\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,tabularx,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={VI-B: Local harmonic Einstein--scalar descent and finite Enter certificates}, pdfsubject={A supported-presentation adaptation on a prescribed conical spacetime} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{3pt} \setlist{itemsep=3pt,topsep=5pt} \allowdisplaybreaks \numberwithin{equation}{section} \setcounter{tocdepth}{1} \urlstyle{same} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\HH}{\mathcal H} \newcommand{\DD}{\mathcal D} \newcommand{\JJ}{\mathcal J} \newcommand{\OO}{\mathcal O} \newcommand{\AA}{\mathcal A} \newcommand{\LL}{\mathcal L} \newcommand{\Mclock}{\mathcal M_C} \newcommand{\Glin}{G^{(1)}} \newcommand{\GR}{G_{\mathrm R}} \newcommand{\GA}{G_{\mathrm A}} \newcommand{\Gzero}{G_{\Box,0}} \newcommand{\Ninv}{\mathcal N} \newcommand{\dd}{\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\formal}{\pmod{\kappa^3}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\proj}[1]{\kb{#1}{#1}} \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{\loc}[1]{\texttt{\detokenize{#1}}} \newcommand{\doi}[1]{\href{https://doi.org/#1}{\nolinkurl{#1}}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Ad}{Ad} \DeclareMathOperator{\WF}{WF} \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{VI-B: Local harmonic Einstein--scalar descent\\ and finite Enter certificates}\\[4pt] \large A supported-presentation adaptation on a prescribed conical spacetime} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We give a self-contained local adaptation of Manuscript VI to the regular part of I-B's prescribed conical spacetime. On an explicitly supported developing presentation, we construct a second-order Einstein--scalar coefficient space and prove descent of a finite-time harmonic relational scalar under formal diffeomorphisms fixed near the initial surface. A constraint-compatible sourced family makes the construction nonempty and gives a strictly positive second-order relational coefficient. The principal obstruction is retained rather than hidden by the adaptation. Replacing the scalar coefficient by a compactly switched retarded response produces an exact first-order gauge defect. A nested-support example has a normalized smeared gap at most \(-1/2\), although the switch equals one near the entire observation. No correction beginning at second coupling order, or at positive \(\hbar\)-order, removes that classical gap while the lower-order prescription and identification are held fixed. The additional contribution is an explicit finite certificate interface: three descended evaluations and the separate plateau-gap witness are encoded in finite diagonal matrices, read by a complete destructive-clock instrument, and transported with untouched references on a target-adapted causal route. Fixed-table approximation has a proved complete-channel limit. This combination may be useful as a reproducible separation between classical descent, switched compatibility, and operational readout; it is not a measurement construction for an unknown continuum field. The local background, external schedule, initial-data prescriptions, and transport permissions are inputs. The regular cone is not globally hyperbolic, and the sourced example is not claimed to have compact metric Cauchy data. No intrinsic conical Green operator, quantum Ward identity, radiative response, or quantum renormalization separator is constructed. The quantum certificate remains RG-open. \end{abstract} \statementbox{Exact scope}{% The positive result is local Grade S, defined below. It is not Grade E: records do not determine the metric, lapse, deficit, or causal cone. The classical descent and compact-switch obstruction are adaptations of the findings of VI, not priority claims over that manuscript. The finite encoding is an interface for already specified classical evaluations, not a finite-qubit field instrument.} \tableofcontents %===================================================================== \section{Introduction: three constructions that must remain distinct} %===================================================================== The present problem has three different maps: \begin{equation} \text{solution jet} \longmapsto \text{harmonic relational scalar}, \qquad \text{free input} \longmapsto \text{switched dressed response}, \qquad \text{finite evaluation table} \longmapsto \text{recorded instrument}. \label{eq:three-maps} \end{equation} The first can descend even when the second fails to intertwine the proposed representative identification. The third can record that failure without repairing it. Manuscript VI identifies this distinction for a second-order Einstein--scalar expansion \cite{CheungVI}. We reprove the required classical statements on a supported local domain. No proof of a companion manuscript is an unprinted premise. Relational perturbation theory and higher-order invariant observables have substantial antecedents \cite{BFHPR,Frob}. In particular, neither relational observables nor gauge cancellations in previously constructed quantum correlation functions are claimed as new here. The question addressed here is compatibility of one fixed finite-time prescription with one fixed compact switch, followed by a precisely limited finite operational interpretation. \subsection{Enter and its philosophical role} Cheung's Enter calculus treats measured quantum names as linear resources: destructive readout consumes those names and returns a classical record, while explicitly unmeasured factors may survive \cite{CheungEnter,CheungI}. The introductory philosophy emphasizes distinctions among readout, quantum recovery, physical implementation, and geometric identification \cite{CheungPhilosophy}. We adopt the specified finite Born instrument and prove its complete channel formula directly. We do not rederive the measurement characterization from an abbreviated list of axioms, and we do not extend it to a quantum field apparatus. I-B supplies the model of a finite Enter interface on a prescribed conical product \cite{CheungIB}. Its geometric time coordinate remains after clock consumption. Accordingly, a phrase such as ``consuming the fourth dimension'' is, in this paper, only an operational analogy for consuming a finite encoding calibrated to an external time coordinate. \begin{definition}[Local Grade S] \label{def:grade} A local Grade S package consists of: \begin{enumerate}[label=\textup{(S\arabic*)}] \item a prescribed Lorentzian background, local rods, and a supported analytical presentation; \item named classical coefficient equations and proved descent statements on that presentation; \item a finite destructive clock with a complete recorded instrument; \item an explicit finite encoding of selected classical evaluations; \item reference-preserving transport whose complete dependencies respect the prescribed causal relation. \end{enumerate} This definition makes no claim that records generate the background or its equations. \end{definition} The supported-laboratory discipline is comparable with the Grade S sheets III-B and V-B \cite{CheungGradeS}. Their results are not substituted for the proofs below. %===================================================================== \section{The prescribed cone and the supported presentation} \label{sec:geometry} %===================================================================== Fix \[ c>0,\qquad 0<\mu<2\pi,\qquad \alpha=1-\frac{\mu}{2\pi}. \] Prescribe \begin{equation} M=\RR_T\times C_\mu^\circ\times\RR_z,\qquad \bar g=-c^2\dd T^2+\dd r^2+\alpha^2r^2\dd\phi^2+\dd z^2, \label{eq:cone} \end{equation} where \(r>0\) and \(\phi\in\RR/(2\pi\mathbb Z)\). The singular locus in the completion is \[ \mathcal W=\RR_T\times\{o\}\times\RR_z. \] It is a worldsheet. No smooth metric chart or ordinary pointwise Einstein tensor is asserted on \(\mathcal W\). \begin{proposition}[Local geometry and proper time] \label{prop:geometry} The metric \eqref{eq:cone} has signature \((-,+,+,+)\). Choose the future orientation defined by \(\partial_T\). For a nonzero future-causal tangent \(a\partial_T+w\), \[ a>0,\qquad |w|_{\rm spatial}\le ca. \] At a static port, \(\dd\tau=\dd T\). On a sufficiently small lifted angular interval, \[ \operatorname{dev}(T,r,\phi,z) =\bigl(T,r\cos(\alpha\phi),r\sin(\alpha\phi),z\bigr) \] is a local isometry to Minkowski space. The global conical gluing remains nontrivial. \end{proposition} \begin{proof} The metric matrix has one negative and three positive diagonal entries. The causal inequality is \(-c^2a^2+|w|_{\rm spatial}^2\le0\), and a nonzero causal tangent cannot have \(a=0\). For \(w=0\), the proper-time formula follows directly. Differentiation of the developing map gives \[ \dd x^2+\dd y^2=\dd r^2+\alpha^2r^2\dd\phi^2. \] A deck shift \(\phi\mapsto\phi+2\pi\) changes the developed angle by \(2\pi-\mu\). In the inverse-deck convention this is rotation by \(\mu\). Thus local flatness does not eliminate the conical identification. \end{proof} \begin{proposition}[Failure of global hyperbolicity] \label{prop:nonGH} The regular spacetime \(M\) has a noncompact causal diamond. \end{proposition} \begin{proof} Fix \(r_0>0\), one angle, and \(z=0\). Let \(p,q\) be the events at that spatial point with times \(-2r_0/c\) and \(2r_0/c\). For any sequence \(r_n\downarrow0\), the events \((0,r_n,\phi,0)\) lie in \(J^+(p)\cap J^-(q)\), by radial causal travel. They have no convergent subsequence in \(M\), because the apex is excluded. \end{proof} Consequently local flatness alone does not justify a global conical Cauchy or Green theorem. Standard global results require appropriate global hypotheses \cite{BGP}. \subsection{The local analytical domain} Choose a regular spatial developing chart \[ \chi:U\longrightarrow V\subset\RR^3 \] with \(V\) convex and containing a closed Euclidean ball \(\overline B_R\) in its interior. Set \[ t=c(T-T_0),\qquad x=\chi(r,\phi,z), \qquad \eta=\diag(-1,1,1,1). \] Fix \(00. \] For a smooth source on \(\DD_{R,b}\), the wave and Klein--Gordon equations with prescribed smooth initial data have unique smooth solutions on that domain. On every compact target they depend only on the data and source in its truncated past. The solution maps are continuous in the local smooth topology. \end{lemma} \begin{proof} For compact spatial data and sources, spatial Fourier transformation gives the cosine and sine evolution, with \[ \Omega_m=(-\Delta+m^2)^{1/2},\qquad S_m(t)=\frac{\sin(t\Omega_m)}{\Omega_m}. \] For example, zero-data solutions of \(Pu=F\) are \[ u(t)=-\int_0^tS_m(t-s)F(s)\,\dd s. \] The multiplier has a continuous massless value at zero frequency. Finite propagation follows by integrating the usual wave energy on a truncated backward cone. For the homogeneous difference of two solutions, the lateral flux has the outward nonnegative sign, and zero initial energy gives zero energy throughout the cone. The same argument includes the nonnegative mass term. For a compact target in \(\DD_{R,b}\), its initial shadow lies strictly inside \(B_R\). Extend its required data and source smoothly with compact cutoffs, solve the compact problem, and restrict. Two choices agree by the cone uniqueness just proved. An exhaustion by compact targets gives a compatible smooth solution on the full domain. Differentiated energy estimates and Sobolev estimates on compact cones give continuous dependence in every smooth seminorm. \end{proof} Write \[ \Gzero F=-\int_0^t\frac{\sin((t-s)|D|)}{|D|}F(s)\,\dd s. \] For compact sources supported after \(0\), the Klein--Gordon retarded inverse is \begin{equation} \GR(t)=-\Theta(t)S_m(t). \label{eq:retarded} \end{equation} These operators are defined on the developing presentation and then pulled back. They are not an intrinsic global conical prescription. \begin{remark}[Two metrics, two roles] The prescribed \(\bar g\) controls the local presentation and the finite apparatus schedule. The formal metric \(\gamma(\kappa)\) below is an unknown of stipulated coefficient equations. We do not identify the apparatus cone with a nonperturbatively constructed causal cone of that unknown metric. \end{remark} %===================================================================== \section{The second-order Einstein--scalar coefficient space} \label{sec:jets} %===================================================================== All indices in coefficient formulas are raised with \(\eta\). For a symmetric tensor \(a\), put \[ \bar a_{\mu\nu} =a_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta a, \qquad D(a)^\nu=\partial_\mu\bar a^{\mu\nu}. \] For a vector \(\xi\), \[ (K\xi)_{\mu\nu} =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu, \qquad X_\xi=\xi^\mu\partial_\mu. \] Set \begin{align} L_a u &=-a^{\mu\nu}\partial_\mu\partial_\nu u -D(a)^\nu\partial_\nu u, \label{eq:L}\\ \Glin(a)_{\mu\nu} &=-\frac12\Box\bar a_{\mu\nu} +\partial_{(\mu}D(a)_{\nu)} -\frac12\eta_{\mu\nu}\partial_\rho D(a)^\rho. \label{eq:G1} \end{align} \begin{lemma}[Linear identities] \label{lem:linear} One has \[ D(K\xi)=\Box\xi,\qquad \Glin(K\xi)=0,\qquad \partial^\mu\Glin(a)_{\mu\nu}=0, \qquad L_{K\xi}=-[P,X_\xi]. \label{eq:linear-identities} \] \end{lemma} \begin{proof} The mixed derivatives of \(\partial_\mu\xi^\mu\) cancel in \(D(K\xi)\). Substituting this result in \eqref{eq:G1} proves the pure-gauge identity, and taking a divergence proves the linearized Bianchi identity. Finally, \[ [P,X_\xi]u =(\Box\xi^\nu)\partial_\nu u +2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu u, \] which is \(-L_{K\xi}u\). \end{proof} \subsection{Normalization and the scalar second variation} Use the action convention \begin{equation} S[\gamma,\varphi] =\frac{2}{\kappa^2}\int\sqrt{-\gamma}\,R[\gamma]\,\dd^4x -\frac12\int\sqrt{-\gamma} \left(\gamma^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi +m^2\varphi^2\right)\dd^4x. \label{eq:action} \end{equation} The Ricci convention is the one yielding \eqref{eq:G1}. Compactly supported inverse-metric variation gives \begin{equation} G_{\mu\nu}(\gamma)=\frac{\kappa^2}{4}T_{\mu\nu}(\gamma,\varphi), \qquad (\Box_\gamma-m^2)\varphi=0, \label{eq:equations} \end{equation} where \[ T_{\mu\nu}(\gamma,\varphi) =\partial_\mu\varphi\partial_\nu\varphi -\frac12\gamma_{\mu\nu} \left(\gamma^{\rho\sigma}\partial_\rho\varphi \partial_\sigma\varphi+m^2\varphi^2\right). \] Indeed, the coefficients of the metric variation are \(2G_{\mu\nu}/\kappa^2-T_{\mu\nu}/2\). Thus the source factor \(1/4\) is fixed, not freely suppressed. Work modulo \(\kappa^3\), without factorials: \begin{equation} \gamma=\eta+\kappa h+\kappa^2j,\qquad \varphi=\phi_0+\kappa\phi_1+\kappa^2\phi_2. \label{eq:expansion} \end{equation} Define \(G^{(2)}[h,h]\) by \[ G(\gamma) =\kappa\Glin(h)+ \kappa^2\left(\Glin(j)+G^{(2)}[h,h]\right) \formal. \] An explicit differential specification of \(G^{(2)}\) appears in Appendix~\ref{app:G2}. Let \[ d_h=\frac12\tr_\eta h,\qquad e_h=\frac18(\tr_\eta h)^2-\frac14h_{\mu\nu}h^{\mu\nu}, \] \[ A_h^{\mu\nu}=d_h\eta^{\mu\nu}-h^{\mu\nu},\qquad C_h^{\mu\nu} =h^\mu{}_\rho h^{\rho\nu}-d_hh^{\mu\nu}+e_h\eta^{\mu\nu}. \] \begin{lemma}[Complete scalar expansion] \label{lem:scalar-expansion} One has \[ \Box_\gamma-m^2 =P+\kappa L_h+\kappa^2(L_j+\mathsf M_h)\formal, \] where \begin{equation} \mathsf M_hu =\partial_\mu(C_h^{\mu\nu}\partial_\nu u) -d_hL_hu-e_h\Box u. \label{eq:M} \end{equation} \end{lemma} \begin{proof} Matrix inversion and the determinant expansion give \[ \gamma^{\mu\nu} =\eta^{\mu\nu}-\kappa h^{\mu\nu} +\kappa^2(h^\mu{}_\rho h^{\rho\nu}-j^{\mu\nu}), \] \[ \sqrt{-\gamma}=1+\kappa d_h+\kappa^2(d_j+e_h). \] Their product is \[ \sqrt{-\gamma}\,\gamma^{\mu\nu} =\eta^{\mu\nu}+\kappa A_h^{\mu\nu} +\kappa^2(A_j^{\mu\nu}+C_h^{\mu\nu}). \] Insert this and the inverse density in \[ \Box_\gamma u =(\sqrt{-\gamma})^{-1} \partial_\mu(\sqrt{-\gamma}\gamma^{\mu\nu}\partial_\nu u). \] The linear term is \(L_h\), the part linear in \(j\) is \(L_j\), and the remaining quadratic term is \[ \partial_\mu(C_h^{\mu\nu}\partial_\nu u) -d_h\partial_\mu(A_h^{\mu\nu}\partial_\nu u) +(d_h^2-e_h)\Box u, \] which equals \eqref{eq:M}. \end{proof} \begin{definition}[Named local solution-jet space] \label{def:jet} Let \(\JJ_{R,b}\) be the real smooth tuples \[ (h,j,\phi_0,\phi_1,\phi_2) \] on the supported presentation satisfying \begin{align} \Glin(h)&=0, \label{eq:jet1}\\ \Glin(j)+G^{(2)}[h,h]&=\frac14T(\eta,\phi_0), \label{eq:jet2}\\ P\phi_0&=0, \label{eq:jet3}\\ P\phi_1&=-L_h\phi_0, \label{eq:jet4}\\ P\phi_2&=-L_h\phi_1-(L_j+\mathsf M_h)\phi_0. \label{eq:jet5} \end{align} Each tuple includes its coefficient initial data. No assertion is made that arbitrary unconstrained initial metric data extend to this space. \end{definition} These are the coefficient equations of \eqref{eq:equations}. They are assumptions defining the model, not equations derived from the Enter records. The exact zero-scalar background is \((\eta,0)\). A nonzero \(\phi_0\) in \eqref{eq:expansion} is a leading coupling-expansion coefficient. It is not asserted to be an exact nonzero Minkowski Einstein--scalar background at nonzero coupling. This distinction is also essential in VI \cite{CheungVI}. \subsection{The initial-surface-preserving gauge group} Let \[ \mathfrak g=C_c^\infty(\DD_{R,b}^{\circ};T\DD_{R,b}), \] and define formal pullbacks \begin{equation} F_{\xi,\zeta}^* =\exp(\kappa\LL_\xi+\kappa^2\LL_\zeta)\formal, \qquad \xi,\zeta\in\mathfrak g. \label{eq:gauge} \end{equation} The generators vanish near \(\Sigma_0\) and near the lateral boundary. Their formal compositions again have compact generators, by the truncated Baker--Campbell--Hausdorff formula. \begin{proposition}[Full coefficient action] \label{prop:gauge-action} The action \eqref{eq:gauge} is \begin{align} h'&=h+K\xi,\nonumber\\ j'&=j+\LL_\xi h+K\zeta+\frac12\LL_\xi^2\eta,\nonumber\\ \phi_0'&=\phi_0,\nonumber\\ \phi_1'&=\phi_1+X_\xi\phi_0,\nonumber\\ \phi_2'&=\phi_2+X_\xi\phi_1+X_\zeta\phi_0 +\frac12X_\xi^2\phi_0. \label{eq:gauge-coefficients} \end{align} It preserves \(\JJ_{R,b}\) and the coefficient initial data. \end{proposition} \begin{proof} Expand the exponential and collect powers of \(\kappa\). The Levi--Civita connection of a pullback metric is the pulled-back connection: it is torsion free and compatible with that metric, which characterizes it uniquely. Its curvature and Einstein tensor therefore pull back. The gradient-divergence definition of the scalar wave operator gives, by the chain rule, \[ (\Box_{F^*\gamma}-m^2)(F^*\varphi) =F^*((\Box_\gamma-m^2)\varphi). \] The stress tensor has the same covariance by its displayed tensor formula. Taking coefficients proves preservation of \eqref{eq:jet1}--\eqref{eq:jet5}. All changes vanish on a neighborhood of the initial surface, including their initial time derivatives. \end{proof} The scalar changes in \eqref{eq:gauge-coefficients} are part of the action. Omitting them gives a different identification. %===================================================================== \section{Finite-time harmonic coordinates and descent} \label{sec:descent} %===================================================================== Define four scalar coordinate functions, indexed by an external label \(\mu=0,1,2,3\): \[ X_\gamma^\mu=x^\mu+\kappa w^\mu+\kappa^2v^\mu\formal. \] Require \begin{equation} \Box_\gamma X_\gamma^\mu=0\formal,\qquad X_\gamma^\mu|_0=x^\mu|_0,\qquad \partial_tX_\gamma^\mu|_0=\delta^\mu_0. \label{eq:coordinate-data} \end{equation} The label \(\mu\) does not make \(X_\gamma^\mu\) the components of a spacetime vector. \begin{proposition}[Coordinate construction] \label{prop:coordinates} For arbitrary smooth \(h,j\), the unique coefficient solution of \eqref{eq:coordinate-data} is \begin{align} w^\mu[h]&=\Gzero D(h)^\mu,\nonumber\\ v^\mu[h,j]&= \Gzero\left(D(j)^\mu-\mathsf M_hx^\mu-L_hw^\mu[h]\right). \label{eq:coordinates} \end{align} The coefficients and their first time derivatives vanish at \(0\). They depend continuously on \(h,j\) on compact truncated pasts. \end{proposition} \begin{proof} Since \(\Box x^\mu=0\) and \(L_hx^\mu=-D(h)^\mu\), the first coordinate equation is \[ \Box w^\mu=D(h)^\mu. \] The second is \[ \Box v^\mu=D(j)^\mu-\mathsf M_hx^\mu-L_hw^\mu. \] Lemma~\ref{lem:wave} proves existence, uniqueness, locality, and continuity for their zero-data problems. \end{proof} \begin{lemma}[Coordinate covariance] \label{lem:coordinate-covariance} Under \eqref{eq:gauge}, \[ X_{\gamma'}=F_{\xi,\zeta}^*X_\gamma\formal. \] In particular, \begin{align} w'^\mu&=w^\mu+\xi^\mu,\nonumber\\ v'^\mu&=v^\mu+\xi^\nu\partial_\nu w^\mu +\zeta^\mu+\frac12\xi^\nu\partial_\nu\xi^\mu. \label{eq:coordinate-transform} \end{align} \end{lemma} \begin{proof} Scalar covariance shows that \(F^*X_\gamma^\mu\) solves the harmonic equation for \(F^*\gamma\). Because \(F\) is the identity near \(\Sigma_0\), it has precisely the data \eqref{eq:coordinate-data}. Coefficientwise uniqueness in Proposition~\ref{prop:coordinates} identifies the solutions. Expansion gives \eqref{eq:coordinate-transform}. \end{proof} The formal inverse coordinate map is \begin{equation} Y_\gamma^\mu(X) =X^\mu-\kappa w^\mu(X) +\kappa^2\left(w^\nu\partial_\nu w^\mu-v^\mu\right)(X) \formal. \label{eq:inverse} \end{equation} Substitution in \(X_\gamma(Y_\gamma(X))\) verifies this identity directly. For real \(f\in C_c^\infty(\DD_{R,b}^{\circ})\), interpreted as a test in label coordinates, define \begin{equation} \OO_f[\gamma,\varphi] =\int f(X)\varphi(Y_\gamma(X))\,\dd^4X \formal. \label{eq:observable} \end{equation} The measure is the fixed label density. It is not a fluctuating spacetime volume element. \begin{theorem}[Second-order local descent] \label{thm:descent} The coefficients of \eqref{eq:observable} are \begin{align} \OO_f^{[0]}&=\int f\phi_0\,\dd^4x,\nonumber\\ \OO_f^{[1]}&=\int f\left(\phi_1-w^\mu\partial_\mu\phi_0\right) \dd^4x,\nonumber\\ \OO_f^{[2]}&=\int f\psi_2\,\dd^4x, \label{eq:observable-coefficients}\\ \psi_2&= \phi_2-w^\mu\partial_\mu\phi_1 +(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0 +\frac12w^\mu w^\nu\partial_\mu\partial_\nu\phi_0. \nonumber \end{align} Each is invariant under \eqref{eq:gauge-coefficients}. Consequently they descend to the orbit set \(\JJ_{R,b}/\mathcal G\) of the stated local formal gauge group. \end{theorem} \begin{proof} Taylor expansion using \eqref{eq:inverse} gives all displayed terms. No factor \(1/2\) is attached to \(\phi_2\), because \eqref{eq:expansion} has no factorials. Lemma~\ref{lem:coordinate-covariance} implies \[ Y_{\gamma'}=F^{-1}\circ Y_\gamma\formal. \] Together with \(\varphi'=\varphi\circ F\), this gives \[ \varphi'\circ Y_{\gamma'} =\varphi\circ F\circ F^{-1}\circ Y_\gamma =\varphi\circ Y_\gamma\formal. \] The same \(f(X)\dd^4X\) is used on both sides, so equality is coefficientwise. All integrals are finite, and the required coordinate coefficients are defined on compact causal pasts. \end{proof} The proof is off shell as a formal coordinate statement. Restricting it to \(\JJ_{R,b}\) gives the named on-shell descent. It does not establish that the orbit set is a global conical reduced phase space, or that the observable has renormalized quantum products. \begin{remark}[Dependence on the coordinate prescription] Changing coordinate Cauchy data is not a transformation of the specified gauge group. If \(w\) is replaced by \(w+a\) with \(\Box a=0\), then \[ \widetilde\OO_f^{[1]}-\OO_f^{[1]} =-\int f a^\mu\partial_\mu\phi_0\,\dd^4x, \] which need not vanish. Descent therefore does not imply independence of the relational prescription. \end{remark} \subsection{Local conical interpretation} Pull the coefficient fields and their fixed chart data back to the regular laboratory. The background chart is an isometry, and the defining differential equations and compact evaluations are identical in developed coordinates. This establishes the same local identities in that specified presentation. It does not assert that the chartwise coefficients \(w^\mu\) are global vector components or that the selected inverse patches consistently across the conical gluing. Such a global statement would require additional construction. %===================================================================== \section{Compact switching: exact defect and plateau obstruction} \label{sec:switch} %===================================================================== Fix a real compact switch \[ g\in C_c^\infty(\DD_{R,b}^{\circ}). \] It is not transformed in this section. The compact minimal scalar vertex has classical density \[ V_1^m(g;h,\varphi) =\frac12\int gh^{\mu\nu}T_{\mu\nu}(\eta,\varphi)\,\dd^4x. \] For \(H=gh\), its scalar Hessian is \begin{equation} \mathsf Q_H =-\partial_\mu(\bar H^{\mu\nu}\partial_\nu) -\frac12m^2\tr_\eta H. \label{eq:Q} \end{equation} Integration by parts shows formal self-adjointness. Expanding the derivatives gives \begin{equation} \mathsf Q_H=L_H+\frac12(\tr_\eta H)P. \label{eq:QL} \end{equation} Hence, for \(P\phi_0=0\), \begin{equation} \mathsf Q_{gh}\phi_0 =gL_h\phi_0 -(\partial_\mu g)\bar h^{\mu\nu}\partial_\nu\phi_0. \label{eq:Qswitch} \end{equation} Define the classical zero-past response \[ \phi_{1,g}[h,\phi_0]=-\GR\mathsf Q_{gh}\phi_0. \] It equals the zero-data solution of \(P\psi=-L_{gh}\phi_0\), by \eqref{eq:QL} and uniqueness. In particular, multiplication by \(g\) occurs inside the full metric operator, not merely in front of \(L_h\). Consider the candidate \begin{equation} \AA_{f,g}^{(1)}(h,\phi_0) =\int f\left(\phi_{1,g}[h,\phi_0] -X_{w[h]}\phi_0\right)\dd^4x \label{eq:candidate} \end{equation} under the unchanged-free-input identification \begin{equation} (h,\phi_0)\sim(h+K\xi,\phi_0), \qquad \xi\in\mathfrak g. \label{eq:free-identification} \end{equation} This is not the full action on \(\phi_1,\phi_2\) in \eqref{eq:gauge-coefficients}. \begin{theorem}[Exact first-order switch defect] \label{thm:switch-defect} For every smooth free scalar \(\phi_0\), \begin{align} &\AA_{f,g}^{(1)}(h+K\xi,\phi_0) -\AA_{f,g}^{(1)}(h,\phi_0)\nonumber\\ &\quad=\int f\left[ (g-1)X_\xi\phi_0+ \GR\left( (\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi_0 -[P,g]X_\xi\phi_0 \right)\right]\dd^4x, \label{eq:defect} \end{align} where \[ [P,g]u=(\Box g)u+2(\partial^\mu g)\partial_\mu u. \] \end{theorem} \begin{proof} The coordinate variation is \[ w[h+K\xi]-w[h]=\Gzero\Box\xi=\xi, \] because \(\xi\) vanishes near the initial surface. By Lemma~\ref{lem:linear} and \eqref{eq:Qswitch}, \[ \mathsf Q_{gK\xi}\phi_0 =-gP(X_\xi\phi_0) -(\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi_0. \] Therefore \[ \delta_\xi\phi_{1,g} =\GR\left(gP(X_\xi\phi_0) +(\partial_\mu g)\overline{K\xi}^{\mu\nu}\partial_\nu\phi_0 \right). \] Use \[ gP(X_\xi\phi_0)=P(gX_\xi\phi_0)-[P,g]X_\xi\phi_0. \] The function \(gX_\xi\phi_0\) is compactly supported after \(0\), so zero-past uniqueness gives \(\GR P(gX_\xi\phi_0)=gX_\xi\phi_0\). Subtract \(X_\xi\phi_0\), the dressing variation, and smear. \end{proof} This is a classical retarded identity. It is not a loop anomaly, a renormalized Ward identity, or a quantum Master Ward identity. \begin{theorem}[A negative plateau gap] \label{thm:plateau-obstruction} An admitted supported packet exists with real compact \(f_w,g,\xi\) and a real compact-data free scalar \(\phi_w\) such that \[ f_w\ge0,\qquad \int f_w\,\dd^4x=1,\qquad g=1\text{ near }\supp f_w,\qquad gK\xi=0, \] while \begin{equation} \Delta:= \AA_{f_w,g}^{(1)}(K\xi,\phi_w) -\AA_{f_w,g}^{(1)}(0,\phi_w) =-\int f_w\partial_1\phi_w\,\dd^4x \le-\frac12. \label{eq:gap} \end{equation} Both metric representatives satisfy \(\Glin(h)=0\). \end{theorem} \begin{proof} Choose \(t_*\in(0,b)\). On the auxiliary flat Cauchy slice \(t=t_*\), prescribe \[ \phi_w(t_*,x)=x^1\chi_0(x),\qquad \partial_t\phi_w(t_*,x)=0, \] where \(\chi_0\) is compact smooth and equals one near zero. Klein--Gordon evolution gives compact smooth initial data at \(0\), and \(\partial_1\phi_w(t_*,0)=1\). Choose a small relatively compact neighborhood \(O\) of \((t_*,0)\) on which \(\partial_1\phi_w\ge1/2\). Let \(f_w\ge0\) have unit integral and support in \(O\). Take a compact smooth \(g=1\) near \(\supp f_w\). Choose \(\chi_\xi\) compactly supported after \(0\), equal to one near the entire \(\supp g\), and put \(\xi=\chi_\xi\partial_1\). All supports can be chosen inside \(\DD_{R,b}^{\circ}\) by enlarging the laboratory first if necessary. Every derivative of \(\xi\) vanishes near \(\supp g\). Thus \(gK\xi=0\), and both switched responses in \eqref{eq:gap} are zero. But \(w[K\xi]=\xi\), and \(\xi=\partial_1\) on \(\supp f_w\). This proves \eqref{eq:gap}. Finally, \(\Glin K=0\). \end{proof} The same \(f_w,g,\phi_w\) is used for both representatives. The strictly negative gap is not produced by changing the interaction switch between comparisons. \begin{corollary}[Coefficientwise order obstruction] \label{cor:order-obstruction} Suppose the first coefficient is fixed by \eqref{eq:candidate}. No choice of the coefficient \(\AA_{f,g}^{(2)}\) makes \[ \int f\phi_0+\kappa\AA_{f,g}^{(1)} +\kappa^2\AA_{f,g}^{(2)} \formal \] invariant under \eqref{eq:free-identification} on the full stated domain. Nor can terms of positive \(\hbar\)-degree cancel the fixed nonzero \(\kappa\hbar^0\) gap in a coefficientwise formal identity. \end{corollary} \begin{proof} Extract the coefficient of \(\kappa\), and then, if applicable, of \(\hbar^0\). The value \(\Delta\ne0\) is unaffected by terms starting at higher orders. \end{proof} \subsection{Why this is consistent with descent} The geometric transformation requires \[ \delta\phi_1=X_\xi\phi_0,\qquad \delta w=\xi. \] Those terms cancel in \(\phi_1-X_w\phi_0\). The plateau witness instead has \[ \delta\phi_{1,g}=0,\qquad \delta w=\xi. \] Thus the switched assignment fails to supply the scalar part of the full coefficient action. It does not disprove Theorem~\ref{thm:descent}. The quantified obstruction retains all four choices: the unswitched coordinate dressing, the fixed compact switch, the full compact gauge-support class, and \eqref{eq:free-identification}. Changing any of them defines another problem. \begin{center} \begin{tabularx}{\linewidth}{@{}p{.31\linewidth}XX@{}} \toprule Gauge-support class & Meaning & Status here\\ \midrule All admitted compact generators & The support need not lie in the switch plateau & Exact defect and negative witness proved\\ Generators supported where \(g=1\) & A smaller, separately specified class & VII-A's intertwiner is a documentary comparison; its second-order construction is not incorporated\\ \bottomrule \end{tabularx} \end{center} VII-A records a classical repair for the restricted class \cite{CheungVIIA}. We neither use that result as a premise nor claim its second-order intertwiner as a theorem of VI-B. \begin{remark}[Green subtractions are prescription dependent] A chartwise subtraction of the retarded remainder in \eqref{eq:defect} changes the specified observable. It does not automatically define a tensorial subtraction on the full cone. Already \[ [P,X_\xi]\ne0 \] for a general compact \(\xi\), so the fixed flat inverse does not commute with arbitrary spacetime pullbacks. A global replacement needs its own covariance and initial-data theorem. This is the relevant warning behind the fixed-data subtraction comparison in V-B \cite{CheungGradeS}. \end{remark} %===================================================================== \section{An explicit nonempty sourced family} \label{sec:sourced} %===================================================================== The metric coefficient in \eqref{eq:jet2} is an unknown, not an external tensor. We now solve a nontrivial subclass. For compact smooth spatial \(b_0\), prescribe the auxiliary flat Newton branch \begin{equation} \Ninv b_0(x) =-\frac1{4\pi}\int_{\RR^3}\frac{b_0(y)}{|x-y|}\,\dd^3y, \qquad \Delta\Ninv b_0=b_0. \label{eq:Newton} \end{equation} The kernel is locally integrable, and derivatives can be transferred to the smooth compact source. This gives a smooth potential. The identity follows from the flux of \(\nabla(1/|x|)\) through a small sphere. This initial-data branch is additional model data. It is not an intrinsic conical inverse. \begin{theorem}[Sourced local family] \label{thm:sourced} Let \(\phi_0\) be a real free Klein--Gordon solution with compact smooth auxiliary Cauchy data. There is a tuple in \(\JJ_{R,b}\) with \[ h=0,\qquad\phi_1=0,\qquad D(j)=0,\qquad \Glin(j)=Q:=\frac14T(\eta,\phi_0). \] The scalar \(\phi_2\) can be chosen with zero initial data. For nonzero scalar data, \(Q\ne0\). \end{theorem} \begin{proof} A direct differentiation of the stress tensor gives \[ \partial^\mu T_{\mu\nu}(\eta,\phi_0) =(P\phi_0)\partial_\nu\phi_0=0. \] Thus \(Q\) is conserved. Finite propagation makes it spatially compact on each bounded time slab. At \(0\), set \[ \rho=Q_{00},\qquad J_i=Q_{0i},\qquad a=-2\Ninv\rho,\qquad V_i=2\Ninv J_i, \] \[ B_{ij}=\partial_iV_j+\partial_jV_i -\delta_{ij}\partial_\ell V_\ell. \] Prescribe the trace-reversed metric data \begin{align} \bar j_{00}|_0&=a,& \bar j_{0i}|_0&=0,& \bar j_{ij}|_0&=0,\nonumber\\ \partial_t\bar j_{00}|_0&=0,& \partial_t\bar j_{0i}|_0&=0,& \partial_t\bar j_{ij}|_0&=B_{ij}. \label{eq:metricdata} \end{align} Solve \begin{equation} \Box\bar j=-2Q. \label{eq:sourcedwave} \end{equation} For example, its componentwise expression is \[ \bar j(t)= \cos(t|D|)\bar j|_0+S_w(t)\partial_t\bar j|_0 +2\int_0^tS_w(t-s)Q(s)\,\dd s, \] with \(S_w(t)=\sin(t|D|)/|D|\). Local smooth existence follows also from Lemma~\ref{lem:wave}. Write \[ d_\nu=-\partial_t\bar j_{0\nu}+\partial_i\bar j_{i\nu}. \] The initial data give \(d_\nu|_0=0\). Furthermore, \[ \Delta a=-2\rho,\qquad \partial_iB_{ij}=\Delta V_j=2J_j. \] Using \eqref{eq:sourcedwave}, these identities imply \(\partial_td_\nu|_0=0\). Conservation gives \(\Box d_\nu=0\), so uniqueness gives \(D(j)=0\). Equation~\eqref{eq:G1} now gives \(\Glin(j)=Q\). Finally solve \[ P\phi_2=-L_j\phi_0 \] with zero initial data. The source is smooth and is spatially compact on finite slabs because it contains derivatives of \(\phi_0\). All five jet equations hold. For nonzero data, \[ \int T_{00}(0,x)\,\dd^3x =\frac12\int\left( |\dot\phi_0|^2+|\nabla\phi_0|^2+m^2|\phi_0|^2\right)(0,x) \dd^3x>0, \] so \(Q\ne0\). \end{proof} \begin{remark}[The necessary support correction] The Newton position data in \eqref{eq:metricdata} generally have tails. They are not compact-Cauchy metric data. Their restrictions to \(\Sigma_0\) are declared before evolution. There is no incoming lateral data for the subsequent wave problem on \(\DD_{R,b}\). That statement does not turn the initial Newton prescription into a local source-acquisition operation. In particular, the construction proves no worldsheet boundary condition and no global conical solution. \end{remark} \begin{theorem}[A positive second-order descended evaluation] \label{thm:sourced-nonzero} For \(m>0\), the sourced family contains a tuple and a nonnegative compact test \(f_s\) such that \[ \OO_{f_s}^{[0]}>0,\qquad \OO_{f_s}^{[1]}=0,\qquad \OO_{f_s}^{[2]}>0. \] \end{theorem} \begin{proof} Choose \(F\in C_c^\infty(\RR^3)\), \(F\ge0\), equal to a constant \(A>0\) near zero, and prescribe \[ \phi_0|_0=F,\qquad\dot\phi_0|_0=0. \] Then \[ J_i=0,\qquad B_{ij}=0,\qquad \rho=\frac18\left(|\nabla F|^2+m^2F^2\right). \] The nonnegative nonzero source gives \(a(0)=-2\Ninv\rho(0)>0\). Trace reversal yields \[ j_{00}|_0=\frac a2,\qquad j_{ij}|_0=\frac a2\delta_{ij},\qquad j_{0i}|_0=0. \] Since \(h=0\) and \(D(j)=0\), the coordinate equations give \(w=v=0\). Hence the second relational density is \(\phi_2\). At the origin, the spatial Hessian of \(F\) is zero and \(\partial_t^2\phi_0(0,0)=-m^2A\). Therefore \[ L_j\phi_0(0,0)=\frac12a(0)m^2A>0. \] The zero initial data of \(\phi_2\) and its equation imply \[ \partial_t^2\phi_2(0,0)=\frac12a(0)m^2A, \qquad \phi_2(t,0)=\frac14a(0)m^2At^2+O(t^3). \] For a sufficiently small positive time, \(\phi_2\) is positive on an open neighborhood, while \(\phi_0>0\) there. Choose a nonzero nonnegative \(f_s\) in that neighborhood. The three assertions follow. \end{proof} This proves nonvacuity and a sourced coefficient inside the stipulated equations. It is not a derivation of Einstein dynamics or of physical backreaction from records. \begin{corollary}[A useful exact scaling family] \label{cor:scaling} For the preceding tuple, let \[ \phi_{0,\lambda}=\lambda\phi_0,\qquad j_\lambda=\lambda^2j,\qquad \phi_{2,\lambda}=\lambda^3\phi_2,\qquad h_\lambda=\phi_{1,\lambda}=0. \] For every real \(\lambda\) this is a jet, and \[ \OO_{f_s}^{[0]}(\lambda)=\lambda a_0,\qquad \OO_{f_s}^{[1]}(\lambda)=0,\qquad \OO_{f_s}^{[2]}(\lambda)=\lambda^3a_2, \] where \(a_0,a_2>0\). \end{corollary} \begin{proof} The stress is quadratic, the metric response and its initial preparation are linear in that stress, and \(L_j\phi_0\) is linear in each argument. The zero-data scalar response therefore scales cubically. Here \(w=v=0\). \end{proof} %===================================================================== \section{Finite certificates and their logical limits} \label{sec:certificate} %===================================================================== Let \(\mathcal B\) be a declared set of test configurations. Suppose real functions \[ q,n_1,\ldots,n_L:\mathcal B\longrightarrow\RR \] are already defined. For an ordered finite sample \(S=(\beta_1,\ldots,\beta_s)\), write \[ q_S=(q(\beta_i))_{i=1}^s,\qquad N_S=(n_a(\beta_i))_{i,a}. \] \begin{theorem}[Finite membership--separation alternative] \label{thm:certificate} Exactly one of the following alternatives holds: \begin{enumerate}[label=\textup{(\roman*)}] \item \(q_S=N_Sc\) for some \(c\in\RR^L\); \item there is \(z\in\RR^s\) such that \[ z^{\mathsf T}N_S=0,\qquad z^{\mathsf T}q_S\ne0. \] \end{enumerate} In the second case \(z\) may be chosen with \(z^{\mathsf T}q_S>0\). \end{theorem} \begin{proof} Let \(W=\im N_S\). If \(q_S\in W\), every vector orthogonal to \(W\) annihilates \(q_S\), so (ii) is impossible. If \(q_S\notin W\), set \(z=q_S-\operatorname{proj}_Wq_S\). Then \(z\ne0\), \(z\perp W\), and \(z^{\mathsf T}q_S=\norm z^2>0\). \end{proof} To interpret (ii) against a physical or renormalization class, one must separately prove that \(n_1,\ldots,n_L\) exhaust that class's response space. A catalogue-completeness hypothesis is not proved by matrix rank. A finite fit is not a global functional identity. For example, sampling \(q(x)=x\) and \(n_1(x)=1\) at \(x=0\) gives a finite fit but no global constant-coefficient identity. Injective evaluation on \(\Span\{q,n_1,\ldots,n_L\}\) is sufficient to promote a finite fit to a global one, since it annihilates no nonzero difference. It is not necessary: \(q=1,n_1=1,n_2=x\) already has a global identity even though evaluation at \(0\) is noninjective. These are the certificate distinctions made in II, II-B, and VI \cite{CheungCertificates}. No contact catalogue is transferred between their different observables. \begin{proposition}[Weighting changes an integration-by-parts quotient] \label{prop:weight} A compact total divergence need not remain zero after multiplication by a nonconstant switch. \end{proposition} \begin{proof} For a compact vector density \(v\), \[ \int\partial_\mu v^\mu\,\dd^4x=0,\qquad \int g\partial_\mu v^\mu\,\dd^4x =-\int(\partial_\mu g)v^\mu\,\dd^4x. \] If \(\partial_1g\ne0\) on a neighborhood, choose \(v^1=\chi\partial_1g\), other components zero, with \(\chi\ge0\) a nonzero compact bump there. The weighted integral is \(-\int\chi(\partial_1g)^2<0\). \end{proof} \begin{proposition}[Approximate annihilation needs coefficient bounds] \label{prop:approximate} If \(z^{\mathsf T}N_S\ne0\), then \[ \sup_{c\in\RR^L}|z^{\mathsf T}N_Sc|=\infty. \] If instead \(\norm c\le C\), the residual is bounded by \(C\norm{z^{\mathsf T}N_S}\) in dual norms. \end{proposition} \begin{proof} Scale any \(c_0\) for which \(z^{\mathsf T}N_Sc_0\ne0\). The bounded assertion is the dual norm inequality. \end{proof} The two-representative plateau table is already an exact classical obstruction certificate: \[ \begin{array}{c|cc} h & \phi_{1,g}[h,\phi_w] & \AA_{f_w,g}^{(1)}(h,\phi_w)\\ \hline 0&0&0\\ K\xi&0&\Delta \end{array} \qquad \Delta\le-\frac12. \] This is not a radiative vector \(q_S\) or a quantum counterterm-response matrix \(N_S\). Neither quantum object is evaluated here. %===================================================================== \section{The distinct conditional quantum expansion} \label{sec:quantum} %===================================================================== For independent smooth fields \(h,\varphi\), set \(v_0[h]=v[h,0]\). The bare geometric coefficients are \begin{align} \OO_f^{(0)}&=\int f\varphi\,\dd^4x,\nonumber\\ \OO_f^{(1)}&=-\int f w^\mu[h]\partial_\mu\varphi\,\dd^4x, \nonumber\\ \OO_f^{(2)}&=\int f\left[ (w^\nu\partial_\nu w^\mu-v_0^\mu)\partial_\mu\varphi +\frac12w^\mu w^\nu\partial_\mu\partial_\nu\varphi \right]\dd^4x. \label{eq:bare} \end{align} Linearity in \(j\) of \eqref{eq:coordinates} gives \[ v[h,j]=w[j]+v_0[h]. \] Substituting the solution expansion in the bare expression therefore reproduces Theorem~\ref{thm:descent}. Round and square superscripts denote different expansions. Suppose, conditionally, that a chosen quantum construction has compatible retarded products \(R_1,R_2\), interaction \(\kappa V_1+\kappa^2V_2\), and the convention \[ R_V(F)=F+R_1(V;F)+\frac12R_2(V,V;F)+\cdots. \] Then the coefficient at \(\kappa^2\) is algebraically \begin{equation} \OO_f^{(2)} +R_1(V_1;\OO_f^{(1)}) +R_1(V_2;\OO_f^{(0)}) +\frac12R_2(V_1,V_1;\OO_f^{(0)}). \label{eq:fourterms} \end{equation} This follows simply by collecting coupling powers. Equation~\eqref{eq:fourterms} does not construct any of its renormalized products. In particular, the last term requires a two-vertex and three-insertion construction; the second contains the nonlocal coordinate kernel. Their existence is not implied by an external smooth-metric response. The retarded-product framework \cite{DF} does not discharge these new domain and extension obligations by notation alone. Neither \(\OO_f^{[2]}\) nor \eqref{eq:fourterms} is identified with V's separately prescribed Newton--wave cubic form \cite{CheungV}. No identity with the inherited characteristic quantum coefficient is asserted. \statementbox{Quantum status}{% No switch-compatible quantum realization, renormalized second-order relational product, complete finite-renormalization response catalogue, or radiative evaluation is constructed. The certificate for that quantum target remains \textbf{RG-open}.} The classical defect \eqref{eq:defect} is not a quantum renormalization separator. The finite-dimensional theorem of Section~\ref{sec:certificate} cannot define a missing product or supply missing catalogue coverage. %===================================================================== \section{A finite evaluation encoding} \label{sec:encoding} %===================================================================== The following construction makes the word ``readout'' precise. Its inputs are finite classical evaluations already specified by the preceding theorems. Choose \(\lambda_a=a+1\), \(a=0,1,2,3\). Use the source family of Corollary~\ref{cor:scaling} on the fixed named test \(f_s\), and the separate plateau family with scalar \(\lambda_a\phi_w\). The four-by-four evaluation table is \begin{equation} b_{ka}= \begin{cases} \lambda_a a_0,&k=0,\\ 0,&k=1,\\ \lambda_a^3a_2,&k=2,\\ \lambda_a\Delta,&k=3. \end{cases} \label{eq:table} \end{equation} Rows \(0,1,2\) are descended solution-jet evaluations. Row \(3\) is the representative gap of a different, explicitly named switched experiment; it is not itself asserted to be a descended observable of that switched identification. Let \[ \HH_X=\CC^4,\qquad A_k=\sum_{a=0}^3b_{ka}\proj a_X. \label{eq:matrices} \] These are finite Hermitian matrices. The labels \(a\) identify classical sample configurations, not one-particle field states. Choose \[ B>2\max_{k,a}|b_{ka}|, \qquad E_{k,y}=\frac12(I+yA_k/B),\qquad M_{k,y}=E_{k,y}^{1/2},\quad y=\pm1. \label{eq:effects} \] The effects have spectrum in \((1/4,3/4)\), and \[ \sum_yE_{k,y}=I,\qquad \sum_yByE_{k,y}=A_k. \label{eq:score} \] \begin{proposition}[Meaning of the finite encoding] \label{prop:encoding} On payload input \(\proj a\) at setting \(k\), the scored outcome \(By\) has expectation \(b_{ka}\). More generally, on a payload density matrix \(\rho_X\), its mean is \(\Tr(A_k\rho_X)\). Gauge-related source jets have the same first three table entries. The fourth row records the nonzero plateau mismatch rather than eliminating it. \end{proposition} \begin{proof} The score identity \eqref{eq:score} gives the mean. Gauge independence of the first three rows follows from Theorem~\ref{thm:descent}; linearity in the witness scalar gives the fourth row. \end{proof} The matrices contain real calibration constants as fixed metadata. A finite outcome word is not an exact determination of an arbitrary real coefficient in one shot. The encoding specifies scored expectations for a finite catalogue. There is no acquisition channel here from an unknown Einstein--scalar configuration to \(\HH_X\). Nor is a coherent payload superposition declared to be a superposition of continuum solution jets. This is an evaluation-table instrument, not a field measurement. The distinction from a local quantum field measurement is substantive: such a measurement needs a specified system--probe coupling and a corresponding dynamical comparison, as in \cite{FV}. %===================================================================== \section{Destructive clock, complete instrument, and dictionaries} \label{sec:instrument} %===================================================================== Let the clock be two qubits, \[ \HH_C=\CC^4,\qquad \{\ket{\tau_k}:k=0,1,2,3\}\text{ orthonormal},\qquad P_k=\proj{\tau_k}. \] A fixed basis-change unitary takes \(\ket{\tau_k}\) to \(\ket k\), followed by destructive computational-basis Enter. Define \[ D_k=\bra{\tau_k}_C\otimes I_X. \] Let \(R\) be the classical clock record. \begin{theorem}[Complete destructive clock and scored instrument] \label{thm:instrument} The clock channel is \begin{align} \Mclock(Z) &=\sum_k\proj k_R\otimes D_kZD_k^\dagger\nonumber\\ &=\sum_k\proj k_R\otimes \Tr_C\left[(P_k\otimes I_X)Z(P_k\otimes I_X)\right]. \label{eq:clock} \end{align} The subsequent code-selected table readout is \begin{equation} \mathcal I(Z)= \sum_{k,y}\proj k_R\otimes\proj y_Y\otimes M_{k,y}D_kZD_k^\dagger M_{k,y}^\dagger. \label{eq:instrument} \end{equation} These maps are completely positive and trace preserving on all input operators, including every untouched reference extension. For \(\sigma_k=D_k\rho D_k^\dagger\) and \(p_k=\Tr\sigma_k>0\), \[ \mathbb E[By\mid k]=\frac{\Tr(A_k\sigma_k)}{p_k}. \] The measured clock names are consumed; they are not quantum outputs of either instrument. \end{theorem} \begin{proof} For \eqref{eq:clock}, use Kraus operators \(\ket k_R\otimes D_k\). Their squared moduli sum to \(\sum_kP_k\otimes I_X=I_{CX}\). Expansion in the clock basis gives the partial-trace identity. For \eqref{eq:instrument}, use the recorded Kraus operators \[ L_{k,y}=\ket k_R\otimes\ket y_Y\otimes M_{k,y}D_k. \] Since \(\sum_yM_{k,y}^\dagger M_{k,y}=I_X\), their squared moduli also sum to \(I_{CX}\). Tensoring these operator identities with any reference identity proves the complete assertions. The conditional score follows from \eqref{eq:score}. The specified implementation consumes \(C\), and its output type contains \(R,Y,X\), not a quantum \(C\). \end{proof} The table reading has an explicit finite implementation. Allocate a fresh probe qubit \(A\) and apply the isometry \[ \psi\longmapsto \sum_{y=\pm1}\ket y_A\otimes M_{k,y}\psi. \] Completeness makes it isometric, so it extends to a unitary on a finite larger space. Destructive probe readout yields \eqref{eq:instrument}. This proves an abstract finite gate realization, not hardware synthesis or experimental implementation. The complete conditioning formula agrees in type with I-B and En09-A \cite{CheungIB,CheungEn09}. We do not assume the additional stationarity and covariance hypotheses needed for a conditional unitary-evolution interpretation in En09-A. That interpretation is not required for this certificate readout. \begin{proposition}[Clock and record limits] \label{prop:clocklimits} The clock channel has no universal left inverse. Its classical record marginal does not determine the quantum residual. No finite clock matrices satisfy \([T_C,H_C]=iI_C\). Discarding the clock record gives \(\Tr_C Z\). \end{proposition} \begin{proof} With the same payload, the clock states \[ \frac{\ket{\tau_0}+\ket{\tau_1}}{\sqrt2}, \qquad \frac{\ket{\tau_0}-\ket{\tau_1}}{\sqrt2} \] have identical destructive outputs. A left inverse cannot recover both. The inputs \(P_0\otimes\rho_X\) and \(P_0\otimes\rho_X'\) have the same deterministic record and different residuals. The commutator claim follows by taking matrix traces. Finally, summing diagonal clock blocks gives the partial trace. \end{proof} Complete channel identities do not mean that every selected reference state is unchanged. Conditioning can steer a correlated reference; its unconditional marginal remains unchanged. \subsection{Three dictionaries} Choose distinct externally prescribed geometric times \[ T_0^{\rm hist}0\Longrightarrow T_k^{\rm hist}=T(e), \] has at most one supported outcome. For a normalized input it therefore has one deterministic outcome, or is impossible. The clock and payload channels do not uniquely determine the history map, rod frame, or conical deficit. \end{proposition} \begin{proof} Distinctness permits at most one index with \(T_k^{\rm hist}=T(e)\), and probabilities sum to one. If the corresponding clock effect has probability one, positivity implies that the input is supported in that clock subspace. For nonuniqueness, leave every finite matrix unchanged and reassign the history times, or rotate the developing rod frame and corresponding port coordinates. The abstract instrument is unchanged. Likewise different prescribed deficits admit isometric small regular laboratories carrying the same finite matrices and schedule. Their global completions differ. \end{proof} The labels agree with the admitted classical evaluations by their explicit table definition. The rod and causal maps agree with the background by construction, not by inference from records. \subsection{A fixed-table approximation limit} \begin{proposition}[Finite compression limit] \label{prop:compressionlimit} Suppose a sequence of admitted smooth data packets converges on common compact dependency regions, with the same observation tests and prescriptions. For the sourced family, require common compact scalar-data support and the same Newton branch. Then the finite table entries converge. If \(A_k^{(n)}\) are the resulting diagonal matrices and \[ \epsilon_n=\max_k\norm{A_k^{(n)}-A_k}\longrightarrow0, \] the complete instruments formed with a common sufficiently large \(B\) satisfy \begin{equation} d_\diamond(\mathcal I_n,\mathcal I) \le\min\{1,4\epsilon_n/B\}\longrightarrow0, \qquad d_\diamond=\frac12\norm{\cdot}_\diamond. \label{eq:instrument-limit} \end{equation} \end{proposition} \begin{proof} Lemma~\ref{lem:wave}, smooth multiplication and differentiation, and compact integration prove continuity of the coordinate and scalar evaluations. For a common compact spatial source support, \[ \partial^\beta\Ninv b_0=\Ninv(\partial^\beta b_0). \] The locally integrable Newton kernel bounds this expression uniformly on each compact target by a source smooth seminorm. Thus the declared initial preparation, its wave evolution, and the sourced evaluations are continuous as well. The gap evaluation is another compact smooth pairing. Choose \(B\) so that all limiting and approximating effects under consideration have spectrum in \([1/4,3/4]\). Because the matrices are diagonal, \[ \norm{M_{k,y}^{(n)}-M_{k,y}} \le\frac{\epsilon_n}{2B}, \] using \(|\sqrt a-\sqrt b|\le |a-b|\) on that interval. For contractions \(L,\widetilde L\), \[ \norm{\Ad_L-\Ad_{\widetilde L}}_\diamond \le2\norm{L-\widetilde L}. \] Sum this estimate over the eight recorded Kraus branches and divide by two. This gives \eqref{eq:instrument-limit}. \end{proof} This is a limit for one fixed finite catalogue. It is not a limit to a field instrument, a finite-dimensional one-particle approximation, or a continuum quantum theory. The complete-channel language is standard \cite{Watrous}; the estimates needed here have been proved above. %===================================================================== \section{Schedule locality, transport, and a resource ledger} \label{sec:transport} %===================================================================== A complete finite schedule includes every quantum register, classical record, guard, and control actually used. At each tick let the channel factor over disjoint declared blocks. Reserve output records as explicit registers. For terminal region \(Y\), define the backward block sets \[ N_n(Y)=Y,\qquad N_s(Y)= \bigcup\{B:B\text{ is a tick-}s\text{ block and } B\cap N_{s+1}(Y)\ne\varnothing\}. \] \begin{theorem}[Complete schedule locality] \label{thm:locality} The complete channel obeys \[ \Phi^*(\mathcal A(Y))\subseteq\mathcal A(N_0(Y)). \] Thus the output reduction on \(Y\) depends only on the input reduction to \(N_0(Y)\). With an untouched reference \(E\), the output on \(YE\) depends only on the input on \(N_0(Y)E\). If every scheduled input-output dependency maps to a future-causal pair in \((M,\bar g)\), every scheduled ancestor of an output lies in its metric past. Operational intervention influence is contained in that ancestry. \end{theorem} \begin{proof} On a block disjoint from the current backward support, the observable is the identity. A trace-preserving channel adjoint preserves it. Other blocks can enlarge support only within their declared union. Reverse-tick induction gives the inclusion. Equality of all pulled-back expectations gives equality of output reductions. Appending a reference identity gives the referenced statement. Causal transitivity gives geometric containment along a dependency path. Changing a complete operation outside that path acts only on an identity factor in the backward calculation and cannot affect the output. \end{proof} This is a dependency theorem of the kind compared in En03-A and En05-A \cite{CheungLocality}. Scheduled ancestry can be larger than actual influence: a replacement channel may ignore an input it receives. The theorem does not derive a metric or an invariant speed. \subsection{Exact target-adapted endpoint transport} Choose sender and receiver positions \(x_A,x_B\in U\) with developed separation \[ L=|\chi(x_B)-\chi(x_A)|>0. \] Assume the full straight developed segment lies in the regular laboratory. Install \(n+1\) sites at \[ x_j=\chi^{-1}\left( \chi(x_A)+\frac jn(\chi(x_B)-\chi(x_A))\right), \qquad j=0,\ldots,n. \] Each site has a two-qubit payload register \(X_j\cong\CC^4\). Prepare \(X_1,\ldots,X_n\) in a fixed pure blank \(\omega\). At hop \(j\), SWAP \(X_{j-1}\) with \(X_j\). The externally prescribed tick duration is \[ \delta T=\frac{L}{nc}. \] \begin{theorem}[Reference-preserving target-adapted transport] \label{thm:transport} After hop \(j\), every payload-reference input has state \[ \rho_{X_jE}\otimes \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{X_i}. \] The logical channel from \(X_0\) to \(X_n\) is exactly \(\id_{\CC^4}\), including every reference. Arrival is at \(T_{\rm d}+L/c\). All tick-boundary communication dependencies are causal in the prescribed background. \end{theorem} \begin{proof} The first SWAP transfers the entire input factor, including all reference correlations, into the next independent blank. Induction proves the displayed state identity. Each hop has spatial length \(L/n=c\delta T\). Its developed straight path is null, and convexity keeps the route in the regular chart. Pullback by the local isometry therefore gives a causal route in \((M,\bar g)\). \end{proof} Read the clock at the sender before departure. The payload may then be correlated with its classical code \(R\) and reference \(E\). The theorem applies with \(RE\) as the untouched comparison system. A receiver operation selected by \(k\) additionally needs that code. Reserve a two-bit classical register at each route site, copy the classical code along the same causal hops, and apply \(M_{k,y}\) only after both payload and code arrive. Copying a diagonal classical register does not clone an unknown quantum state. The complete received instrument is exactly \eqref{eq:instrument}, with \(X\) relocated. No instantaneous remote classical control is assumed. For a route of \(n\) hops, the ledger is: \[ \begin{array}{l|l} \text{resource}&\text{charge}\\ \hline \text{payload registers}&n+1\text{ registers of two qubits}\\ \text{clock}&2\text{ qubits, consumed before final probe allocation}\\ \text{table-readout probe}&1\text{ fresh qubit, then consumed}\\ \text{payload transfer}&n\text{ register SWAPs}\\ \text{classical route storage}&2(n+1)\text{ bits}\\ \text{final score record}&1\text{ bit} \end{array} \] With all payload registers reserved initially, the quantum laboratory peak is bounded by \begin{equation} Q_{\rm lab}=2(n+1)+2, \qquad C_{\rm lab}\le2(n+1)+1. \label{eq:resource} \end{equation} External reference storage is charged separately. The exact result is for an installed target-adapted chain, not every null endpoint on one previously fixed lattice. En08-B's null-boundary approximation is a different contract \cite{CheungEn08}. \begin{remark}[Operational implementation boundary] The finite schedule prescribes complete block channels and their causal input-output ports. It does not construct a continuous-time relativistic Hamiltonian implementing a spatially extended SWAP during a tick. No physical field-acquisition or apparatus-localization theorem is inferred from the endpoint channel. \end{remark} For implementations on matching interfaces, replacing one channel factor at a time and using diamond-norm contraction gives \[ d_\diamond(\widetilde\Phi,\Phi) \le\min\left\{ 1,\epsilon_{\rm enc}+\sum_j\epsilon_j+\epsilon_{\rm dec} \right\}. \] This is a conditional complete-channel budget, not a report of hardware errors or an executed experiment. %===================================================================== \section{Release, novelty, and non-identifications} \label{sec:release} %===================================================================== \begin{theorem}[VI-B local Grade S release] \label{thm:release} For the supported presentation and finite interface specified above: \begin{enumerate}[label=\textup{(\roman*)}] \item the local Einstein--scalar jet has the displayed second-order equations and the compact initial-preserving formal gauge action; \item the finite-time harmonic relational scalar descends through second order; \item the compactly switched candidate has the exact classical defect \eqref{eq:defect}, the negative plateau gap \eqref{eq:gap}, and the coefficientwise order obstruction; \item the explicit constraint-compatible family is nonempty and has a positive second-order descended evaluation; \item the finite membership--separation alternative holds with its stated coverage and sampling limits; \item the finite evaluation encoding has the complete destructive-clock instrument, explicit dictionaries, fixed-table approximation limit, and reference-preserving causal endpoint transport; \item the quantum certificate remains RG-open. \end{enumerate} Items (i)--(vi) satisfy Definition~\ref{def:grade}. No cited companion theorem is a premise of their proofs. \end{theorem} \begin{proof} The coefficient equations and action are Definition~\ref{def:jet} and Proposition~\ref{prop:gauge-action}. The coordinate construction and Theorem~\ref{thm:descent} prove (ii). Theorems~\ref{thm:switch-defect} and \ref{thm:plateau-obstruction}, with Corollary~\ref{cor:order-obstruction}, prove (iii). Theorems~\ref{thm:sourced} and \ref{thm:sourced-nonzero} prove (iv). Theorem~\ref{thm:certificate} and the subsequent quantifier audit prove (v). Sections~\ref{sec:encoding}--\ref{sec:transport} prove (vi). Section~\ref{sec:quantum} constructs neither the required quantum products nor a complete evaluated response catalogue, so the proposed quantum certificate has not acquired a closed status. This is (vii), not a quantum nonexistence theorem. \end{proof} \subsection{Contribution relative to the source} The classical descent, sourced family, and switching obstruction reproduce the mechanisms of VI \cite{CheungVI}. The additional content of this adaptation is: \begin{itemize} \item an explicit supported domain separating local propagation from a nonexistent global conical Cauchy premise; \item a distinction between the background cone and the formal unknown metric coefficients; \item a finite table combining descended evaluations with a separately typed representative-gap witness; \item a complete destructive-clock readout, its approximation bound, and a fully charged payload-and-code route. \end{itemize} The potential impact is methodological rather than a new gravitational dynamics claim: a classical obstruction can be preserved in an operational certificate without being relabelled as an anomaly, while a valid classical quotient can be represented without pretending that an unknown field has been measured. \subsection{Six absent Grade E implications} No theorem above: \begin{enumerate}[label=\textup{(E\arabic*)}] \item reconstructs a spacetime manifold or global topology from records; \item determines the conical deficit or Lorentzian signature; \item determines lapse, proper-time scale, or a causal metric without the external calibration; \item derives Einstein--scalar equations or physical backreaction from a residual matrix response; \item constructs a continuum field-acquisition instrument or an interacting Hilbert-space realization; \item supplies the full inherited characteristic quantum map, its output theorem, or its coefficient admission. \end{enumerate} \subsection{Companion separations} The following are documentary comparisons, not mechanisms silently inserted into VI-B. \begin{longtable}{@{}p{.24\linewidth}p{.70\linewidth}@{}} \toprule Source or track & Separation retained\\ \midrule \endfirsthead \toprule Source or track & Separation retained\\ \midrule \endhead \bottomrule \endfoot III and III-B & Their equation spaces and the sourced successor S1--S3 are not identified with \(\JJ_{R,b}\). A vacuum metric row retained in III is not the sourced equation \eqref{eq:jet2}. \cite{CheungIII,CheungGradeS}\\[4pt] IV and IV-B & Their represented apparatuses and Model 3's prescribed Newton--wave response are different objects. A timestamp does not implement a global field apparatus. \cite{CheungIV}\\[4pt] VII-A & The plateau-restricted classical intertwiner has a different gauge-support contract. It is reserved for a separate local adaptation. It supplies no quantum product here. \cite{CheungVIIA}\\[4pt] VII-B--VII-D; IX--XI & Sharp insertion, mixed products, finite parts, the unresolved nonlocal one-loop construction, and specified free-shell obstructions do not define \eqref{eq:fourterms}. \cite{CheungProductTracks}\\[4pt] VII-E--VII-J; VIII-A--VIII-C & Observation boundaries, particle slicing events, finite networks, free susceptibility, and fixed-presentation matching are not this Einstein--scalar jet. A particle event is not the conical worldsheet. No variable-deficit dictionary selects \eqref{eq:cone}. \cite{CheungOperational}\\[4pt] XVI-A and XVI-B & A smooth multiplier satisfies \(\WF(gu)\subseteq\WF(u)\), with equality locally where \(g\ne0\). This is not a construction of a retarded coefficient. Late source equality is not response equality, and \(\partial_\mu(gS^\mu)=g\partial_\mu S^\mu+ (\partial_\mu g)S^\mu\) retains a weighted residual. No pure-trace witness is imported. \cite{CheungPrepared}\\[4pt] Em02-E, XX-A, XIX-A & A classical relational lapse, harmonic coordinate coefficient, finite pointer, and event-time label remain different prescriptions. \cite{CheungClocks}\\[4pt] En08, En10, XXII & Schedule consistency does not select this cone. Null-boundary approximation is not the exact target-adapted chain above. Source-weight estimates and conditional cone-extraction templates are not this relational jet. \cite{CheungOtherInterfaces}\\[4pt] Oai282-A and the OpenAI catalogue & The finite Ward contract has its own improvement data and contacts. No OpenAI family supplies our cone, jet, clock, or defect. A static area law is not this descent or a Page curve. All such citations are separation records, not premises or verification. \cite{CheungOai282,OpenAIRecord}\\[4pt] Hawk, Psy, and Em packet sheets & Their named access, statistical, or packet interfaces are not identified with a relational Einstein--scalar jet. No Page curve, island, interior, consciousness assignment, or gravitational field acquisition follows here. \cite{CheungOtherTracks}\\ \end{longtable} The elementary smooth-multiplier wavefront observation does not explain away the compact-switch defect. The defect is already an identity among smooth classical solutions and their prescriptions, not a wavefront enlargement. \subsection{Analytical and standing boundaries} The inherited characteristic programme remains distinct. XIV-H's historical H2 stop, XIV-I's narrower constructed I2 internal amplitude, and the still-unconstructed full joint maps at I3/J2 must not be conflated \cite{CheungAnalytic}. Saying that VI-B leaves that programme untouched does not withdraw the narrower I2 construction. For this release, \[ \mathrm{O.1}=\loc{NOT_CLOSED},\qquad \text{XV-B quantum-admission exit: not opened},\qquad \text{XVI-N: not discharged}. \] Existence of a technical manuscript called XV-B is not the opening of that admission exit. Separately scoped subsector statements do not update the broader standing boundary \cite{CheungStops}. No inherited characteristic quantum coefficient is admitted. This paper also constructs no quantum relational coefficient of its own. Its finite matrices and complete instruments are finite operational objects, not quantum field coefficients. There is no spacetime emergence, Grade E theorem, Page curve, island, interior, quantized relational time without external scheduling, ultraviolet completion, adiabatic limit, radiative calculation, interacting Hilbert-space construction, or global conical Green theorem. The source equations are included, not derived. The clock encoding is not the time axis. A channel formula does not specify switching work, a bath, reset work, or a heat ledger. No compilation or physical implementation is asserted. %===================================================================== \appendix \section{Explicit curvature definition of the quadratic metric term} \label{app:G2} %===================================================================== This appendix makes the symbol \(G^{(2)}[h,h]\) a fully specified differential expression. For a symmetric \(h\), let \[ H_{\mu\nu\sigma} =\partial_\mu h_{\nu\sigma} +\partial_\nu h_{\mu\sigma} -\partial_\sigma h_{\mu\nu}, \] \[ \Gamma_1^\rho{}_{\mu\nu} =\frac12\eta^{\rho\sigma}H_{\mu\nu\sigma}, \qquad \Gamma_2^\rho{}_{\mu\nu} =-\frac12h^{\rho\sigma}H_{\mu\nu\sigma}. \] Define \[ R_{1,\mu\nu} =\partial_\rho\Gamma_1^\rho{}_{\mu\nu} -\partial_\nu\Gamma_1^\rho{}_{\mu\rho}, \] \begin{align*} R_{2,\mu\nu} ={}&\partial_\rho\Gamma_2^\rho{}_{\mu\nu} -\partial_\nu\Gamma_2^\rho{}_{\mu\rho}\\ &+\Gamma_1^\rho{}_{\rho\lambda} \Gamma_1^\lambda{}_{\mu\nu} -\Gamma_1^\rho{}_{\nu\lambda} \Gamma_1^\lambda{}_{\mu\rho}. \end{align*} Then \begin{equation} \begin{split} G^{(2)}[h,h]_{\mu\nu} ={}&R_{2,\mu\nu} -\frac12\eta_{\mu\nu} \left(\eta^{ab}R_{2,ab}-h^{ab}R_{1,ab}\right)\\ &-\frac12h_{\mu\nu}\eta^{ab}R_{1,ab}. \end{split} \label{eq:G2explicit} \end{equation} These are the coefficients obtained by inserting \(\eta+\kappa h\) into the coordinate Christoffel and Ricci formulas. Terms linear in \(j\) give exactly \(\Glin(j)\). Thus no hidden second-order gravitational source normalization is contained in the notation. %===================================================================== \section{Domain, prescription, and status audit} %===================================================================== \subsection{What zero-past uniqueness does and does not say} The identities \[ \Gzero\Box\xi=\xi,\qquad \GR P(gX_\xi\phi_0)=gX_\xi\phi_0 \] use fields vanishing near the initial surface. They are not identities for arbitrary Cauchy data. For example, for a general smooth scalar \(u\), \[ G_{P,0}Pu=u-H_m(u|_0,\dot u|_0), \] where \[ H_m(A,B)=\cos(t\Omega_m)A+S_m(t)B. \] Both sides solve the same equation with zero data. Dropping the homogeneous subtraction changes the prescription. Nor is homogeneous uniqueness of a wave equation a claim that every zero-past solution of an unrestricted gauge-degenerate Einstein equation vanishes. Compact pure gauges already refute that stronger assertion. \subsection{Why the source preparation is not a transport claim} The Newton branch determines initial metric data by a spatial integral. Its use is logically prior to the finite apparatus schedule. The subsequent local wave problem has finite propagation, but this does not prove causal acquisition of the Newton data from a distributed physical scalar source. All finite certificate preparation is treated as declared input. The transport theorem concerns only that finite payload and its classical code. It does not transport a continuum solution jet. \subsection{The finite certificate has no hidden catalogue} The finite linear-algebra alternative applies to any already defined \(q_S,N_S\). It does not select: \begin{itemize} \item a field algebra or state; \item the relevant time-ordered and retarded products; \item the admissible quantum contact class; \item the complete response columns; \item a gauge-compatible switched realization. \end{itemize} Consequently no fit, separator, or measured table in this paper can be interpreted as the missing quantum RG certificate. \subsection{A separate successor, not a stronger claim here} A natural next construction is a local Grade S treatment of VII-A's plateau-restricted intertwiner on the same supported presentation. It would need to reprove its own second-order covariance, retain the larger-support defect, and leave the quantum product gate separate. That successor is not part of the release of VI-B. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Authorship.} The author field is intentionally empty for anonymous review. Authorized author and affiliation metadata are required before a signed submission. Citation of N.~Cheung and attribution of registry source records do not assign authorship of this adaptation. \paragraph{Provenance and novelty.} Manuscript VI, version 1, supplies the source problem and the classical results adapted here. I-B supplies the documentary geometric and operational pattern. The mathematical arguments needed for this release are written explicitly. The claimed adaptation concerns domain control and the finite certificate interface, not a new discovery of relational observables or a new quantum-gravity calculation. \paragraph{Classical and quantum boundaries.} The finite channels use declared Born semantics. No field quantization, renormalized relational composite, radiative coefficient, physical state-acquisition channel, or interacting gravitational Ward identity is constructed. The intended quantum certificate remains RG-open. \paragraph{Verification.} No successful \texttt{pdflatex} execution, proof-assistant check, independent symbolic computation, numerical execution, physical circuit compilation, experimental realization, independent referee certification, or journal acceptance is asserted. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} \bibitem{CheungVI} N.~Cheung, \emph{Relational Einstein--scalar observables at second order: classical descent and a compact-switch obstruction}. Manuscript VI, \loc{VI.tex}, version 1, Enter Calculus Companion, unpublished source record. The registered source has an empty author field; the attribution here follows the companion registry. Relevant locators include \loc{def:jet-space}, \loc{thm:descent}, \loc{thm:switch-defect}, \loc{thm:plateau-obstruction}, \loc{cor:order-obstruction}, \loc{thm:sourced-nonzero}, and \loc{thm:finite-certificate}. \url{https://enter.grok.me/browse/VI}. \bibitem{CheungEnter} N.~Cheung, ``The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,'' \emph{Frontiers in Physics} 14 (2026), article 1954943. DOI: \doi{10.3389/fphy.2026.1954943}. The consulted publisher record identifies an accepted article with final formatting forthcoming. \bibitem{CheungI} N.~Cheung, \emph{I}, \loc{I.tex}, version 2. Enter Calculus Companion, unpublished manuscript. Characterization inside a stated finite operational class; not a field-instrument theorem. \url{https://enter.grok.me/browse/I}. \bibitem{CheungPhilosophy} N.~Cheung, \emph{The Philosophy of Enter: Destructive Readout, Quantum Recovery, and the Refusal of Identification}. \loc{EntPhil-v2.tex}, version 2. Introductory essay, Enter Calculus Companion. \url{https://enter.grok.me/enter}. \bibitem{CheungIB} N.~Cheung, \emph{I-B-v1: A Finite Enter Spacetime---Calibrated Destructive Time Readout on a Prescribed Conical Product}. \loc{I-B-v1.tex}, version 1. Enter Calculus Companion, unpublished manuscript. \url{https://enter.grok.me/browse/I-B}. \bibitem{BFHPR} R.~Brunetti, K.~Fredenhagen, T.-P.~Hack, N.~Pinamonti, and K.~Rejzner, ``Cosmological perturbation theory and quantum gravity,'' \emph{Journal of High Energy Physics} 08 (2016), 032. DOI: \doi{10.1007/JHEP08(2016)032}. \url{https://arxiv.org/abs/1605.02573}. \bibitem{Frob} M.~B.~Fr\"ob, ``Gauge-invariant quantum gravitational corrections to correlation functions,'' \emph{Classical and Quantum Gravity} 35 (2018), 055006. DOI: \doi{10.1088/1361-6382/aaa74c}. \url{https://arxiv.org/abs/1710.00839}. \bibitem{BGP} C.~B\"ar, N.~Ginoux, and F.~Pf\"affle, \emph{Wave Equations on Lorentzian Manifolds and Quantization}. ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007. \url{https://arxiv.org/abs/0806.1036}. \bibitem{CheungGradeS} N.~Cheung, \emph{III-B}, \loc{III-B-v1.tex}, version 1; and \emph{V-B}, \loc{V-B-v1.tex}, version 1. Enter Calculus Companion, supplied unpublished source records. Local Grade S and supported-presentation comparisons only. \bibitem{CheungVIIA} N.~Cheung, \emph{VII-A}, \loc{VII-A.tex}, version 1. Enter Calculus Companion, supplied unpublished manuscript. Locators \loc{thm:first}, \loc{thm:second}, \loc{prop:defect}, and \loc{cor:nohigherrepair}. Plateau-restricted classical intertwiner and the larger-support boundary. \url{https://enter.grok.me/browse/VII-A}. \bibitem{CheungCertificates} N.~Cheung, \emph{II}, \loc{II.tex}, version 2; and \emph{II-B}, \loc{II-B-v1.tex}, version 1. Enter Calculus Companion, supplied unpublished manuscripts. Finite contact-span certificates and their non-transfer limits. No response catalogue is imported. \bibitem{DF} M.~D\"utsch and K.~Fredenhagen, ``Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity,'' \emph{Reviews in Mathematical Physics} 16 (2004), 1291--1348. DOI: \doi{10.1142/S0129055X04002266}. \url{https://arxiv.org/abs/hep-th/0403213}. \bibitem{CheungV} N.~Cheung, \emph{Admissible currents and first retarded corrections for a fixed-time scalar response}. Manuscript V, \loc{V.tex}, version 1. Enter Calculus Companion, unpublished manuscript. Current, first-retarded, oscillator, and distinct cubic-form objects are not identified with the present second-order jet. \url{https://enter.grok.me/browse/V}. \bibitem{FV} C.~J.~Fewster and R.~Verch, ``Quantum fields and local measurements,'' \emph{Communications in Mathematical Physics} 378 (2020), 851--889. DOI: \doi{10.1007/s00220-020-03800-6}. \url{https://arxiv.org/abs/1810.06512}. \bibitem{CheungEn09} N.~Cheung, \emph{En09-A: Realization of Relational Time by Destructive Enter on a Clock Register}. \loc{En09-A.tex}, version 1. Enter Calculus Companion, supplied unpublished manuscript. Complete conditioning and additional covariance--stationarity hypotheses. \url{https://enter.grok.me/browse/En09-A}. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author's book interface and errata: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. \bibitem{CheungLocality} N.~Cheung, \emph{En03-A}, \loc{En03-A.tex}, version 1; and \emph{En05-A}, \loc{En05-A.tex}, version 1. Enter Calculus Companion, supplied unpublished source records. Complete-channel schedule locality on separately prescribed interfaces. \bibitem{CheungEn08} N.~Cheung, \emph{En08-B}, \loc{En08-B.tex}, version 1. Enter Calculus Companion, supplied unpublished manuscript. Resource-controlled approximation toward a prescribed null boundary; comparison only. \bibitem{CheungIII} N.~Cheung, \emph{III}, \loc{III.tex}, version 2. Enter Calculus Companion, supplied unpublished manuscript. Separate equation-space quotients and the unconstructed sourced successor S1--S3. \bibitem{CheungIV} N.~Cheung, \emph{IV}, \loc{IV.tex}, version 1; and \emph{IV-B}, \loc{IV-B-v1.tex}, version 1. Enter Calculus Companion, supplied unpublished manuscripts. Represented models and their declared apparatus interfaces; documentary comparisons. \bibitem{CheungProductTracks} N.~Cheung, \emph{VII-B}, \emph{VII-C}, \emph{VII-D}; \emph{IX-A}, \emph{IX-D}, \emph{IX-E}; \emph{X-G}, \emph{XI-F}, and \emph{XI-G}. Enter Calculus Companion, supplied version-specific unpublished records. Sharp-admission, product, finite-part, variational, and free-shell questions remain separately scoped. No theorem from these tracks is a premise here. \bibitem{CheungOperational} N.~Cheung, \emph{VII-E} through \emph{VII-J}; and \emph{VIII-A}, \emph{VIII-B}, \emph{VIII-C}, version 3. Enter Calculus Companion, supplied unpublished records. Observation, slicing, fixed-presentation, and causal-network comparisons only. \bibitem{CheungPrepared} N.~Cheung, \emph{XVI-A}, \loc{XVI-A.tex}, version 1; and \emph{XVI-B}, \loc{XVI-B.tex}, version 1. Enter Calculus Companion, supplied unpublished manuscripts. Conditional wavefront and prepared-source comparison interfaces; no inherited retarded coefficient is supplied by those comparisons. \bibitem{CheungClocks} N.~Cheung, \emph{Em02-E}, version 1; \emph{XX-A}, version 1; and \emph{XIX-A}, version 1. Enter Calculus Companion, supplied unpublished records. Classical relational lapse, original rods, and clock-prescription separations. \bibitem{CheungOtherInterfaces} N.~Cheung, \emph{En08-A}, \emph{En10-A}, \emph{En10-B}, and \emph{XXII-A} through \emph{XXII-F}. Enter Calculus Companion, supplied unpublished source records. Conditional operational and geometric comparisons, not inputs to this construction. \bibitem{CheungOai282} N.~Cheung, \emph{Oai282-A}, \loc{Oai282-A.tex}, version 1. Enter Calculus Companion, supplied unpublished source record. Finite local Ward residual for its own improvement and contact contract. Not a premise and not a verification of Family 282. \bibitem{OpenAIRecord} OpenAI, \emph{Supplied mathematics catalogue}. Separate documentary material attributed to OpenAI in the provided companion record. No family proof, publication-status claim, formalization certificate, or physical identification is imported or verified. \bibitem{CheungOtherTracks} N.~Cheung, \emph{Hawk}, \emph{Psy}, and \emph{Em} packet-sheet records named in the supplied dossier. Enter Calculus Companion, unpublished documentary comparisons. Their models are not identified with \(\JJ_{R,b}\). \bibitem{CheungAnalytic} N.~Cheung, \emph{XIV-H}, version 1; \emph{XIV-I}, version 1; \emph{XIV-J}, version 1; and technical \emph{XV-B}, version 2. Enter Calculus Companion, supplied unpublished manuscripts. Historical H2, constructed I2 internal amplitude, unresolved I3/J2 full joint map, and relative-lifting interfaces remain distinct. \bibitem{CheungStops} N.~Cheung, \emph{XVI-N}, revision r12; \emph{XVI-O}, revision o70-sync; and \emph{XVI-Q}, release Q11. Enter Calculus Companion, supplied unpublished source records. Conditional insertion obstruction and separately scoped subsector reports. VI-B updates none of their release conditions. \end{thebibliography} \end{document}