gives the graph extension. Let \(A\) be either operator in the theorem. There is \(C_A<\infty\) such that \(\|A\Psi\|\le C_A\|(N+2)\Psi\|\), and \(A\) raises total particle number by at most two. If \(\Psi\) has sectors at most \(M\), induction gives \begin{align} \norm{A^k\Psi} &\le C_A^k\prod_{j=0}^{k-1}(M+2j+2)\norm\Psi \nonumber\\ &=(2C_A)^k \frac{\Gamma(k+M/2+1)}{\Gamma(M/2+1)}\norm\Psi. \label{eq:analytic} \end{align} Thus every vector of \(\Dmix\) is analytic for \(A\). Nelson's analytic-vector criterion \cite{Nelson} proves essential self-adjointness. \end{proof} In particular, \[ \norm{\mathcal R_{g,f}\Pi_{\le M}} \le C_{g,f}(M+2). \] The closed operator agrees on \(\Dom(N)\) with the number-graph extension, by core approximation and closedness. The switched bilinear and the retarded coefficient have the same common-core action and hence the same closure. The remainder satisfies \begin{equation} \norm{(\mathcal R_{g,f}-\widehat B_f^{0,t_0})\Pi_{\le M}} \le\sqrt2(M+2) \left(\norm{v_e-v_e^0}+\norm{v_r-v_r^0}\right). \label{eq:remainderbound} \end{equation} No numerical constant calculated from an unswitched observable is transferred without controlling these differences. Figure~\ref{fig:switched-remainder} separates the exact switched identity from the controlled comparison with the unswitched fixed-time response. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[pbox, text width=5.4cm] (sw) {% switched: compact $g$,\ $t_0$ before its support\\ $R_{1,1}(V(g),\phi(f))=\widehat B_f^{g,t_0}$\\ $L_{ga}$ includes derivatives of $g$}; \node[mbox, text width=5.4cm, right=8mm of sw] (un) {% unswitched\\ $\widehat B_f^{0,t_0}=Q\Phi(v_e^0)+\Pi\Phi(v_r^0)$}; \node[cbox, text width=11.4cm, below=6mm of sw, xshift=3.1cm] (rem) {% $\mathcal R_{g,f}-\widehat B_f^{0,t_0} =Q\Phi(v_e-v_e^0)+\Pi\Phi(v_r-v_r^0)$\\[1mm] fixed-number bound of \eqref{eq:remainderbound}\\ switching/initial-time remainder, not renormalization ambiguity}; \draw[arr] (sw.south)--(rem.north west); \draw[arr] (un.south)--(rem.north east); \node[cbox, below=of rem] (bad) {replace $L_{ga}$ by $gL_a$}; \draw[corange, thick] (bad.north west)--(bad.south east); \draw[corange, thick] (bad.north east)--(bad.south west); \node[sbox, text width=8.5cm, right=4mm of bad] (miss) {$-\sum_a O_a\int u_f(\partial_i g)a^{ij}\partial_j\phi$}; \end{tikzpicture} \caption{Switched identity and unswitched comparison. The first retarded coefficient equals the quantized switched Duhamel bilinear when the initial time precedes the vertex support. Comparison with the unswitched response has an explicit mode-vector remainder. Derivatives of the switch are essential, and the remainder is not an extension ambiguity.} \label{fig:switched-remainder} \end{figure} \subsection{Same-scalar coherent displacement} For a one-particle vector \(z\), let \[ \mathsf E(z)=e^{-\|z\|^2/2} \bigoplus_{n\ge0}\frac{z^{\otimes n}}{\sqrt{n!}}. \] Coherent vectors are not in the algebraic core unless \(z=0\), but belong to \(\Dom(N^k)\) for every \(k\). For Schwartz \(z\), their number truncations lie in \(\Dmix\) and converge in number-graph norms. \begin{theorem}[Mixed coherent witness] \label{thm:coherent} There exist compact real \(f,g\) and a nonzero scalar Schwartz vector \(z\) such that, for every real \(s\ne0\), the preparations \[ \Psi_0=\mathsf E(0)\otimes\mathsf E(z),\qquad \Psi_s=\mathsf E(s/\sqrt2)\otimes\mathsf E(z) \] satisfy \[ \pair{\Psi_s}{A_f^g(\kappa)\Psi_s} -\pair{\Psi_0}{A_f^g(\kappa)\Psi_0} =\kappa s\,d,\qquad d>0. \] The scalar preparation is the same in both states. The choice can have \(\phi(f)=0\). \end{theorem} \begin{proof} Apply Theorem~\ref{thm:externalwitness} with \(h=e\). It supplies \(f=P\rho\) and compact \(g\) for which \(v_e=\Kmap\Qop_{ge}u_f\ne0\). Take \(z=v_e\). Then \[ \langle\Phi(v_e)\rangle_{\mathsf E(z)} =2\operatorname{Re}\pair{v_e}z =2\norm{v_e}^2=:d>0. \] For the real oscillator displacement, \[ \langle Q\rangle_{\mathsf E(s/\sqrt2)}=s,\qquad \langle\Pi\rangle_{\mathsf E(s/\sqrt2)}=0. \] Both means are zero at oscillator vacuum. Use \eqref{eq:mixed}. All expectations are legitimate by the graph extension in Theorem~\ref{thm:ESA}. \end{proof} This is a witness for the represented polynomial. It is not a measurement implementation or a finite-coupling remainder theorem for an interacting theory. Figure~\ref{fig:analytic-coherent-witness} connects the number-graph estimates to the polynomial closure and its coherent witness. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[gbox, text width=6.2cm] (core) {% common core $\mathscr D_{\mathrm{mix}}$\\ finite occupied sectors\\ no universal particle ceiling}; \node[pbox, text width=6.2cm, below=of core] (bd) {% $\|\mathcal R_{g,f}\Psi\|\le C_{g,f}\|(N+2)\Psi\|$\\ number shifts of at most two}; \node[pbox, text width=6.2cm, below=of bd] (an) {iterated bound \eqref{eq:analytic}\\ dense analytic vectors}; \node[pbox, below=of an] (esa) {essential self-adjointness}; \node[gbox, font=\scriptsize, below=of esa] (app) {applies to $\mathcal R_{g,f}$ and $A_f^g(\lambda)$, not the mixed cubic vertex}; \draw[arr] (core)--(bd); \draw[arr] (bd)--(an); \draw[arr] (an)--(esa); \draw[arr] (esa)--(app); \node[sbox, text width=5.6cm, right=8mm of core] (psi) {% $\Psi_0=\mathsf E(0)\otimes\mathsf E(z)$\\ $\Psi_s=\mathsf E(s/\sqrt2)\otimes\mathsf E(z)$\\ $z=v_e$,\ $d=2\|v_e\|^2>0$\\ $\Delta\langle A_f^g(\kappa)\rangle=\kappa s d$}; \node[gbox, font=\scriptsize, below=of psi] (note) {same scalar factor; represented polynomial witness; no measurement instrument}; \end{tikzpicture} \caption{Polynomial closure and coherent witness. Number-graph bounds and shifts of at most two give analytic vectors on the common core, yielding essential self-adjointness by Nelson's criterion \cite{Nelson}. Coherent vectors lie in the number graph domain and exhibit a nonzero same-scalar displacement witness. Neither conclusion constructs an interacting Hamiltonian or a measurement implementation.} \label{fig:analytic-coherent-witness} \end{figure} \subsection{External gauge comparison and compact-switch continuity} Theorem~\ref{thm:intersection} shows that a nonzero pure gauge is not an internal change within the declared TT plane. An external replacement \[ \widehat h\mapsto\widehat h+(K\xi)\one \] is nevertheless a defined change of the switched vertex. Its coefficient changes by \(\one_g\) times \eqref{eq:gaugedefect}, or \eqref{eq:gaugesmeared} after smearing. It is not represented as a unitary change of the oscillator coordinates. For fixed compact supports, \((g,f)\mapsto v_e,v_r\) is continuous into Schwartz space. Thus the mixed coefficient is continuous in the number-graph bound and in each fixed particle-cutoff norm. This does not control \(g\to1\). %===================================================================== \section{Contrast: the Newton--wave cubic form} \label{sec:contrast} %===================================================================== The object in this section is not a retarded product. It is obtained by first inserting a classical scalar stress into a specified nonlocal metric inverse, then applying the fixed-time scalar response, and finally fully Wick ordering the resulting cubic kernel. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.19\linewidth}XX@{}} \toprule & External or finite-mode \(R_{1,1}\) & Newton--wave \(\cC_f^{(3)}\)\\ \midrule Metric & Prescribed \(h\) or \(Qe+\Pi r\) & Reconstructed \(\cR^A_{t_0}T(\phi)\)\\ Quantum degree & One, or mixed two & Three\\ Kernel issue & Two-point contractions, \(\sd\le3\) & Newton transfer \(1/|\bq|^2\)\\ Extension & Unique coefficientwise minimal extension & Locally integrable fully Wick kernel\\ Witness & Vacuum--one-particle or coherent displacement & Vacuum--three-particle away from transfers\\ Time control & Compact switch \(g\) & Sharp \(t_0\); separate response-edge regulator\\ Bilinear comparison & Mixed coefficient equals switched bilinear & A different composed observable\\ \bottomrule \end{tabularx} \end{center} We provide the definitions and kernel estimates needed for this contrast, rather than invoking a result from a companion manuscript. \subsection{Option A inverse and its constraints} For \(P\phi=0\), use the stress from Section~\ref{sec:external} and its polarization \[ \cT_{\mu\nu}(v,w) =\partial_{(\mu}v\,\partial_{\nu)}w -\frac12\eta_{\mu\nu} \left(\partial_\alpha v\,\partial^\alpha w+m^2vw\right). \] Differentiation gives \[ \partial^\mu\cT_{\mu\nu}(v,w) =\frac12\left((Pv)\partial_\nu w+(Pw)\partial_\nu v\right). \] Thus it is conserved for free scalar solutions. Take these solutions to have Schwartz Cauchy data. On bounded time intervals the polarized sources are smooth Schwartz functions in space. Fix the Newton inverse \[ \Delta^{-1}b(x) =-\frac1{4\pi}\int\frac{b(y)}{|x-y|}\dd^3y, \qquad \widehat{\Delta^{-1}b}(\bq) =-\frac{\widehat b(\bq)}{|\bq|^2}, \] with its locally integrable multiplier and no added zero-supported Fourier distribution. For a conserved source \(Q_{\mu\nu}\), let \[ \rho=Q_{00}(t_0),\quad J_i=Q_{0i}(t_0),\quad a=-2\Delta^{-1}\rho,\quad V_i=2\Delta^{-1}J_i, \] \[ B_{ij}=\partial_iV_j+\partial_jV_i -\delta_{ij}\partial_\ell V_\ell. \] Specify trace-reversed metric data by \[ \mathsf a_{00}=a,\quad \mathsf a_{0i}=\mathsf a_{ij}=0, \qquad \mathsf b_{ij}=B_{ij},\quad \mathsf b_{00}=\mathsf b_{0i}=0. \] Let \(\mathsf r\) denote trace reversal. Define \begin{align} \cR^{A,\init}_{t_0}Q(t) &=\mathsf r\left[ \cos((t-t_0)|D|)\mathsf a+\Sw(t-t_0)\mathsf b \right], \nonumber\\ \cR^{A,\src}_{t_0}Q(t) &=2\mathsf r\int_{t_0}^t\Sw(t-s)Q(s)\dd s, \qquad j=\cR^A_{t_0}Q. \label{eq:RA} \end{align} Then \(\Box\bar j=-2Q\). \begin{lemma}[Constraint-compatible inverse] \label{lem:RAconstraints} For the above conserved sources, \(D(j)=0\). In this gauge \(\LE j=Q\). \end{lemma} \begin{proof} The initial data give \(D_\nu(j)(t_0)=0\). Also \[ \Delta a=-2\rho,\qquad \partial_iB_{ij}=\Delta V_j=2J_j. \] Using \(\partial_t^2\bar j=\Delta\bar j+2Q\) gives \(\partial_tD_\nu(j)(t_0)=0\). Conservation implies \(\Box D_\nu(j)=0\). Wave uniqueness proves the first assertion; the second follows from \eqref{eq:LEidentities}. \end{proof} The classical composition and its polarization are \begin{equation} \cC_f(\phi)=B_f(\cR^A_{t_0}T(\phi),\phi), \qquad \cC_f(\phi_1,\phi_2,\phi_3) =\frac13\sum_{\mathrm{cyc}} B_f(\cR^A_{t_0}\cT(\phi_1,\phi_2),\phi_3). \label{eq:cubicclassical} \end{equation} Here \(f\in\Cc((t_0,T)\times\R^3)\). Newton potentials of Schwartz sources are smooth with polynomially bounded derivatives. Finite-time wave propagation preserves such bounds. Multiplication by derivatives of a Schwartz scalar solution gives Schwartz response sources. The finite Duhamel integrals therefore define \eqref{eq:cubicclassical}. Figure~\ref{fig:Option-A-composition} displays the inverse used in the cubic composition and the distinct comparison prescription. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[sbox] (q) {conserved scalar stress $Q=T(\phi)$}; \node[mbox, text width=5.0cm, below left=8mm and 1mm of q] (a) {% Option A, used\\ position $(a,0,0)$\\ velocity $(0,0,B_{ij})$}; \node[gbox, text width=5.0cm, below right=8mm and 1mm of q] (b) {% Option B, comparison only\\ position $(a,-V_i,0)$\\ velocity $(-\partial_i V_i,0,0)$}; \draw[arr] (q)--(a); \draw[arrg] (q)--(b); \node[gbox, font=\scriptsize, below=3mm of b] (same) {same sourced wave integral\\ different initial data, generally different inverse maps\\ scalar responses not labeled equal}; \node[mbox, below=of a] (j) {$j=\mathcal R^A T(\phi)$\\ degree two}; \node[sbox, below=of j] (bf) {$B_f(j,\phi)$\\ degree three}; \node[sbox, font=\scriptsize, below=of bf] (split) {$j=j^{\mathrm{init}}+j^{\mathrm{src}}$\\ $D(j)=0$,\ $L_{\mathrm E}j=Q$}; \draw[arr] (a)--(j); \draw[arr] (j)--(bf); \draw[arr] (bf)--(split); \node[cbox, text width=6.4cm, right=5mm of split] (not) {not the external metric\\ not $Qe+\Pi r$\\ not $R_{1,1}$}; \end{tikzpicture} \caption{Specified inverse and cubic composition. Option A combines constraint-compatible Newton initial data with the sourced wave integral. Feeding scalar stress into this inverse and then into the scalar response gives a cubic functional. Option B has different initial data and is not substituted into the theorem. Neither reconstructed metric is the metric argument defining the earlier retarded coefficients.} \label{fig:Option-A-composition} \end{figure} \subsection{Explicit signed kernels} Let \[ N_{\bp}=(2\pi)^{-3/2}(2\omega_{\bp})^{-1/2},\qquad u_\sigma(x;\bp)=N_{\bp}e^{i\sigma\omega_{\bp}t-i\sigma\bp\cdot\bx}, \] with \(a_+=a^\dagger,\ a_-=a\). For a signed triple define \[ k_{a\mu}=(\sigma_a\omega_a,-\sigma_a\bp_a),\quad e_{12}=\sigma_1\omega_1+\sigma_2\omega_2,\quad \bq=\sigma_1\bp_1+\sigma_2\bp_2,\quad r=|\bq|, \quad \bw=\bq+\sigma_3\bp_3. \] The stress numerator is \[ n_{\mu\nu} =-k_{1(\mu}k_{2\nu)} +\frac12\eta_{\mu\nu}(k_1\cdot k_2-m^2). \] In particular, \begin{align} \rho_{12}=n_{00} &=\frac12\left[ m^2-\sigma_1\sigma_2(\omega_1\omega_2+\bp_1\cdot\bp_2) \right], \nonumber\\ J_{12,i}=n_{0i} &=\frac{\sigma_1\sigma_2}{2} (\omega_1p_{2i}+\omega_2p_{1i}), \qquad e_{12}n_{0\nu}+q_i n_{i\nu}=0. \label{eq:numerators} \end{align} Remove the common mode factors and the spatial wave \(e^{-i\bq\cdot\bx}\) from the trace-reversed metric. Writing \(E_0=e^{ie_{12}t_0}\) and \(s_0(\tau,r)=\sin(\tau r)/r\), with value \(\tau\) at zero, the remaining amplitudes are \begin{align} Z^{\init}_{00}(s) &=\frac{2E_0\rho_{12}}{r^2}\cos((s-t_0)r), \qquad Z^{\init}_{0i}=0, \nonumber\\ Z^{\init}_{ij}(s) &=\frac{2iE_0}{r^2}s_0(s-t_0,r) \left(q_iJ_{12,j}+q_jJ_{12,i} -\delta_{ij}\bq\cdot\mathbf J_{12}\right), \nonumber\\ Z^{\src}_{\mu\nu}(s) &=2n_{\mu\nu}\int_{t_0}^s s_0(s-u,r)e^{ie_{12}u}\dd u. \label{eq:Z} \end{align} For either part set \[ d_\nu[Z]=-\partial_sZ_{0\nu}-iq_iZ_{i\nu}, \qquad \mathcal L_3[Z] =(\mathsf rZ)^{\mu\nu}k_{3\mu}k_{3\nu} -i\,d[Z]^\nu k_{3\nu}. \] These retain the full \(D(j)\) term before summation. Directly from \eqref{eq:numerators}, \begin{equation} d_\nu[Z^{\init}] =2E_0s_0(s-t_0,r)n_{0\nu} =-d_\nu[Z^{\src}]. \label{eq:dcancel} \end{equation} For the spatial part use \(q_i(q_iJ_j+q_jJ_i-\delta_{ij}q\cdot J)=r^2J_j\); for the source part integrate once in \(u\). Thus the individual divergence terms are bounded and cancel in the total. Let \[ F_f(t,\bk)=\int e^{-i\bk\cdot\bx}f(t,\bx)\dd^3x, \qquad s_m(\tau,\bw)=\frac{\sin(\tau\omega_{\bw})}{\omega_{\bw}}. \] For each of the eight signs define \begin{align} U_f^\sigma ={}&N_{\bp_1}N_{\bp_2}N_{\bp_3} \int_{t_0}^T\dd t\int_{t_0}^t\dd s\, F_f(t,\bw)s_m(t-s,\bw)e^{i\sigma_3\omega_3s} \nonumber\\ &\hspace{9em}\times \mathcal L_3[Z^{\init}+Z^{\src}](s), \nonumber\\ K_f^\sigma ={}&\frac13\left( U_f^{\sigma_1\sigma_2\sigma_3}(\bp_1,\bp_2,\bp_3) +U_f^{\sigma_2\sigma_3\sigma_1}(\bp_2,\bp_3,\bp_1) +U_f^{\sigma_3\sigma_1\sigma_2}(\bp_3,\bp_1,\bp_2) \right). \label{eq:cubickernel} \end{align} This is a finite-integral definition of the complete symmetric kernel, including all mode normalizations. The first two arguments of \(U\) are symmetric; the cyclic average is therefore fully symmetric in signed arguments. \subsection{Transfer behavior} \begin{lemma}[Transfer singularity and polynomial bound] \label{lem:cubicbound} Put \[ R=\left(1+|\bp_1|^2+|\bp_2|^2+|\bp_3|^2\right)^{1/2}. \] There is \(C_f<\infty\) such that \begin{equation} |K_f^\sigma| \le C_fR^5 \left(1+\sum_{i] (0.2,-1.3) -- (1.8,-0.4) node[midway, above, font=\scriptsize] {$\sigma_1\mathbf p_1$}; \draw[mblue, ->] (0.2,-1.3) -- (1.8,-2.2) node[midway, below, font=\scriptsize] {$\sigma_2\mathbf p_2$}; \node[mbox, font=\scriptsize] at (2.5,-1.3) {$\mathbf q=\sigma_1\mathbf p_1+\sigma_2\mathbf p_2$\\ $r=|\mathbf q|$}; \draw[sbox, ->] (3.9,-1.3) -- (5.5,-1.3) node[midway, above, font=\scriptsize] {$\sigma_3\mathbf p_3$}; \node[sbox, font=\scriptsize, right] at (5.7,-1.3) {$\mathbf w=\mathbf q+\sigma_3\mathbf p_3$}; \node[mbox, text width=5.2cm, anchor=north west] at (0,-2.8) {% initial and source amplitudes\\ $d[Z^{\mathrm{init}}]+d[Z^{\mathrm{src}}]=0$}; \node[pbox, text width=5.4cm, anchor=north west] at (5.6,-2.8) {% cyclic symmetrization $\to K_f^\sigma$\\[1mm] equal signs: $\rho_{12}=O(r^2)$,\ $J_{12}=O(r)$\\ locally bounded\\[1mm] opposite signs: a generally surviving $r^{-2}$\\ locally integrable in three transverse dimensions}; \node[gbox, text width=11.0cm, anchor=north west] at (0,-5.7) {% $|K_f^\sigma|\le C_f R^5\bigl(1+\sum_{i0\) and \(t_00. \] The normalized states \[ \Psi_\pm=\frac{\vac\pm\varepsilon\chi_3} {\sqrt{1+\varepsilon^2}} \] have equal expectations for every defined polynomial form of degree at most two, but their cubic expectations differ. \end{theorem} \begin{proof} Choose \(\bp_*\ne0\), all three signs positive, and all three momenta equal to \(\bp_*\). Every pair transfer is \(2\bp_*\ne0\). Here \[ \rho_{12}=-|\bp_*|^2,\qquad r^2=4|\bp_*|^2. \] At \(s=t_0\), the source and spatial initial amplitudes vanish. Their total divergence vanishes as well. Therefore \[ \mathcal L_3[Z^{\init}+Z^{\src}](t_0) =-\frac{m^2+2|\bp_*|^2}{4}e^{2i\omega_*t_0}. \] The ordered response, with the three mode normalizations, has short-time expansion \[ -\frac{N_*^3(m^2+2|\bp_*|^2)}8 e^{3i\omega_*t_0}\tau^2+O(\tau^3). \] All cyclic summands agree at this point, so this is already the symmetrized coefficient. Choose an even nonnegative compact spatial bump \(b\) with \(\int b(x)e^{-3i\bp_*\cdot x}\dd^3x>0\), possible by taking its support sufficiently small. Choose a nonnegative \(g_0\in\Cc((1,2))\) of integral one and let \[ f_h(t,x)=h^{-1}g_0((t-t_0)/h)b(x). \] For sufficiently small \(h>0\), the value \(K_{f_h}^{+++}(\bp_*,\bp_*,\bp_*)\) has a nonzero leading term of order \(h^2\), hence is nonzero. Fix such \(f_h\). In a small symmetric neighborhood avoiding all transfers the kernel is smooth and nonzero. Take a nonnegative symmetric bump \(\beta\) there and set \[ \chi_3= \frac{\beta K_{f_h}^{+++}}{\|\beta K_{f_h}^{+++}\|_2}. \] The vacuum block gives \[ \cC_{f_h}^{(3)}(\chi_3,\vac) =\frac{\sqrt6}{\|\beta K_{f_h}^{+++}\|_2} \int\beta|K_{f_h}^{+++}|^2>0. \] A polynomial of degree at most two cannot connect vacuum and three particles, so its expectations in \(\Psi_\pm\) agree. The cubic difference is \[ \frac{4\varepsilon}{1+\varepsilon^2} \cC_{f_h}^{(3)}(\chi_3,\vac)>0. \] \end{proof} The singularity used in the form estimate is a spatial Newton transfer, not a time-ordered two-point extension at a spacetime diagonal. The cutoff theorem therefore proves nothing about the extension normalization of \(R_{1,1}\), and the converse is equally invalid. Figure~\ref{fig:cubic-degree-witness} displays the regular-region three-particle witness and its separation from lower-degree polynomial forms. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[font=\scriptsize, sgray] (sch) {schematic}; \node[mbox, text width=5.6cm, below=2mm of sch] (mom) {% three-creation sector\\ $\mathbf p_1=\mathbf p_2=\mathbf p_3=\mathbf p_*\neq 0$\\ every pair transfer is $2\mathbf p_*\neq 0$\\ small symmetric compact neighborhood\\ avoids all three transfer sets}; \node[sbox, text width=5.4cm, below=of mom] (chi) {% $\chi_3=\dfrac{\beta K_f^{+++}}{\|\beta K_f^{+++}\|_2}$\\ $\mathcal C_f^{(3)}(\chi_3,\Omega)>0$}; \draw[arr] (mom)--(chi); \node[pbox, text width=5.2cm, right=6mm of mom] (psi) {% $\Psi_\pm=\dfrac{\Omega\pm\varepsilon\chi_3}{\sqrt{1+\varepsilon^2}}$}; \node[gbox, text width=5.2cm, below=of psi] (low) {no vacuum--three-particle edge\\ for degree $\le 2$}; \node[sbox, text width=5.2cm, below=of low] (cub) {% cubic edge\\ $\Delta\langle\mathcal C_f^{(3)}\rangle =\dfrac{4\varepsilon}{1+\varepsilon^2}\mathcal C_f^{(3)}(\chi_3,\Omega)>0$\\ for $\varepsilon>0$}; \draw[arrg] (psi)--(low); \draw[arr] (low)--(cub); \node[gbox, font=\scriptsize, below=4mm of chi, xshift=3.1cm] (note) {% analytic form witness away from transfers; not a record experiment}; \end{tikzpicture} \caption{Cubic witness away from transfer singularities. A compact symmetric three-particle wavefunction is selected in a regular region where the three-creation kernel is nonzero. The relative sign in the vacuum--three-particle superpositions is invisible to defined polynomial forms of degree at most two, but produces a nonzero cubic expectation difference.} \label{fig:cubic-degree-witness} \end{figure} %===================================================================== \section{Successor problems and final separation of conclusions} \label{sec:successors} %===================================================================== The closed statements are: \[ B_{\mathrm{raw}}\text{ on }\Sol \quad\Longleftrightarrow\quad \bfH=0, \] \[ B_R\text{ on }\Solret,\quad B_0\text{ on }\Sol,\quad B_{\mathrm{raw}}\text{ on }\SoldD \quad\Longleftrightarrow\quad \E_1\bfH=0, \] \[ B_f|_{\Vfin}=F_{J_c}|_{\Vfin}, \] and, at compact \(g\), \[ R_{1,1}(V(g;h),\phi(f))=\Phi(\Kmap\Qop_{gh}(-G_Af)), \] \[ R_{1,1}(V(g),\phi(f)) =Q\otimes\Phi(v_e)+\Pi\otimes\Phi(v_r) =\widehat B_f^{g,t_0}. \] The independently defined Newton--wave composition has the form and witness of Section~\ref{sec:contrast}. The following four questions remain separate. \begin{enumerate}[label=\arabic*.] \item \textbf{Dense-TT representability.} Does there exist \(J\in\Test\) representing the raw or sectioned pairing on a specified infinite-dimensional TT space? The 2-plane current and the \(\Sol\) classification do not decide this. \item \textbf{Adiabatic switch.} Existence or obstruction for \(g\to1\) in the external and one-oscillator retarded coefficients. Compact-support continuity is not that limit. \item \textbf{Second vertex product.} Construct \(T_2(V,V)\) and its retarded subtraction. The absence of a degree-four parameter in \(R_{1,1}(V,\phi(f))\) says nothing about this product. \item \textbf{Graviton loops and a local metric field.} Replace the oscillator-valued wave by a local linearized-gravity field and construct the corresponding causal products. The finite-mode commutator \eqref{eq:nonlocalcomm} already fails the required locality. \end{enumerate} None of these questions asks to identify the Newton--wave cubic form with either retarded coefficient. Their metric arguments, quantum degrees, and construction steps are different by definition. Figure~\ref{fig:closed-and-successor} separates the established conclusions from the four stated successor problems. \begin{figure}[htbp] \centering \begin{tikzpicture}[node distance=2.4mm and 4mm] \node[sbox, text width=3.5cm] (r1) {raw classification\\ $\mathbf H=0$}; \node[sbox, text width=3.7cm, right=2mm of r1] (r2) {harmonic classification\\ $\mathsf E_1\mathbf H=0$}; \node[mbox, text width=3.6cm, right=2mm of r2] (r3) {compact representation\\ on the fixed TT two-plane}; \node[pbox, text width=5.4cm, below=of r1] (r4) {external and one-oscillator\\ first retarded coefficients}; \node[sbox, text width=5.6cm, right=3mm of r4] (r5) {independent Newton--wave cubic form\\ and joint cutoff limit}; \node[gbox, text width=11.6cm, below=5mm of r4] (wall) {scope boundary}; \node[gbox, text width=2.7cm, below left=4mm and -1mm of wall] (s1) {dense-TT\\ representability}; \node[gbox, text width=2.5cm, right=2mm of s1] (s2) {adiabatic\\ $g\to 1$}; \node[gbox, text width=2.7cm, right=2mm of s2] (s3) {second vertex\\ $T_2(V,V)$}; \node[gbox, text width=2.8cm, right=2mm of s3] (s4) {local metric field\\ and graviton loops}; \draw[arrg] (wall)--(s1); \draw[arrg] (wall)--(s2); \draw[arrg] (wall)--(s3); \draw[arrg] (wall)--(s4); \node[cbox, font=\scriptsize, below=of s2, xshift=1.4cm] (need) {separate definitions and proofs required}; \node[cbox, text width=5.5cm, below left=3mm and -8mm of need] (w1) {two-plane interpolation is not dense-TT representability}; \node[cbox, text width=5.5cm, right=2mm of w1] (w2) {one scalar insertion is not the second vertex product}; \end{tikzpicture} \caption{Closed conclusions and successor problems. The manuscript establishes classical admissibility criteria, a restricted finite-plane construction, first retarded coefficients at compact switch, and an independent cubic form theorem. Dense-TT representation, the adiabatic limit, second vertex products, and local gravitational causal products remain separate tasks.} \label{fig:closed-and-successor} \end{figure} %===================================================================== \appendix \clearpage \section{Sign, initial-data, and Duhamel audit} \label{app:signs} %===================================================================== \subsection{One response, two notations} All body formulas use \[ P=-(\partial_t^2+\Om^2),\qquad L_h=-h^{\mu\nu}\partial_\mu\partial_\nu-D(h)^\nu\partial_\nu, \qquad PS=-L_h\phi. \] If \[ L_h^+=h^{\mu\nu}\partial_\mu\partial_\nu+D(h)^\nu\partial_\nu, \] then \(L_h^+=-L_h\), so the alternative notation \(PS=L_h^+\phi\) has exactly the same right-hand side. The two expressions \[ -\int_{t_0}^t\frac{\sin((t-s)\Om)}{\Om}L_h^+\phi(s)\dd s, \qquad \int_{t_0}^t\frac{\sin((t-s)\Om)}{\Om}L_h\phi(s)\dd s \] are identical. An extra sign reversal of \(J_{\mathrm{raw}}\) would be an error. An actual change from \(PS=-L_h\phi\) to \(PS=+L_h\phi\), while keeping \(L_h\) fixed, would negate the response, the current, and the defect. It would leave the zero conditions \(\bfH=0\) and \(\E_1\bfH=0\) unchanged. For \(u_f=-G_Af\), the raw adjoint identity is \[ B_f(h,\phi)=\int_{t_0}^T u_fL_h\phi. \] Integrating the metric derivative by parts gives \[ J_{\mathrm{raw}}^{\mu\nu} =\partial^{(\mu}(\Theta_{t_0}u_f)\partial^{\nu)}\phi -\frac12\eta^{\mu\nu} \partial_\alpha(\Theta_{t_0}u_f\partial^\alpha\phi). \] Its initial-surface components are \[ j_\Sigma^{00}=\frac12u_0\dot\phi_0,\qquad j_\Sigma^{0i}=-\frac12u_0\partial_i\phi_0,\qquad j_\Sigma^{ij}=\frac12\delta^{ij}u_0\dot\phi_0. \] The canonical distribution formula is \[ P(\Theta_{t_0}u_f) =-f-\delta(t-t_0)\dot u_0-\delta'(t-t_0)u_0. \] There is only one \(\dot u_0\) in the displayed delta coefficient. An intermediate expression with \(2\dot u\delta\) must also expand the product \(u(t)\delta'\), which removes one copy. Figure~\ref{fig:sign-audit} summarizes the conventions audited in this appendix. \subsection{Adjoints and propagation} The gauge parameter is taken as a vector, the defect as a covector: \[ (K^*J)_\nu=-2\eta_{\nu\beta}\partial_\mu J^{\mu\beta}. \] If instead both are written with opposite index placement, the component equations must be raised or lowered, not silently copied. For classification, \[ \E_r=\mathsf G_{r,A}-\mathsf G_{r,R}, \] so its massless kernel is \(+\Sw(t)\). For the scalar commutator, \[ \Delta_m=G_R-G_A, \] whose spatial Fourier kernel is \(-\sin(t\Om)/\Om\). Thus the two uses of a causal propagator have opposite orientation by explicit notation, not by an implicit change within a proof. The positive current construction is \[ q=\E_2\overline{\flatT C}, \qquad J_C=2\sharpT{\LE(\chi q)}. \] Both index raising operations are included. Changing to a retarded-minus-advanced \(q\) would require a corresponding minus sign in \(J_C\). \newpage \subsection{The retarded coefficient sign} The scalar contractions obey \[ W_m^{\mathrm F}(y-x)-W_m(y-x)=iG_R(x-y). \] With \[ R_{1,1}=i(T_2-V\phi), \] the contraction in the retarded coefficient is \(-G_R\). After the compact vertex integration by parts, \[ \phi^{(1)}_{g,h}=-G_R\Qop_{gh}\phi. \] Smearing gives \[ -\int fG_R\Qop_{gh}\phi =-\int(G_Af)\Qop_{gh}\phi =\int u_f\Qop_{gh}\phi =\phi(\Qop_{gh}u_f). \] This is why the compact test is \[ q_{g,h,f}=+\Qop_{gh}(-G_Af). \] The minus sign in \(-G_R\Qop_{gh}\phi\) has already been absorbed into \(u_f=-G_Af\); inserting another minus in \(q\) would reverse the coefficient. For a TT tensor, \(\Qop_{gh}=-\partial_i(gh^{ij}\partial_j)\). Thus the equivalent explicit formula is \[ q_{g,h,f}=-\partial_i(gh^{ij}\partial_j u_f). \] The leading minus here belongs to the differential operator \(\Qop\), not to an additional convention for the retarded product. \subsection{Switching, free-null tests, and initial data} On a free scalar, \[ \Qop_{gh}\phi=L_{gh}\phi =gL_h\phi-(\partial_\mu g)\bar h^{\mu\nu}\partial_\nu\phi. \] All derivatives of \(g\) are retained. The equality \[ -G_R\Qop_{gh}\phi=S_{t_0}(gh)\phi \] holds when \(t_0\) precedes the compact vertex support. It does not identify that expression with the unswitched zero-data response \(S_{t_0}(h)\phi\). The witness \(f=P\rho\) has \[ \phi(f)=0,\qquad G_AP\rho=\rho,\qquad u_f=-\rho. \] The retarded insertion is nevertheless \[ \phi(-\Qop_{gh}\rho), \] because its off-shell input label was retained before representation. The fixed contact \(PW_m^{\mathrm F}=i\delta\) is essential to this conclusion. For a pure gauge, \[ L_{K\xi}=-[P,X_\xi],\qquad S_{t_0}(K\xi)\phi =X_\xi\phi-\mathsf H_m(\Data_{t_0}(X_\xi\phi)). \] The homogeneous subtraction is the initial-data contribution. It is absent only when those data vanish. A general gauge transformation of the triangular scalar response does not preserve its zero-data condition. \subsection{Independence of auxiliary cutoffs} In the smooth-current construction, two temporal cutoffs \(\chi_1,\chi_2\) differ only in a bounded slab. For a spacelike-compact \(q\), \[ r=(\chi_1-\chi_2)q \] is smooth and compact, and the representative difference is \(2\sharpT{\LE r}\). It pairs to zero with every vacuum solution by formal self-adjointness. Thus auxiliary cutoff independence is on shell. By contrast, the physical interaction switch \(g\) changes the retarded observable. It cannot be discarded using this auxiliary-cutoff argument. Equation~\eqref{eq:remainder} is the explicit comparison required for that change. The auxiliary cutoff in Figure~\ref{fig:smoothing-classification} changes the current only by an on-shell annihilator. It is distinct from the physical interaction switch in Figure~\ref{fig:switched-remainder}. \clearpage \section{Option A and Option B are different inverses} \label{app:AB} %===================================================================== All entries in the following table refer to the trace-reversed metric. Let \[ \rho=Q_{00}(t_0),\qquad J_i=Q_{0i}(t_0), \] and use the same Newton inverse in both columns. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.27\linewidth}XX@{}} \toprule Datum & Option A, used in Section~\ref{sec:contrast} & Option B, comparison only\\ \midrule \(\bar j_{00}(t_0)\) & \(-2\Delta^{-1}\rho\) & \(-2\Delta^{-1}\rho\)\\ \addlinespace \(\bar j_{0i}(t_0)\) & \(0\) & \(-2\Delta^{-1}J_i\)\\ \addlinespace \(\bar j_{ij}(t_0)\) & \(0\) & \(0\)\\ \addlinespace \(\partial_t\bar j_{00}(t_0)\) & \(0\) & \(-2\Delta^{-1}\partial_iJ_i\)\\ \addlinespace \(\partial_t\bar j_{0i}(t_0)\) & \(0\) & \(0\)\\ \addlinespace \(\partial_t\bar j_{ij}(t_0)\) & \(\partial_iV_j+\partial_jV_i-\delta_{ij}\partial_\ell V_\ell\), \(V_i=2\Delta^{-1}J_i\) & \(0\)\\ \addlinespace Wave-source part & \(2\int_{t_0}^t\Sw(t-s)Q(s)\dd s\) & The same\\ \bottomrule \end{tabularx} \end{center} Figure~\ref{fig:Option-A-composition} displays the different initial-data prescriptions and identifies the option used in the cubic theorem. The two sourced solutions have the same wave forcing, so their difference is homogeneous. In general their Cauchy data differ, and therefore their inverses differ. Homogeneity of the difference does not identify their scalar responses. Even a separately proved pure-gauge relation would require the fixed-time boundary-data calculation in \eqref{eq:boundarygauge}. With a scalar response constrained to zero Cauchy data, a gauge change can alter the selected functional. The contrast theorem uses Option A alone. Neither option is used to define the external metric of Section~\ref{sec:external} or the independent oscillator-valued metric of Section~\ref{sec:mixed}. Replacing those metrics by \(\cR^AT(\phi)\) changes the object and its field degree. \section*{Declarations} \paragraph{Scope of the proofs.} All special formulas needed from the supplied companion manuscripts have been written out and proved here. No identification with an unspecified series construction, record observable, or implementation is used. \paragraph{Data and computation.} No empirical data or numerical integration is used. The existence and nonzero witnesses are analytic. No code execution or successful \texttt{pdflatex} compilation is asserted. \paragraph{Foundational inputs.} The only external functional-analytic theorem used to close an operator argument is the stated analytic-vector criterion. The compact-current argument is proved directly from flat wave Green identities. References below identify the standard observable class and perturbative terminology; no supplied companion paper is cited. \begin{thebibliography}{9} \bibitem{FH} C.~J.~Fewster and D.~S.~Hunt, \emph{Quantization of linearized gravity in cosmological vacuum spacetimes}, Reviews in Mathematical Physics \textbf{25} (2013), 1330003. arXiv:1203.0261v4. doi:10.1142/S0129055X13300033. The compact conserved test-tensor class is used as terminology. No radical-exhaustion theorem is applied to Minkowski spacetime. \bibitem{BF} R.~Brunetti and K.~Fredenhagen, \emph{Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds}, Communications in Mathematical Physics \textbf{208} (2000), 623--661. arXiv:math-ph/9903028. doi:10.1007/s002200050004. The subcritical extension criterion is Theorem 5.2. \bibitem{DF} M.~D\"utsch and K.~Fredenhagen, \emph{Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity}, Reviews in Mathematical Physics \textbf{16} (2004), 1291--1348. arXiv:hep-th/0403213. doi:10.1142/S0129055X04002266. Used for the off-shell retarded-product framework; the single-insertion calculations are explicit in the text. \bibitem{Nelson} E.~Nelson, \emph{Analytic vectors}, Annals of Mathematics (2) \textbf{70} (1959), 572--615. doi:10.2307/1970331. Used for the criterion that a symmetric operator with a dense domain of analytic vectors is essentially self-adjoint. \end{thebibliography} \end{document}