\documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,longtable,tabularx} \usepackage{enumitem} \usepackage{microtype} \usepackage{xurl} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={XVIII-B: A prepared massive-scalar stress-response benchmark}, pdfauthor={}, pdfsubject={Version 2; spectral evaluation, transverse contacts, linear Ward identities, and preparation residual}, pdfkeywords={massive scalar, stress tensor, spectral density, retarded response, contact terms, Ward identities, closed time path, preparation, Enter readout} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{0.25em} \setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt} \setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt} \renewcommand{\arraystretch}{1.12} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{specification}[theorem]{Specification} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\eps}{\varepsilon} \newcommand{\id}{\mathrm{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\sd}{\operatorname{sd}} \newcommand{\Disc}{\operatorname{Disc}} \newcommand{\PV}{\operatorname{PV}} \newcommand{\D}{\mathcal D} \newcommand{\Sch}{\mathcal S} \newcommand{\T}{\mathsf T} \newcommand{\QT}{\mathsf Q_2} \newcommand{\QS}{\mathsf Q_0} \newcommand{\PT}{\mathsf P_2} \newcommand{\PS}{\mathsf P_0} \newcommand{\Ein}{\mathsf E} \newcommand{\PiR}{\Pi^{\mathrm R}} \newcommand{\PiA}{\Pi^{\mathrm A}} \newcommand{\sigeps}{\sigma_\eps} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\sourceid}[1]{\texttt{\detokenize{#1}}} \newcommand{\NOTCLOSED}{\texttt{NOT\_CLOSED}} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}} \newcommand{\scopebox}[2]{% \begin{center} \fbox{\begin{minipage}{0.92\linewidth} \textbf{#1}\par\smallskip #2 \end{minipage}} \end{center} } \title{ \textbf{Landmark XVIII-B}\\[3pt] A Prepared Massive-Scalar Stress-Response Benchmark:\\ Spectral Evaluation, Transverse Contact Freedom,\\ and a Fixed-Preparation Residual } % Insert responsible author and affiliation information before submission. \author{} \date{Version 2} \begin{document} \maketitle \begin{abstract} We evaluate one external-source benchmark on four-dimensional Minkowski space. The source is classical, and the quantum field is a free real scalar of fixed mass \(m>0\), with its minimally coupled canonical stress tensor Wick ordered in the Minkowski vacuum. A two-particle calculation gives the exact spin-two and spin-zero spectral densities. A once-subtracted dispersion prescription for the reduced form factors defines a real, causal, tempered response distribution of scaling degree at most eight. We classify the complete parity-even, Lorentz-covariant, pair-symmetric, doubly transverse contact freedom in that extension class: one quadratic and two quartic polynomial tensors. A zero-momentum condition fixes all three coefficients uniquely. Both linear Ward identities hold distributionally. The four closed-time-path kernels are constructed consistently at quadratic source order. Smooth preparation is incorporated by multiplying the external source by a fixed time ramp. The resulting response is finite on arbitrary compact joint tests. We calculate its transformation residual, prove plateau-supported intertwining, and construct a late, contact-independent detecting test for an unrestricted preparation-layer transformation. Spectral detection is not a measurement instrument, a classical record, or a quantum-return decoder. Documentary comparisons retain their individual models and versions and are not premises of the calculation. No relational scalar two-point coefficient, inherited characteristic graph, graviton--ghost inventory, interacting BRST/BV identity, bulk restoration, or ultraviolet completion is established. \end{abstract} \scopebox{Evaluated target and proof boundary}{% The completed object is the connected, renormalized vacuum two-stress response of the \emph{printed linear external-source model}, at quadratic source order. The tensor source and Minkowski metric are classical. The only quantum field is one free scalar with \(m>0\). No theorem identifies this response with the second metric variation of a nonlinear gravitational action or with the companion's prepared relational scalar. Gravitons, ghosts, nonlinear metric seagulls, and relational coordinate insertions are outside the target. No sharp-preparation, massless, adiabatic, or ultraviolet completion limit is taken.} \subsection*{Editorial provenance} The supplied material contains the revision brief and companion excerpts, but not the original XVIII-B source. Accordingly, this document prints a complete independent specification rather than asserting identity with an unavailable version. Its choice of the canonical minimally coupled stress tensor, Fourier convention, source sign, and extension class is explicit below. Authorship and registration are not inferred. \subsection*{Enter Philosophy: response, readout, and quantum return} The coefficient \(\mathcal R_\eps(j,h)\) is a response, not a classical record; a record claim would require a separately declared measurement instrument, and Manuscript I's reported destructive-readout characterization in a stated class---where record compatibility is not residual compatibility---is only a documentary comparison, not an imported uniqueness theorem \cite{CheungI}. A decoder is a quantum-return map from a declared accessible factor; the preservation/recovery distinctions of Hawk04-A/B do not identify either such a map or a measurement channel with the quadratic closed-time-path bookkeeping of Proposition~\ref{prop:ctp} \cite{CheungHawk04A,CheungHawk04B}. The positive vacuum norm in Proposition~\ref{prop:detector} detects spectral weight, not an Enter instrument or an observation boundary; VII-E's separately specified spectator, record, residual-instrument, and continuation requirements are a comparison explaining why a pairing alone is not observation \cite{CheungVIIE}. This benchmark does not reconstruct spacetime, derive a metric, identify an entanglement wedge, or construct a Page curve or an island, and it imports no reconstruction map from the separately identified XVII-A finite-matrix AdS$_2$ benchmark \cite{CheungXVIIA}. \paragraph{Standing stops.} The inherited XV-B quantum-admission exit is not opened. XVI-N and Route B are not discharged. O.1 remains \NOTCLOSED. No quantum coefficient of the inherited companion response is admitted. No successful compilation is asserted. These are scope statements, not nonexistence theorems \cite{CheungXIVJ,CheungXVB,CheungXVIN,CheungXVIO,CheungXVIP}. \tableofcontents %===================================================================== \section{Frozen data and the selected response} \label{sec:model} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}P{.27\linewidth}Y@{}} \toprule Datum & Frozen choice\\ \midrule Spacetime & \(M=\R^4\), \(\eta=\operatorname{diag}(1,-1,-1,-1)\).\\ Quantum field & One free real scalar, \((\Box+m^2)\phi=0\), \(m>0\).\\ State & Minkowski Fock vacuum \(\Omega\).\\ Stress insertion & Canonical minimally coupled tensor, vacuum Wick ordered; no improvement parameter is varied.\\ External source & A real compact smooth symmetric contravariant tensor.\\ Coupling & \(g=\kappa/2\), \(V_H=-g\,\T(H)\).\\ Order & Connected vacuum source degree two; response proportional to \(\kappa^2\hbar\).\\ Preparation & Fixed smooth time ramp \(\sigeps\), fixed \(\eps>0\); prepared source \(H=\sigeps h\).\\ Reference prescription & One subtraction at \(z=0\) in each reduced spectral form factor.\\ Allowed finite changes & Point-supported, scaling degree \(\le8\), real, parity-even, Lorentz-covariant, pair-symmetric, and transverse in both stress slots.\\ Calibration & The full momentum-space response has vanishing Taylor terms of total degrees two and four at \(p=0\).\\ Operational output & A distributional response coefficient; no measurement instrument is constructed.\\ \bottomrule \end{tabularx} \end{center} Indices are raised and lowered with \(\eta\). Write \(p^2=\eta_{ab}p^ap^b\) and use \begin{equation} \widehat f(p)=\int_M e^{\ii p\cdot x}f(x)\dd^4x, \qquad f(x)=\int_{\R^4}\frac{\dd^4p}{(2\pi)^4} e^{-\ii p\cdot x}\widehat f(p). \label{eq:Fourier} \end{equation} Thus \(\partial_a\) has multiplier \(-\ii p_a\). The stress tensor is \begin{equation} T_{ab} = :\partial_a\phi\,\partial_b\phi -\frac12\eta_{ab} \bigl(\partial^c\phi\,\partial_c\phi-m^2\phi^2\bigr):, \qquad \T(j)=\int_M j^{ab}T_{ab}\dd^4x. \label{eq:stress} \end{equation} Wick ordering is with respect to \(\Omega\), so \(\langle T_{ab}\rangle_\Omega=0\). The free field equation gives \(\partial^aT_{ab}=0\) as an operator-valued distribution on the usual finite-particle domain. \begin{definition}[The selected response] \label{def:response} Let \(\PiR_{ab\,cd}\) be a renormalized retarded splitting of the vacuum stress commutator, with convention \begin{equation} \PiR_{ab\,cd}(x) = \frac{\ii}{\hbar}\Theta(x^0) \langle[T_{ab}(x),T_{cd}(0)]\rangle_\Omega \quad\text{away from }x=0. \label{eq:Kubo} \end{equation} For compact smooth symmetric \(j,h\), set \begin{equation} \begin{split} \mathcal R_\eps(j,h) &= g^2\int_{M^2} j^{ab}(x)\PiR_{ab\,cd}(x-y) \sigeps(y^0)h^{cd}(y)\dd^4x\dd^4y\\ &= g^2\pair{\PiR_{ab\,cd}}{F_{j,\sigeps h}^{ab\,cd}}, \qquad F_{j,H}^{ab\,cd}(r) = \int_M j^{ab}(y+r)H^{cd}(y)\dd^4y . \end{split} \label{eq:response} \end{equation} This is the linear response of the insertion \(-g\T(j)\) to the source \(V_{\sigeps h}\), hence it has quadratic source order. The corresponding first-order stress response is \(-g\,\PiR*(\sigeps h)\). \end{definition} The double integral in~\eqref{eq:response} is distributional. The function \(F_{j,H}\) is compact and smooth; no pointwise multiplication of singular kernels is being used. \paragraph{Inventory.} The selected connected source-degree-two coefficient consists of the two Wick contractions between the two normal-ordered stress insertions, together with its completely classified local extension freedom. \begin{center} \small \begin{tabularx}{\linewidth}{@{}P{.43\linewidth}Y@{}} \toprule Contribution & Status in this target\\ \midrule Two scalar lines between the two stress insertions & Included, with both identical-particle contractions.\\ Vacuum disconnected one-point factors & Zero by the specified Wick ordering.\\ Finite transverse local contacts & Included and classified in Theorem~\ref{thm:contacts}.\\ Graviton loops and graviton external quantum legs & Absent: the source is classical.\\ Ghost and auxiliary-field contributions & Absent: no gravitational gauge-fixed quantum theory is specified.\\ Nonlinear metric seagulls & Outside the model: the coupling is defined to be linear.\\ Relational coordinate and scalar completion insertions & Outside the model: the insertion is~\eqref{eq:stress}.\\ Higher source cumulants & Outside the quadratic-order claim.\\ \bottomrule \end{tabularx} \end{center} The distinction between a two-stress correlator and the full second variation of a metric-dependent action, including local variation terms, is important in the external linear-response literature \cite{AMP,Mottola}. Those works motivate a comparison; they do not insert seagulls into the present linear-source definition. %===================================================================== \section{Polynomial tensor structures} \label{sec:tensors} %===================================================================== For \(s=p^2\neq0\), define \[ \theta_{ab}(p)=\eta_{ab}-\frac{p_ap_b}{s}, \] \begin{align} (\PS)_{ab\,cd} &=\frac13\theta_{ab}\theta_{cd}, \label{eq:P0}\\ (\PT)_{ab\,cd} &=\frac12(\theta_{ac}\theta_{bd}+\theta_{ad}\theta_{bc}) -\frac13\theta_{ab}\theta_{cd}. \label{eq:P2} \end{align} These are the scalar and traceless symmetric projectors on the three-dimensional transverse space for timelike \(p\). To avoid any division by \(p^2\) in the final kernel, put \[ L_{ab}(p)=s\eta_{ab}-p_ap_b \] and define the polynomial tensors \begin{align} (\QS)_{ab\,cd} &=\frac13L_{ab}L_{cd}=s^2(\PS)_{ab\,cd}, \label{eq:Q0}\\ (\QT)_{ab\,cd} &=\frac12(L_{ac}L_{bd}+L_{ad}L_{bc}) -\frac13L_{ab}L_{cd} =s^2(\PT)_{ab\,cd}. \label{eq:Q2} \end{align} The quadratic polynomial tensor is \begin{equation} \begin{split} \Ein_{ab\,cd} ={}& s\left[ \frac12(\eta_{ac}\eta_{bd}+\eta_{ad}\eta_{bc}) -\eta_{ab}\eta_{cd}\right]\\ &-\frac12\left( \eta_{ac}p_bp_d+\eta_{ad}p_bp_c +\eta_{bc}p_ap_d+\eta_{bd}p_ap_c\right)\\ &+\eta_{ab}p_cp_d+\eta_{cd}p_ap_b. \end{split} \label{eq:E} \end{equation} For \(s\neq0\), \[ \Ein=s(\PT-2\PS). \] Equation~\eqref{eq:E}, not this rational representation, defines \(\Ein\) also on \(s=0\). \begin{lemma}[Polynomial transversality] \label{lem:transverse} Every tensor \(\Ein,\QT,\QS\) is symmetric in each index pair and under exchange of the pairs. Each satisfies \[ p^aX_{ab\,cd}=0,\qquad p^cX_{ab\,cd}=0 \] as a polynomial identity on all of \(\R^4\). \end{lemma} \begin{proof} For \(\QT,\QS\), use \(p^aL_{ab}=0\). For \(\Ein\), contraction of~\eqref{eq:E} cancels the terms proportional to \(p_b\eta_{cd}\), \(p_c\eta_{bd}+p_d\eta_{bc}\), and \(p_bp_cp_d\). The symmetries follow from the displayed formulas. \end{proof} %===================================================================== \section{The exact two-particle spectral calculation} \label{sec:spectrum} %===================================================================== \subsection{Field and state normalization} Let \(\omega_{\mathbf k}=\sqrt{|\mathbf k|^2+m^2}\) and \[ \dd\mu_m(k)=\frac{\dd^3\mathbf k}{(2\pi)^3\,2\omega_{\mathbf k}}, \qquad k=(\omega_{\mathbf k},\mathbf k). \] Use \[ \phi(x)=\sqrt{\hbar}\int\dd\mu_m(k) \bigl(a(k)e^{-\ii k\cdot x}+a^\dagger(k)e^{\ii k\cdot x}\bigr), \] \[ [a(k),a^\dagger(q)] =(2\pi)^3\,2\omega_{\mathbf k}\, \delta^3(\mathbf k-\mathbf q), \qquad a(k)\Omega=0. \] The unordered two-particle completeness relation has factor \(1/2!\). This factor is retained below. Define \[ W^>_{ab\,cd}(x) =\langle T_{ab}(x)T_{cd}(0)\rangle_\Omega, \qquad W^<_{ab\,cd}(x)=W^>_{cd\,ab}(-x). \] \begin{lemma}[Vacuum-to-two-particle stress matrix element] \label{lem:matrixelement} For on-shell positive-energy \(k,q\), define \[ \tau_{ab}(k,q) =-(k_aq_b+k_bq_a)+\eta_{ab}(k\cdot q+m^2). \] The vacuum matrix element is \(\hbar\tau_{ab}\), and \begin{equation} \widehat W^>_{ab\,cd}(p) = \frac{\hbar^2}{2} \int\dd\mu_m(k)\dd\mu_m(q)\, (2\pi)^4\delta^4(p-k-q)\, \tau_{ab}(k,q)\tau_{cd}(k,q). \label{eq:phaseformula} \end{equation} Moreover \((k+q)^a\tau_{ab}=0\). \end{lemma} \begin{proof} The two annihilation operators in~\eqref{eq:stress} give the two derivative contractions \(-(k_aq_b+k_bq_a)\). The trace term gives \(\eta_{ab}(k\cdot q+m^2)\). The factor \(\hbar\) comes from the two field factors. Inserting the two-particle identity gives \eqref{eq:phaseformula} and its factor \(1/2\). Using \(k^2=q^2=m^2\), contraction with \(k+q\) cancels the two terms. \end{proof} \begin{lemma}[Two-body phase space and angular moments] \label{lem:phase} For timelike future \(p\), \(s=p^2>4m^2\), write \[ v(s)=\sqrt{1-\frac{4m^2}{s}}, \qquad K^2=\frac{s}{4}-m^2. \] In the rest frame of \(p\), \[ \int\dd\mu_m(k)\dd\mu_m(q)\, (2\pi)^4\delta^4(p-k-q) =\frac{v(s)}{8\pi}, \] and the angular distribution of \(\mathbf k=K\mathbf n\) is uniform. Its normalized moments are \[ \langle n_in_j\rangle=\frac{\delta_{ij}}3, \qquad \langle n_in_jn_kn_l\rangle =\frac{\delta_{ij}\delta_{kl} +\delta_{ik}\delta_{jl} +\delta_{il}\delta_{jk}}{15}. \] \end{lemma} \begin{proof} The spatial delta sets \(\mathbf q=-\mathbf k\). The energy delta \(\delta(\sqrt s-2\omega_{\mathbf k})\) has radial Jacobian \(\omega_{\mathbf k}/(2K)\). Combining the measure factors gives \(v/(8\pi)\), with normalized angular measure \(\dd\Omega/(4\pi)\). Rotation invariance makes the second moment a multiple of \(\delta_{ij}\), determined by \(|\mathbf n|^2=1\). The fourth moment is a multiple of the three displayed Kronecker products; double contraction determines \(1/15\). \end{proof} \begin{theorem}[Evaluated stress spectral densities] \label{thm:spectral} Set \(a=4m^2\) and, for \(u\ge a\), define \begin{align} r_2(u) &=\frac{1}{960\pi^2} \left(1-\frac{a}{u}\right)^{5/2}, \label{eq:r2}\\ r_0(u) &=\frac{1}{96\pi^2} \left(1-\frac{a}{u}\right)^{1/2} \left(1+\frac{2m^2}{u}\right)^2. \label{eq:r0} \end{align} Extend them by zero below \(a\). Then \begin{equation} \widehat W^>_{ab\,cd}(p) = 2\pi\hbar^2\Theta(p^0) \left[ (\QT)_{ab\,cd}(p)r_2(p^2) +(\QS)_{ab\,cd}(p)r_0(p^2) \right]. \label{eq:Wspec} \end{equation} Both densities are strictly positive above threshold. The corresponding projector densities are \[ \rho_2(u)=u^2r_2(u),\qquad \rho_0(u)=u^2r_0(u). \] The formula defines a positive, tempered vacuum two-point distribution. \end{theorem} \begin{proof} In the rest frame \(p=(\sqrt s,\mathbf0)\), \(\tau_{00}=\tau_{0i}=0\) and \[ \tau_{ij}=2K^2n_in_j-\frac{s}{2}\delta_{ij}. \] Its spatial trace is \(-(s+2m^2)\). The traceless part is \(2K^2(n_in_j-\delta_{ij}/3)\). Lemma~\ref{lem:phase} therefore gives \[ \langle\tau_{ij}\tau_{kl}\rangle = \frac{8K^4}{15}(\PT)_{ij\,kl} +\frac{(s+2m^2)^2}{3}(\PS)_{ij\,kl}. \] Multiplying by \(\hbar^2v/(16\pi)\) from~\eqref{eq:phaseformula} gives \[ \hbar^2\frac{s^2v^5}{480\pi}\PT +\hbar^2\frac{v(s+2m^2)^2}{48\pi}\PS. \] This is~\eqref{eq:Wspec}. Lorentz covariance and transversality extend the rest-frame calculation to the future two-particle cone. Positivity also follows directly from the squared matrix element in~\eqref{eq:phaseformula}. The polynomial tensor growth is at most quartic, while \(r_2,r_0\) are bounded. Hence the spectral distribution is tempered. \end{proof} The spectral normalization is derived here from the printed field convention and the identical-particle factor. For comparison, the same spin decomposition and massive-scalar spectral functions appear in the external linear-response calculation of Anderson, Molina-Par\'is, and Mottola \cite[Appendix B]{AMP}. No semiclassical metric equation or stability theorem from that work is used here. \subsection{Spectral detection, not observation} \begin{proposition}[A compact spectral detector] \label{prop:detector} For every nonzero \(f\in C_c^\infty(M)\), take \(j^{ab}=\eta^{ab}f\). Then \begin{equation} \begin{split} \|\T(j)\Omega\|^2 &= \int\frac{\dd^4p}{(2\pi)^4} \overline{\widehat j^{ab}(p)} \widehat W^>_{ab\,cd}(p)\widehat j^{cd}(p)\\ &= 6\pi\hbar^2 \int_{\substack{p^0>0\\p^2>a}} \frac{\dd^4p}{(2\pi)^4} (p^2)^2r_0(p^2)|\widehat f(p)|^2 >0. \end{split} \label{eq:vacnorm} \end{equation} This detects the spectral coefficient and is independent of the finite retarded contact prescription. It is not a finite measurement instrument, an Enter destructive readout, a classical record, or a decoder. \end{proposition} \begin{proof} The norm follows by vacuum positivity and the Fourier representation of \(W^>\). Since \(\eta^{ab}\eta^{cd}\QT_{ab\,cd}=0\) and \(\eta^{ab}\eta^{cd}\QS_{ab\,cd}=3(p^2)^2\), the second line follows. The Fourier transform of a nonzero compact smooth function is real analytic on real momentum space and cannot vanish on the entire open future threshold region. The positive integrand is therefore nonzero on an open set. Rapid Fourier decay makes the integral finite. A finite change in the retarded splitting does not change the already fixed vacuum Wightman distribution. No apparatus or selective map is constructed by this norm. \end{proof} The interpretation is deliberately spectral. The supplied readout and observation-boundary manuscripts I and VII-E concern different interfaces \cite{CheungI,CheungVIIE}. They are comparisons, not premises of~\eqref{eq:vacnorm}. %===================================================================== \section{The once-subtracted retarded prescription} \label{sec:dispersion} %===================================================================== \subsection{Reduced form factors} For \(i\in\{2,0\}\) and \(z\in\C\setminus[a,\infty)\), define \begin{equation} F_i(z) = \int_a^\infty r_i(u) \left(\frac1{u-z}-\frac1u\right)\dd u = z\int_a^\infty\frac{r_i(u)}{u(u-z)}\dd u. \label{eq:subtracted} \end{equation} The subtraction is in the \emph{reduced} form factor. It is not a once-subtracted integral of the growing projector density \(\rho_i=u^2r_i\). \begin{lemma}[Analyticity and discontinuity] \label{lem:dispersion} The functions \(F_i\) are analytic off the cut, real for real \(za\), \begin{equation} F_i(s+\ii0)-F_i(s-\ii0)=2\pi\ii r_i(s). \label{eq:disc} \end{equation} Their real boundary values are continuous at threshold and grow at most logarithmically at infinity. \end{lemma} \begin{proof} The densities are bounded. On compact subsets off the cut, the subtracted integrand is \(O(u^{-2})\), uniformly with its required \(z\) derivatives. This proves convergence and analyticity. Reality and \(F_i(0)=0\) follow directly. The distributional identity \[ \frac1{u-s-\ii0} =\PV\frac1{u-s}+\ii\pi\delta(u-s) \] gives~\eqref{eq:disc}. Near threshold, the densities vanish at least as \((u-a)^{1/2}\), so their Cauchy transforms have finite continuous boundary values there. The bounded limiting densities at large \(u\) give logarithmic growth. Explicit formulas below also establish these boundary statements directly. \end{proof} Define the retarded boundary value by analytic continuation from \(\operatorname{Im}p^0>0\): \[ F_i^{\mathrm R}(p) = F_i\bigl((p^0+\ii0)^2-|\mathbf p|^2\bigr). \] On real momenta this is \(F_i(p^2+\ii0\,\operatorname{sgn}p^0)\) at the spectral cut. The advanced value reverses the sign. The reference response is \begin{equation} \widehat\PiR_{\mathrm{ref},ab\,cd}(p) = \hbar\left[ (\QT)_{ab\,cd}(p)F_2^{\mathrm R}(p) +(\QS)_{ab\,cd}(p)F_0^{\mathrm R}(p) \right]. \label{eq:referencePi} \end{equation} The tensor structures in this formula are polynomials on all momenta; there is no singular projector evaluation at \(p^2=0\). \subsection{Explicit evaluation of the reduced integrals} Put \(w=z/a\) and \[ J_n(w)=\int_0^1 \frac{v^{2n}}{1-w+wv^2}\dd v. \] These functions are analytic on the same slit plane. The change \(u=a/(1-v^2)\) in~\eqref{eq:subtracted} gives \begin{align} F_2(z) &=\frac{w}{480\pi^2}J_3(w), \label{eq:F2closed}\\ F_0(z) &=\frac{w}{192\pi^2} \bigl(9J_1(w)-6J_2(w)+J_3(w)\bigr). \label{eq:F0closed} \end{align} For \(n\ge1\), \begin{equation} wJ_n(w) =\frac1{2n-1}-(1-w)J_{n-1}(w). \label{eq:Jrecursion} \end{equation} The regular value at \(w=0\) is \(J_n(0)=1/(2n+1)\). For real \(w<1\), \(w\neq0\), \begin{equation} J_0(w)= \begin{cases} \displaystyle \frac{\operatorname{artanh}\sqrt{-w/(1-w)}} {\sqrt{-w(1-w)}}, &w<0,\\[3mm] \displaystyle \frac{\arctan\sqrt{w/(1-w)}}{\sqrt{w(1-w)}}, &01\), set \(v_*=\sqrt{1-1/w}\). Then \begin{equation} J_0(w+\ii0\,\operatorname{sgn}p^0) = \frac{ \log\bigl((1-v_*)/(1+v_*)\bigr) +\ii\pi\,\operatorname{sgn}p^0} {2wv_*}. \label{eq:J0above} \end{equation} The apparent singularities of the intermediate \(J_0\) representation cancel in the combinations \eqref{eq:F2closed}--\eqref{eq:F0closed} at \(w=0,1\). \begin{proposition}[Evaluated normalization checks] \label{prop:checks} The printed form factors obey \begin{align} F_2'(0)&=\frac1{13440\pi^2m^2}, & F_0'(0)&=\frac{17}{6720\pi^2m^2}, \label{eq:lowchecks}\\ F_2(a)&=\frac1{2400\pi^2}, & F_0(a)&=\frac3{80\pi^2}. \label{eq:thresholdchecks} \end{align} Their imaginary parts above threshold are \[ \operatorname{Im}F_i^{\mathrm R}(p) =\pi\,\operatorname{sgn}(p^0)\,r_i(p^2). \] \end{proposition} \begin{proof} At \(w=0\), use \(J_n(0)=1/(2n+1)\) in \eqref{eq:F2closed}--\eqref{eq:F0closed} and \(a=4m^2\). In the scalar combination, \[ \frac93-\frac65+\frac17=\frac{68}{35}. \] At \(w=1\), \(J_1=1\), \(J_2=1/3\), and \(J_3=1/5\), which gives~\eqref{eq:thresholdchecks}. The imaginary term in~\eqref{eq:J0above}, together with the recursion, gives the stated imaginary parts; equivalently use Lemma~\ref{lem:dispersion}. \end{proof} These are analytic checks, not reported computer execution. \subsection{Causality, extension, and scaling degree} \begin{lemma}[Convergent subtracted spectral kernel] \label{lem:retardedexistence} For either bounded density \(r_i\), the momentum-space integral \[ \int_a^\infty r_i(u) \left( \frac1{u-(p^0+\ii0)^2+|\mathbf p|^2} -\frac1u \right)\dd u \] exists in \(\Sch'(\R^4)\). Its inverse Fourier transform is supported in \(\overline V_+\). \end{lemma} \begin{proof} For fixed \(u>0\), the first multiplier is the Fourier transform of the retarded Klein--Gordon fundamental solution of mass \(\sqrt u\). The second is the transform of \(-\delta_0/u\), so their difference has future causal support. We verify convergence on Schwartz tests. On \(|p|_{\mathrm E}\le\sqrt u/2\), the denominator is bounded away from zero, and subtraction gives a bound \(C|p|_{\mathrm E}^2/u^2\). Its test pairing is therefore \(O(u^{-2})\). On the complementary region, the poles occur at \(p^0=\pm\sqrt{u+|\mathbf p|^2}\). Partial fractions reduce them to one-dimensional principal values and delta terms, with prefactor \(1/(2\sqrt{u+|\mathbf p|^2})\). The principal-value estimate obtained by subtracting the test value at the pole uses only a test and its first derivative. After the rescaling \(p=\sqrt u\,k\), Schwartz decay on \(|k|_{\mathrm E}\ge1/2\) dominates the Jacobian and these finite-order distribution bounds. Choosing a sufficiently high, fixed Schwartz seminorm makes this region \(O(u^{-2})\) as well. Thus the whole pairing is bounded by \(Cu^{-2}p_N(\psi)\) for some finite Schwartz seminorm \(p_N\), uniformly for large \(u\). Since \(r_i\) is bounded, integration converges in \(\Sch'\). Its finite spectral integrals have future causal support, and that support property is preserved by distributional convergence. \end{proof} \begin{theorem}[Constructed retarded extension] \label{thm:retarded} Equation~\eqref{eq:referencePi} defines a real, tempered, future-causal tensor distribution. It agrees with~\eqref{eq:Kubo} on \(M\setminus\{0\}\) and has \[ \sd_0(\PiR_{\mathrm{ref}})\le8. \] \end{theorem} \begin{proof} Lemma~\ref{lem:retardedexistence} gives the reduced retarded distributions. Applying the constant-coefficient polynomial differential operators corresponding to \(\QT,\QS\) preserves causal support and temperedness. The retarded-minus-advanced difference is, by \eqref{eq:disc} and~\eqref{eq:Wspec}, \[ \PiR_{\mathrm{ref}}-\PiA_{\mathrm{ref}} =\frac{\ii}{\hbar}(W^>-W^<). \] It is therefore a retarded splitting of the prescribed commutator. Away from the origin, causal support and the splitting identity give~\eqref{eq:Kubo}. Possible equal-time derivative terms from representing the stress tensor by polynomial differential operators are supported at the origin and are part of the extension prescription, not omitted nonlocal terms. The boundary-value relation \(F_i^{\mathrm R}(-p)=\overline{F_i^{\mathrm R}(p)}\), with real even tensor polynomials, proves reality. Finally, the explicit form factors are bounded by \(C\log(2+|p|_{\mathrm E})\). Thus \[ |\widehat\PiR_{\mathrm{ref}}(p)| \le C(1+|p|_{\mathrm E})^4 \log(2+|p|_{\mathrm E}). \] Under \(x\mapsto\lambda x\), its pairing with a fixed Schwartz test is bounded by \(C_\psi\lambda^{-8}(1+|\log\lambda|)\). Multiplication by \(\lambda^\omega\) therefore gives zero as \(\lambda\downarrow0\) for every \(\omega>8\). This is the claimed scaling-degree bound. \end{proof} The standard scaling-degree extension framework is discussed in~\cite{BF}; the wavefront and product distinctions are reviewed in~\cite{BDH}. Here the spectral construction supplies the actual distribution before any extension comparison is made. The improper spectral integral is a definition and convergence proof within an already specified free continuum field theory. It is not a regulator-removal theorem for a family of interacting gravitational cutoff theories. %===================================================================== \section{Complete contact freedom and zero-momentum calibration} \label{sec:contacts} %===================================================================== \begin{definition}[Permitted extension class] \label{def:contacts} A permitted response is a tempered retarded extension agreeing with~\eqref{eq:Kubo} off the origin, with scaling degree at most eight, and satisfying: \begin{enumerate}[label=\textup{(\roman*)}] \item real position-space coefficients; \item parity-even Lorentz covariance; \item symmetry in each stress pair and under exchange of the index pairs; \item transversality in both stress slots. \end{enumerate} The mass and stress insertion are fixed. This is an extension class for the selected two-stress response, not a classification of all gravitational counterterms. \end{definition} \begin{theorem}[The complete three-parameter contact family] \label{thm:contacts} Every permitted response differs from \(\PiR_{\mathrm{ref}}\) by exactly a tensor of the form \begin{equation} \widehat C_{ab\,cd}(p) = \hbar\left[ \alpha\,\Ein_{ab\,cd}(p) +\beta\,(\QT)_{ab\,cd}(p) +\gamma\,(\QS)_{ab\,cd}(p)\right], \qquad \alpha,\beta,\gamma\in\R. \label{eq:contacts} \end{equation} The three tensors are linearly independent. Here \(\alpha\) has mass dimension two and \(\beta,\gamma\) are dimensionless in the stated four-dimensional convention. Conversely, every contact~\eqref{eq:contacts} gives a permitted extension. \end{theorem} \begin{proof} Two extensions agreeing off zero differ by a distribution supported at zero. A point-supported distribution is a finite sum of delta derivatives. The scaling-degree bound restricts its derivative order to at most four, so its Fourier transform is a tensor polynomial of degree at most four. Parity-even Lorentz covariance and the pair symmetries give the general polynomial form \[ \begin{split} P_{ab\,cd}={}& A(s)\eta_{ab}\eta_{cd} +B(s)(\eta_{ac}\eta_{bd}+\eta_{ad}\eta_{bc})\\ &+C(s)(\eta_{ab}p_cp_d+\eta_{cd}p_ap_b)\\ &+D(s)(\eta_{ac}p_bp_d+\eta_{ad}p_bp_c +\eta_{bc}p_ap_d+\eta_{bd}p_ap_c)\\ &+E\,p_ap_bp_cp_d, \end{split} \] where \(A,B\) have degree at most two in \(s\), \(C,D\) at most one, and \(E\) is constant. Contracting with \(p^a\) gives the independent equations \[ A+sC=0,\qquad B+sD=0,\qquad C+2D+sE=0. \] Writing \(C=c_0+c_1s\), \(D=d_0+d_1s\), these require \[ c_0=-2d_0,\qquad E=-c_1-2d_1. \] There are therefore exactly three free real coefficients. The degree-two solution is proportional to \(\Ein\). At degree four the independent spin-two and spin-zero solutions are \(\QT,\QS\). Their independence follows on a timelike rest frame: \(\QT,\QS\) are independent projectors multiplied by \(s^2\), while \(\Ein\) has degree two. This proves~\eqref{eq:contacts}. Each displayed contact is real, local, transverse, pair-symmetric, and of scaling degree at most eight. Its support at zero is compatible with retarded support, proving the converse. \end{proof} Transversality is a normalization requirement of this extension class. An arbitrary unnormalized extension can contain additional nontransverse local terms. The theorem neither counts them as physical parameters nor claims that the Ward condition holds automatically for every such extension. \begin{corollary}[Unique zero-momentum prescription] \label{cor:unique} Among the permitted extensions, exactly one satisfies \begin{equation} \widehat\PiR(p)=O(|p|_{\mathrm E}^6) \quad\text{in its analytic neighborhood of }p=0. \label{eq:zeronorm} \end{equation} It is \(\PiR_{\mathrm{ref}}\). Equivalently, calibration sets the degree-two and degree-four Taylor tensors to zero. This conclusion uses \(m>0\). No \(m\to0\) limit is taken; it does not discharge XVI-N, produce a massless susceptibility, or match VII-H. \end{corollary} \begin{proof} The gap \(a>0\) makes \(F_i(z)\) analytic near zero. Since \(F_i(0)=0\), each term \(\mathsf Q_i(p)F_i(p^2)\) begins at momentum degree six. Thus the reference satisfies~\eqref{eq:zeronorm}. For a different extension, the degree-two term is \(\hbar\alpha\Ein\), forcing \(\alpha=0\). Independence of \(\QT,\QS\) then forces \(\beta=\gamma=0\) at degree four. \end{proof} The massless Wick-square susceptibility in the supplied VII-H record has a different insertion, quantum order, and contact classification \cite{CheungVIIH}. Even a future massless limit here would require a connecting theorem before it could be compared with that benchmark. Choosing \(m>0\) is not a completion of the massless special-family problem in XVI-N \cite{CheungXVIN}. \paragraph{Final normalized kernel.} For the zero-momentum prescription, the completed response is \begin{equation} \boxed{ \widehat\PiR_{ab\,cd}(p) = \hbar\left[ (\QT)_{ab\,cd}(p) F_2\bigl((p^0+\ii0)^2-|\mathbf p|^2\bigr) + (\QS)_{ab\,cd}(p) F_0\bigl((p^0+\ii0)^2-|\mathbf p|^2\bigr) \right]. } \label{eq:finalPi} \end{equation} Equations~\eqref{eq:F2closed}--\eqref{eq:J0above} evaluate its two scalar functions completely. %===================================================================== \section{Both linear Ward identities and the nonlocal invariant} \label{sec:ward} %===================================================================== \begin{theorem}[Distributional linear Ward identities] \label{thm:ward} Every extension in Theorem~\ref{thm:contacts} satisfies \begin{equation} \partial^a\PiR_{ab\,cd}=0, \qquad \partial^c\PiR_{ab\,cd}=0 \quad\text{in }\Sch'(M). \label{eq:ward} \end{equation} Let \[ (K\xi)^{ab}=\partial^a\xi^b+\partial^b\xi^a \] for a compact smooth vector field \(\xi\). The unprepared bilinear response satisfies \[ \mathcal R(j,K\xi)=0,\qquad \mathcal R(K\xi,h)=0. \] These are complete linear identities for the printed kernel, all compact tests, and the stated contact family. They are not interacting gravitational BRST/BV identities. \end{theorem} \begin{proof} Lemma~\ref{lem:transverse} annihilates each polynomial tensor in the reference and contact formulas. Multiplication by the scalar boundary values preserves the polynomial zero identities. Fourier inversion proves~\eqref{eq:ward}. Compact integration by parts in either source slot gives the two bilinear identities. \end{proof} \begin{remark}[Coverage is model-specific] Oai282-A/B provide documentary examples of finite free-scalar Ward audits and homogeneous stress coverage. Oai282-B's improvement scalar \(L=-Q/2\) on its ordered Wick algebra is not any of the three coefficients here; its two probes do not cover unrestricted tensors \cite{CheungOai282A,CheungOai282B}. Conservation of a conformal-Killing current does not by itself prove Ward action on every field. No free conformal action, unrestricted stress classification, or verification of Family 282 is imported \cite{Catalogue}. The supplied Oai069-A finite free-scalar Ward-defect summary is a second comparison only, not this kernel's identity or a chiral-to-four-dimensional bridge \cite{CheungOai069A}. \end{remark} \begin{theorem}[Contact-independent nonlocal response] \label{thm:nonlocal} For \(p^2>a\), \begin{equation} \widehat\PiR(p)-\widehat\PiA(p) = 2\pi\ii\hbar\,\operatorname{sgn}(p^0) \bigl[\QT(p)r_2(p^2)+\QS(p)r_0(p^2)\bigr]. \label{eq:absorptive} \end{equation} This discontinuity is unchanged by every permitted contact. It is nonzero, so no permitted contact can make the response local or identically zero. \end{theorem} \begin{proof} The advanced form reverses the boundary value, while the real local polynomial is unchanged. Lemma~\ref{lem:dispersion} gives~\eqref{eq:absorptive}. The positive densities and nonzero rest-frame projectors show nonvanishing. A distribution supported only at zero has polynomial Fourier transform and hence no spectral discontinuity. Thus the nonzero discontinuity excludes locality and cannot be cancelled by the polynomial family. \end{proof} On positive-energy momenta, the norm in Proposition~\ref{prop:detector} can equivalently be read from \[ \widehat W^>(p)=2\hbar\,\operatorname{Im}\widehat\PiR(p). \] This relation identifies spectral detection; it does not construct an instrument. The contact-independent discontinuity is a scheme-stable feature of this benchmark, not a renormalization-group residue, physical beta function, or the graph-specific curvature-span statement in XVI-M. VI's RG-open quantum certificate and XVI-M's separate scale-matching boundary remain documentary comparisons \cite{CheungVI,CheungXVIM}. %===================================================================== \section{Quadratic closed-time-path bookkeeping} \label{sec:ctp} %===================================================================== The closed-time-path construction concerns the same vacuum and two-stress insertion. It is not an all-orders source-evolution theorem. Use the zero-momentum contacts, or transport a common contact \(C\) consistently as described below. Define \[ D^> =\frac{\ii}{\hbar}W^>, \qquad D^< =\frac{\ii}{\hbar}W^<, \] and in Fourier space \begin{align} \widehat D^{++}(p) &= \hbar\bigl[\QT(p)F_2(p^2+\ii0) +\QS(p)F_0(p^2+\ii0)\bigr]+\widehat C(p), \label{eq:Dpp}\\ \widehat D^{--}(p) &= -\hbar\bigl[\QT(p)F_2(p^2-\ii0) +\QS(p)F_0(p^2-\ii0)\bigr]-\widehat C(p),\\ D^{+-}&=D^<,\qquad D^{-+}=D^>. \label{eq:ctp} \end{align} Away from coincident points, these are respectively \(\ii/\hbar\) times the time-ordered, anti-time-ordered, lesser, and greater correlators. The minus sign in \(D^{--}\) follows from its anti-time-ordering boundary value after multiplication by \(\ii/\hbar\); it is not optional. The branch signs in the quadratic functional are separate from these kernel definitions. For sources \(H_+,H_-\), let \[ H_{\mathrm c}=\frac{H_++H_-}{2}, \qquad H_{\mathrm d}=H_+-H_-, \qquad N=\frac12(W^>+W^<). \] All contractions below include spacetime integration and the two tensor slots. \begin{proposition}[Consistent quadratic CTP coefficient] \label{prop:ctp} The four kernels satisfy \begin{equation} D^{++}+D^{--}=D^>+D^<, \qquad D^{++}-D^<=\PiR, \qquad D^{++}-D^>=\PiA. \label{eq:ctpidentities} \end{equation} The quadratic connected CTP cumulant, with \(V_H=-g\T(H)\), is \begin{equation} \begin{split} \Gamma_{\mathrm{CTP}}^{[2]} =\frac{g^2}{2}\bigl( &H_+D^{++}H_+ +H_-D^{--}H_-\\ &-H_-D^>H_+ -H_+D^+\widehat D^<\). Subtracting the negative-energy or positive-energy Wightman component from \(D^{++}\) yields the retarded or advanced boundary prescription. The contacts have been transported with opposite signs on the two diagonal branches, so the identities retain them. The vacuum source-degree-two cumulant has the displayed two self-branch terms and the two cross-branch terms. Substitute \(H_\pm=H_{\mathrm c}\pm H_{\mathrm d}/2\). Use the first identity to cancel the \(H_{\mathrm c}H_{\mathrm c}\) term and \(D^>(x-y)=D^<(y-x)\) with exchanged tensor pairs. The mixed term is \(g^2H_{\mathrm d}\PiR H_{\mathrm c}\); the remaining difference-source term is \(\ii g^2H_{\mathrm d}NH_{\mathrm d}/(2\hbar)\). For a real test, \(H_{\mathrm d}NH_{\mathrm d}\) equals a nonnegative vacuum norm, proving the sign. No selective maps or quantum output registers occur in this calculation. \end{proof} This is a specified quadratic-source use of the CTP framework; the external literature discusses its causal role in stress response~\cite{AMP}. Hawk04-B's classical-record versus quantum-return distinction is only a documentary non-identification: \eqref{eq:ctp} is neither output interface \cite{CheungHawk04B}. %===================================================================== \section{Prepared response and its calculated residual} \label{sec:preparation} %===================================================================== Fix a real smooth time ramp with \begin{equation} \sigeps(t)=0\quad(t\le0),\qquad \sigeps(t)=1\quad(t\ge\eps), \qquad \eps>0. \label{eq:ramp} \end{equation} It is not required to be a ultraviolet regulator. Take \(h,j\in C_c^\infty(M;S^2TM)\) and \(H=\sigeps h\). Then \(H\) is compact and smooth. \begin{proposition}[Admitted compact-test prepared coefficient] \label{prop:prepared} Equation~\eqref{eq:response} is a continuous bilinear functional of \(j,h\) on each fixed compact-support test-function space. It is real for real data and retarded: only points of \(H\) in the causal past of the observation test can contribute. \end{proposition} \begin{proof} The map \((j,H)\mapsto F_{j,H}\) is continuous into a fixed compact-support smooth test space. Pairing with the constructed distribution is continuous. Smooth multiplication by the fixed ramp is continuous. Reality follows from Theorem~\ref{thm:retarded}. The support of \(\PiR\) supplies the causal statement. \end{proof} No sharp-preparation limit is needed for this construction. \subsection{Exact transformation formula} For a compact smooth vector field \(\xi\), put \begin{equation} B_\eps(\xi)^{ab} = (\partial^a\sigeps)\xi^b +(\partial^b\sigeps)\xi^a. \label{eq:Bres} \end{equation} It is compact and supported in the preparation layer where the ramp derivative meets \(\supp\xi\). The product rule gives \begin{equation} \sigeps K\xi=K(\sigeps\xi)-B_\eps(\xi). \label{eq:productrule} \end{equation} \begin{theorem}[Fixed-preparation residual and plateau intertwining] \label{thm:residual} For every permitted contact prescription, \begin{equation} \mathcal R_\eps(j,h+K\xi)-\mathcal R_\eps(j,h) = -g^2\pair{j}{\PiR*B_\eps(\xi)}. \label{eq:residual} \end{equation} If \(\xi\) is supported in the plateau \(t>\eps\), the right side is zero. Thus multiplication by the fixed ramp intertwines that plateau-supported linear action, not the unrestricted action. For every nonconstant ramp~\eqref{eq:ramp} and every \(T>\eps\), there exist real compact \(\xi\) in the preparation layer and a real compact tensor test \(j\) supported in \(t>T\) such that the residual is nonzero. For that detecting pair its value is unchanged by all contacts~\eqref{eq:contacts}. The residual concerns this prepared external source; it is not the companion's density-weighted cubic source. \end{theorem} \begin{proof} By~\eqref{eq:productrule}, the change in the source is \(K(\sigeps\xi)-B_\eps(\xi)\). The first term is annihilated by Theorem~\ref{thm:ward}, proving~\eqref{eq:residual}. On the plateau \(\partial\sigeps=0\) on \(\supp\xi\), so \(B_\eps(\xi)=0\). It remains to prove a late nonzero detector. Choose a real nonzero \(\chi\in C_c^\infty((0,\eps))\) such that \(b(t)=\sigeps'(t)\chi(t)\) is nonzero. Take \[ \xi^1(t,\mathbf x)=\chi(t)\psi(\mathbf x), \qquad \xi^0=\xi^2=\xi^3=0, \] where \(\psi\) is real, compact, and chosen with \(\widehat\psi(0,q,0)\neq0\) for some \(q\neq0\). Then \(B_\eps(\xi)\) has only its \(01\) and \(10\) components, equal to \(b(t)\psi(\mathbf x)\). Fix spatial momentum \(\mathbf p=(0,q,0)\). For timelike \(p=(\omega,0,q,0)\), the transverse trace of this tensor is zero because \(p^1=0\). Its spin-two projection is nonzero when its scalar Fourier factor is nonzero and \(q\neq0\). The entire function \(\widehat b(\omega)\) is not identically zero; it therefore does not vanish on every interval above \(\omega_*=\sqrt{q^2+a}\). Consequently the vector-valued function \[ G(\omega)= \widehat\PiR(\omega,0,q,0)\, \widehat B_\eps(\xi)(\omega,0,q,0) \] has a nonzero discontinuity across an open portion of \(\omega>\omega_*\), by~\eqref{eq:absorptive}. Local real contacts have no such discontinuity. Suppose \(u=\PiR*B_\eps(\xi)\) had compact support. Its Fourier transform would then be entire, including its restriction to this complex \(\omega\) line. On \(|\omega|<\omega_*\), it agrees with the analytic spectral expression. The identity theorem on \[ \C\setminus \bigl((-\infty,-\omega_*]\cup[\omega_*,\infty)\bigr) \] would make that entire function agree with both boundary values of the spectral expression, forcing its discontinuity to vanish. This contradicts the nonzero spin-two discontinuity. Thus \(u\) is not compactly supported. Retarded support gives \(\supp u\subset J^+(\supp B_\eps(\xi))\). If its support were bounded above in time, this causal containment and the compact source support would make it compact in spacetime. Hence its support reaches times greater than every \(T\). There is a compact real tensor test \(j\) supported in \(t>T\) with \(\pair{j}{u}\neq0\). The reality of \(u\) permits a real detector. A contact changes \(u\) by a finite differential operator applied to \(B_\eps(\xi)\), supported in the preparation layer. Such a change is invisible to this late \(j\). The nonzero residual is therefore contact independent for the constructed detector. \end{proof} \begin{remark}[Why this is not a scheme change] Changing \(\eps\), the ramp, or the preparation changes the source \(\sigeps h\). It is not an admissible finite renormalization of a fixed source experiment. For overlapping source and observation supports, a residual pairing can still contain local calibration dependence; the late detector in Theorem~\ref{thm:residual} is the separate construction that removes all such contacts. \end{remark} The supplied XVI-B source/response distinction and XII-C's retained ramp derivatives are documentary counterparts of the product-rule mechanism \cite{CheungXVIB,CheungXIIC}. The fixed switched convolution of X-A-conv has its own first-order covariance obstruction and is not identified here as a successful ultraviolet regulator \cite{CheungXAconv}. No formula equates \(\sigeps h\) with any of those manuscripts' cubic sources. %===================================================================== \section{Five dependences kept separate} \label{sec:dependences} %===================================================================== \begin{enumerate}[label=\textbf{D\arabic*.}] \item \textbf{Nonlocal response.} The threshold and discontinuity \eqref{eq:absorptive} are fixed by the printed insertion, mass, and vacuum. Permitted local contacts cannot change them. \item \textbf{Local calibration.} The three real coefficients \(\alpha,\beta,\gamma\) are the entire freedom in Definition~\ref{def:contacts}. Condition~\eqref{eq:zeronorm} fixes them uniquely. It is a calibration choice, not a theorem that every physical application must choose zero local coefficients. \item \textbf{Preparation.} The map \(h\mapsto\sigeps h\), its width, and its ramp shape are part of the external source experiment. Varying \(\eps\) or the physical preparation changes the source, not the finite renormalization prescription. The residual~\eqref{eq:residual} keeps that distinction visible. \item \textbf{Subtraction prescription.} For a fixed \(\mu>0\), replacing \(1/u\) in \eqref{eq:subtracted} by \(1/(u+\mu^2)\) gives \[ F_{i,\mu}(z)=F_i(z)+c_i(\mu), \qquad c_i(\mu)= \int_a^\infty r_i(u) \left(\frac1u-\frac1{u+\mu^2}\right)\dd u. \] The difference is finite and independent of \(z\). It changes only the quartic contact coefficients. Transporting the local calibration recovers the same normalized response. This subtraction-point comparison is not, by itself, a physical beta function. \item \textbf{State.} The Minkowski vacuum is explicitly fixed. A different state generally changes the stress two-point function and destroys this particular Lorentz-invariant spectral description. No state change is absorbed into the three vacuum contact constants. \end{enumerate} These distinctions do not introduce a ultraviolet cutoff, a gravitational RG flow, or a continuum-matching theorem. A convergent subtracted dispersion integral is not a proof of ultraviolet completeness. %===================================================================== \section{Documentary comparisons and the sibling contract} \label{sec:documentary} %===================================================================== All entries below are version-qualified comparisons. They are not premises of the spectral, contact, Ward, CTP, or residual proofs. A reported source theorem retains its own hypotheses and is not independently verified by this manuscript. \begingroup \small \begin{longtable}{@{}P{.47\linewidth}P{.47\linewidth}@{}} \toprule Comparison used & Conclusion not imported\\ \midrule \endfirsthead \toprule Comparison used & Conclusion not imported\\ \midrule \endhead \bottomrule \endfoot \textbf{XVIII-A, v2} \cite{SiblingXVIIIA}. The sibling contract separates witness recovery, kernel admission, normalization, and ultraviolet matching. & Its conditional assembly and matching theorems are not premises of this spectral calculation. This benchmark does not discharge the original relational target's gates. \\[4pt] \textbf{I, v2} \cite{CheungI}. Destructive-readout semantics distinguish a classical record from a surviving measured quantum name. & No measurement uniqueness or field instrument is imported from a positive vacuum pairing. \\[4pt] \textbf{VII-E, v1} \cite{CheungVIIE}. Observation boundaries retain complete records, spectators, residual instruments, and continuations. & The spectral detector is not such a boundary or a scheduled apparatus realization. \\[4pt] \textbf{Hawk04-A, v1} \cite{CheungHawk04A}. Complete preservation and recovery through one declared output are different tasks. & No complete-access inverse or radiation decoder is constructed by the stress response. \\[4pt] \textbf{Hawk04-B, v2} \cite{CheungHawk04B}. Classical readout is distinguished from quantum return. & The CTP kernels are neither a classical-output channel nor a quantum-return register. \\[4pt] \textbf{XVII-A, v1} \cite{CheungXVIIA}. The separately identified finite-matrix bounded-effect reconstruction remains a different benchmark. & No reconstruction map, boundary CFT, entanglement wedge, Page curve, or metric identification is transferred. \\[4pt] \textbf{VII-H, v1} \cite{CheungVIIH}. A free massless Wick-square response has its own one-parameter contact freedom and invariant detector. & Its \(\hbar\delta_0\) contact and insertion are not this massive two-stress classification. No massless connecting theorem is supplied. \\[4pt] \textbf{XVI-B, v1} \cite{CheungXVIB}. Late source agreement is not response agreement; source, response, and class kernel remain separate. & The preparation product rule does not identify \(\sigeps h\) with the companion cubic source. \\[4pt] \textbf{XII-C, v1} \cite{CheungXIIC}. The cubic-only density freeze retains actual preparation derivatives. & No cubic density, source array, or coupled Einstein jet is imported into the linear external-source model. \\[4pt] \textbf{X-A-conv, v1} \cite{CheungXAconv}. One fixed switched convolution has its own first-order covariance defect. & No successful ultraviolet-regulator identification, universal convolution no-go, or shared source law. \\[4pt] \textbf{Oai282-A, v1} \cite{CheungOai282A}. Finite Ward residuals require their declared contacts and test coverage. & No unrestricted field identity, gravitational counterterm map, or Family 282 verification. \\[4pt] \textbf{Oai282-B, v1} \cite{CheungOai282B}. Its \(L=-Q/2\), local Wick action, and two probes belong to one homogeneous free stress class. & No identification of \(L\) with \(\alpha,\beta,\gamma\), interacting improvement, or unrestricted tensor coverage. \\[4pt] \textbf{Oai069-A, supplied summary} \cite{CheungOai069A}. A second described finite free-scalar Ward-defect audit. & No full theorem from an unavailable source, chiral-to-four-dimensional bridge, positive-channel interpretation, or identity with this kernel. \\[4pt] \textbf{OpenAI catalogue / Family 282, supplied record} \cite{Catalogue}. A recorded local-Ward scope comparison. & No attribution audit, proof verification, global conformal action, fixed point, or ultraviolet match. \\[4pt] \textbf{VI, v1} \cite{CheungVI}. Classical relational descent and the RG-open quantum certificate remain separate. & A subtraction point or spectral discontinuity here is not that certificate or a physical beta function. \\[4pt] \textbf{XVI-M, supplied v1 record} \cite{CheungXVIM}. Its degree-four curvature-span statement is graph-specific. & No curvature-span identification, full scale tensor, or broad matched-scale release. \\[4pt] \textbf{XVI-H, supplied v1 record} \cite{CheungXVIH}. Partial witness recovery does not report full-kernel wavefront or prepared quantum admission. & The local stress insertion is not its three-slot relational kernel, and no root construction is imported. \\[4pt] \textbf{XIV-H/I/J, v1} \cite{CheungXIVH,CheungXIVI,CheungXIVJ}. Their internal maps, I2 advance, and supported reference retain different H2/I3/J2 boundaries. & The simple two-stress spectrum does not construct their full grouped characteristic map or its output theorem. \\[4pt] \textbf{XV-B, v2} \cite{CheungXVB}. Technical lifting and normal-jet interfaces retain the full-map construction boundary. & No inherited quantum-admission exit is opened by the present local benchmark. \\[4pt] \textbf{XVI-N, r12} \cite{CheungXVIN}. The conditional massless special-family obstruction has its own insertion and Route A/Route B domains. & Choosing a different massive local insertion does not discharge the theorem or repair Route B. \\[4pt] \textbf{XVI-O/P, supplied revisions} \cite{CheungXVIO,CheungXVIP}. Kernel subsectors remain distinct from bulk restoration. & No bulk BV restoration or closure of O.1. \\[4pt] \textbf{Em02-E, v1} \cite{CheungEmIIE}. Its incomplete quantum audit lists covariance, state, counterterm, remainder, and gauge obligations for a different background and observable. & No regulator-removal theorem for that observable is supplied. Its classical lapse is not this stress response. \\ \end{longtable} \endgroup \subsection{XVIII-A is a sibling, not a parent} XVIII-A version 2 states interface obligations without evaluating the inherited gravitational coefficient. XVIII-B evaluates one narrower coefficient from its own printed free-field data. Neither is a proof premise of the other. A successor of the original relational target still needs the full kernel, characteristic operation, complete inventory, normalization, representation, and matching obligations set out in the sibling contract. Equation~\eqref{eq:finalPi} can be a regression check only after a comparison map has been proved to preserve the insertion, source normalization, quantum order, state, and contact prescription. Shared spin-projector notation does not supply that map. \subsection{External field-theory comparisons} The massive scalar spectral calculation is a familiar external-source structure, not a newly claimed gravitational discovery \cite{AMP,Mottola}. The present contribution is the explicit convention-bound proof package, complete transverse finite comparison, consistent quadratic CTP prescription, and prepared residual. Fr\"ob's relational scalar correlator contains a different observable and gravitational contributions \cite{FrobCorrelation}. No match to it is asserted. A connecting calculation would have to include the relational-coordinate insertions and the relevant quantum gravitational inventory, not merely relabel \eqref{eq:finalPi}. The external mathematical references supply standard distributional tools on their stated domains. They do not verify any Cheung sheet or the supplied OpenAI catalogue. \subsection{Source availability} Generic manuscript identifiers are retained when a complete bibliographic title was not supplied. Oai069-A is cited only through its supplied summary. The authorship of XVIII-A was not specified and is not assigned to N.~Cheung. The supplied catalogue attribution is not independently authenticated here. No corpus-wide completeness claim, catalogue statistic, or current registry-status claim is used in a proof. %===================================================================== \section{Debugging corrections and exact release} \label{sec:release} %===================================================================== The following corrections are incorporated: \begin{enumerate} \item The unavailable original XVIII-B source is not represented as having been inspected or revised. The reconstructed benchmark is frozen explicitly. \item The identical-particle factor \(1/2!\), field factors of \(\hbar\), source factor \(g=\kappa/2\), and connected-cumulant factor \(1/2\) are kept distinct. \item ``Once subtracted'' refers to the bounded reduced densities \(r_i\), not the growing densities \(\rho_i=u^2r_i\). \item The final tensor structures are polynomial; no rational projector is evaluated at \(p^2=0\). \item The contact classification is complete only for the declared transverse extension class. Nontransverse raw contacts are not silently admitted and then counted as physical parameters. \item Zero-momentum normalization fixes degree-two and degree-four Taylor terms; the massive reference starts at degree six. \item The anti-time-ordered kernel and its local contact have the sign needed for the CTP cancellation identity. \item Vacuum spectral positivity is not promoted to instrument construction or observation. \item The preparation residual is computed for an unrestricted transformation before any plateau restriction is applied. \item The late detecting residual is separated from overlapping-support contact dependence. \item The fixed mass is not treated as a massless completion, and preparation width is not a scheme parameter or a ultraviolet cutoff. \item Linear transversality, a complete external-source inventory, and an inherited gravitational BRST/BV identity are not identified. \end{enumerate} \begin{theorem}[Exact benchmark release] \label{thm:release} For the frozen model of Section~\ref{sec:model}, this manuscript establishes: \begin{enumerate}[label=\textup{(\roman*)}] \item the evaluated positive spin-two and spin-zero vacuum spectral densities \eqref{eq:r2}--\eqref{eq:Wspec}; \item the explicit once-subtracted reduced form factors and real, causal, tempered retarded extension \eqref{eq:finalPi}, with scaling degree at most eight; \item the complete three-parameter transverse contact freedom and unique zero-momentum calibration; \item both distributional linear Ward identities on all compact tests in the declared class; \item the nonzero contact-independent spectral discontinuity and a positive compact vacuum detector; \item consistent four-kernel CTP bookkeeping at quadratic source order; \item the continuous prepared response \(\mathcal R_\eps(j,h)\); \item the exact preparation residual, plateau-supported intertwining, and a late contact-independent nonzero residual detector. \end{enumerate} These statements do not establish the companion's relational scalar two-point coefficient, its characteristic graph, a graviton--ghost inventory, nonlinear metric seagulls, an interacting BRST/BV identity, a match to Fr\"ob, bulk restoration of O.1, a measurement instrument, or ultraviolet completeness. No inherited companion quantum coefficient is admitted. \end{theorem} \begin{proof} Item (i) is Theorem~\ref{thm:spectral}. Item (ii) is Lemma~\ref{lem:dispersion}, Proposition~\ref{prop:checks}, and Theorem~\ref{thm:retarded}. Item (iii) is Theorem~\ref{thm:contacts} and Corollary~\ref{cor:unique}. Item (iv) is Theorem~\ref{thm:ward}. Item (v) is Theorem~\ref{thm:nonlocal} and Proposition~\ref{prop:detector}. Item (vi) is Proposition~\ref{prop:ctp}. Item (vii) is Proposition~\ref{prop:prepared}. Item (viii) is Theorem~\ref{thm:residual}. The excluded objects are not among the frozen fields, vertices, insertions, or interfaces. No proof above constructs or identifies them. \end{proof} \subsection*{Successor package} A successor may use the printed spectrum, normalization checks, contact basis, CTP identities, and prepared residual as regression data after proving a same-object comparison. It does not receive: \begin{enumerate} \item a relational-coordinate map or completed relational insertion; \item a quantum metric, ghost action, or gravitational equation ideal; \item the full inherited characteristic contraction; \item a complete interacting Ward or BV anomaly problem; \item a regulator-removal or ultraviolet connecting theorem; \item a calibrated field instrument or quantum-return decoder. \end{enumerate} Those omissions are not repaired by the success of the external-source benchmark. \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Analytical scope.} The proofs concern one free massive scalar in the specified vacuum and one linear classical tensor-source coupling. The spectral functions are standard structures and are not claimed as new foundational results. \paragraph{Source scope.} Companion citations identify supplied records and versions. Their full proofs and artifact packages are not independently certified. Actual submission should provide stable archival access to unpublished comparisons. \paragraph{Operational scope.} No finite measurement instrument, apparatus realization, destructive-readout implementation, or decoder is constructed. The response, vacuum norm, and CTP kernels are not records. \paragraph{Physical scope.} The metric and background are inputs. No Einstein dynamics, spacetime emergence, holographic dictionary, Page curve, island, de Sitter state, or evaporation endpoint is derived. \paragraph{Limits.} The calculation defines a convergent subtracted dispersion integral within the printed free theory. It takes no \(m\to0\), \(\eps\downarrow0\), adiabatic, interacting cutoff-removal, or ultraviolet-completion limit. \paragraph{Verification and submission.} No successful pdfLaTeX build, numerical or symbolic execution, machine proof, independent referee approval, registration, or journal acceptance is asserted. Responsible author metadata and independent technical review remain necessary. \paragraph{Authorship.} Citation of N.~Cheung does not attribute authorship of this reconstruction to that author. Authorship authorization must be supplied by the submitting author. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} % Documentary comparisons: supplied manuscripts or summaries. % No invented journal metadata or independent verification. \bibitem{SiblingXVIIIA} \emph{XVIII-A: Prepared Relational Observables: Non-Determination, Normalization Compatibility, and Conditional Ultraviolet Matching}, version 2, supplied manuscript draft, n.d. Author not specified. Sibling interface contract; not a premise of the present calculation. \bibitem{CheungI} N.~Cheung, \emph{Landmark I}, version 2, unpublished manuscript, n.d. Supplied \sourceid{I.tex}; destructive-readout characterization and its stated structural class. Enter Calculus Companion, \url{https://enter.grok.me}. \bibitem{CheungVIIE} N.~Cheung, \emph{Landmark VII-E}, version 1, unpublished manuscript, n.d. Supplied observation-boundary and prescribed-schedule comparisons; \sourceid{thm:locality} and \sourceid{cor:local-readout}. \url{https://enter.grok.me}. \bibitem{CheungHawk04A} N.~Cheung, \emph{Hawk04-A}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:global}, \sourceid{thm:exact}, and \sourceid{thm:release}; complete preservation versus declared-output recovery. \url{https://enter.grok.me}. \bibitem{CheungHawk04B} N.~Cheung, \emph{Hawk04-B}, version 2, unpublished manuscript, n.d. Supplied \sourceid{thm:global}, \sourceid{thm:classical}, and complete classical-record/quantum-return distinctions. \url{https://enter.grok.me}. \bibitem{CheungXVIIA} N.~Cheung, \emph{XVII-A: One boundary reconstruction is not holographic emergence}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:main}, \sourceid{lem:timeband}, and \sourceid{thm:release}; separate finite-matrix AdS$_2$ benchmark. \url{https://enter.grok.me/browse/XVII-A}. \bibitem{CheungVIIH} N.~Cheung, \emph{VII-H: Normalization-independent connected quantum responses: complete ambiguity, observable descent, and a nonzero free-field benchmark}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:ambiguity} and \sourceid{thm:release}; free massless Wick-square susceptibility. \url{https://enter.grok.me}. \bibitem{CheungXVIB} N.~Cheung, \emph{Landmark XVI-B}, version 1, unpublished manuscript, n.d. Supplied fixed-width source identities and source/response/class comparison boundaries. \url{https://enter.grok.me}. \bibitem{CheungXIIC} N.~Cheung, \emph{XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question}, version 1, unpublished manuscript, n.d. Supplied cubic-only density freeze and actual source derivatives. \url{https://enter.grok.me}. \bibitem{CheungXAconv} N.~Cheung, \emph{X-A-conv}, version 1, unpublished manuscript, n.d. Supplied fixed switched-convolution construction and first-order covariance obstruction. \url{https://enter.grok.me}. \bibitem{CheungOai282A} N.~Cheung, \emph{Oai282-A}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:ward}; finite local-Ward insertion contract, contact bookkeeping, and coverage qualifications. \url{https://enter.grok.me}. \bibitem{CheungOai282B} N.~Cheung, \emph{Oai282-B}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:coverage} and \sourceid{thm:release}; ordered free-scalar Wick action and homogeneous stress class. \url{https://enter.grok.me}. \bibitem{CheungOai069A} N.~Cheung, \emph{Oai069-A}, unpublished manuscript, n.d.; version not supplied. Cited only through the supplied scope summary of a finite free-scalar Ward-defect audit. No full theorem or decoder construction is assumed. \url{https://enter.grok.me}. \bibitem{Catalogue} \emph{OpenAI mathematics catalogue / Family 282}, attributed to OpenAI in the supplied companion record. Documentary scope summary only; attribution and mathematical proofs are not independently authenticated here. \url{https://enter.grok.me/OpenAI}. \bibitem{CheungVI} N.~Cheung, \emph{Landmark VI}, version 1, unpublished manuscript, n.d. Supplied classical relational-descent theorem and RG-open quantum finite-renormalization audit. \url{https://enter.grok.me}. \bibitem{CheungXVIM} N.~Cheung, \emph{Landmark XVI-M}, supplied version 1 record, unpublished manuscript, n.d. Supplied \sourceid{thm:wpB-dp}, \sourceid{cor:wpB-MC}, and top-versus-full scale-tensor qualifications. Artifact reports are not independently executed here. \url{https://enter.grok.me}. \bibitem{CheungXVIH} N.~Cheung, \emph{XVI-H: Compact polarized kernel and witness recovery}, supplied version 1 record, unpublished manuscript, n.d. Supplied compact-kernel construction and retained full-wavefront/class-admission boundary. \url{https://enter.grok.me/browse/XVI-H}. \bibitem{CheungXIVH} N.~Cheung, \emph{XIV-H: Internal kernel relations on the operator route}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:H2}, exact internal maps, and version-local amplitude boundary. \url{https://enter.grok.me}. \bibitem{CheungXIVI} N.~Cheung, \emph{Landmark XIV-I}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:I2}, \sourceid{thm:I3}, and \sourceid{thm:release}. \url{https://enter.grok.me}. \bibitem{CheungXIVJ} N.~Cheung, \emph{Landmark XIV-J}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:reference}, \sourceid{thm:J2}, and \sourceid{cor:noadmission}. \url{https://enter.grok.me}. \bibitem{CheungXVB} N.~Cheung, \emph{Landmark XV-B}, version 2, unpublished manuscript, n.d. Supplied relative-lifting and finite normal-jet interfaces; full characteristic-map boundary retained. \url{https://enter.grok.me}. \bibitem{CheungXVIN} N.~Cheung, \emph{Landmark XVI-N}, revision r12, unpublished manuscript, n.d. Supplied Theorem 11, \sourceid{thm:massless-obstruction}, and Route A/Route B qualifications. \url{https://enter.grok.me}. \bibitem{CheungXVIO} N.~Cheung, \emph{Landmark XVI-O}, revision o70-sync, unpublished manuscript, n.d. Supplied kernel-prescription subsector verdict and separate unresolved bulk O.1 status. \url{https://enter.grok.me}. \bibitem{CheungXVIP} N.~Cheung, \emph{Landmark XVI-P}, supplied p23 record, unpublished manuscript, n.d. Selected two-point sector, external-leg scheme, and retained bulk-restoration boundary. \url{https://enter.grok.me}. \bibitem{CheungEmIIE} N.~Cheung, \emph{Em02-E}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:lapse}, \sourceid{thm:release}, and incomplete quantum regulator-removal audit. \url{https://enter.grok.me}. % External primary references. \bibitem{AMP} P.~R.~Anderson, C.~Molina-Par\'is, and E.~Mottola, ``Linear response, validity of semiclassical gravity, and the stability of flat space,'' \emph{Physical Review D} \textbf{67} (2003), 024026. \url{https://arxiv.org/abs/gr-qc/0209075}. External spectral and CTP comparison; no metric stability theorem is imported. \bibitem{Mottola} E.~Mottola, ``Gravitational Vacuum Polarization: Decoupling and the Conformal Anomaly,'' arXiv:2607.18180 (2026). \url{https://arxiv.org/abs/2607.18180}. External distinction between the stress correlator and the complete second metric variation with local terms. \bibitem{BF} R.~Brunetti and K.~Fredenhagen, ``Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds,'' \emph{Communications in Mathematical Physics} \textbf{208} (2000), 623--661. \url{https://arxiv.org/abs/math-ph/9903028}. \bibitem{BDH} C.~Brouder, N.~V.~Dang, and F.~H\'elein, ``A smooth introduction to the wavefront set,'' \emph{Journal of Physics A: Mathematical and Theoretical} \textbf{47} (2014), 443001. \url{https://arxiv.org/abs/1404.1778}. \bibitem{FrobCorrelation} M.~B.~Fr\"ob, ``Gauge-invariant quantum gravitational corrections to correlation functions,'' \emph{Classical and Quantum Gravity} \textbf{35} (2018), 055006. \url{https://arxiv.org/abs/1710.00839}. Different relational scalar observable; no match is asserted. \end{thebibliography} \end{document}