% XIX-D.tex % Self-contained pdfLaTeX source. % No external bibliography, figures, code, or input files are required. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XIX-D.tex % pdflatex -interaction=nonstopmode -halt-on-error XIX-D.tex % % No successful compilation or proof-assistant execution is asserted. % Anonymous submission: supply authorized authorship metadata before % a signed submission. Citation of N. Cheung does not establish % authorship of this manuscript. \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={XIX-D: Quantum-Certificate Inventory for Separated Clock Prescriptions}, pdfauthor={}, pdfsubject={Quantum input gates, non-determination, and certificate nontransfer}, pdfkeywords={relational observable, scalar clock, quantum certificate, counterterm image, Ward identity, finite register, continuum limit} } \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{\N}{\mathbb N} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\im}{\operatorname{im}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\sd}{\operatorname{sd}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\lvert #1\rangle\langle #2\rvert} \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{\sid}[1]{\texttt{\detokenize{#1}}} \newcommand{\CV}{\mathsf C_{\mathrm{VI}}} \newcommand{\CE}{\mathsf C_{\mathrm E}} \newcommand{\CI}{\mathsf C_{\mathrm{VIII}}} \newcommand{\RX}{\mathsf R_{\mathrm X}} \newcommand{\NN}{\mathcal N_{\mathrm E}} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\CT}{\mathscr C_{\mathrm{ct}}} \newcommand{\Def}{\mathscr V_{\mathrm{def}}} \newcommand{\dtrace}{d_{\mathrm{tr}}} \newcommand{\ddiam}{d_\diamond} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \title{\textbf{XIX-D}\\[3pt] Quantum-Certificate Inventory for Separated Clock Prescriptions\\ \large Non-Determination, Counterterm Images, and Unclosed Quantum Gates} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} We give a quantum-certificate inventory for four previously separated time-related prescriptions: a harmonic relational scalar, an original-rod scalar-clock lapse, a prescribed incidence register, and a coordinate-density ramp. The ramp remains a prescribed non-clock. No common quantum observable is selected by the labels. The inventory is supported by self-contained non-implication arguments. A classical operator and its causal commutator do not select a two-point state; a prescribed singularity class does not establish positivity. Individually defined distributions need not have a raw product. Pure-trace evaluations do not determine a three-slot kernel, its microlocal admission, or its free-shell functional. Late source agreement does not determine retarded response. Exact affine repair requires membership in the actual counterterm image, not agreement on finite probes or membership in its closure. Positive formal powers of \(\hbar\) cannot cancel a fixed nonzero classical coefficient, and a projected identity is not a full identity. We also distinguish records from residual instruments and show that even exact subdivision of one finite channel need not select a unique continuous-time evolution. These arguments are audit criteria, not constructions of any native gravitational quantum coefficient. The original-rod lapse retains its five quantum input obligations; VI's certificate remains RG-open in the supplied record. The reported XVI-Q two-point check is recorded only within its stated prescription and scope. It supplies neither a clock certificate nor a de Sitter state. No inherited characteristic coefficient, quantum evolution without external time, compilation, or downstream quantum-admission conclusion is released. \end{abstract} \noindent\textbf{Keywords:} quantum certificate; relational scalar; scalar-clock lapse; counterterm image; Ward identity; state selection; finite register; continuum non-determination. \tableofcontents %===================================================================== \section{Question, exact claim, and debugging conventions} \label{sec:scope} %===================================================================== The question is what would be needed to upgrade each of the separated classical prescriptions to a specified quantum observable with a proved retained-order gauge identity and, where requested, a controlled regulator-removal or continuum limit. This paper is an inventory and a collection of non-determination results. It does not close either the VI quantum certificate or the Em02-E lapse audit. Its mathematical claims are the explicit arguments below. Its source-status statements are version-qualified documentary reports. The comparison is motivated by Cheung's classical and quantum interface records \cite{CheungVI,CheungEm02E,CheungXVIIIA,CheungXVIIIB, CheungXVIQ,CheungVIIIA,CheungVIIIB,CheungVIIIC}. General constructions of relational observables, perturbative gravity, local Wick polynomials, and time-ordered products provide context \cite{BFHPR,Frob,BFR,BF,HWlocal,HWexistence}. Those theories are not automatically realizations of the particular coordinate data, spatial rods, state conditions, or insertion kernels audited here. \subsection{The classical papers are comparisons, not hidden premises} XIX-A labelled four time-related objects and supplied classical separation calculations \cite{XIXA}. XIX-B excluded one specified direct comparison of VI's unswitched first coordinate coefficient with XVI-D's prepared candidate \cite{XIXB}. XIX-C analyzed finite incidence interfaces and their failure to determine a continuum evolution \cite{XIXC}. None of their detecting witnesses is imported into a quantum calculation in this paper. Where an elementary principle is needed, it is proved below. The direct mismatch of XIX-B does not show that VI lacks its own harmonic coordinate prescription. It shows that one declared map does not identify that prescription with another candidate. Consequently, there is no common coordinate field inherited \emph{through that comparison}. An independent quantum construction of either native object remains a different possible undertaking. Similarly, finite-register non-determination does not show that finite quantum channels are absent. XIX-C explicitly defines complete finite channel realizations. The missing target is a specified continuum or autonomous physical evolution of the incidence package, not the existence of every finite quantum update. \subsection{Three corrections to the proposed audit language} \paragraph{Missing is not impossible.} ``Not constructed here'' means that this release contains no realization and proof of the specified object. It is not a universal nonexistence theorem. An exclusion theorem must name its candidate class and prove that every candidate in that class is excluded. \paragraph{An old gate label is not a denial of a later partial advance.} XIV-H's H2 stop is the stop within that supplied sheet. XIV-I reports an internal I2 amplitude construction and a remaining I3 joint-map gap. XIV-J reports a supported-sector reference and a remaining J2 full-map gap \cite{CheungXIVH,CheungXIVI,CheungXIVJ}. We do not describe the internal amplitude as simultaneously unconstructed after reporting that later advance. No full characteristic map or quantum admission follows from it. \paragraph{Catalogue attribution and manuscript authorship are distinct.} The supplied OpenAI family catalogue is attributed to OpenAI. The supplied comparison sheets Oai282-B and Oai069-A are cited as manuscripts by N.~Cheung \cite{Catalogue,CheungOai282B,CheungOai069A}. Their names do not make them OpenAI-authored proofs or verify the catalogue families. This distinction is retained throughout. \subsection{Standing release boundary} No quantum evolution without an external time is constructed. No quantum coefficient of the inherited characteristic graph is admitted. No VI radiative certificate, Em02-E quantum gauge identity, native incidence continuum limit, ultraviolet completion, or compilation is asserted. The reported scoped XVI-Q conclusion is not erased, but it is not extended. Outside that separately specified scope, \sid{O.1 = NOT_CLOSED} remains the supplied standing record \cite{CheungXVIQ}. XVI-N, Route B, and the downstream XV-B quantum-admission exit are not discharged by this paper. %===================================================================== \section{Four objects and a precisely typed quantum audit} \label{sec:objects} %===================================================================== \subsection{The four labelled entries} The entries are \begin{equation} \boxed{\CV,\qquad \CE,\qquad \CI,\qquad \RX.} \label{eq:labels} \end{equation} They are not four already verified quantum clocks. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.16\linewidth}YY@{}} \toprule Entry & Classical prescription being audited & Data retained\\ \midrule \(\CV\) & \(T\circ H_q^{-1}\), with a specified harmonic-coordinate Cauchy problem & Metric and scalar jets, equation space, initial surface, coordinate-time data, and gauge group\\[3pt] \(\CE\) & \(N[q,T]\circ\alpha_{q,T}\), using the original \(T\)-slice harmonic spatial rods & Timelike branch, clock calibration, elliptic boundary condition, relational map, and background interval\\[3pt] \(\CI\) & Prescribed incidence parameter together with a finite reading register and declared updates & Installed packet, exact guards, supplied token, complete record and residual interfaces, and schedule\\[3pt] \(\RX\) & Fixed smooth profile in a coordinate density & External coordinate time, profile, width, placement, and its transformation convention\\ \bottomrule \end{tabularx} \end{center} For the scalar-clock entry, the raw lapse is \[ N[q,T]=\bigl(-q^{-1}(\dd T,\dd T)\bigr)^{-1/2}, \] on its positive timelike branch. Its nonlinear formula does not by itself define an operator inverse or an inverse square root of a quantum distribution. For the ramp entry, a typical profile is \[ \sigma_\varepsilon(t) =\chi\!\left(\frac{t-t_0}{\varepsilon}\right), \] held fixed as external data. With that convention its field variation vanishes. Giving it a transforming-background role, or introducing a new quantum field in its place, changes the declared object. \subsection{A certificate is more than a formula} \begin{definition}[Declared quantum target] \label{def:target} A declared quantum target specifies: \begin{enumerate}[label=\textup{(\roman*)}] \item its classical configuration or equation space, background, observable prescription, and gauge conventions; \item its quantum field content, graded test spaces, contraction data, and operator or functional domain; \item the coupling and \(\hbar\) orders retained; \item the products, extensions, normalization conditions, and counterterm freedom permitted at those orders; \item the identity, expectation, or limit that is actually requested. \end{enumerate} An external-source response, a bulk two-point identity, a relational composite, and a complete measurement instrument are different targets unless comparison maps identify them. \end{definition} \begin{definition}[Five audit slots] \label{def:slots} For a target requiring a perturbative field-theoretic construction, the five audit slots are: \begin{description}[style=nextline,leftmargin=2.8em] \item[Q1: Gauge-fixed covariance.] Field multiplet, gauge fermion or other fixing prescription, auxiliary and ghost sectors, full coupled quadratic operator, its domain, Green inverses, causal distribution, and component normalizations. \item[Q2: State or parametrix.] An identified two-point matrix or parametrix with the singular, field-equation, symmetry, and tail properties needed by the target. A physical expectation additionally requires an actual state on the relevant physical algebra and its positivity. \item[Q3: Products and counterterm class.] A realization of the required Wick, time-ordered, retarded, pullback, and composite operations; the admissible local, boundary, switch, and insertion corrections; their images in the observable construction; and the normalization convention. \item[Q4: Controlled remainder and limit.] The retained formal order, the topology of every claimed limit, and estimates for the complete remainder on the declared tests. Formal truncation and numerical or regulator error are distinct. \item[Q5: Retained-order gauge identity.] A defined gauge differential or Ward operation on the constructed domain, an evaluated complete breaking term, and a proved identity or admissible repair at the retained order. \end{description} \end{definition} The order in which these slots are solved may vary in a particular construction. The distinction among them does not. A state is not necessary for every algebraic statement, but a state-dependent expectation cannot be evaluated by naming a parametrix alone. For a finite register, Q1--Q5 are not imposed by pretending that a register already has a graviton quadratic operator. Its native quantum input problem is a complete finite channel or instrument. A claimed field-theoretic or continuum interpretation adds the corresponding comparison and limit obligations. \begin{convention}[Meaning of certificate closure] \label{conv:closure} Closure in this paper means that all operations and conclusions required by one declared target have been supplied on matching domains, with its allowed changes and verification scope stated. It does not mean all-orders renormalizability, an RG fixed point, ultraviolet completeness, or a quantum theory of spacetime. \end{convention} %===================================================================== \section{Covariance, state selection, and quantum products} \label{sec:state-products} %===================================================================== \subsection{An equation and a singularity condition do not select a state} The distinction between field equations, microlocal singularities, state positivity, and local composite normalization is standard in quantum field theory on classical curved spacetime \cite{HWreview,HWlocal,HWexistence}. The following conditional construction proves the elementary non-determination needed here. \begin{proposition}[Smooth bisolutions preserve algebraic data but not positivity] \label{prop:state} Let \(P\) be a real scalar field operator on a smooth spacetime. Suppose a bidistribution \(W\) is a positive two-point function satisfying \[ P_xW=P_yW=0, \qquad W(x,y)-W(y,x)=iE(x,y), \] with a specified wavefront set. Let \(u\) be a nonzero real smooth solution of \(Pu=0\). For real \(a\), put \[ W_a=W+a\,u\otimes u. \] Then: \begin{enumerate}[label=\textup{(\roman*)}] \item every \(W_a\) has the same field equations, antisymmetric part, and wavefront set as \(W\); \item every \(W_a\) with \(a\ge0\) is positive; \item for some \(a<0\), \(W_a\) is not positive. \end{enumerate} Thus those algebraic and singularity data do not uniquely select a positive covariance, and do not by themselves establish positivity. \end{proposition} \begin{proof} The added term is smooth, symmetric, and a bisolution. Adding a smooth distribution does not change the wavefront set. This proves (i). For a complex compact test \(f\), \[ W_a(\overline f,f) =W(\overline f,f)+a\abs{\pair{u}{f}}^2. \] This is nonnegative when \(a\ge0\). Since \(u\ne0\), choose \(f\) with \(\pair{u}{f}\ne0\). The displayed expression becomes negative when \[ a<-\frac{W(\overline f,f)}{\abs{\pair{u}{f}}^2}. \] This proves (iii). \end{proof} The proposition assumes a scalar covariance for its comparison. It constructs no Einstein--scalar covariance for Em02-E. Additional invariance, infrared, decay, or state-selection conditions may eliminate some of the freedom; those conditions must be stated and proved. Positivity also has a graded and gauge-qualified meaning in a gauge-fixed theory. One does not impose ordinary positive scalar state conditions on every ghost or unphysical metric component. The physical algebra, its involution, and its state must be identified. Linearized Einstein--Klein--Gordon quantization on on-shell backgrounds has a separate construction with its own field and gauge hypotheses \cite{Hack}. That reference does not choose the two-point matrix, initial state, or spatial-tail estimates for an independently specified original-rod lapse. \subsection{The classical Hessian is not its contraction} For a formal bosonic multiplet \(\varphi^A\), a quadratic classical observable may be written schematically as \begin{equation} O^{[2]}[\varphi] =\frac12\sum_{A,B} \pair{K_{AB}}{\varphi^A\otimes\varphi^B}. \label{eq:hessian} \end{equation} If a covariance and all pairings have been supplied, a prospective contraction has the form \begin{equation} \frac{\hbar}{2}\sum_{A,B}\pair{K_{AB}}{W^{AB}}. \label{eq:prospective} \end{equation} Equation~\eqref{eq:prospective} is only a displayed target until the product, pullback, or pairing between its distributions is defined. It is not evaluated by knowing \(K_{AB}\) classically. Fermionic fields require the corresponding graded signs. A common bosonic-looking formula is not a generator-level sign check for a BRST or BV multiplet. \begin{proposition}[Individually defined distributions need not have a raw product] \label{prop:product} There are smooth regularizations of two individually defined distributions whose ordinary products have no distributional limit. \end{proposition} \begin{proof} Take a nonzero real \(\rho\in C_c^\infty(\R)\) with \(\int\rho=1\), and put \[ \delta_\epsilon(t)=\epsilon^{-1}\rho(t/\epsilon). \] Each \(\delta_\epsilon\) converges to \(\delta_0\). For a compact test \(f\) with \(f(0)>0\), \[ \pair{\delta_\epsilon^2}{f} =\epsilon^{-1}\int \rho(v)^2f(\epsilon v)\,\dd v =\epsilon^{-1}f(0)\int\rho(v)^2\,\dd v+O(1). \] The leading coefficient is positive, so the pairing has no finite limit. \end{proof} This proves failure of the raw regularized product, not nonexistence of every possible renormalized prescription. Distributional extension and normalization frameworks address specified products under additional hypotheses \cite{BF,HWexistence,Dang}. They cannot be applied to an unnamed kernel or an unproved composition merely by citing the framework. \begin{remark}[A sufficient product test is not a universal obstruction] The ordinary wavefront criterion excludes opposing covectors when defining a product by that calculus. Failure to meet a sufficient criterion does not by itself prove that every generalized or renormalized construction fails. Conversely, separate existence of the factors is not a proof of their ordinary product. \end{remark} \subsection{Formal remainders and regulator limits} For a formal series \[ O=\sum_{r,s\ge0}\kappa^r\hbar^s O_{r,s}, \] a statement modulo \(\kappa^{R+1}\) or \(\hbar^{S+1}\) is coefficientwise. It is not an estimate at numerical values of the parameters. Likewise, an upper bound \[ \abs{\pair{O_\epsilon}{f}}\le C_f\epsilon^{-r} \] is neither convergence nor an evaluated divergence. The zero family and a family proportional to \(\epsilon^{-r}\) can satisfy the same upper bound. The elementary family \[ Z_\epsilon=\epsilon^{-1} \] is smooth and invariant under every action on its configuration arguments, yet \(\pair{Z_\epsilon}{f}\) diverges for a test with nonzero integral. Thus classical covariance does not imply regulator removal. %===================================================================== \section{Witness data do not determine kernels or admission} \label{sec:kernels} %===================================================================== \subsection{The elementary fiber test} \begin{lemma}[Factorization through declared data] \label{lem:fiber} For maps \(\pi:D\to B\) and \(F:D\to V\), a map \(\overline F:\pi(D)\to V\) satisfying \(F=\overline F\circ\pi\) exists if and only if \(F\) is constant on each fiber of \(\pi\). \end{lemma} \begin{proof} A composition with \(\pi\) is constant on fibers. Conversely, define \(\overline F(\pi(d))=F(d)\). Fiber constancy makes this definition independent of \(d\). \end{proof} This criterion is also used in XIX-A and XIX-C \cite{XIXA,XIXC}. No predecessor proof is required here. \subsection{Pure-trace tests have an explicit blind sector} Use the flat signature \(\eta=\operatorname{diag}(1,-1,-1,-1)\) for this subsection. A compact three-slot kernel has two symmetric tensor slots and one scalar slot: \[ K^{\mu\nu,\rho\sigma}(x_1,x_2,x_3). \] Its pure-trace evaluation is its pairing with \[ (\eta_{\mu\nu}\psi_1(x_1)) \otimes(\eta_{\rho\sigma}\psi_2(x_2)) \otimes\phi(x_3). \] \begin{proposition}[Pure-trace data do not determine the smooth kernel or its free-shell functional] \label{prop:trace} There is a nonzero smooth compact kernel symmetric under exchange of its tensor slots, invisible to every pure-trace pair. It can have a nonzero evaluation on a declared free metric--scalar solution family. \end{proposition} \begin{proof} Let \(A^{12}=A^{21}=1\), with all other components zero. Then \(A^{\mu\nu}\eta_{\mu\nu}=0\). Choose nonnegative nonzero \(b,c\in C_c^\infty(\R^4)\), and set \[ K^{\mu\nu,\rho\sigma} =A^{\mu\nu}A^{\rho\sigma} b(x_1)b(x_2)c(x_3). \] Every pure-trace evaluation vanishes, whereas \(K\ne0\). For an explicit free-shell comparison, let each metric input have constant components \(h_{12}=h_{21}=1\) and all other components zero. These are smooth solutions of \(\Box h_{\mu\nu}=0\). Let \[ \phi(t,\mathbf x)=\cos(mt),\qquad m>0, \] so \((\Box+m^2)\phi=0\). Choose the time support of \(c\) where \(\cos(mt)>0\). The evaluation is \[ 4\left(\int b\,\dd^4x\right)^2 \int c(x)\cos(mt)\,\dd^4x>0. \] Thus zero and \(K\) also define different functionals on this specified free solution family. \end{proof} The free solution family in the proof is a comparison domain, not the complete VI solution-jet domain or the inherited XI-G class. The proposition does not assert nonuniqueness of a kernel constructed from a complete set of root inputs. \subsection{The same witness data do not decide microlocal admission} For a polynomial functional, fix the usual comparison admission condition that the wavefront set of its \(n\)-slot kernel avoid tuples whose covectors all lie in the same closed future or past causal cone, with the total zero tuple removed. This is a stated test domain for the following witness, not an admission theorem for a native clock. \begin{proposition}[Pure-trace equality does not decide the stated wavefront condition] \label{prop:microcausal} There is a compact distributional three-slot kernel, invisible to all pure-trace pairs, which violates the stated wavefront condition. Therefore identical pure-trace data can occur for both the zero kernel and a nonadmitted kernel. \end{proposition} \begin{proof} Retain \(A\) from Proposition~\ref{prop:trace}. Choose a time \(t_c\), nonzero compact spatial \(a\), a compact smooth \(b\) whose time support is disjoint from \(t_c\), and nonzero compact smooth \(c\). Define \[ d(x)=\delta(t-t_c)a(\mathbf x), \] and \[ K_{\mathrm{sing}}^{\mu\nu,\rho\sigma} =\frac12 A^{\mu\nu}A^{\rho\sigma} \bigl(d(x_1)b(x_2)+b(x_1)d(x_2)\bigr)c(x_3). \] Trace contractions vanish. At a point with \(t_1=t_c\), \(a(\mathbf x_1)\ne0\), \(b(x_2)\ne0\), and \(c(x_3)\ne0\), the second summand vanishes in a neighborhood because \(b\) is supported away from \(t_c\). The first summand has wavefront covectors \[ (\lambda\,\dd t_1,0,0),\qquad \lambda\ne0, \] with zero covectors in the other slots. For \(\lambda>0\) these lie in the product of closed future causal cones; for \(\lambda<0\) in the corresponding past product. The required exclusion fails. \end{proof} This is a comparison witness, not a calculation of the wavefront set of XVI-H's kernel. The supplied XVI-H record separately reports its complete-kernel and admission boundaries \cite{CheungXVIH}. \subsection{Finite tests and class restrictions} \begin{lemma}[Finite linear probes leave an infinite-dimensional blind sector] \label{lem:finite-tests} Let \(W\subset\R^d\) be nonempty and open, with \(d\ge1\). For any \(m\) linear functionals \(\ell_1,\ldots,\ell_m\) on \(C_c^\infty(W)\), there is a nonzero \(k\in C_c^\infty(W)\) with \(\ell_i(k)=0\) for every \(i\). \end{lemma} \begin{proof} Choose \(m+1\) nonzero smooth functions with pairwise disjoint compact supports in \(W\). Their span has dimension \(m+1\). The evaluation map from that span to \(\C^m\) has a nonzero kernel. \end{proof} Finite probes can be complete on a \emph{specified finite-dimensional class}; they cannot be treated as complete on all compact kernels without that restriction and a coverage theorem. The witness, class-kernel, and admission separations recorded in XVIII-A retain this quantifier \cite{CheungXVIIIA}. %===================================================================== \section{Counterterm images, finite cancellation, and projection} \label{sec:counterterms} %===================================================================== \subsection{Exact repair is an actual-image statement} \begin{definition}[Affine repair problem] \label{def:repair} Let \(\CT,\Def\) be complex vector spaces. A linear map \[ B:\CT\longrightarrow\Def \] is the change in a defined defect produced by an admissible counterterm. For a fixed defect \(D\in\Def\), the repair problem is \begin{equation} D+Bc=0,\qquad c\in\CT. \label{eq:repair} \end{equation} The obstruction class is \[ [D]\in\Def/\im B. \] The quotient is algebraic unless a different meaning is explicitly declared. \end{definition} \begin{proposition}[Exact affine cancellation criterion] \label{prop:repair} Equation~\eqref{eq:repair} has a solution if and only if \([D]=0\) in \(\Def/\im B\). If \(c_*\) is one solution, all solutions are \(c_*+\ker B\). For a linear probe map \(T:\Def\to\C^m\), the tested equation \[ T(D)+TBc=0 \] does not generally imply the full equation. It does imply the full equation if \(T\) is injective on \(\operatorname{span}\{D\}+\im B\). \end{proposition} \begin{proof} The repair equation is equivalent to \(-D\in\im B\). Since \(\im B\) is a vector space, this is equivalent to \(D\in\im B\), proving the first assertion. Subtracting two solutions proves the affine description. If the tested equation holds, \(D+Bc\in\ker T\). Injectivity on the displayed subspace forces \(D+Bc=0\). Without injectivity, take \(\Def=\C^2\), \(B=0\), \(D=(0,1)\), and \(T(x,y)=x\). The tested equation holds but the full one fails. \end{proof} No physical counterterm space or clock-specific map \(B\) is constructed by this proposition. Its application requires the complete admissible class, the actual defect, and the actual observable images. This is the affine-cancellation shape recorded in XVIII-A and the relative-lifting comparison of XV-B \cite{CheungXVIIIA,CheungXVB}. \subsection{Closure of the image is not exact membership} \begin{proposition}[Approximate and finite-probe cancellation need not be exact] \label{prop:dense-image} There is a continuous linear repair map with dense image for which every finite coordinate test can be cancelled exactly, while the full defect cannot be cancelled. \end{proposition} \begin{proof} Let \(\Def=\ell^2(\N)\), let \(\CT=c_{00}\) carry the inherited \(\ell^2\) norm, and let \(B\) be the inclusion. Its image is dense but not all of \(\ell^2\). Take \[ D=(1/n)_{n\ge1}. \] Then \(D\in\ell^2\setminus c_{00}\), so no \(c\in\CT\) satisfies \(D+Bc=0\). For the probe onto the first \(m\) coordinates, choose \[ c^{(m)}_n= \begin{cases} -1/n,&n\le m,\\ 0,&n>m. \end{cases} \] It cancels those probes exactly. Moreover, \[ \norm{D+Bc^{(m)}}_2^2=\sum_{n>m}n^{-2}\longrightarrow0. \] Neither fact puts \(-D\) in the actual image. \end{proof} A completion of the allowed counterterm class could change this example. That is a changed repair class, not exact cancellation within the original one. \subsection{Contact-supported changes cannot repair an off-contact witness} \begin{lemma}[Support separation] \label{lem:contact} Let all permitted corrections be distributions supported in a closed set \(S\). If a defect distribution \(D\) has a nonzero pairing with a test supported in the complement of \(S\), no permitted correction cancels \(D\). \end{lemma} \begin{proof} Every permitted correction pairs to zero with that test. The nonzero defect pairing therefore survives. \end{proof} A graph-specific application still has to establish the complete defect and the complete permitted support class. This lemma is not an evaluated obstruction for the inherited graph. \subsection{Projected identities and projected fixed points} \begin{proposition}[Projection leaves a precise unresolved component] \label{prop:projection} Let \(P:E\to E\) satisfy \(P^2=P\). For \(d\in E\), the equality \(Pd=0\) implies only \(d\in\ker P\). A projected fixed-point equation \[ P F(x)=Px \] implies only \(F(x)-x\in\ker P\), not \(F(x)=x\). \end{proposition} \begin{proof} Both assertions follow from the kernel definition. For a nonzero example, let \[ E=\C^2,\qquad P(x,y)=(x,0),\qquad F(x,y)=(x,y+1). \] Every point satisfies the projected equation, but none is a fixed point of \(F\). \end{proof} A projected one-loop identity, a full identity on a declared finite sector, and a fixed point of an RG map are therefore different assertions. No RG map or physical beta function is constructed by giving a subtraction scale a name. %===================================================================== \section{Retarded memory, formal order, and gauge closure} \label{sec:ward} %===================================================================== \subsection{Late source agreement does not determine retarded response} \begin{proposition}[A compact preparation difference can leave a late response] \label{prop:memory} There are two smooth sources agreeing identically after a fixed preparation interval whose equal-zero-data retarded responses differ at every later time. \end{proposition} \begin{proof} Fix \(0<\epsilon0,\quad M_1=\int_0^\epsilon s b(s)\,\dd s<\epsilon M_0. \] Hence \(d(t)>0\) for \(t\ge\epsilon\), although the sources agree there. \end{proof} The example is a homogeneous wave-mode comparison, not XIX-B's prepared source or coordinate calculation. It proves the general reason that a late plateau is not the same as equal forcing on the full causal past. A field-theoretic version must retain its actual operators, source domains, and Cauchy data. \subsection{Positive formal quantum order cannot remove a fixed classical coefficient} \begin{lemma}[Formal-order persistence] \label{lem:formal} Let \(E\) be a complex vector space and \(D_0\ne0\). For every \(D_+\in\hbar E[[\hbar]]\), \[ D_0+D_+\ne0 \qquad\text{in }E[[\hbar]]. \] The same conclusion holds at a fixed coupling coefficient \(\kappa^r\) in a doubly formal series. \end{lemma} \begin{proof} The coefficient of \(\hbar^0\) is \(D_0\), unchanged by \(D_+\). For the doubly formal statement, extract the coefficient of \(\kappa^r\hbar^0\). \end{proof} This is a coefficientwise statement. It makes no claim about numerical cancellation at one chosen nonzero value of \(\hbar\), or a singular regrading that introduces inverse powers of \(\hbar\). It applies to a fixed nonzero classical defect only after that defect and its domain have been established. The classical switch defect of VI, the direct coordinate mismatch of XIX-B, and the source-preserving upgrade obstruction reported for X-G are separate objects \cite{CheungVI,XIXB,CheungXG}. No identity among their witnesses is asserted. \subsection{Free closure is not interacting closure} \begin{proposition}[A free closed observable need not remain closed after a differential deformation] \label{prop:free-interacting} There is a graded algebra, a nilpotent free differential, and a nilpotent formal deformation for which a free closed element is not closed under the deformed differential. \end{proposition} \begin{proof} Let \[ \mathscr A=\C[x,y]\otimes\Lambda(c), \qquad \operatorname{gh}(x)=\operatorname{gh}(y)=0,\quad \operatorname{gh}(c)=1. \] Define a degree-one graded derivation \(s_\kappa\) by \[ s_\kappa x=c,\qquad s_\kappa y=\kappa c,\qquad s_\kappa c=0. \] It squares to zero on generators and hence on the whole algebra. At \(\kappa=0\), \(s_0y=0\); at positive formal coupling, \(s_\kappa y=\kappa c\ne0\). \end{proof} The example is algebraic, not a proposed gravitational BRST multiplet. It proves that an interacting identity requires the interacting differential and the corresponding observable completion. A free complex or a linear transversality identity does not supply them. \subsection{A linear transverse contact does not fix an interacting identity} Use the same flat \(\eta\) and define \[ D_{\mu\nu} =\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu, \qquad C_{\mu\nu\rho\sigma} =D_{\mu\nu}D_{\rho\sigma}\delta_0 \quad\text{on }\R^4. \] \begin{proposition}[Transversality leaves contact freedom] \label{prop:transverse} The nonzero contact distribution \(C\) is symmetric in each tensor pair, symmetric under pair exchange, and satisfies \[ \partial^\mu C_{\mu\nu\rho\sigma}=0, \qquad \partial^\rho C_{\mu\nu\rho\sigma}=0. \] It has scaling degree eight at the origin. Consequently, a defined linearly transverse response kernel \(\Pi\) and \(\Pi+aC\) have the same off-origin values and linear transversality for every \(a\in\C\), although their local contact normalizations differ. \end{proposition} \begin{proof} Constant-coefficient derivatives commute, and \[ \partial^\mu D_{\mu\nu} =\partial_\nu\Box-\Box\partial_\nu=0. \] This proves the transverse identities and the symmetries. The Fourier transform of \(C\) is a nonzero homogeneous degree-four tensor polynomial, so \(C\ne0\). A nonzero order-four derivative combination of \(\delta_0\) in four dimensions scales with degree \(4+4=8\). The remaining statements follow by linearity and the support of \(C\). \end{proof} In this one-parameter family, the Fourier polynomial vanishes to order four at zero momentum. Its jet through order three does not distinguish \(a\); an appropriate order-four jet does. This observation is not a complete stress-contact classification and is not transferred to a relational lapse. In particular, the equation \(k^\mu\widehat\Pi_{\mu\nu\rho\sigma}=0\) is linear transversality. It is not, without further construction, a nonlinear Slavnov--Taylor identity, a BV master equation, or Q5 for one of the labelled clocks. %===================================================================== \section{Finite instruments and continuum quantum non-determination} \label{sec:finite} %===================================================================== \subsection{Record agreement can conceal different residual channels} A finite instrument is a family of completely positive maps whose sum is trace preserving. Its record-only probabilities do not specify its selective residual maps. \begin{proposition}[Identical records for all inputs, different residuals] \label{prop:record} There are two complete recorded channels with identical classical record laws on every input but different residual outputs on one qubit input. \end{proposition} \begin{proof} Let \(P_0=\kb{0}{0}\), \(P_1=\kb{1}{1}\), and define \[ \mathcal D(\rho)=P_0\rho P_0+P_1\rho P_1. \] Use one record symbol \(*\) and the channels \[ \mathcal G(\rho)=\kb{*}{*}_R\otimes\rho,\qquad \mathcal F(\rho)=\kb{*}{*}_R\otimes\mathcal D(\rho). \] Both always report \(*\). For \(\rho=\kb{+}{+}\), where \(\ket{+}=(\ket{0}+\ket{1})/\sqrt2\), \[ \mathcal D(\rho)=I_2/2. \] The residual difference is \(X/2\), with eigenvalues \(1/2,-1/2\). Therefore \[ \dtrace(\mathcal G(\rho),\mathcal F(\rho)) =\frac12\norm{\mathcal G(\rho)-\mathcal F(\rho)}_1 =\frac12. \] \end{proof} This independently proves the record--residual shape reported in VIII-A \cite{CheungVIIIA}. Equality of effects is weaker than equality of complete instruments, including their residual ownership and reference correlations. Finite destructive-readout semantics and operational characterization results are separate from field-theoretic instrument construction \cite{CheungI,CheungEnter}. A correlator or formal coefficient is not automatically a positive effect or an apparatus coupling. \subsection{A token table does not acquire an unknown expectation} \begin{lemma}[Finite-token and fixed-copy refusals] \label{lem:token} \begin{enumerate}[label=\textup{(\roman*)}] \item A finite token with a fixed numerical decoder cannot represent exactly an expectation taking infinitely many values on its preparation class. \item For two positive definite density matrices, no measurement on a fixed finite number of copies can deterministically output different exact labels. \end{enumerate} \end{lemma} \begin{proof} A finite decoder has finite image, proving (i). For (ii), if \(\rho,\sigma>0\), then their \(n\)-fold tensor powers are positive definite. Every nonzero positive effect \(E\) has strictly positive pairing with both: \[ \tr(E\rho^{\otimes n})>0,\qquad \tr(E\sigma^{\otimes n})>0. \] Thus no label effect can have probability one for one input and zero for the other. \end{proof} Orthogonal promises, approximation, copy-dependent limits, and separately calibrated acquisition instruments are not excluded. They are extra constructions rather than consequences of a token replacement table. \subsection{Even exact subdivision does not select a unique evolution} The following strengthens the simplest finite-member non-determination pattern by imposing exact coarse agreement at every subdivision. \begin{proposition}[One coarse channel with two exactly consistent continuous-time completions] \label{prop:evolution} The identity qubit channel has two different continuous-time channel groups with the same time-one channel. Each group has exact subdivision channels whose \(n\)-fold composition equals the common coarse channel. The corresponding tick approximations converge uniformly on compact time intervals in diamond norm, but their limits differ. \end{proposition} \begin{proof} Let \[ H_0=0,\qquad H_1=2\pi\kb{1}{1}, \qquad U_a(t)=e^{-itH_a}, \qquad \mathcal T_t^a(\rho)=U_a(t)\rho U_a(t)^\dagger. \] Both families are continuous channel groups. Since \(U_0(1)=U_1(1)=I\), \[ \mathcal T_1^0=\mathcal T_1^1=\id. \] At refinement \(n\), use the tick channel \[ \Phi_n^a=\mathcal T_{1/n}^a. \] Then \((\Phi_n^a)^n=\id\) exactly. Define \[ \mathcal C_n^a(t) =(\Phi_n^a)^{\lfloor nt\rfloor} =\mathcal T_{\lfloor nt\rfloor/n}^a. \] For real \(s,t\), \[ \norm{U_a(t)-U_a(s)} \le\norm{H_a}\abs{t-s}. \] For an arbitrary reference and a trace-class input \(M\), expanding the difference of the two unitary conjugations gives a trace-norm bound \(2\norm{U_a(t)-U_a(s)}\norm M_1\). Hence \[ \norm{\mathcal C_n^a(t)-\mathcal T_t^a}_\diamond \le\frac{2\norm{H_a}}{n}, \] uniformly in \(t\). At \(t=1/2\), \(\mathcal T_{1/2}^0\) leaves \(\kb{+}{+}\) unchanged, whereas \(\mathcal T_{1/2}^1\) sends it to \(\kb{-}{-}\). The two outputs have trace distance one. The limiting evolutions therefore differ. \end{proof} These are elementary finite-dimensional comparison channels, not Hamiltonians or continuum realizations of the incidence package. They show that even exact root consistency at the coarse time does not uniquely select a generator. Additional spectral, calibration, locality, or interpolation restrictions may select one; those restrictions must be declared. Diamond-norm comparison preserves arbitrary references \cite{Watrous}. It does not by itself establish a physical metric, an invariant speed, or an autonomous geometry. \paragraph{Correct finite-register boundary.} It would be false to infer from this inventory that no quantum update of a finite register exists. Complete channels, permutations, and finite instruments are legitimate quantum operations. What is not supplied here is a uniquely selected continuum evolution of the native incidence construction, a field-theoretic source acquisition, or quantum evolution without external time. %===================================================================== \section{Object-by-object inventory} \label{sec:inventory} %===================================================================== \subsection{\texorpdfstring{\(\CV\)}{CVI}: harmonic relational scalar} The supplied VI record distinguishes classical second-order descent from its uncompleted quantum finite-renormalization certificate \cite{CheungVI}. The missing native constructions include: \begin{enumerate}[label=\textup{(\arabic*)}] \item an interacting BRST/BV or other explicitly equivalent gauge prescription on the declared observable domain; \item renormalized products containing the actual coordinate and scalar-response kernels; \item complete action, composite, boundary, and switch counterterm classes at the selected quantum order; \item their defined images in the observable's complete defect space; \item a specified state or radiative response and an evaluated quantum coefficient; \item the complete target vector and response matrix for any claimed finite radiative certificate; \item an actual rank, fit, separator, or gauge-repair conclusion. \end{enumerate} No such native response matrix or radiative calculation is supplied by this paper. Proposition~\ref{prop:repair} is only the criterion that would apply after those objects had been constructed. The classical compact-switch defect concerns a specified switched assignment on unchanged free inputs. It is not a proof that the unswitched coordinate Cauchy problem is obstructed. The direct prepared-coordinate mismatch of XIX-B is another classical comparison, not a loop anomaly \cite{XIXB}. Neither closes the RG-open certificate. If a retained classical coefficient is nonzero, Lemma~\ref{lem:formal} states why a positive-\(\hbar\) correction cannot erase it while that classical prescription is held fixed. \subsection{\texorpdfstring{\(\CE\)}{CE}: the original-rod relational lapse} Retain the original inflaton-slice harmonic rods and the fixed clock calibration of the supplied Em02-E record \cite{CheungEm02E}. The classical composition \[ \NN[q,T]=N[q,T]\circ\alpha_{q,T} \] and its second variation remain classical input data. Modified rods cannot be substituted silently; the rod-dependent quadratic distinction is recorded in XIX-A \cite{XIXA}. All five audit slots remain unsupplied for this native target: \begin{enumerate}[label=\textup{(\roman*)}] \item \textbf{Q1:} no gauge fermion, complete coupled metric--scalar--ghost operator, operator domain, or normalized covariance matrix is constructed here; \item \textbf{Q2:} no state or suitable parametrix with the required singular and spatial-tail properties is selected here; \item \textbf{Q3:} no complete Wick/time-ordered realization, admissible graph-counterterm classification, or spatial image of those corrections is calculated here; \item \textbf{Q4:} no complete quantum remainder or regulator-removal theorem is proved here; \item \textbf{Q5:} no retained-order quantum breaking term or admissible repair is evaluated here. \end{enumerate} A useful schematic target would be the contraction of the classical quadratic lapse Hessian against the appropriate coupled two-point matrix, together with every other contribution at the same retained order. Equation~\eqref{eq:prospective} does not define that target until its distributions and operations have been supplied. The background interval is kept strictly away from \(t=0\). No limit toward \(t=0\), de Sitter state, or quantum gravitational evolution on that cosmological background is inferred. \subsection{\texorpdfstring{\(\CI\)}{CVIII}: incidence parameter and finite register} The supplied VIII-A/B/C records concern declared packets, matching, residual interfaces, promised source tokens, and guarded composition \cite{CheungVIIIA,CheungVIIIB,CheungVIIIC}. During incidence the packet remains frozen; regular source replacement is a separate command. To claim a continuum quantum interpretation, a successor must add at least: \begin{enumerate}[label=\textup{(\arabic*)}] \item a refinement family, time calibration, common state and observable comparison interfaces; \item complete source-acquisition instruments when the source is not an externally supplied promise; \item controlled convergence of complete channels or observables, including their reference extensions; \item a physical causal realization if scheduled support is to be interpreted as physical causality; \item a matter--geometry law if the cone is to evolve dynamically; \item frame-covariance data if an invariant physical speed is claimed. \end{enumerate} None is constructed for the native incidence package here. The record and token refusals, and Proposition~\ref{prop:evolution}, show why finite data alone do not select them. There is no field regulator family to remove by renaming a register refinement. A finite update can be a quantum channel; it is not thereby a continuum clock or a field-theoretic instrument. \subsection{\texorpdfstring{\(\RX\)}{RX}: the prescribed ramp} The ramp remains a prescribed non-clock. Scalar quantum fields coupled to classical ramp-dressed backgrounds, classical metric trees, and free gauge multiplets are different field-content choices. The supplied X-B/X-C/X-D records report scalar-only and fixed-background constructions on their stated domains \cite{CheungXB,CheungXC,CheungXD}. The supplied X-E/X-F records report selected tree constructions and \(\kappa^2\hbar^0\) identities on their own gauges and domains \cite{CheungXE,CheungXF}. Those positive statements are not denied. They do not make the profile a quantum clock or close the native VI or Em02-E audit. X-G's stated source-preserving variational-upgrade obstruction has its own quantifier, and its complete nonlocal one-loop target remains separate \cite{CheungXG}. Introducing a dynamical profile field, an independently quantized metric, or a different switching prescription is a changed quantum target. No such upgrade of \(\RX\) is constructed here. %===================================================================== \section{Benchmark shapes and the scope of reported quantum checks} \label{sec:benchmarks} %===================================================================== \subsection{XVIII-B: a free stress-response benchmark, not a clock certificate} The supplied XVIII-B, version 2, record describes one free massive scalar stress response on classical Minkowski spacetime: spin-two and spin-zero vacuum spectral densities, its stated once-subtracted retarded prescription, a scaling-degree bound at most eight, transverse contact normalization through the specified momentum jet, linear transversality, and a preparation-layer residual vanishing on the stated plateau \cite{CheungXVIIIB}. This paragraph reports that benchmark's scope. No spectral density, dispersion integral, subtraction convention, or normalization coefficient is recomputed in the present paper. In particular, the number and location of subtractions must be read within the benchmark's actual form-factor and tensor conventions, not assigned to an unidentified relational kernel. The comparison is useful because it names: \[ \text{field and state} \;+\; \text{retarded prescription} \;+\; \text{contact class} \;+\; \text{identity scope}. \] It is not a construction of the companion harmonic scalar, original-rod lapse, interacting gravitational product, or inherited characteristic coefficient. The input Minkowski metric is not derived. The stress response is neither Fr\"ob's relational correlator nor VII-H's free massless susceptibility \cite{Frob,CheungVIIH}. Choosing \(m>0\) in this benchmark does not discharge the conditional massless insertion obstruction of XVI-N or its spatially compact Route B \cite{CheungXVIN}. \subsection{XVI-Q: the scoped two-point record} The supplied Q11 reading record makes a specific report \cite[\sid{rem:o1}]{CheungXVIQ}. Its scope is: \begin{enumerate}[label=\textup{(\roman*)}] \item one loop and order \(\kappa^2\); \item the bulk two-point sector of \(S_{\mathrm{inv}}\); \item zero mass and gauge fermion held fixed; \item QdS external-leg conventions; \item XVI-P's prescription, with the enlarged evanescent counterterm domain, not XVI-O's prescription. \end{enumerate} The reported counts must not be conflated: \[ \boxed{ \begin{gathered} 11\text{ four-dimensional directions over }\mathbb Q(i),\\ 10\text{ p22 evanescent operators},\\ 21\text{ counterterm directions in total},\\ 23\text{ bulk-identity rows}. \end{gathered}} \] The supplied report states cancellation of those 23 rows and full \(\sigma_0\) consistency on the bulk 11 directions. Twenty-three is a row count, not the dimension of the counterterm domain. Q02 is reported to have confirmed the scoped closure through a rebuild; Q03 records \sid{O1_closure_confirmed}. The same-computer qualification remains. D01 and D03 were not rederived from fully independent Feynman rules in those reviews; the source reports cross-checks through the \(c\cdot h\) defect, and later agreement of Q05 rules with the stated two-point poles. The author rebuild and pinned package gates are not an independent full derivation or a series-wide certificate. We neither rerun nor authenticate those calculations. The documentary conclusion retained here is \[ \text{reported scoped closure} \quad\text{for the above two-point sector only}. \] It is not a VI or lapse certificate, bulk BV restoration, an insertion completion, a higher-loop theorem, or ultraviolet completeness. The supplied Q11 reading also distinguishes the partial \(\kappa^3\) three-point analysis: 43 of 104 directions remain unclassified, no direction is certified physical, and the evanescent operator identification and consistency questions are still open \cite[\sid{rem:p6},\sid{rem:classification}]{CheungXVIQ}. This inventory does not reopen or resolve that calculation. QdS is an external-leg scheme, not a selected de Sitter state, not \(\NN\), and not horizon entropy \cite[\sid{rem:qds}]{CheungXVIQ}. The Q11 prescription does not update XVI-O's different prescription or its separately recorded status \cite{CheungXVIO,CheungXVIP}. Outside the specified Q11 scope, \sid{O.1 = NOT_CLOSED} remains the standing source record. \subsection{Why a scoped identity does not transfer to an insertion} \begin{lemma}[Transport of a defined identity requires a matching map] \label{lem:transport} Let \(s_1,s_2\) be linear gauge operations on declared spaces \(A_1,A_2\), and let \(I:A_1\to A_2\). If \[ s_2I=Is_1,\qquad O_2=IO_1,\qquad s_1O_1=0, \] then \(s_2O_2=0\). For affine repair maps \(B_i:C_i\to V_i\), maps \(J:V_1\to V_2\) and \(L:C_1\to C_2\) transport a specified repair if \[ D_2=JD_1,\qquad JB_1=B_2L. \] No completeness of the target correction class follows merely from that transport. \end{lemma} \begin{proof} The first conclusion is \(s_2O_2=s_2IO_1=Is_1O_1=0\). For a source repair \(D_1+B_1c=0\), \[ D_2+B_2Lc=J(D_1+B_1c)=0. \] The hypotheses identify one transported family only. They do not assert that \(L\) covers every target correction. \end{proof} A physical comparison also has to match products, domains, normalization, state-dependent data when relevant, and the observable itself. Lemma~\ref{lem:transport} is an elementary conditional interface, not a supplied map from \(S_{\mathrm{inv}}\) to a clock insertion. \subsection{Oai282-B, Oai069-A, and the OpenAI catalogue} The supplied Oai282-B record describes a free local conformal Ward construction for a massless scalar in a declared homogeneous Minkowski class, including a two-dimensional residual family and finite-probe coverage within that class \cite{CheungOai282B}. Coverage inside a finite class does not cover unrestricted conserved tensors. Local Ward action and integration to a global group action remain distinct. The supplied Oai069-A record describes a finite free-scalar Ward-defect comparison and a categorical/local-system recovery benchmark \cite{CheungOai069A}. A category of local systems is not a density-matrix state space or a clock channel merely because finite labels occur in both. Neither comparison sheet is used as a theorem premise. Their residual families are not copied into \(\CT\) for VI, the original-rod lapse, or the incidence register. No gravitational dictionary, RG fixed point, or family verification is imported from them. \begin{proposition}[Stress constraints alone do not imply an Einstein equation] \label{prop:einstein} Symmetry, conservation, and vanishing trace of a tensor do not, without a metric--matter equation, imply \(G_{\mu\nu}=8\pi G_{\mathrm N}T_{\mu\nu}\). \end{proposition} \begin{proof} On any Lorentzian geometry take \(T_{\mu\nu}=0\). It is symmetric, conserved, and trace free. Choose a geometry whose Einstein tensor is nonzero, for example a four-dimensional constant-curvature geometry with \(R_{\mu\nu}=3H^2g_{\mu\nu}\), \(H\ne0\). Then \[ G_{\mu\nu} =3H^2g_{\mu\nu}-\frac12(12H^2)g_{\mu\nu} =-3H^2g_{\mu\nu}\ne0. \] The zero-cosmological-constant Einstein equation with that \(T_{\mu\nu}\) therefore fails. \end{proof} This is a kinematic counterexample to a logical implication. It is not a construction of a de Sitter state or a solution of the native Em02-E quantum problem. The supplied OpenAI Family 282 catalogue has its own author attribution, hypotheses, and local/global scope \cite{Catalogue}. Its status is documentary here. No independent family verification or gravitational transfer is asserted. %===================================================================== \section{The inherited characteristic map remains a separate gate} \label{sec:graph} %===================================================================== \subsection{Reconstruction descent is necessary but not admission} \begin{proposition}[Descent through a reconstruction map] \label{prop:descent} Let \(\Rec:P\to K\) be a surjective linear reconstruction map. A linear candidate \(F:P\to V\) has the form \(F=M\Rec\) for a map \(M:K\to V\) if and only if \[ \ker\Rec\subseteq\ker F. \] When it exists, \(M\) is unique. \end{proposition} \begin{proof} If \(F=M\Rec\), every reconstruction-null element has zero image. Conversely, equal reconstructions have a difference in \(\ker\Rec\). The kernel condition therefore makes \(M(\Rec p)=F(p)\) well defined. Surjectivity gives uniqueness. \end{proof} This is the reconstruction compatibility shape in XIV-J and XV-B \cite{CheungXIVJ,CheungXVB}. It creates no missing distributional product. \begin{lemma}[A normal null identity mixes presentation degrees] \label{lem:null} For any nonzero distribution \(R\), \[ q\delta'(q)\otimes R+\delta(q)\otimes R=0. \] The two summands are separately nonzero. \end{lemma} \begin{proof} For a test \(\psi\), \[ \pair{q\delta'}{\psi} =-\left.\frac{\dd}{\dd q}(q\psi(q))\right|_{q=0} =-\psi(0). \] Hence \(q\delta'=-\delta\). Tensor with \(R\); nonvanishing of \(\delta\otimes R\) follows by choosing separated tests with nonzero pairings. \end{proof} A highest polynomial degree of a chosen amplitude presentation is therefore not automatically an intrinsic symbol of the reconstructed kernel. This does not invalidate estimates on a named presentation module. It prevents promoting a splitting to a reconstruction-independent joint symbol without proving descent and the relevant calculus. \subsection{What the partial references do not supply} The supplied analytic records distinguish: \begin{enumerate}[label=\textup{(\roman*)}] \item exact internal tree maps and regular charts; \item the internal amplitude construction reported at I2; \item the common precomposition sum \[ K_{\mathrm{ret}}=K_{47}+K_5+K_6+K_8+K_9; \] \item a supported-sector reference for \(K_6+K_8+K_9\) away from the characteristic locus; \item the missing complete characteristic operation on the full retained sum. \end{enumerate} These are reported scopes, not newly reconstructed kernels \cite{CheungXIVH,CheungXIVI,CheungXIVJ}. A reference for the supported sector does not define the characteristic compositions of \(K_{47}\) or either placement inside \(K_5\). A conditional coefficient-extension equivalence does not select extension coefficients. The remaining complete-target obligations are: \[ \begin{gathered} \text{full joint characteristic map},\\ \text{joint section or specified prescription and complete remainder},\\ \text{uniform transverse estimates},\\ \text{contact compatibility},\\ \text{output wavefront and normalization theorem},\\ \text{admission in the stated output space}. \end{gathered} \] No step is supplied by the present inventory. The notation \[ Q_HR_1^{(1)}\in\mathscr E_H \] is retained only as the source's unasserted admission target. It is not an equality or membership claim established here. No gauge differential is applied to an undefined inherited coefficient, and no anomaly is calculated from it. \subsection{The other product and ramp tracks do not fill this gate} The IX records separate classical sharp-insertion carriers, partial free products, selected wordwise associativity, and time-ordered or causal retarded extensions \cite{CheungIXA,CheungIXC,CheungIXFree, CheungIXFreeTwo,CheungIXD,CheungIXE}. The later IX-Free-2 relative partial-product result is not denied by retaining the earlier IX-Free stop. Conversely, a selected word through \(\hbar^3\) does not release sharp time ordering or the complete product domain of another target. The X records separate scalar-only fixed-background maps, tree-level quantum-metric constructions, and the free graded multiplet from the specified interacting upgrade and loop obligations \cite{CheungXA,CheungXB,CheungXC, CheungXE,CheungXF,CheungXG}. An order-\(\kappa^2\hbar^0\) tree identity is not an order-\(\hbar\) loop identity. The technical XV-B manuscript exists and proves its stated lifting and extension interfaces \cite{CheungXVB}. Saying that the downstream quantum-admission exit is not opened does not deny the existence of that manuscript. It means the missing complete coefficient and its admission are not supplied by this release. %===================================================================== \section{Main nontransfer theorem and exact inventory release} \label{sec:main} %===================================================================== \begin{theorem}[Quantum-certificate nontransfer and non-determination] \label{thm:main} On the explicit comparison domains defined in this paper, the following implications are invalid without additional hypotheses: \begin{enumerate}[label=\textup{(\roman*)}] \item classical field equations, causal commutator data, and a prescribed wavefront condition imply a selected positive state or a completed composite contraction; \item pure-trace or finitely many kernel evaluations imply a unique full kernel, its free-shell functional, or its microlocal admission; \item late source equality implies equality of retarded responses; \item finite-probe cancellation, image-closure membership, or a projected identity implies exact repair in the actual counterterm class; \item a positive formal quantum correction cancels a fixed nonzero classical coefficient; \item free gauge closure implies interacting closure; \item identical classical record laws or exact subdivision of one finite channel determines complete residual instruments or a unique continuous-time evolution. \end{enumerate} Exact affine repair is instead the condition \([D]=0\) in \(\Def/\im B\). A defined identity transports by the matching equations of Lemma~\ref{lem:transport}; no such map is furnished between the native clocks or by the reported external benchmarks. These conclusions are separation results, not universal nonexistence theorems for the native quantum targets. \end{theorem} \begin{proof} For (i), Proposition~\ref{prop:state} gives both covariance nonuniqueness and a failure of positivity with unchanged algebraic and singularity data. Proposition~\ref{prop:product} shows why separate factor definitions do not give a raw product. For (ii), Propositions~\ref{prop:trace} and \ref{prop:microcausal}, together with Lemma~\ref{lem:finite-tests}, give explicit blind sectors. Proposition~\ref{prop:memory} proves (iii). Propositions~\ref{prop:repair}, \ref{prop:dense-image}, and \ref{prop:projection} prove (iv). Lemma~\ref{lem:formal} proves (v). Proposition~\ref{prop:free-interacting} proves (vi). Proposition~\ref{prop:record} and Proposition~\ref{prop:evolution} prove (vii). The exact repair criterion and conditional transport statement are proved in Proposition~\ref{prop:repair} and Lemma~\ref{lem:transport}. No predecessor quantum calculation is used in these proofs. \end{proof} \subsection{The release is an inventory, not a newly earned certificate} \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.17\linewidth}YY@{}} \toprule Entry & Missing native construction retained & Ending here\\ \midrule \(\CV\) & Interacting gauge prescription, coordinate-kernel products, complete counterterm images, state-dependent radiative target, response matrix, and quantum gauge certificate & RG-open in the supplied VI record; no native radiative calculation performed\\[4pt] \(\CE\) & Q1 covariance, Q2 state or parametrix, Q3 product and counterterm class, Q4 complete remainder or removal limit, Q5 retained-order quantum gauge identity & All five slots remain unsupplied by this manuscript; original rods and interval away from \(t=0\) retained\\[4pt] \(\CI\) & Native refinement and continuum realization, source acquisition, regulator-removal target where relevant, physical causal embedding, autonomous geometry and invariant-speed dictionary & Finite quantum updates are not denied; no native continuum or clock-defined quantum evolution is constructed\\[4pt] \(\RX\) & Any newly declared dynamical or quantum profile target, with its own action, state, products, and gauge law & Prescribed non-clock retained; no quantum-clock upgrade\\ \bottomrule \end{tabularx} \end{center} \paragraph{Other scopes are not changed.} The free benchmarks are not invalidated. The reported Q11 two-point check remains reported within its 21-direction, 23-row prescription and stated verification limits. It is not adopted as Q5 for any entry in the table. The sheet-specific H2 stop, later internal I2 advance, remaining I3/J2 full-map boundaries, and technical XV-B interface results retain their respective scopes. None is closed or promoted by this inventory. \[ \boxed{\text{Declared path: null.}} \] This is a relationship convention: no comparison label makes Em02-E, VI, XVI-Q, the incidence papers, or the OpenAI catalogue a theorem premise of another native quantum construction. It is not a claim that every future connecting map must vanish. \subsection{Standing status} \[ \begin{gathered} \text{VI quantum certificate: RG-open in the supplied record},\\ \text{Em02-E quantum gauge identity: not verified here},\\ \text{native incidence continuum limit: not constructed here},\\ \sid{O.1 = NOT_CLOSED} \quad\text{outside the separately reported Q11 scope},\\ \text{XVI-N and spatially compact Route B: not discharged},\\ \text{downstream XV-B quantum-admission exit: not opened},\\ \text{inherited characteristic quantum coefficient: not admitted},\\ \text{quantum evolution without external time: not constructed}. \end{gathered} \] These source-status comparisons retain their supplied qualifiers \cite{CheungVI,CheungEm02E,CheungXVIQ, CheungXVIN,CheungXIVJ,CheungXVB,CheungVIIIC}. They are not new nonexistence claims. %===================================================================== \section{Successor contract and conclusion} \label{sec:conclusion} %===================================================================== A successor must first select one target rather than upgrading the word ``clock.'' For a field-theoretic target it must identify the actual classical observable, quantized fields, gauge prescription, domains, contraction matrix, insertion products, and retained orders. It must then construct the operations in Q1--Q5 for that target, or explicitly restrict the requested conclusion so that a slot is not needed. A physical expectation needs a state; a purely algebraic identity need not claim one. A continuum register interpretation needs its refinement and complete-channel comparison, not a graviton propagator assigned by analogy. A claim of exact counterterm cancellation must use the actual admissible image. A claim based on probes must prove coverage of the declared class. A claim of gauge closure must concern the same complete observable, differential, and retained order. The central conclusion is therefore: \[ \boxed{ \begin{gathered} \text{classical separation is not quantum certification},\\ \text{a free response benchmark is not a relational-clock product},\\ \text{a scoped bulk identity is not an insertion identity},\\ \text{a finite record is not a continuum quantum clock},\\ \text{the quantum slots remain unclosed in this release}. \end{gathered}} \] No later ultraviolet or three-point calculation is opened. No new action, state, graviton covariance, native quantum coefficient, field instrument, or compilation is supplied. The contribution is the precise inventory and the proved reasons that the proposed shortcuts do not fill it. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} %===================================================================== \paragraph{Mathematical scope.} The new proofs are conditional distributional comparisons, linear-algebraic repair criteria, formal-order statements, and explicit finite-dimensional counterexamples. They construct no native gravitational quantum coefficient. Elementary comparison channels and contact distributions are not released as clock quantizations. \paragraph{Source scope.} The cited unpublished versions and supplied excerpts provide documentary scope and motivation. Their unprinted proofs, artifact outputs, archive hashes, and rebuild reports are not premises of Theorem~\ref{thm:main}. No exhaustive scan of the manuscript series is claimed. \paragraph{Attribution.} The comparison manuscripts bearing Oai identifiers are cited under their supplied N.~Cheung attribution. The separate catalogue is cited under its supplied OpenAI attribution. No identity of those authorship records or independent catalogue authentication is asserted. \paragraph{Bibliographic qualification.} The Enter-calculus journal metadata are reproduced from the supplied accepted-article record. Independent authentication of that publisher record is not claimed here. A signed submission should check its final publication metadata and provide stable copies of all cited unpublished versions. \paragraph{Verification.} The source contains one internal bibliography and uses standard pdfLaTeX packages without external figures or input files. No successful pdfLaTeX run, proof-assistant verification, independent symbolic or numerical execution, native program compilation, referee decision, or acceptance is asserted. \paragraph{Authorship.} The submission format is anonymous. Citation of N.~Cheung does not establish authorship of this manuscript. A signed submission must supply authorized author and affiliation metadata. \paragraph{Unchanged boundaries.} No quantum evolution without an external time, invariant physical speed, derived metric, Einstein dynamics, de Sitter state, ultraviolet completion, or inherited characteristic quantum admission is concluded. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} %===================================================================== \bibitem{XIXA} \emph{XIX-A: Clock Separation Before Coordinate Identification: Four Labelled Objects and Five Non-Identifications}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-A.tex}, n.d. Classical comparison only; no quantum certificate is inferred. \bibitem{XIXB} \emph{XIX-B: A Witness Mismatch Between Two Coordinate Cauchy Prescriptions: Unswitched Harmonic Coordinates and Prepared Coordinate Completion}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-B.tex}, n.d. The declared direct first-order mismatch is not a quantum anomaly or a premise of the present proofs. \bibitem{XIXC} \emph{XIX-C: Finite Incidence Registers and the Missing Continuum Evolution: Matching, Records, Promised Tokens, and Scheduled Support}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-C.tex}, n.d. Finite quantum channels and native continuum non-determination remain distinct. \bibitem{CheungVI} N.~Cheung, \emph{Relational Einstein--scalar observables at second order: classical descent and a compact-switch obstruction}, Landmark VI, version 1, supplied unpublished manuscript, \sid{VI.tex}, n.d. Enter Calculus Companion. Relevant documentary section: ``Static audit of the proposed quantum certificate.'' The quantum finite-renormalization certificate remains RG-open in the supplied record. \bibitem{CheungEm02E} N.~Cheung, \emph{Em02-E}, version 1, supplied unpublished manuscript, \sid{Em02-E-v1.tex}, n.d. Enter Calculus Companion. Relevant locators: \sid{thm:lapse}, \sid{thm:release}, and ``The quantum input gate and the first contraction target.'' Original inflaton-slice rods and the 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. Enter Calculus Companion. Witness, response, kernel, normalization, fixed-point, and formal-order comparison interfaces. \bibitem{CheungXVIIIB} N.~Cheung, \emph{Landmark XVIII-B}, version 2, supplied unpublished manuscript, \sid{XVIII-B-v2.tex}, n.d. Enter Calculus Companion. Free massive scalar stress-response benchmark; relevant supplied locator: \sid{thm:spectral}. The local benchmark is not a native relational-clock certificate. \bibitem{CheungVIIIA} N.~Cheung, \emph{Landmark VIII-A}, version 3, supplied unpublished manuscript, \sid{VIII-A-v3.tex}, n.d. Enter Calculus Companion. Relevant supplied locator: \sid{A:thm:residual}. Record agreement and complete residual-instrument agreement remain distinct. \bibitem{CheungVIIIB} N.~Cheung, \emph{Landmark VIII-B}, version 3, supplied unpublished manuscript, \sid{VIII-B-v3.tex}, n.d. Enter Calculus Companion. Finite source, packet, token, and parameter--lapse non-identification interfaces. \bibitem{CheungVIIIC} N.~Cheung, \emph{Landmark VIII-C}, version 3, supplied unpublished manuscript, \sid{VIII-C-v3.tex}, n.d. Enter Calculus Companion. Frozen-packet incidence, regular source replacement, complete-output scope, and missing mixed-germ, acquisition, physical-embedding, and dynamics interfaces. \bibitem{CheungI} N.~Cheung, \emph{Landmark I}, version 2, supplied unpublished manuscript, \sid{I.tex}, n.d. Enter Calculus Companion. Finite measurement semantics and structural characterization; not a field-theoretic instrument construction. \bibitem{CheungEnter} N.~Cheung, ``The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,'' \emph{Frontiers in Physics} \textbf{14} (2026), article 1954943. DOI: \sid{10.3389/fphy.2026.1954943}. Metadata as supplied in the accepted-article record, accepted 28 September 2026. Finite operational context only; publisher authentication is not claimed by this manuscript. \bibitem{CheungVIIH} N.~Cheung, \emph{Landmark VII-H}, version 1, supplied unpublished manuscript, \sid{VII-H.tex}, n.d. Enter Calculus Companion. Free massless scalar susceptibility, retarded prescription, contact ambiguity, and invariant-test benchmark. Not the massive stress kernel or the inherited characteristic graph. \bibitem{CheungXVIQ} N.~Cheung, \emph{XVI-Q}, release Q11, supplied unpublished manuscript, \sid{XVI-Q.tex}, n.d. Enter Calculus Companion. Relevant supplied locators: \sid{rem:o1}, \sid{rem:qds}, \sid{rem:p6}, \sid{rem:classification}, \sid{rem:bulk}, \sid{rem:insertion}, and \sid{rem:package}. Scoped one-loop bulk two-point report and its explicit verification and nontransfer qualifications. \bibitem{CheungXVIP} N.~Cheung, \emph{XVI-P}, version p23, supplied unpublished manuscript, \sid{XVI-P.tex}, n.d. Enter Calculus Companion. Declared two-point prescription, ghost-orientation correction, external-leg scheme, and separately scoped identity flags. \bibitem{CheungXVIO} N.~Cheung, \emph{XVI-O}, version o70-sync, supplied unpublished manuscript, \sid{XVI-O.tex}, n.d. Enter Calculus Companion. Kernel two-point Ward subsector under its own prescription; not the enlarged Q11 prescription or bulk BV restoration. \bibitem{CheungXVIN} N.~Cheung, \emph{XVI-N}, version r12, supplied unpublished manuscript, \sid{XVI-N.tex}, n.d. Enter Calculus Companion. Relevant supplied locator: \sid{thm:massless-obstruction}. Conditional massless special-family insertion obstruction; spatially compact Route B is not discharged. \bibitem{CheungXVIH} N.~Cheung, \emph{Landmark XVI-H: Compact polarized kernel and witness recovery}, version 1, supplied unpublished manuscript, \sid{XVI-H.tex}, n.d. Enter Calculus Companion. Compact kernel and partial witness recovery under stated root inputs; admission and prepared quantum-product boundaries remain distinct. \bibitem{CheungOai282B} N.~Cheung, \emph{Oai282-B}, version 1, supplied unpublished manuscript, \sid{Oai282-B.tex}, n.d. Enter Calculus Companion. Declared homogeneous free-scalar local Ward class, finite coverage, and local/global nontransfer comparisons. Not an OpenAI family verification. \bibitem{CheungOai069A} N.~Cheung, \emph{Oai069-A}, version 1, supplied unpublished manuscript, \sid{Oai069-A.tex}, n.d. Enter Calculus Companion. Finite Ward-defect comparison and finite-monodromy/local-system benchmark; relevant supplied locator: \sid{thm:category}. No gravitational or positive-channel dictionary is imported. \bibitem{Catalogue} \emph{OpenAI catalogue and companion scope records}, attributed to OpenAI in the supplied documentary record, including Families 271, 282, and 069, record dated 7 October 2026. Enter Calculus Companion. Family 271's reported Lean scope names 271-04 only; VI is not 271-04. Attribution, proof validity, and Lean execution are not independently authenticated here. \bibitem{CheungXIVH} N.~Cheung, \emph{Landmark XIV-H: Internal kernel relations on the operator route}, version 1, supplied unpublished manuscript, \sid{XIV-H.tex}, n.d. Relevant supplied locators: \sid{prop:K5exact}, \sid{thm:H2}, and \sid{thm:release}. Sheet-specific internal-amplitude stop. \bibitem{CheungXIVI} N.~Cheung, \emph{Landmark XIV-I}, version 1, supplied unpublished manuscript, \sid{XIV-I.tex}, n.d. Enter Calculus Companion. Relevant supplied locators: \sid{thm:I2}, \sid{thm:I3}, and \sid{thm:release}. Internal amplitude construction and remaining joint-map boundary. \bibitem{CheungXIVJ} N.~Cheung, \emph{Landmark XIV-J}, version 1, supplied unpublished manuscript, \sid{XIV-J.tex}, n.d. Enter Calculus Companion. Relevant supplied locators: \sid{prop:descent}, \sid{lem:null}, \sid{thm:reference}, \sid{thm:J2}, and \sid{cor:noadmission}. Supported-sector reference and full-characteristic-map separation. \bibitem{CheungXVB} N.~Cheung, \emph{XV-B}, version 2, supplied unpublished manuscript, \sid{XV-B.tex}, n.d. Enter Calculus Companion. Relative lifting, finite normal jets, and extension interfaces. The inherited complete map and downstream quantum admission are not supplied by those interfaces alone. \bibitem{CheungIXA} N.~Cheung, \emph{Landmark IX-A}, version 1, supplied unpublished manuscript, \sid{IX-A.tex}, n.d. Enter Calculus Companion. Classical sharp-insertion carrier and topology; not a completed quantum deformation. \bibitem{CheungIXC} N.~Cheung, \emph{Landmark IX-C}, version 1, supplied unpublished manuscript, \sid{IX-C.tex}, n.d. Enter Calculus Companion. One-word binary-coherence reductions and necessary tests; not completed binary factorization. \bibitem{CheungIXFree} N.~Cheung, \emph{IX-Free}, version 1, supplied unpublished manuscript, \sid{IX-Free.tex}, n.d. Enter Calculus Companion. Version-qualified partial results and its stated unresolved descent and overlap boundary. \bibitem{CheungIXFreeTwo} N.~Cheung, \emph{IX-Free-2}, version 1, supplied unpublished manuscript, \sid{IX-Free-2.tex}, n.d. Enter Calculus Companion. One relative truncated partial product on its declared tensor domain and coherence of a selected word through \(\hbar^3\); not time ordering or a complete quantum algebra. \bibitem{CheungIXD} N.~Cheung, \emph{Landmark IX-D}, version 3, supplied unpublished manuscript, \sid{IX-D-v3.tex}, n.d. Enter Calculus Companion. Partial free coherence, time-ordered extension, and normalized retarded maps remain distinct. \bibitem{CheungIXE} N.~Cheung, \emph{Landmark IX-E}, version 1, supplied unpublished manuscript, \sid{IX-E.tex}, n.d. Enter Calculus Companion. Relevant supplied locator: \sid{thm:obstruction}. Fixed-domain conditional causal-factorization obstruction; not a universal clock-product obstruction. \bibitem{CheungXA} N.~Cheung, \emph{X-A}, version 1, supplied unpublished manuscript, \sid{X-A.tex}, n.d. Enter Calculus Companion. Density ramp and its separately stated classical covariance and prerequisite interfaces. \bibitem{CheungXB} N.~Cheung, \emph{X-B}, version 3, supplied unpublished manuscript, \sid{X-B-v3.tex}, n.d. Enter Calculus Companion. Scalar-only quantum field content with classical metric backgrounds; a quantized-metric branch is separate. \bibitem{CheungXC} N.~Cheung, \emph{X-C}, version 3, supplied unpublished manuscript, \sid{X-C-v3.tex}, n.d. Enter Calculus Companion. Fixed-width scalar response maps through second coupling order on the declared off-shell linear tests. \bibitem{CheungXD} N.~Cheung, \emph{X-D}, version 3, supplied unpublished manuscript, \sid{X-D-v3.tex}, n.d. Enter Calculus Companion. Coefficient-test limit at fixed input in its stated quotient topology; not joint sharp quantum admission. \bibitem{CheungXE} N.~Cheung, \emph{X-E}, version 1, supplied unpublished manuscript, \sid{X-E.tex}, n.d. Enter Calculus Companion. One-tree quantum-metric benchmark and its stated order boundary. \bibitem{CheungXF} N.~Cheung, \emph{X-F}, version 1, supplied unpublished manuscript, \sid{X-F.tex}, n.d. Enter Calculus Companion. Three-tree background identity through \(\kappa^2\hbar^0\); no loop identity is transferred. \bibitem{CheungXG} N.~Cheung, \emph{X-G}, version 1, supplied unpublished manuscript, \sid{X-G.tex}, n.d. Enter Calculus Companion. Free graded multiplet, specified source-preserving variational upgrade obstruction, and separate unresolved one-loop target. \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; no native 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 local stress benchmark or an imported clock certificate. \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. Separate linearized field and gauge construction on on-shell backgrounds. \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. Locally covariant perturbative gravity and renormalized BV framework; no ultraviolet-completeness conclusion is imported. \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. Distributional products and extension framework. \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 covariance and normalization ambiguities on the stated field-theoretic domain. \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. Local time-ordered-product construction; not a theorem for an unnamed nonlocal clock kernel. \bibitem{HWreview} S.~Hollands and R.~M.~Wald, ``Quantum fields in curved spacetime,'' \emph{Physics Reports} \textbf{574} (2015), 1--35. DOI: \sid{10.1016/j.physrep.2015.02.001}. arXiv:1401.2026. States, microlocal spectrum conditions, local composites, and perturbative construction on classical backgrounds. \bibitem{Dang} N.~V.~Dang, ``The extension of distributions on manifolds, a microlocal approach,'' \emph{Annales Henri Poincar\'e} \textbf{17} (2016), 819--859. arXiv:1412.2808. Extension framework only; no application to an unconstructed inherited characteristic target is asserted. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}, Cambridge University Press, Cambridge, 2018. DOI: \sid{10.1017/9781316848142}. Complete channels, reference systems, trace distance, and diamond-norm conventions. \end{thebibliography} \end{document}