% XX-A.tex % One self-contained pdfLaTeX source; no external inputs or bibliography. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XX-A.tex % pdflatex -interaction=nonstopmode -halt-on-error XX-A.tex % % No successful compilation, numerical execution, or proof-assistant % verification is asserted. % % Principal corrections to the proposed plan: % 1. A programme for a quantum observable is not its construction. % This first artifact freezes and proves a classical prescription % and gives an exact quantum admission contract. Quantum admission % remains unresolved at Q1. % 2. The specified target is an independently frozen original-rod % variant. Exact identification with every equation and convention % of Em02-E is not asserted from the excerpts alone. % 3. XIX-B excludes one cross-prescription coordinate identification; % it does not prove a defect intrinsic to every VI construction. % 4. Pure-trace kernel non-determination is attributed to XVIII-A. % XVIII-B is a separately frozen massive stress-response benchmark. % 5. Elliptic rods retain their spatial boundary condition. Hyperbolic % uniqueness without spatial decay does not remove that condition. % 6. The quadratic observable obeys a coupled classical gauge identity; % it is not separately invariant under the linear gauge operation. % 7. An uncomputed Ward breaking is marked undefined, not zero. % 8. Failure of a sufficient wavefront criterion is not nonexistence % of every possible extension or generalized product. \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,array,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage{xurl} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={XX-A: Original-Rod Lapse: Classical Freeze and Quantum Admission Contract}, pdfauthor={}, pdfsubject={Track A, classical relational lapse, quantum input gates}, pdfkeywords={relational observable, harmonic rods, Einstein scalar, quantum certificate, Ward identity, renormalization} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{3pt} \setlength{\parindent}{15pt} \numberwithin{equation}{section} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{convention}[theorem]{Convention} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\Lie}{\mathcal L} \newcommand{\im}{\operatorname{im}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\pair}[2]{\langle #1,#2\rangle} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\sid}[1]{\texttt{\detokenize{#1}}} \newcommand{\CE}{\mathsf C_{\mathrm E}} \newcommand{\CV}{\mathsf C_{\mathrm{VI}}} \newcommand{\CI}{\mathsf C_{\mathrm{VIII}}} \newcommand{\RX}{\mathsf R_{\mathrm X}} \newcommand{\G}{\mathsf G} \newcommand{\Op}{\mathcal O} \newcommand{\CT}{\mathcal C_{\mathrm{adm}}} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \title{\textbf{XX-A}\\[3pt] Original-Rod Lapse: Classical Freeze and Quantum Admission Contract\\ \large The First Artifact of Track A on a Fixed Cosmological Background} \author{} \date{} \begin{document} \maketitle \begin{abstract} Track A seeks a renormalized relational observable on a fixed external-time background. This first artifact does not announce that quantum construction. It specifies an independent original-rod lapse prescription on an explicit on-shell power-law Einstein--scalar background and proves its complete classical expansion through second perturbative order. The clock calibration, positive lapse branch, slice-harmonic rod equations, spatial boundary condition, perturbation convention, and gauge group are fixed before calculation. The second spatial rod is constructed, and its absence from the quadratic lapse is proved. An explicit compact perturbation distinguishes the original rods from lapse-weighted modified rods. The complete linear observable and quadratic Hessian satisfy a coupled classical gauge identity; the Hessian alone is not declared linearly gauge invariant. We also isolate the instantaneous kernel and spatial-tail issues that a quantum construction must address. A regularized polynomial has a conditional finite-order Wick realization once contraction data are supplied, but this fact does not construct the unsmoothed lapse. An exact affine Ward-repair criterion is proved without assuming that its target breaking or counterterm images have been calculated. The native quantum milestone remains unresolved at Q1: a complete coupled gauge-fixed realization, Green distributions, and normalized contraction data have not been supplied. No retained-order quantum Ward identity, regulator removal, inherited characteristic quantum coefficient, external-time-free evolution, or compilation is asserted. All predecessor comparisons retain declared path null, and the standing insertion, route, and admission boundaries are unchanged. \end{abstract} \noindent\textbf{Keywords:} relational lapse; harmonic spatial rods; Einstein--scalar background; classical gauge identity; quantum admission; nonlocal composite; Ward repair. \tableofcontents %===================================================================== \section{Question, choice of target, and exact release} \label{sec:intro} %===================================================================== The proposed Track A milestone is a well-defined quantum relational observable whose rod contributions, product domains, counterterm images, and retained-order gauge identity are all supplied. A plan naming these objects does not construct them. XX-A is the first classical-prescription and admission-contract artifact of that track. Its mathematical release is the printed classical construction and the comparison and repair theorems below. Its quantum ending is explicitly unresolved. \subsection{Target and source discipline} The motivating target is Cheung's original-rod lapse in Em02-E \cite{CheungEm02E}. The supplied record distinguishes: \[ \text{raw matter-clock lapse} \quad\hbox{from}\quad \text{the lapse pulled back using the original spatial rods}. \] It also distinguishes those rods from modified rods. The available excerpts do not reproduce every native background constant, rod formula, domain, and normalization needed for a verbatim reconstruction of Em02-E. Consequently, this paper freezes an explicit \emph{original-rod variant} in Sections~\ref{sec:background}--\ref{sec:lapse}. Every equation used in its proofs is displayed here. An exact identification with Em02-E would require a separate comparison of all defining data; it is not asserted. The variant retains the geometric original-rod rule: harmonic spatial coordinates on the actual matter-clock slices, with asymptotically fixed spatial labels. It does not insert a lapse weight into that rule. The five quantum audit slots of XIX-D are used as a checklist \cite{XIXD}. No unprinted predecessor proof, catalogue claim, reported artifact, or rebuild result is a theorem premise. Published mathematical frameworks are cited for context and for the precise analytic results identified where used. \subsection{Corrections to the target-selection rationale} XIX-B's reported forcing residual \[ \partial_t\bigl((\sigma-1)V_\lambda^{0\mu}\bigr) \] excludes one direct identification between two specified coordinate prescriptions \cite{XIXB,CheungXVID}. It does not show that every independently frozen VI harmonic prescription is defective. Choosing the original-rod lapse avoids importing that particular identification; it does not establish that the lapse is easier to quantize. Likewise, Em02-E's supplied record already specifies a classical observable and classical covariance. Its incomplete \emph{quantum} audit must not be renamed as failure of all its classical results \cite{CheungEm02E}. The classical construction below is proved independently on a narrower, explicitly stated domain. Pure-trace kernel non-determination is the relevant shape of XVIII-A \cite{CheungXVIIIA}. XVIII-B supplies a different massive stress-response benchmark \cite{CheungXVIIIB}; it is not silently renamed as that kernel theorem. \subsection{Standing boundary} Throughout, \[ \boxed{\text{Declared path: null.}} \] This is a relationship convention: the comparisons do not supply a cross-programme identification or import unproved source theorems. It does not mean that a displayed mathematical map is zero. The work remains on a fixed cosmological background and uses an external cosmic-time coordinate. Field-dependent labels do not remove that background from the quantum construction. %===================================================================== \section{An explicit on-shell background and the perturbative freeze} \label{sec:background} %===================================================================== \subsection{Action, conventions, and background} Use signature \((-,+,+,+)\), a reduced Planck mass \(M>0\), and the minimally coupled action density \begin{equation} S[g,\phi] =\int \sqrt{-g}\left( \frac{M^2}{2}R[g] -\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi -V(\phi)\right)\dd^4x. \label{eq:action} \end{equation} On noncompact spatial slices the background action need not be finite. We use its local Euler equations and compactly supported variations, not an asserted finite total background action. Choose \(p>1\), \(t_*>0\), \(\phi_*\in\R\), and set \begin{equation} A=\sqrt{2p}\,M,\qquad a(t)=\left(\frac{t}{t_*}\right)^p,\qquad \bar\phi(t)=\phi_*+A\log\left(\frac{t}{t_*}\right), \label{eq:background-data} \end{equation} \begin{equation} \bar g=-\dd t^2+a(t)^2\delta_{ij}\dd x^i\dd x^j, \qquad V(\phi)=\frac{M^2p(3p-1)}{t_*^2} \exp\left(-\frac{2(\phi-\phi_*)}{A}\right). \label{eq:potential} \end{equation} Work on \[ \mathcal M=I\times\R^3,\qquad I=(t_-,t_+),\qquad 00\). \end{proof} \subsection{Configurations, labels, and orders} Let \(s\) be a dimensionless formal perturbation parameter. It is not automatically an independently normalized gravitational coupling. Freeze the factorial convention \begin{equation} g_s=\bar g+s h+\frac{s^2}{2}k,\qquad \phi_s=\bar\phi+s\varphi+\frac{s^2}{2}\psi \pmod{s^3}. \label{eq:jet} \end{equation} All tensor components are retained. The classical coefficients are smooth and, on each compact time interval used in the calculation, have a common compact spatial support. They are not required to solve the perturbed Euler equations. This is an off-shell classical observable prescription on an on-shell background. Choose a compact label-time interval \(J\Subset I\), and perform the construction on a slightly larger interval \(J_+\Subset I\). This leaves room for the formal time displacement. The external label coordinates are \((\tau,\mathbf x)\). The clock calibration is fixed by \begin{equation} T_s=F(\phi_s),\qquad F(z)=t_*\exp\left(\frac{z-\phi_*}{A}\right), \qquad F(\bar\phi(t))=t. \label{eq:clock} \end{equation} The raw lapse is calibrated to the \emph{matter scalar} \(\phi_s\), not to a Euclidean norm and not to a separately rescaled clock: \begin{equation} N_s=\left( -g_s^{\mu\nu}\partial_\mu\phi_s\partial_\nu\phi_s \right)^{-1/2}. \label{eq:raw-lapse} \end{equation} The positive timelike branch is used. Formal square roots and inverses are well defined around the positive background coefficient. No assertion of a nonlinear global coordinate chart for arbitrary large perturbations is needed. The proposed quantum target is the coefficient range \[ s^0,s^1,s^2,\qquad \hbar^0,\hbar^1. \] A formal truncation is not a numerical error estimate. The classical local expressions below use at most second derivatives of the elementary fields before the displayed spatial inverses. A proposed local counterterm search may be restricted to four derivatives, but such a restriction is not declared exhaustive for this nonlocal observable. \subsection{Clock-slice inversion} Write \[ T_s=t+s\chi+\frac{s^2}{2}\zeta,\qquad \theta_s(\tau,\mathbf x) =\tau+s u_1+\frac{s^2}{2}u_2, \] where \(T_s(\theta_s(\tau,\mathbf x),\mathbf x)=\tau\). Taylor expansion gives \begin{align} \chi&=\frac{t}{A}\varphi, & \zeta&=\frac{t}{A}\psi+\frac{t}{A^2}\varphi^2, \label{eq:clock-coeff}\\ u_1&=-\chi, & u_2&=2\chi\dot\chi-\zeta. \label{eq:time-inverse} \end{align} All coefficients on the right of the inverse formula are evaluated at \(t=\tau\). \begin{proof}[Verification of the inverse coefficients] Substitution yields the first coefficient \(u_1+\chi=0\). The second coefficient is \(u_2+2u_1\dot\chi+\zeta=0\), which gives \eqref{eq:time-inverse}. Expansion of the exponential in \eqref{eq:clock} gives \eqref{eq:clock-coeff}. \end{proof} The inverse \(\theta_s\) is algebraic and local in time to this order. It is not a retarded Green operator. %===================================================================== \section{Original slice-harmonic rods} \label{sec:rods} %===================================================================== \subsection{The actual slice metric} Embed a clock slice by \[ \iota_s(\tau,\mathbf x) =(\theta_s(\tau,\mathbf x),\mathbf x). \] Its induced spatial metric is \begin{equation} \gamma_{s,ij} =(g_s)_{ij}(\theta_s,\mathbf x) +(g_s)_{0i}(\theta_s,\mathbf x)\partial_j\theta_s +(g_s)_{0j}(\theta_s,\mathbf x)\partial_i\theta_s +(g_s)_{00}(\theta_s,\mathbf x) \partial_i\theta_s\partial_j\theta_s. \label{eq:induced} \end{equation} Write \[ \gamma_s=a^2\delta+s\gamma_1+\frac{s^2}{2}\gamma_2. \] Direct expansion gives \begin{align} (\gamma_1)_{ij} &=h_{ij}+2a^2H u_1\delta_{ij}, \label{eq:gamma1}\\ (\gamma_2)_{ij} &=k_{ij}+2u_1\dot h_{ij} +u_2\partial_t(a^2\delta_{ij}) +u_1^2\partial_t^2(a^2\delta_{ij}) \notag\\ &\quad +2h_{0i}\partial_j u_1+2h_{0j}\partial_i u_1 -2\partial_i u_1\partial_j u_1. \label{eq:gamma2} \end{align} In particular, scalar gradients, mixed metric components, and the perturbation of the embedding have not been omitted. \subsection{Rod equation and spatial boundary condition} \begin{definition}[Frozen original rods] \label{def:original-rods} On each clock slice, the original rods satisfy \begin{equation} \partial_i\left( \sqrt{\det\gamma_s}\,\gamma_s^{ij} \partial_j R_s^a\right)=0, \qquad a=1,2,3, \label{eq:rod-exact} \end{equation} with \[ R_s^a(\tau,\mathbf x)-x^a\longrightarrow0 \quad\text{as }\abs{\mathbf x}\longrightarrow\infty. \] The condition is imposed coefficientwise through \(s^2\). There is no lapse weight in \eqref{eq:rod-exact}. \end{definition} This is a slice-elliptic prescription, not a hyperbolic coordinate Cauchy problem. There are no rod position and velocity data on an initial time surface to replace its spatial boundary condition. Normalize its density by the background spatial scale: \[ B_s^{ij} =a(\tau)^{-1}\sqrt{\det\gamma_s}\,\gamma_s^{ij} =\delta^{ij}+sB_1^{ij}+\frac{s^2}{2}B_2^{ij}. \] Let \[ C=a^{-2}\gamma_1,\qquad D=a^{-2}\gamma_2, \] with spatial indices raised by \(\delta\). Then \begin{align} B_1&=\frac12(\tr C)I-C, \label{eq:B1}\\ B_2&= \left(\frac12\tr D+\frac14(\tr C)^2 -\frac12\tr(C^2)\right)I -D-(\tr C)C+2C^2. \label{eq:B2} \end{align} These formulas follow by multiplying the second-order expansions of the determinant square root and matrix inverse. Use the Newton inverse \begin{equation} (\G f)(\mathbf x) =-\frac1{4\pi}\int_{\R^3} \frac{f(\mathbf y)}{\abs{\mathbf x-\mathbf y}} \dd^3y,\qquad \Delta\G f=f. \label{eq:newton} \end{equation} For \[ R_s^a=x^a+sY_1^a+\frac{s^2}{2}Y_2^a, \] the complete coefficient equations and solutions are \begin{align} \Delta Y_1^a&=-\partial_i B_1^{ia}, & Y_1^a&=-\G\partial_iB_1^{ia}, \label{eq:Y1}\\ \Delta Y_2^a &=-\partial_i\left(B_2^{ia} +2B_1^{ij}\partial_jY_1^a\right), & Y_2^a &=-\G\partial_i\left(B_2^{ia} +2B_1^{ij}\partial_jY_1^a\right). \label{eq:Y2} \end{align} \begin{proposition}[Existence, uniqueness, and smooth parameter dependence] \label{prop:rods} Equations \eqref{eq:Y1}--\eqref{eq:Y2} construct unique smooth decaying rod coefficients on the frozen classical domain. On compact time intervals their spatial decay is \(O(\abs{\mathbf x}^{-2})\). The coefficient maps are continuous in the smooth topology on fixed compact spatial supports. \end{proposition} \begin{proof} The tensors \(B_1,B_2\) are smooth and compactly supported in space. The first source is a divergence of a compact smooth vector. Convolution with the locally integrable fundamental solution \eqref{eq:newton} is smooth; derivatives can be transferred to the compact source. The vector inside the second divergence is also compact: the possibly noncompact \(\partial Y_1\) is multiplied by \(B_1\). The same argument constructs \(Y_2\). For a compact smooth vector \(b^i\), \[ \G\partial_i b^i =\partial_i\G b^i. \] Outside a sufficiently large ball the derivative of \(1/\abs{\mathbf x-\mathbf y}\) is \(O(\abs{\mathbf x}^{-2})\), uniformly for \(\mathbf y\) in the source support. This proves the decay and its compact-time uniformity. A difference of two first-order solutions is an entire harmonic function tending to zero at infinity. The maximum principle on large balls gives zero. Once \(Y_1\) is fixed, the same argument proves uniqueness of \(Y_2\). On compact spatial sets, convolution and transferred derivatives are bounded by finitely many source seminorms. Outside such sets the explicit kernel estimates give the stated uniform bounds. The sources are finite smooth algebraic expressions in the input coefficients and the previously constructed \(Y_1\). This proves continuity and smooth time dependence. \end{proof} \begin{remark}[The spatial boundary condition cannot be discarded] Without decay, constants or other harmonic functions can be added to a homogeneous rod coefficient. Smooth uniqueness for a zero-data hyperbolic problem without spatial decay is therefore not a uniqueness theorem for \eqref{eq:rod-exact}. No such transfer from XVI-D is made. \end{remark} %===================================================================== \section{Complete raw and relational lapse through second order} \label{sec:lapse} %===================================================================== \subsection{Raw coefficient calculation} Raise four-dimensional indices using \(\bar g\), and put \(v_\mu=\partial_\mu\bar\phi\). The inverse metric coefficients are \[ g_s^{\mu\nu} =\bar g^{\mu\nu}-s h^{\mu\nu} +\frac{s^2}{2} \left(2h^\mu{}_\rho h^{\rho\nu}-k^{\mu\nu}\right). \] Let \[ W_s=-g_s^{-1}(\dd\phi_s,\dd\phi_s) =W_0+sW_1+\frac{s^2}{2}W_2. \] Its coefficients are \begin{align} W_0&=\frac{A^2}{t^2}, \label{eq:W0}\\ W_1&=h^{\mu\nu}v_\mu v_\nu -2\bar g^{\mu\nu}v_\mu\partial_\nu\varphi, \label{eq:W1}\\ W_2&= \left(k^{\mu\nu}-2h^\mu{}_\rho h^{\rho\nu}\right)v_\mu v_\nu +4h^{\mu\nu}v_\mu\partial_\nu\varphi \notag\\ &\quad -2\bar g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi -2\bar g^{\mu\nu}v_\mu\partial_\nu\psi. \label{eq:W2} \end{align} For \(N_s=N_0+sN_1+s^2N_2/2\), \begin{align} N_0&=W_0^{-1/2}=\frac tA, \label{eq:N0}\\ N_1&=-\frac12W_0^{-3/2}W_1, \label{eq:N1}\\ N_2&=\frac34W_0^{-5/2}W_1^2 -\frac12W_0^{-3/2}W_2. \label{eq:N2} \end{align} These follow from two derivatives of \(w^{-1/2}\). The cosmic-clock raw lapse is a different normalization: \[ L_s=\left(-g_s^{-1}(\dd T_s,\dd T_s)\right)^{-1/2}. \] Since \(F'(\phi_s)=T_s/A\), \begin{equation} N_s=\frac{T_s}{A}L_s. \label{eq:normalization} \end{equation} Neither normalization is silently substituted for the other. \subsection{The complete inverse relational map} The forward label map is \begin{equation} \mathcal C_s(t,\mathbf x) =\left(T_s(t,\mathbf x), R_s(T_s(t,\mathbf x),\mathbf x)\right). \label{eq:forward} \end{equation} Let its formal inverse be \[ \alpha_s=\id+s\alpha_1+\frac{s^2}{2}\alpha_2. \] First invert \(R_s\) at fixed \(\tau\), then insert the resulting spatial argument into \(\theta_s\). This gives \begin{align} \alpha_1^0&=u_1, & \alpha_1^i&=-Y_1^i, \label{eq:alpha1}\\ \alpha_2^0&=u_2-2Y_1^i\partial_i u_1, & \alpha_2^i&=2Y_1^j\partial_jY_1^i-Y_2^i. \label{eq:alpha2} \end{align} \begin{definition}[Frozen relational lapse] \label{def:rel-lapse} The observable scalar is \[ n_s=N_s\circ\alpha_s. \] For \(f\in C_c^\infty(J\times\R^3)\), fixed in label space, its smeared functional is \begin{equation} \Op_f(s)=\int f(\tau,\mathbf x)n_s(\tau,\mathbf x) \dd\tau\,\dd^3x. \label{eq:smeared} \end{equation} The smearing uses the fixed label density, not a fluctuating spacetime volume element. \end{definition} \begin{theorem}[Complete classical original-rod lapse] \label{thm:lapse} The frozen relational lapse has the expansion \[ n_s=N_0+s n_1+\frac{s^2}{2}n_2\pmod{s^3}, \] where \begin{align} n_1&=N_1+\dot N_0u_1, \label{eq:n1}\\ n_2&=N_2+2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2 -2Y_1^i\partial_i n_1. \label{eq:n2} \end{align} All terms are supplied by \eqref{eq:clock-coeff}--\eqref{eq:Y2} and \eqref{eq:W0}--\eqref{eq:N2}. The second spatial rod \(Y_2\) is constructed but does not enter \(n_2\). For this background, \begin{equation} n_1= -\frac{t}{2A}h_{00} -\frac{t^2}{A^2}\dot\varphi -\frac{t}{A^2}\varphi, \qquad \ddot N_0=0. \label{eq:n1-explicit} \end{equation} \end{theorem} \begin{proof} Composition with a formal inverse gives \[ n_1=N_1+\alpha_1^\mu\partial_\mu N_0, \] \[ n_2=N_2+2\alpha_1^\mu\partial_\mu N_1 +\alpha_2^\mu\partial_\mu N_0 +\alpha_1^\mu\alpha_1^\nu \partial_\mu\partial_\nu N_0. \] The background \(N_0\) depends only on time. Substitution of \eqref{eq:alpha1}--\eqref{eq:alpha2} gives \[ N_2+2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2 -2Y_1^i\partial_iN_1-2\dot N_0Y_1^i\partial_i u_1, \] which is \eqref{eq:n2}. The coefficient \(\alpha_2^i\), containing \(Y_2\), multiplies \(\partial_iN_0=0\). Since \(v_0=A/t\), \(\bar g^{00}=-1\), and \(h^{00}=h_{00}\), \[ W_1=\frac{A^2}{t^2}h_{00}+\frac{2A}{t}\dot\varphi. \] Equations \eqref{eq:N1}, \eqref{eq:time-inverse}, and \(\dot N_0=1/A\) give \eqref{eq:n1-explicit}. \end{proof} The familiar shape of \eqref{eq:n1}--\eqref{eq:n2} agrees with the supplied composition formula in Em02-E \cite{CheungEm02E}. The agreement of that shape is not a proof that every native definition has been identified. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.27\linewidth}Y@{}} \toprule Origin & Terms retained\\ \midrule Raw metric and scalar & \(N_1,N_2\), including all components and scalar-gradient terms\\ Clock inversion & \(\dot N_0u_1\), \(2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2\)\\ Original spatial rods & \(-2Y_1^i\partial_i n_1\)\\ Second spatial rods & Constructed; absent from this coefficient for the proved reason\\ Gauge fixing & Absent from the classical definition; relevant to quantum realization\\ \bottomrule \end{tabularx} \end{center} %===================================================================== \section{Classical gauge identity and rod-detection tests} \label{sec:gauge} %===================================================================== \subsection{Frozen gauge group} Admit spacetime diffeomorphisms with compact support in \(\mathcal M\), preserving orientation and the perturbative timelike branch. They are the identity in the spatial asymptotic region and near the time ends of their support. They therefore preserve the fixed asymptotic rod labels. The group acts by pullback on \(g,\phi\). The background is held as the reference for the perturbative expansion, not treated as a newly dynamical quantum variable. \begin{theorem}[Coefficientwise classical covariance] \label{thm:classical-gauge} The formal original-rod construction is gauge invariant through second order: \[ n[\rho^*g,\rho^*\phi]=n[g,\phi]\pmod{s^3} \] for the admitted formal diffeomorphisms. Consequently, every \(\Op_f\) is invariant to that order. \end{theorem} \begin{proof} The scalar \(T=F(\phi)\) transforms by composition with \(\rho\). The diffeomorphism maps its clock slices to the corresponding old slices. Pullback preserves the induced metric and the slice-harmonic scalar equation. The pullback of an old rod is therefore a solution of the new rod equation with the same asymptotic labels. The uniqueness argument of Proposition~\ref{prop:rods}, applied coefficientwise, identifies it with the new rod through second order. Thus \[ \mathcal C[\rho^*g,\rho^*\phi]=\mathcal C[g,\phi]\circ\rho. \] The raw lapse is a scalar formed by contraction of a covector with the inverse metric, so \(N'=N\circ\rho\). The inverse coordinate map is \(\alpha'=\rho^{-1}\circ\alpha\). Hence \(N'\circ\alpha'=N\circ\alpha\). The fixed label-space smearing preserves the equality. \end{proof} \subsection{The linear observable and Hessian must be coupled} Let \(z=(h,\varphi)\), \(\ell=(k,\psi)\). Define \[ F_{1,f}(z)=\int f n_1[z],\qquad F_{2,f}(z,z)=\int f n_2[z,0], \] and polarize \(F_{2,f}\) bilinearly. The complete expansion is \begin{equation} \Op_f(s)=\Op_{0,f} +sF_{1,f}(z) +\frac{s^2}{2} \bigl(F_{1,f}(\ell)+F_{2,f}(z,z)\bigr). \label{eq:observable-jet} \end{equation} For a compact vector field \(\xi\), put \[ K\xi=(\Lie_\xi\bar g,\Lie_\xi\bar\phi). \] \begin{corollary}[Coupled classical Ward identity] \label{cor:coupled} For every admitted \(\xi\) and every classical \(z\), \begin{align} F_{1,f}(K\xi)&=0, \label{eq:linear-ward}\\ F_{2,f}(z,K\xi)+F_{1,f}(\Lie_\xi z)&=0. \label{eq:quadratic-ward} \end{align} The second equation does not assert \(F_{2,f}(z,K\xi)=0\) separately. \end{corollary} \begin{proof} Use the infinitesimal pullback generated by \(s\xi\). The variation of the first jet is \(K\xi\), and the variation of the second jet is \(2\Lie_\xi z\). Differentiate \eqref{eq:observable-jet} and use Theorem~\ref{thm:classical-gauge}. The coefficients of \(s\) and \(s^2\) give the two equations. As a direct first-order check, \[ (K\xi)_{00}=-2\dot\xi^0,\qquad \Lie_\xi\bar\phi=\frac A t\xi^0. \] Substitution in \eqref{eq:n1-explicit} gives zero. \end{proof} This is an off-shell kinematic identity. It does not construct an interacting quantum BRST operation on the free solution quotient. In particular, tangency of the nonlinear gauge operation to that quotient is not inferred. \subsection{A compact witness distinguishing modified rods} For comparison only, define lapse-weighted rods by \begin{equation} \partial_i\left( \frac{N_s\circ\iota_s}{N_0} B_s^{ij}\partial_j\widehat R_s^a\right)=0, \qquad \widehat R_s^a-x^a\longrightarrow0. \label{eq:modified-rods} \end{equation} This is a different prescription. \begin{proposition}[Original and lapse-weighted rods are detected] \label{prop:rod-detection} There is a compact metric perturbation for which the quadratic relational lapses formed with \eqref{eq:rod-exact} and \eqref{eq:modified-rods} differ. \end{proposition} \begin{proof} Take a real \(w\in C_c^\infty(J\times\R^3)\), not identically zero, and choose \[ h_{00}=-2w,\qquad h_{0i}=h_{ij}=0,\qquad \varphi=k=\psi=0. \] Then \(T=t\), \(\gamma_s=a^2\delta\), and the original rods are \(R_s^a=x^a\) through every displayed order. The raw lapse is \[ N_s=N_0\sqrt{1+2sw}, \qquad N_1=N_0w. \] The first coefficient of \eqref{eq:modified-rods} gives \[ \widehat Y_1^a=-\G\partial_a w. \] The composition formula therefore gives \[ \widehat n_2-n_2 =2N_0\sum_a(\G\partial_a w)\partial_a w. \] At each time, integration by parts and \(\Delta\G w=w\) yield \[ \sum_a\int_{\R^3}(\G\partial_a w)\partial_a w\,\dd^3x =-\int_{\R^3}(\G w)\Delta w\,\dd^3x =-\int_{\R^3}w^2\,\dd^3x. \] The difference is supported where \(\nabla w\) is supported. Choose a compact spatial smearing equal to one there, and a nonnegative time smearing detecting a time where \(w\ne0\). The smeared difference is strictly negative. \end{proof} The witness is off shell. It is not asserted to be a solution of the perturbed Einstein--scalar equations. It independently verifies a prescription distinction rather than importing XIX-A's witness as a premise. \subsection{Residual checks for any future replacement} A future rod replacement \(Y_1\mapsto Y_1+Z_1\) on the same clock slices changes the quadratic lapse, with other data fixed, by \begin{equation} \Delta n_2=-2Z_1^i\partial_i n_1. \label{eq:rod-residual} \end{equation} A claim of identity must either prove this expression vanishes on the named domain or supply the compensating changes in the other coefficients. Equality of selected source contractions is not that proof. Equation \eqref{eq:rod-residual} is the present residual-check template. The prepared source \(S_\lambda\) and the Cauchy residual of XIX-B are documentary warnings, not cosmological rod inputs. %===================================================================== \section{Quantum field input: the operators that are specified and those that remain missing} \label{sec:quantum-input} %===================================================================== \subsection{The actual coupled linearized equations} Let \[ E_{\mu\nu}=M^2G_{\mu\nu}-T_{\mu\nu},\qquad E_\phi=\Box_g\phi-V'(\phi), \] and define \(Pz=(\delta E_{\mu\nu},\delta E_\phi)\) at the displayed background. This defines the coupled linearized operator without discarding mixed metric--scalar terms. For completeness, write \begin{align} (\delta\Gamma)^\rho_{\mu\nu} &=\frac12\bar g^{\rho\sigma} \left(\nabla_\mu h_{\nu\sigma} +\nabla_\nu h_{\mu\sigma} -\nabla_\sigma h_{\mu\nu}\right), \label{eq:dGamma}\\ \delta R_{\mu\nu} &=\nabla_\rho(\delta\Gamma)^\rho_{\mu\nu} -\nabla_\nu(\delta\Gamma)^\rho_{\mu\rho}, \label{eq:dRic}\\ \delta R &=-h^{\mu\nu}\bar R_{\mu\nu} +\bar g^{\mu\nu}\delta R_{\mu\nu}, \label{eq:dR}\\ \delta G_{\mu\nu} &=\delta R_{\mu\nu}-\frac12h_{\mu\nu}\bar R -\frac12\bar g_{\mu\nu}\delta R. \label{eq:dG} \end{align} Here \(\nabla\) is the background Levi-Civita derivative. The matter part is \begin{align} \delta T_{\mu\nu} &=\partial_\mu\varphi\,v_\nu+v_\mu\partial_\nu\varphi -\frac12h_{\mu\nu}(v^\rho v_\rho+2V) \notag\\ &\quad -\frac12\bar g_{\mu\nu} \left(-h^{\rho\sigma}v_\rho v_\sigma +2v^\rho\partial_\rho\varphi +2V'\varphi\right), \label{eq:dT}\\ \delta E_\phi &=\Box_{\bar g}\varphi-V''\varphi -h^{\mu\nu}\nabla_\mu\nabla_\nu\bar\phi \notag\\ &\quad -\left(\nabla_\mu h^{\mu\nu} -\frac12\nabla^\nu h^\rho{}_\rho\right)v_\nu. \label{eq:dScalar} \end{align} The formulas follow by differentiating the metric inverse, connection, curvature, and scalar stress tensor. \begin{lemma}[Gauge kernel of the unfixed operator] \label{lem:PK} The displayed operator satisfies \(PK\xi=0\). It cannot therefore be used as an invertible gauge-fixed covariance operator. \end{lemma} \begin{proof} The exact Euler equations transform tensorially under pullback. Differentiate \(E[\rho^*\bar g,\rho^*\bar\phi]=\rho^*E[\bar g,\bar\phi]=0\) along the flow of \(\xi\). This gives \(PK\xi=0\). Compact vector fields with \(K\xi\ne0\) exist, so the kernel is nontrivial. \end{proof} \subsection{A frozen quadratic gauge-fixing candidate} The field content for the proposed construction is \[ (h_{\mu\nu},\varphi,c^\mu,\bar c^\mu,B^\mu). \] The scalar fluctuation defining the clock and slices is retained. The ghosts and auxiliary variables are gauge-fixing fields, not additional physical clocks. Let \(S_2(z)=\tfrac12 S^{(2)}_{\bar g,\bar\phi}(z,z)\), and define the linear differential \[ s_0h=\Lie_c\bar g,\qquad s_0\varphi=c^\mu v_\mu,\qquad s_0c=0,\qquad s_0\bar c=B,\qquad s_0B=0. \] At this linear level \(s_0^2=0\). Freeze the gauge-fixing candidate \begin{equation} \mathcal F_\nu(z) =\nabla^\mu h_{\mu\nu} -\frac12\nabla_\nu h^\mu{}_\mu -M^{-2}v_\nu\varphi, \qquad \Psi=\int\sqrt{-\bar g}\, \bar c^\nu\left(\mathcal F_\nu+\frac{\beta}{2}B_\nu\right), \label{eq:gauge-fermion} \end{equation} where \(\beta>0\) is fixed. Then \begin{equation} S_{2,\mathrm{gf}}=S_2+s_0\Psi =S_2+\int\sqrt{-\bar g} \left(B^\nu\mathcal F_\nu+\frac{\beta}{2}B^\nu B_\nu -\bar c^\nu(Qc)_\nu\right), \label{eq:gauge-action} \end{equation} with \begin{equation} (Qc)_\nu =\Box_{\bar g}c_\nu+\bar R_{\nu\rho}c^\rho -M^{-2}v_\nu v_\rho c^\rho. \label{eq:ghost-operator} \end{equation} The sign of the ghost term follows from the odd Leibniz rule. Equation \eqref{eq:ghost-operator} follows by applying \(\mathcal F\) to \(Kc\) and commuting background derivatives. These formulas specify a quadratic candidate and its normalization. They do not prove that its complete reduced metric--scalar realization has the intended Green properties, nor select a state. Eliminating \(B\), changing field variables, or using another gauge requires a checked normalization and comparison, not an implicit substitution. Hack's coupled linearized Einstein--Klein--Gordon construction is relevant framework \cite{Hack}. It is not used as a premise that the particular \(\mathcal F\), covariance, rods, spatial tails, or composite in this paper have been constructed. Relational quantum-observable and perturbative BV frameworks likewise do not supply this particular realization by name \cite{BFHPR,Frob,BFR}. \subsection{Required inverses and contraction data} A Q1 completion must supply a specified realization \(\widetilde P\) of the coupled gauge-fixed operator, its domain, and distributions \[ G^{\mathrm{ret}},\qquad G^{\mathrm{adv}},\qquad E=G^{\mathrm{ret}}-G^{\mathrm{adv}} \] with the actual left and right inverse, support, normalization, and field-equation identities. The graded ghost and auxiliary blocks must be included. A Q2 completion must additionally specify a compatible two-point matrix or subtraction parametrix. In a bosonic physical sector, a state-dependent two-point function \(W\) must satisfy the appropriate field equations, Hermiticity, positivity, and \[ W-W^{\mathrm T}=iE \] under its declared component pairing; factors of \(\hbar\) are then restored in the field expectation. The Hadamard wavefront condition uses the positive-frequency null relation \[ \mathcal C_+ =\{(x,k;y,-k'): (x,k)\sim(y,k'),\ k\text{ future null}\}. \] Its scalar characterization and appropriate vector-valued extensions have precise hypotheses \cite{Radzikowski,SV}. They are not claims of positivity for the entire ghost-extended field space. \begin{lemma}[Singularity data do not select a state] \label{lem:state} Suppose a bosonic physical two-point function \(W\) has been constructed, and \(v\) is a smooth real solution of its linear field equation. For every \(c\ge0\), \[ W_c=W+c\,v\otimes v \] has the same antisymmetric part and singular wavefront data, and is positive whenever \(W\) is positive. Distinct \(c\)'s can therefore give different states with the same stated local singularity condition. \end{lemma} \begin{proof} The addition is a smooth symmetric bisolution. Its pairing with a conjugate test pair is \(c\abs{\pair{v}{f}}^2\ge0\). It changes neither the antisymmetric part nor the wavefront set modulo smooth terms. \end{proof} The lemma assumes a physical \(W\); it does not construct one for the present coupled system. Nor does local Hadamard behavior impose the weighted spatial-tail bounds needed for the rods. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.12\linewidth}YY@{}} \toprule Slot & Specified here & Still required\\ \midrule Q1 & Background, complete unfixed linearized equations, graded quadratic gauge-fixing candidate & Checked coupled realization, inverses, normalization, and gauge-compatible contraction identities\\ Q2 & Required singularity, field-equation, and positivity tests & Selected state or suitable parametrix and spatial-tail bounds\\ \bottomrule \end{tabularx} \end{center} The first unclosed quantum gate is Q1. It is not bypassed by calling a quadratic action a covariance. %===================================================================== \section{Observable kernels, products, and spatial images} \label{sec:products} %===================================================================== \subsection{Compact output smearing is not compact functional support} The first-order rod operator is instantaneous in clock time. Its scalar Newton kernel has the spacetime form \begin{equation} K_{\mathrm{sp}}(\tau,\mathbf x;\tau',\mathbf y) =-\frac{\delta(\tau-\tau')} {4\pi\abs{\mathbf x-\mathbf y}}. \label{eq:instantaneous} \end{equation} The actual rod kernels contain its derivatives and the explicitly specified coefficient matrices. For compact \(b\) of nonzero spatial integral, \[ \partial_i\G b(\mathbf x) =\frac{\left(\int b\right)x_i}{4\pi\abs{\mathbf x}^3} +O(\abs{\mathbf x}^{-3}). \] Thus the first rod can have a nonzero \(1/r^2\) tail. \begin{proposition}[Classical rod images are spatially nonlocal] \label{prop:tail} There are compact perturbations for which \(Y_1\) is not compactly supported. Consequently, the rod part of the Hessian of a compactly label-smeared lapse is not automatically a compact joint field kernel. \end{proposition} \begin{proof} Take \(\varphi=h_{00}=h_{0i}=0\) and \(h_{ij}=2a^2b\,\delta_{ij}\), where \(b\) is compact and has nonzero spatial integral at some time. Then \(C=2bI\), \(B_1=bI\), and \(Y_1^i=-\G\partial_i b\). The displayed asymptotic shows its noncompact tail. Polarize the rod term \(-2Y_1^i\partial_i n_1\) against a second perturbation having \(h_{00}\ne0\) near the label-smearing support. Its dependence on the first perturbation is through the Newton kernel even at distant spatial points. No compact support in those first-input variables follows from compact support of the output test. \end{proof} The classical integrals are defined on the frozen spatially compact input class. Quantum fields are distributions, not compactly supported smooth configurations. Extension to their actual smearing spaces needs its own tail theorem. \subsection{The instantaneous wavefront audit} \begin{proposition}[Off-diagonal equal-time singularities] \label{prop:instantWF} On \(\mathbf x\ne\mathbf y\), \begin{equation} \WF(K_{\mathrm{sp}}) = \left\{ (\tau,\mathbf x;\tau,\mathbf y; \lambda,0;-\lambda,0):\lambda\ne0 \right\}. \label{eq:instantWF} \end{equation} At the full diagonal there are, in addition, conormal directions from the spatial Newton singularity. In particular, the kernel cannot be treated as smooth merely because its spatial endpoints are distinct. \end{proposition} \begin{proof} Away from the spatial diagonal the Newton coefficient is smooth and nonzero. Multiplication by it preserves the wavefront set of \(\delta(\tau-\tau')\), which is the conormal set displayed in \eqref{eq:instantWF}. At coincidence, the relative-coordinate kernel is the tensor product of a temporal delta and the spatial Newton distribution. The latter is singular at its spatial origin. Its Fourier transform has inverse-quadratic spatial behavior rather than rapid decay; the temporal delta supplies arbitrary temporal frequencies. These produce conormal directions at the full diagonal. \end{proof} These intermediate kernel directions are not, by themselves, a proof that every final observable fails a microcausal domain test. That test concerns the covectors in the final field slots after the actual operations and smearing. \subsection{Precise analytic acceptance tests} For any proposed kernel operations, Q3 must check: \begin{enumerate}[label=\textup{(\roman*)}] \item \textbf{Products.} The component factors \(u,v\) must satisfy the appropriate product condition; a standard sufficient condition excludes \((x,k)\in\WF(u)\) and \((x,-k)\in\WF(v)\) simultaneously \cite[Theorem 8.2.10]{Hormander}. \item \textbf{Pullbacks.} A proposed restriction or composition \(F^*u\) must avoid the normal set \[ N_F=\{(F(x),\eta):(\dd F_x)^{\mathrm T}\eta=0\} \] in its input wavefront set \cite[Theorem 8.2.4]{Hormander}. \item \textbf{Pushforwards and tails.} Properness on the support, or a separately proved nonproper convergence estimate, is required. Compact external smearing alone does not supply it. \item \textbf{Extensions.} The actual extension locus, reference distribution, scaling bounds, normalization, and allowed corrections must be specified before an extension theorem is applied. \end{enumerate} Local Wick and time-ordered-product existence results have their own local field hypotheses \cite{HWlocal,HWexistence}. Their conclusions are not an automatic extension theorem for \eqref{eq:instantaneous} composed with the complete nonlocal lapse Hessian. Brunetti--Fredenhagen's finite-scaling-degree extension criterion concerns an already defined reference distribution and an actual scaling bound \cite[Theorem 5.2]{BF}; it does not create the absent reference product. In particular, the phrase ``extend across the light cone'' is not one universal operation. Some positive-frequency products already exist by the wavefront criterion, whereas time-ordered or nonlocal characteristic compositions may need extensions. The locus and the operation must be identified separately. Failure of a sufficient ordinary-product criterion is not a theorem that every renormalized or generalized realization fails. \subsection{What fixed smooth regularization can prove} Let \(P_{\epsilon,R}\) be a fixed, properly supported smoothing operator whose kernel is smooth and whose relevant input and output supports are compact. A radius \(R\) and a smoothing width \(\epsilon>0\) are independent external parameters. \begin{lemma}[Conditional regular polynomial realization] \label{lem:regular-wick} Suppose normalized free contraction data \(W,E\) on the required graded test space have been supplied. If a linear and quadratic observable have smooth compact coefficient kernels \(l,k\), then all finite Wick contractions needed for that regular polynomial are defined. For \[ F(z)=c+\pair{l}{z}+\frac12\pair{k}{z\otimes z}, \] one bosonic Wick convention is \[ :F:_W=F-\frac{\hbar}{2}\pair{k}{W}. \] Its finite polynomial star products are defined by the corresponding finite contraction sums. \end{lemma} \begin{proof} Every contraction pairs a distribution or a tensor product of distributions with a smooth compact joint test kernel. The tensor products exist, and the pairings are finite by distributional continuity. A polynomial has only finitely many nonzero functional derivatives, so the contraction sums terminate. The displayed subtraction removes the unique self-contraction of the quadratic term in that convention. \end{proof} Applied to a sufficiently smoothed and spatially cut off \(F_{1,f},F_{2,f}\), this lemma gives a regular comparison polynomial \emph{conditional on} Q1--Q2 data. It does not prove a realization of the original noncompact Hessian, a regulator-removal limit, or a quantum gauge identity. A fixed smoothing operator generally changes the classical prescription. Under a linear gauge transformation, the regularized linear functional has variation \[ F_{1,f}(P_{\epsilon,R}K\xi) =F_{1,f}\bigl((P_{\epsilon,R}-\id)K\xi\bigr), \] using \eqref{eq:linear-ward}. There is no reason for this residual to vanish unless the particular smoothing comparison is proved compatible. Thus ordinary field-independent smoothing cannot be advertised simultaneously as harmless classical identification and as proved gauge-preserving regularization. \subsection{Counterterms must be transported through the observable} A prospective counterterm calculation must distinguish: \[ \text{local action or insertion correction} \quad\longrightarrow\quad \text{its actual image under clock and rod construction}. \] A correction supported on an internal diagonal can acquire a spatial tail after a Newton inverse. No final support claim is made before that image is calculated. The suggested four-derivative local search is only a declared search class. If the computed target requires other structures, one must report that fact or explicitly revise the class. A finite list copied from a different bulk two-point problem does not establish exhaustive coverage. %===================================================================== \section{Formal quantum coefficient and the exact Ward-repair contract} \label{sec:ward-contract} %===================================================================== \subsection{A coefficient template is not an evaluated coefficient} The complete classical insertion is \eqref{eq:observable-jet}. Its homogeneous nonlinear part includes all of \eqref{eq:n2}, not only \(N_2\). A possible renormalized coefficient would have the form \begin{equation} \Op_f^{\mathrm{ren}} =J_{\mathcal R}\!\left[ \Op_{0,f}+sF_{1,f} +\frac{s^2}{2}F_{2,f}\right] +\hbar C_f(s) \pmod{s^3,\hbar^2}, \label{eq:quantum-template} \end{equation} where \(J_{\mathcal R}\), its domain, and \(C_f\) must actually be constructed. Equation \eqref{eq:quantum-template} is a specification of the target, not its definition by fiat. If interactions are included with the amplitude normalization \eqref{eq:jet}, their action expansion must also be retained. For compact variational arguments around the on-shell background, \[ s^{-2}\bigl(S[\bar g+s h,\bar\phi+s\varphi]-S[\bar g,\bar\phi]\bigr) =\frac12S^{(2)}(z,z) +\frac{s}{6}S^{(3)}(z,z,z) +\frac{s^2}{24}S^{(4)}(z,z,z,z) +O(s^3). \] This is a local variational expansion, not a finite subtraction of infinite total background actions. The cubic and quartic vertices, ghosts, gauge-fixing terms, observable vertices, and their prescribed retarded operations must enter any claimed interacting calculation at matching orders. A single free contraction of \(F_{2,f}\) is not the complete interacting one-loop inventory. \subsection{Positive quantum powers do not repair a classical residual} \begin{lemma}[Formal-order separation] \label{lem:hbar} Let \(V\) be a vector space. If a Ward variation has the formal form \[ D=D_{\mathrm{cl}}+\hbar D_1+\hbar^2D_2+\cdots \quad\text{in }V[[\hbar]], \] and \(D_{\mathrm{cl}}\ne0\), no addition belonging to \(\hbar V[[\hbar]]\) can make it vanish. \end{lemma} \begin{proof} The coefficient of \(\hbar^0\) is unchanged by such an addition. Equality of formal series is coefficientwise equality. \end{proof} This applies, for example, to an uncorrected classical smoothing or rod-replacement residual. It does not compute a quantum anomaly. \subsection{The repair criterion} The following theorem concerns one \emph{constructed} operator domain. Its hypotheses are not claimed for the native lapse. \begin{definition}[Same-target Ward-repair data] \label{def:repair} Suppose: \begin{enumerate}[label=\textup{(\roman*)}] \item the complete insertion \(J_{\mathcal R}\Op_{\mathrm{cl}}\) and the retained Ward operation \(\mathcal D_{\mathcal R}\) are defined on one specified domain; \item their classical leading Ward variation is zero; \item the complete retained quantum breaking is a defined element \(B\) of a vector space \(V\) of gauge-parameter-dependent operator coefficients: \[ \mathcal D_{\mathcal R} (J_{\mathcal R}\Op_{\mathrm{cl}}) =\hbar B\pmod{s^3,\hbar^2}; \] \item the admissible quantum corrections form a specified vector space \(\CT\), including their actual observable images, and their Ward image is the linear map \[ L:\CT\longrightarrow V. \] \end{enumerate} All coefficients are taken through \(s^2\). \end{definition} \begin{theorem}[Exact retained-order Ward repair] \label{thm:repair} For the data of Definition~\ref{def:repair}, \[ \mathcal D_{\mathcal R} \left(J_{\mathcal R}\Op_{\mathrm{cl}}+\hbar C\right) =\hbar(B+LC)\pmod{s^3,\hbar^2}. \] An admissible correction proving the identity exists if and only if \begin{equation} -B\in\im L. \label{eq:repair-exact} \end{equation} If \(C_*\) is one solution, all solutions are \(C_*+\ker L\). The class of \(B\) in \(V/\im L\) is unchanged by compatible admissible finite corrections. \end{theorem} \begin{proof} At order \(\hbar\), the Ward variation of the added term is its leading image \(LC\). Adding it to the defined breaking gives the first formula. Vanishing is exactly \(LC=-B\), which is solvable precisely under \eqref{eq:repair-exact}. The difference of two solutions belongs to \(\ker L\), and adding an image \(LC\) leaves the quotient class unchanged. \end{proof} No closure of \(\im L\) replaces its actual image. No finite-dimensionality or exhaustive counterterm parameterization is assumed without proof. No BRST cohomology classification follows unless a valid nilpotent quantum differential and its domain have separately been established. \begin{corollary}[Finite probes are not automatically a field identity] \label{cor:probes} Let \(P:V\to\C^m\) be a finite probe map. The equation \(P(B+LC)=0\) implies the complete Ward identity only if the remaining residual is known to lie in a subspace on which \(P\) is injective. \end{corollary} \begin{proof} Without injectivity, a nonzero residual in \(\ker P\) satisfies all the finite equations. Under the stated restriction, injectivity forces that residual to vanish. \end{proof} For the present target, \(B\) and the complete native image map \(L\) have not been constructed. They are marked \emph{undefined for the native quantum insertion}, not zero. Theorem~\ref{thm:repair} is therefore an admission criterion, not a cancellation theorem for the lapse. %===================================================================== \section{Negative controls and nontransfer map} \label{sec:map} %===================================================================== \subsection{Pure-trace evaluations do not determine a complete kernel} \begin{proposition}[A smooth pure-trace blind sector] \label{prop:trace-blind} Pure-trace tests do not determine an unrestricted compact smooth kernel with two symmetric metric slots and one scalar slot. \end{proposition} \begin{proof} In a local Lorentzian frame choose a nonzero constant symmetric tensor \(A^{\mu\nu}\) with \(A^{\mu\nu}\eta_{\mu\nu}=0\). For nonzero compact smooth functions \(b,c\), set \[ K^{\mu\nu,\rho\sigma}(x,y,z) =A^{\mu\nu}A^{\rho\sigma}b(x)b(y)c(z). \] This is a nonzero compact smooth kernel, symmetric in its metric slots. Its contraction with \((\eta\psi_1)\otimes(\eta\psi_2)\otimes\varphi\) vanishes for every test choice. Thus zero and \(K\) have identical pure-trace evaluations. \end{proof} This proves non-determination on the displayed unrestricted kernel class. It does not prove ambiguity of a kernel derived from complete root data, nor invalidate a full identity actually proved on all declared slots. Pure-trace tests alone cannot justify such an identity. \subsection{One-page source-to-refusal map} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.20\linewidth}YY@{}} \toprule Source record & Shape or checklist retained & Claim refused\\ \midrule XIX-A & Four labelled prescriptions; raw/relational and original/modified distinctions & Identification of \(\CE\) with \(\CV,\CI,\RX\); transfer from selected values\\[3pt] XIX-B & Residual-forcing check for a changed coordinate prescription & An intrinsic defect of every VI target; a quantum anomaly; use of \(S_\lambda\) as a cosmological rod source\\[3pt] XIX-C & Complete finite channels, records, guards, and refinement checks & Continuum observable or physical causal relation from a finite approximation alone\\[3pt] XIX-D & Q1--Q5 and same-target closure discipline & Quantum certificate from classical covariance or a named parametrix alone\\[3pt] XIX-E & Unchanged release boundary & Opening a later characteristic, ultraviolet, or continuum track\\[3pt] Em02-E & Original-rod lapse target and incomplete quantum audit & Automatic identification with this variant; native state, tails, products, or quantum Ward identity\\[3pt] XVIII-A & Pure-trace, formal-order, and actual-image controls & Complete kernel or Ward cancellation from selected witnesses\\[3pt] XVIII-B & Separately frozen stress-response benchmark & Relational lapse, graviton--ghost inventory, or inherited characteristic coefficient\\[3pt] XVI-Q, Q11 & Scoped bulk two-point report & This observable's counterterm class, three-point closure, bulk restoration, or enlarged O.1 scope\\[3pt] XVI-N; XV-B & Separate insertion, route, and technical lifting boundaries & Discharge of Route B or opening of the quantum-admission exit\\[3pt] Oai069-A; Oai282-B & Finite Ward and local-action comparison patterns & Gravitational dictionary, unrestricted coverage, or catalogue verification\\[3pt] OpenAI 069/282/271 & Documentary catalogue scope & Proof premise, clock, counterterm value, or Lean verification of this manuscript\\ \bottomrule \end{tabularx} \end{center} The comparisons use the supplied records \cite{XIXA,XIXB,XIXC,XIXD,XIXE,CheungEm02E, CheungXVIIIA,CheungXVIIIB,CheungXVIQ,CheungXVIN, CheungXVB,CheungOai069,CheungOai282,Catalogue}. They are not a dependency graph or an exhaustive corpus audit. The Oai comparison manuscripts are cited under their supplied N.~Cheung attribution. The separate catalogue is cited under its supplied OpenAI attribution. Those authorship records are not merged. %===================================================================== \section{Gate audit, optional limits, and mathematical release} \label{sec:release} %===================================================================== \subsection{Q1--Q5 audit for XX-A} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.10\linewidth}p{.18\linewidth}Y@{}} \toprule Gate & Ending & Exact statement\\ \midrule Classical freeze & Constructed & Explicit on-shell background, calibrated clock, original rods, boundary condition, complete coefficient, and coupled classical identity\\[4pt] Q1 & Unresolved & Unfixed equations and quadratic gauge-fixing candidate are specified; the complete checked coupled realization and normalized Green/contraction data are not supplied\\[4pt] Q2 & Unresolved & No physical state, native covariance, subtraction parametrix, or required spatial-tail theorem selected\\[4pt] Q3 & Unresolved & Classical kernel issues and precise analytic tests identified; regular-polynomial realization is conditional, not native renormalized insertion admission\\[4pt] Q4 & Unresolved & Formal coefficient range fixed; no complete quantum remainder estimate or regulator-removal theorem\\[4pt] Q5 & Unresolved & Complete classical coupled identity proved; native quantum breaking and actual admissible cancellation are not calculated\\ \bottomrule \end{tabularx} \end{center} A quantum successor cannot replace an unresolved entry by a theorem name. It must supply its object and proof on the same frozen target. \subsection{Regulator removal is optional and not entered} The milestone does not require regulator removal. If removal is later attempted, the two independent parameters \(\epsilon\) and \(R\) must be retained, with a declared topology, uniform complete estimates, and a comparison between admissible refinement families. Convergence at each fixed \(R\) is not uniformity as \(R\to\infty\). Likewise, existence of one limit is not independence of the regularization prescription. A finite-mode or lattice approximation remains labelled as such until its same-target continuum comparison is proved. No numerical, spectral, or finite-channel refinement is performed in this paper. \begin{theorem}[XX-A mathematical release] \label{thm:release} For the prescription explicitly frozen in this manuscript: \begin{enumerate}[label=\textup{(\roman*)}] \item the power-law background is an on-shell Einstein--scalar background with a nondegenerate calibrated matter clock; \item the original spatial rods exist uniquely as decaying smooth coefficient fields through second order; \item the complete raw and relational lapse coefficients are \eqref{eq:N1}--\eqref{eq:N2} and \eqref{eq:n1}--\eqref{eq:n2}; \item the second spatial rod is constructed, and its absence from the quadratic lapse is proved; \item the complete linear functional and Hessian obey \eqref{eq:linear-ward}--\eqref{eq:quadratic-ward}; \item a compact off-shell witness distinguishes the original rods from the specified lapse-weighted rods; \item instantaneous singularities, spatial tails, regular Wick comparisons, and exact Ward-repair criteria have the scopes stated in the corresponding propositions; \item no native quantum covariance, renormalized lapse insertion, complete quantum breaking, admissible cancellation, or retained-order quantum Ward identity is constructed. \end{enumerate} The classical freeze is constructed. The quantum milestone is unresolved at Q1. \end{theorem} \begin{proof} Lemma~\ref{lem:on-shell} proves (i). Proposition~\ref{prop:rods} proves (ii). Theorem~\ref{thm:lapse} proves (iii) and (iv). Theorem~\ref{thm:classical-gauge} and Corollary~\ref{cor:coupled} prove (v). Proposition~\ref{prop:rod-detection} proves (vi). Propositions~\ref{prop:tail} and~\ref{prop:instantWF}, Lemma~\ref{lem:regular-wick}, and Theorem~\ref{thm:repair} prove the mathematical assertions in (vii). For (viii), the formulas defining the classical target, unfixed operator, and gauge-fixing candidate do not supply the missing Q1 realization and contraction data. The regular Wick lemma assumes those data, and the Ward-repair theorem assumes a defined quantum insertion and breaking. Neither assumption is discharged for the native target here. The gate ending is therefore unresolved, not constructed and not a universal impossibility theorem. \end{proof} \subsection{Unchanged standing boundary} The supplied narrow XVI-Q report concerns only its one-loop order-\(\kappa^2\) bulk two-point sector, with its zero-mass, fixed-gauge, QdS external-leg and enlarged-prescription hypotheses \cite{CheungXVIQ}. It is not this cosmological lapse construction. Its reported rebuild is not rerun or enlarged here. The standing release remains \[ \boxed{ \begin{gathered} \text{Declared path: null},\\ \text{quantum relational-lapse milestone: unresolved at Q1},\\ \text{quantum evolution without external time: not constructed},\\ \text{inherited characteristic quantum coefficient: not admitted},\\ \sid{O.1 = NOT_CLOSED} \quad\text{outside the narrow XVI-Q scope},\\ \text{XVI-N and spatially compact Route B: not discharged},\\ \text{XV-B downstream quantum-admission exit: not opened},\\ \text{compilation and independent predecessor verification: not asserted}. \end{gathered}} \] No ultraviolet completeness, three-point closure, de Sitter state, derived metric, or spacetime-emergence theorem is claimed. %===================================================================== \section{Conclusion} %===================================================================== A submission-ready boundary artifact must distinguish what its proofs establish from what its programme intends. XX-A fixes one original-rod lapse variant, derives every classical term through second order, proves its coupled classical identity, and supplies detecting and analytic checks that prevent silent replacement of that target. It also specifies the exact conditions under which a later quantum coefficient could be admitted. The remaining quantum work is not a corollary of that classical construction. It begins with the actual coupled realization and contraction data, continues through the nonlocal insertion and its counterterm images, and ends with an evaluated complete breaking and an admissible cancellation. Only after those constructions would Track A have its stated quantum milestone. Even that milestone would remain a result inside a fixed-background, external-time setting. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} %===================================================================== \paragraph{Mathematical scope.} All classical coefficients and their gauge comparison are derived from the displayed freeze. The quantum realization and Ward-repair theorems are expressly conditional and are not announced as native quantum constructions. \paragraph{Prescription identification.} The manuscript fixes an original-rod variant independently. The supplied Em02-E excerpts motivate its target and audit. They do not establish equality of every native defining datum. \paragraph{Source scope.} Prior series manuscripts supply version-qualified shapes, checklists, and separations, not unprinted theorem premises. Catalogue attribution, proof validity, Lean scope, artifact hashes, and rebuild reports are not independently authenticated. No exhaustive scan of the series is claimed. \paragraph{Bibliography.} Published primary literature is cited for the stated frameworks and analytic results. Unpublished series records are cited by supplied version and filename; descriptive titles are not expanded where complete metadata were unavailable. \paragraph{Verification.} The source requires standard pdfLaTeX packages and contains one internal bibliography. No successful pdfLaTeX run, proof-assistant execution, independent numerical or symbolic calculation, native programme compilation, referee decision, or journal acceptance is asserted. \paragraph{Authorship and availability.} The manuscript is anonymous. Citation of N.~Cheung does not establish authorship of this note. A signed submission must supply authorized author metadata and stable archival copies or locations for the cited unpublished records. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} %===================================================================== \bibitem{CheungEm02E} N.~Cheung, \emph{Em02-E}, version 1, supplied unpublished manuscript, \sid{Em02-E-v1.tex}, n.d. Enter Calculus Companion. Relevant supplied locators: \sid{thm:lapse}, \sid{thm:release}, ``The quantum input gate and the first contraction target,'' and ``Checks that prevent false quantum conclusions.'' Original inflaton-slice rods, classical lapse, and incomplete quantum audit. \bibitem{CheungXVIIIA} N.~Cheung, \emph{Landmark XVIII-A: Prepared Relational Observables: Non-Determination, Normalization Compatibility, and Conditional Ultraviolet Matching}, version 2, supplied unpublished manuscript, \sid{XVIII-A-v2.tex}, n.d. Relevant supplied locators: \sid{thm:witness}, \sid{cor:smoothwitness}, \sid{prop:hbar}, \sid{thm:ward}, and \sid{thm:release}. Pure-trace, formal-order, and actual-image controls. \bibitem{CheungXVIIIB} N.~Cheung, \emph{Landmark XVIII-B}, version 2, supplied unpublished manuscript, \sid{XVIII-B-v2.tex}, n.d. Relevant supplied locators: \sid{thm:spectral}, \sid{thm:ward}, \sid{thm:residual}, and \sid{thm:release}. Separately frozen massive stress-response benchmark; not the original-rod lapse. \bibitem{CheungXVID} N.~Cheung, \emph{Landmark XVI-D: Prepared witness response and completion domains}, version 1, supplied unpublished manuscript, \sid{XVI-D.tex}, n.d. Relevant supplied locators: \sid{eq:Slambda}, \sid{lem:Xcand}, \sid{prop:coordinate-compatibility}, and \sid{thm:response-package}. Smooth witness response and conditional coordinate compatibility; not slice-elliptic rod uniqueness. \bibitem{CheungXVIN} N.~Cheung, \emph{XVI-N}, version r12, supplied unpublished manuscript, \sid{XVI-N.tex}, n.d. Relevant supplied locator: \sid{thm:massless-obstruction}. Conditional massless special-family obstruction and separate undischarged spatially compact Route B. \bibitem{CheungXVIQ} N.~Cheung, \emph{XVI-Q}, release Q11, supplied unpublished manuscript, \sid{XVI-Q.tex}, n.d. Relevant supplied locators: \sid{rem:o1}, \sid{rem:qds}, \sid{rem:sigma}, \sid{rem:classification}, \sid{rem:insertion}, and \sid{rem:package}. Scoped bulk two-point report and its explicit nontransfer and verification qualifications. \bibitem{CheungXVB} N.~Cheung, \emph{XV-B}, version 2, supplied unpublished manuscript, \sid{XV-B.tex}, n.d. Relevant supplied locators: \sid{def:interface}, \sid{thm:lifting}, \sid{thm:jetnull}, and \sid{thm:release}. Technical lifting and extension interfaces; no inherited full-map or quantum-admission conclusion is imported. \bibitem{CheungOai069} N.~Cheung, \emph{Oai069-A}, version 1, supplied unpublished manuscript, \sid{Oai069-A.tex}, n.d. Finite Ward-defect and local-system comparison scopes only. \bibitem{CheungOai282} N.~Cheung, \emph{Oai282-B}, version 1, supplied unpublished manuscript, \sid{Oai282-B.tex}, n.d. Local free-scalar Ward action and declared finite coverage; not a gravitational dictionary or catalogue verification. \bibitem{XIXA} \emph{XIX-A: Clock Separation Before Coordinate Identification: Four Labelled Objects and Five Non-Identifications}, version 1, supplied manuscript record, \sid{XIX-A.tex}, n.d. Classical comparison and original/modified rod distinctions. No independent source verification or authorship authentication is inferred. \bibitem{XIXB} \emph{XIX-B: A Witness Mismatch Between Two Coordinate Cauchy Prescriptions: Unswitched Harmonic Coordinates and Prepared Coordinate Completion}, version 1, supplied manuscript record, \sid{XIX-B.tex}, n.d. Declared direct first-coordinate mismatch, not a quantum anomaly or an intrinsic obstruction to every VI prescription. \bibitem{XIXC} \emph{XIX-C: Finite Incidence Registers and the Missing Continuum Evolution: Matching, Records, Promised Tokens, and Scheduled Support}, version 1, supplied manuscript record, \sid{XIX-C.tex}, n.d. Finite interfaces and continuum non-determination comparison. \bibitem{XIXD} \emph{XIX-D: Quantum-Certificate Inventory for Separated Clock Prescriptions}, version 1, supplied manuscript record, \sid{XIX-D.tex}, n.d. Relevant supplied locator: \sid{def:slots}. Q1--Q5 and same-target quantum admission discipline. \bibitem{XIXE} \emph{XIX-E: Boundary of the Clock-Separation Programme: Nontransfer, Physical Interpretation, and Unchanged Obligations}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-E.tex}, n.d. Boundary inventory, not a universal impossibility theorem. \bibitem{Catalogue} \emph{OpenAI catalogue and companion scope records}, attributed to OpenAI in the supplied documentary record, dated 7 October 2026. Relevant comparisons: Families 069, 282, and 271. The reported Lean scope names 271-04 only. Attribution, proof validity, and execution are not independently authenticated here. \bibitem{Hack} T.-P.~Hack, ``Quantization of the linearised Einstein--Klein--Gordon system on arbitrary backgrounds and the special case of perturbations in inflation,'' \emph{Classical and Quantum Gravity} \textbf{31} (2014), 215004. DOI: \sid{10.1088/0264-9381/31/21/215004}. arXiv:1403.3957. Coupled linearized field and gauge framework; not this observable's covariance or quantum admission. \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} \textbf{08} (2016), 032. DOI: \sid{10.1007/JHEP08(2016)032}. arXiv:1605.02573. Relational-observable framework and cosmological perturbative context; no native rod-prescription identification is inferred. \bibitem{Frob} M.~B.~Fr\"ob, ``Gauge-invariant quantum gravitational corrections to correlation functions,'' \emph{Classical and Quantum Gravity} \textbf{35} (2018), 055006. DOI: \sid{10.1088/1361-6382/aaa74c}. arXiv:1710.00839. Relational quantum-correlator context; not the original-rod lapse calculation. \bibitem{BFR} R.~Brunetti, K.~Fredenhagen, and K.~Rejzner, ``Quantum gravity from the point of view of locally covariant quantum field theory,'' \emph{Communications in Mathematical Physics} \textbf{345} (2016), 741--779. DOI: \sid{10.1007/s00220-016-2676-x}. arXiv:1306.1058. Perturbative gravity and BV framework; no native identity or ultraviolet-completeness result is imported. \bibitem{Hormander} L.~H\"ormander, \emph{The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis}, second edition, Springer, 1990. Theorems 8.2.4 and 8.2.10: pullbacks and products under their wavefront hypotheses. \bibitem{Radzikowski} M.~J.~Radzikowski, ``Micro-local approach to the Hadamard condition in quantum field theory on curved space-time,'' \emph{Communications in Mathematical Physics} \textbf{179} (1996), 529--553. DOI: \sid{10.1007/BF02100096}. Theorem 5.1 and its scalar-field hypotheses; not a selection of a cosmological state. \bibitem{SV} H.~Sahlmann and R.~Verch, ``Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime,'' \emph{Reviews in Mathematical Physics} \textbf{13} (2001), 1203--1246. DOI: \sid{10.1142/S0129055X01001010}. arXiv:math-ph/0008029. Vector-valued microlocal framework under the stated field and quantization hypotheses. \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. DOI: \sid{10.1007/s002200050004}. arXiv:math-ph/9903028. Finite-scaling-degree extension framework; not applied to an undefined native nonlocal product. \bibitem{HWlocal} S.~Hollands and R.~M.~Wald, ``Local Wick polynomials and time ordered products of quantum fields in curved spacetime,'' \emph{Communications in Mathematical Physics} \textbf{223} (2001), 289--326. DOI: \sid{10.1007/s002200100540}. arXiv:gr-qc/0103074. Local Wick products and normalization analysis. \bibitem{HWexistence} S.~Hollands and R.~M.~Wald, ``Existence of local covariant time ordered products of quantum fields in curved spacetime,'' \emph{Communications in Mathematical Physics} \textbf{231} (2002), 309--345. DOI: \sid{10.1007/s00220-002-0719-y}. arXiv:gr-qc/0111108. Existence for the stated local scalar-field construction; not automatic admission of the present nonlocal lapse. \end{thebibliography} \end{document}