% XVIII-A-v2.tex % Standalone pdfLaTeX source with an internal bibliography. % No external figures, bibliography database, or shell escape required. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XVIII-A-v2.tex % pdflatex -interaction=nonstopmode -halt-on-error XVIII-A-v2.tex % % IDENTIFIER: % The working designation is XVIII-A, version 2. % This file does not replace the separately identified XVII-A % finite-matrix AdS2 reconstruction manuscript. % No registration or authorship authorization is asserted. % % PROOF BOUNDARY: % Independent interface results and conditional implications are proved. % The concrete gravitational characteristic product, complete one-loop % coefficient, bulk BV restoration, and ultraviolet completion remain % unconstructed. % % VERIFICATION: % No successful compiler execution, machine proof, numerical execution, % independent technical review, or journal acceptance is asserted. % Insert responsible author metadata before actual submission. % % Principal revisions: % - Noncolliding identifier and consistent version metadata. % - Explicit Enter readout/quantum-return/observable distinctions. % - Smooth witness non-determination stated as a corollary. % - Class change separated from microcausal admission. % - Exact Ward solvability distinguished from approximate cancellation. % - Finite-probe coverage and affine-family qualifications made explicit. % - Formal-order and normalized-retarded-order bookkeeping clarified. % - Preparation residual retained in observable-level matching. % - Two-column documentary constraint ledger. % - A non-identification boundary accompanies every research gate. % - No catalogue-wide completeness or verification claim. \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{array,booktabs,longtable,tabularx} \usepackage{enumitem} \usepackage{microtype} \usepackage{xurl} \usepackage[hidelinks]{hyperref} \usepackage[nameinlink,noabbrev]{cleveref} \hypersetup{ pdftitle={XVIII-A: Prepared relational observables, normalization compatibility, and ultraviolet matching}, pdfauthor={}, pdfsubject={Version 2; independent interface proofs and conditional gravitational construction criteria}, pdfkeywords={relational observables, Einstein scalar theory, microlocal analysis, renormalization, Ward identities, ultraviolet matching, readout, reconstruction descent} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{0.25em} \setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt} \setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt} \renewcommand{\arraystretch}{1.12} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{requirement}[theorem]{Requirement} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \crefname{requirement}{Requirement}{Requirements} \Crefname{requirement}{Requirement}{Requirements} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\dd}{\,\mathrm d} \newcommand{\ii}{\mathrm i} \newcommand{\eps}{\varepsilon} \newcommand{\id}{\mathrm{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\WF}{\operatorname{WF}} \newcommand{\im}{\operatorname{im}} \newcommand{\Ker}{\operatorname{ker}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Rec}{\operatorname{Rec}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\D}{\mathcal D} \newcommand{\K}{\mathcal K} \newcommand{\Isol}{\mathcal I_{\mathrm{sol}}} \newcommand{\Kmc}{\mathcal K_{\mathrm{mc}}} \newcommand{\Pc}{\mathsf P} \newcommand{\ER}{E_0^{\mathrm R}} \newcommand{\JD}{J_{\mathrm D}} \newcommand{\Ds}{\mathcal D_\sigma} \newcommand{\sigeps}{\sigma_\eps} \newcommand{\UV}{\mathrm{UV}} \newcommand{\EFT}{\mathrm{EFT}} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\ket{#1}\bra{#2}} \newcommand{\sourceid}[1]{\texttt{\detokenize{#1}}} \newcommand{\NOTCLOSED}{\texttt{NOT\_CLOSED}} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}} \title{ \textbf{Landmark XVIII-A}\\[3pt] Prepared Relational Observables:\\ Non-Determination, Normalization Compatibility,\\ and Conditional Ultraviolet Matching } % Responsible author and affiliation information must be supplied. \author{} \date{Version 2} \begin{document} \maketitle \begin{abstract} We separate construction of a fixed-order interacting relational observable in a specified Einstein--scalar model from its matching to a specified ultraviolet candidate. The contribution is a self-contained collection of interface results. Pure-trace witness contractions do not determine a compact three-slot kernel, its free-shell class, or its microcausal admission; non-determination already occurs among smooth compact kernels. Density weighting retains preparation derivatives, and late source agreement does not imply agreement of retarded responses. We prove compact balanced-forest composition, reconstruction descent, conditional class-obstruction transfer, and coefficientwise separation of classical defects from positive formal orders in Planck's constant. For a completely specified affine Ward-normalization problem, exact cancellation is equivalent to vanishing in the quotient by the actual image of the admissible counterterm map. Finite probes, bounded normal order, and projected RG fixed points do not supply the corresponding unrestricted conclusions. Conditional assembly and continuum matching preserve a specified symmetry identity once their analytical hypotheses are supplied. The operational framing distinguishes destructive classical readout from quantum-return decoding, complete preservation from subsystem recovery, and observable matching from spacetime or holographic reconstruction. Version-qualified comparisons with Cheung's manuscripts and a supplied OpenAI catalogue record are documentary constraints, not theorem premises. No complete gravitational characteristic coefficient, evaluated interacting one-loop observable, bulk BV restoration, or ultraviolet completion is established. \end{abstract} \paragraph{Identifier.} The working designation is \emph{XVIII-A}, version 2. The supplied companion separately identifies \emph{XVII-A} as a finite-matrix AdS$_2$ reconstruction benchmark \cite{CheungRegisteredXVIIA}. The present manuscript neither replaces that sheet nor imports its reconstruction maps. The designation here does not assert registration or authorization of authorship. \paragraph{Logical status.} Every unconditional mathematical result below is proved from its printed definitions. Every conditional application states its hypotheses explicitly. A source record, comparison, continuation, or quoted stopping rule is not an independently verified theorem premise. An unconstructed object is described as unresolved, not as universally nonexistent. \tableofcontents %===================================================================== \section{Targets and the Enter non-identification discipline} \label{sec:target} %===================================================================== \subsection{Two different construction targets} The programme separates definition of relational observables, construction and normalization of their interacting products, and ultraviolet completion. A classical identity, an admitted free product, or a numerical finite part of one distribution does not by itself construct the interacting observable \cite{CheungVI,CheungVIIC,CheungVIID,CheungXG}. Locally covariant perturbative gravity provides an external effective-theory framework \cite{BFR}. Relational-coordinate contributions can be essential in an observable calculation: Fr\"ob's scalar correlator calculation is an external example \cite{FrobCorrelation}. Neither reference verifies an Enter manuscript or identifies its observables with the objects here. \begin{definition}[Fixed-order and ultraviolet targets] \label{def:targets} The \emph{fixed-order target} is one named renormalized interacting relational correlator or response in a frozen Einstein--scalar specification, at stated orders in the gravitational coupling and $\hbar$, with: \begin{enumerate}[label=\textup{(\roman*)}] \item an admitted coefficient in a declared output space; \item a complete contribution inventory at that order; \item a compatible and completely parameterized counterterm prescription; \item a proved gauge, background, or BRST/BV identity on its stated domain; \item a specified physical state or representation whenever an expectation is claimed. \end{enumerate} If a classical record is claimed, a separately declared measurement instrument and its comparison to that correlator or response are additionally required. The \emph{ultraviolet target} is a continuum theory with a controlled high-scale construction, a physically acceptable state space and evolution, finitely many independent essential physical inputs, and a connecting theorem recovering the same prepared low-energy observable with a stated remainder. \end{definition} The second target includes obligations not discharged by the first. Conversely, proposing a ultraviolet action does not define the composite relational observable needed for the first. The mathematical contribution has eight components, summarized in \cref{thm:release}. It includes independent separation and compatibility results, together with conditional assembly and matching implications. It does not evaluate the concrete gravitational endpoint. \subsection{Enter is readout, not quantum return} \label{sec:philosophy} The \emph{Philosophy of Enter} is used here as a discipline of interfaces and non-identification, not as a gravitational theorem. Under the finite computational-basis convention, destructive readout consumes the measured quantum name and returns a classical record. For a measured factor $S$ and surviving complement $B$, its record--residual interface can be written \begin{equation} \mathcal M_S(\rho) = \sum_z\kb{z}{z}_R\otimes (\bra{z}_S\otimes I_B)\rho(\ket{z}_S\otimes I_B). \label{eq:readout} \end{equation} The output names are $R$ and $B$, not the consumed factor $S$. This is a specified finite readout convention, not a field-theoretic instrument construction or a measurement uniqueness theorem \cite{CheungI}. A \emph{decoder} instead is a quantum-return map from a declared accessible factor to a declared quantum code, with correctness tested on all required inputs and reference correlations. A classical record is not such a return interface \cite{CheungHawk04B}. Complete unitary or isometric preservation does not imply recovery from a selected subsystem \cite{CheungHawk04A}. For example, the isometry \[ \ket{\psi}\longmapsto\ket{0}_Q\otimes\ket{\psi}_E \] preserves every input globally while its $Q$ output is constant. Recovering from $Q$ alone is a different task. The fixed-order target is therefore a correlator or response; its operational interpretation, if separately realized, is a classical record after a declared measurement. It is not a decoder, a quantum-return map, or subsystem code recovery. Moreover, a formal perturbative coefficient is not by itself a positive probability law or a self-adjoint measurable effect. Representation, instrument construction, and control of the operational approximation remain separate obligations. The ultraviolet connecting theorem compares that same prepared observable. It does not reconstruct spacetime, derive a metric, identify an entanglement wedge, or construct a Page curve or an island. The bounded-effect reconstruction and time-band statement of the separately identified XVII-A benchmark are not imported as any of those identifications \cite{CheungRegisteredXVIIA}. \paragraph{Documentary comparisons.} Every comparison with a Cheung sheet or an OpenAI family is an analogy retaining its own version, model, domain, and allowed changes. It is not a theorem premise here. Similar notation does not identify distinct kernels, source laws, states, normalizations, or gauge complexes. \paragraph{Unchanged boundaries.} This manuscript does not open the inherited XV-B quantum-admission exit, discharge XVI-N or its Route B, restore the missing bulk BV construction, or admit the inherited characteristic quantum coefficient. The separately recorded status O.1 remains \NOTCLOSED. These are scope statements, not new nonexistence theorems. No successful compilation is asserted, and the package is not spacetime emergence. %===================================================================== \section{The frozen specification contract} \label{sec:contract} %===================================================================== \subsection{Fields and independent kernel conventions} For the independent kernel results, use \[ M=\R^4,\qquad \eta=\operatorname{diag}(1,-1,-1,-1),\qquad \Box=\partial_t^2-\Delta_{\mathbf x},\qquad P_m=\Box+m^2,\quad m\ge0. \] Tensor components and covectors at different points are compared in the fixed Minkowski trivialization. Distributions are paired using the fixed coordinate density. No corresponding identification on a general curved spacetime is assumed. A future interacting application must specify Einstein--scalar field content together with the ghosts and auxiliary fields of its gauge prescription. Changing the scalar coupling, field parametrization, or gauge complex changes that application. \begin{requirement}[Frozen application data] \label{req:contract} Before constructing an interacting coefficient, specify: \begin{enumerate}[label=\textbf{C\arabic*.}] \item the action, density representatives, field parametrization, gauge fixing, graded signs, and inverse normalization; \item the background, equations, configuration domains, initial or final data, and gauge-parameter support class; \item one preparation prescription, a fixed $\eps>0$, its switch $\sigeps$, and the compact observation smearing; \item the quantum two-point product data and their transport, separately from the physical state; \item the bare relational observable and its complete expansion; \item interaction switches and perturbative-order conventions; \item the output space, topology, equation quotient, and any representation domain; \item admissible bulk, insertion, preparation, and switch counterterms, with their coherence and descent relations; \item the exact symmetry identity and its normalization conditions; \item any ultraviolet regularization, cutoff-removal limit, and intended observable-level matching theorem. \end{enumerate} A measurement claim also requires the separately specified instrument and surviving-output interface of \cref{def:targets}. \end{requirement} \subsection{Parameters are not interchangeable} Keep \begin{equation} \eps\ \text{(preparation width)},\qquad \mu\ \text{(subtraction scale)},\qquad \Lambda_{\UV}\ \text{(ultraviolet cutoff)} \label{eq:threescales} \end{equation} distinct. Below, $k$ is an RG coarse-graining scale and $\lambda$ is a classical witness frequency. Preparation width is not automatically a ultraviolet regulator. A cutoff is not automatically an RG scale. None is identified with a witness frequency. A positive-parameter damping used to represent an existing distribution is not automatically a regularization prescription for a new graph product. Each role and any comparison map must be stated explicitly. The designation ``regulator width'' in a documentary source does not override the role fixed in the present specification. No joint large-$\lambda$, small-$\eps$, or large-cutoff limit is inferred from separate one-parameter statements. \subsection{Documentary normalization, not a reconstructed model} For comparison only, the supplied XVI-A convention records \begin{align} S_{\mathrm{EH}} &=-\frac{2}{\kappa^2}\int\sqrt{-q}\,R, & G_{\mathrm{gr}}^{\mathrm R}&=-\Pc\ER, \label{eq:docnormalization}\\ S_{\mathrm{grav,D}} &=S_{\mathrm{EH}}+\kappa\int(\sigeps-1)\ell_3, & \JD&=\sigeps J_3+\Ds, \label{eq:docsource}\\ r&=-u_{\mathrm D}, & (\Pc a)_{\mu\nu} &=a_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta a. \label{eq:docsign} \end{align} Here $q$ is the source's metric notation, $R$ its curvature scalar, and $\ell_3$ its fixed unintegrated cubic density. These displayed conventions are not a definition here of the full cubic component arrays or gauge complex \cite{CheungXVIA,CheungXIIC}. The cubic-only density weight matters: weighting the entire Einstein--Hilbert density would alter lower orders and would not preserve the same frozen specification \cite{CheungXIIC}. For a time-only weight, the documentary source residual is \begin{equation} \Ds^{\mu\nu} = 2C_2^{\mu\nu00}\sigeps'' -2M_2^{\mu\nu0}\sigeps'. \label{eq:doclayer} \end{equation} The arrays $C_2,M_2$ are source-defined coefficients, not derived from their names in this paper. Nor are these conventions equated with the X-track inverse $-4\Pc\ER$. Comparing the two normalizations requires an explicit transport of fields, vertices, products, and observables \cite{CheungXE,CheungXF}. \subsection{Observable coefficients and physical states} Write \begin{equation} O_\eps(\kappa) = O_{0,\eps}+\kappa O_{1,\eps}+\kappa^2O_{2,\eps}+\cdots. \label{eq:observableexpansion} \end{equation} Every coefficient must distinguish quantum fields, classical backgrounds, and external preparation data. Smooth evaluation on one configuration is not a distributional definition on arbitrary joint tests. Hadamard two-point data used to define products do not alone select a physical state. In a gauge-fixed theory, positivity must be formulated on an appropriate physical algebra or representation, not assumed for the full ghost--auxiliary multiplet \cite{FRBV,BFR,CheungXG}. An expectation and a complete measurement output are different interfaces. %===================================================================== \section{Witness contractions do not determine the kernel} \label{sec:witness} %===================================================================== Let $E_h=S^2T^*M$. Let $\K$ consist of compactly supported distributional kernels on $M^3$ with two dual symmetric-tensor slots and one scalar slot, symmetric under exchange of the tensor slots. Fixed trivializations make the component description unambiguous. Compact support allows pairing with arbitrary smooth configurations. Let $\widetilde V_\pm$ be the future or past causal covector cone with zero adjoined. Define \begin{equation} \Kmc= \left\{ u\in\K: \WF(u)\cap \bigl(\widetilde V_+^{\,3}\cup \widetilde V_-^{\,3}\bigr)=\varnothing \right\}. \label{eq:microcausal} \end{equation} This is a wavefront admission condition, not a gauge identity. For $|\mathbf n|=1$ and $\lambda>0$, put \[ k_1=(1,\mathbf n),\qquad k_2=(1,-\mathbf n),\qquad \gamma_a^\lambda(x)=\eta e^{\ii\lambda k_a\cdot x}, \] where $k_a\cdot x$ denotes the covector pairing. The covectors are null, so $\Box\gamma_a^\lambda=0$. The scalar partial contraction is \begin{equation} \pair{j_{u,\lambda,\mathbf n}}{\phi} = \pair{u}{ \gamma_1^\lambda\otimes\gamma_2^\lambda\otimes\phi}. \label{eq:witnessmap} \end{equation} The pairing is defined by compact support. It does not assert existence of a quantum contraction. Define the free-shell null space \begin{equation} \begin{split} \Isol=\bigl\{w\in\K:\ &\pair{w}{H_1\otimes H_2\otimes\phi}=0\\ &\text{for all smooth }\Box H_a=0,\quad P_m\phi=0 \bigr\}. \end{split} \label{eq:Isol} \end{equation} It is a vector subspace. Its quotient class is $[u]_{\mathrm{sol}}=u+\Isol$. These free equations are not the coupled Einstein equations. \begin{theorem}[Pure-trace witness non-determination] \label{thm:witness} The complete family \cref{eq:witnessmap}, for all $\lambda>0$, unit $\mathbf n$, and scalar compact smooth tests, does not determine: \begin{enumerate}[label=\textup{(\roman*)}] \item the compact three-slot kernel; \item its membership in $\Kmc$; \item its class modulo $\Isol$. \end{enumerate} \end{theorem} \begin{proof} Choose a nonzero constant contravariant symmetric tensor $A$ such that $A^{\mu\nu}\eta_{\mu\nu}=0$, for example $A^{12}=A^{21}=1$ with other components zero. Fix $y_0\in M$. Choose a nonzero nonnegative $v\in C_c^\infty(M)$ supported where $\phi_m(t,\mathbf x)=\cos(mt)>0$; for $m=0$, take $\phi_0=1$. Define \begin{equation} w^{\mu\nu,\rho\sigma}(x_1,x_2,x_3) = A^{\mu\nu}A^{\rho\sigma} \delta_{y_0}(x_1)\delta_{y_0}(x_2)v(x_3). \label{eq:invisiblekernel} \end{equation} This is a nonzero symmetric compact kernel. Since $A:\eta=0$, every pure-trace witness contraction of $w$ vanishes. Thus $w$ and zero have identical witness data, proving (i). At a point where $v(x_3)\neq0$, the wavefront set of the two point deltas contains two nonzero future causal covectors in the tensor slots and zero in the scalar slot. Consequently $w\notin\Kmc$, while $0\in\Kmc$, proving (ii). Choose a constant tensor $H$ with $A:H\neq0$. Then $\Box H=0$ and $P_m\phi_m=0$, but \[ \pair{w}{H\otimes H\otimes\phi_m} = (A:H)^2\int_M v(x)\phi_m(x)\dd^4x\neq0. \] Hence $w\notin\Isol$. Its class differs from the zero class, proving (iii). \end{proof} \begin{corollary}[Smooth compact non-determination] \label{cor:smoothwitness} The witness map has nontrivial kernel even among $C_c^\infty$ kernels, all of which satisfy \cref{eq:microcausal}. \end{corollary} \begin{proof} Replace both deltas in \cref{eq:invisiblekernel} by the same nonzero nonnegative $f\in C_c^\infty(M)$. The resulting nonzero smooth compact kernel is still annihilated by every pure-trace witness. Its wavefront set is empty. \end{proof} \begin{remark}[Documentary counterparts, not transferred conclusions] XVI-H records a general-test compact kernel under its stated root inputs and a partial contraction equal to $j_\lambda$. It does not report the full-kernel wavefront or class-admission calculation \cite{CheungXVIH}. The theorem above does not show that its complete root construction is nonunique. It only excludes reconstruction from the displayed witness family alone. XI-G's class obstruction has different, frozen quantifiers: its detecting witnesses are free-shell tests, not a decision on the Einstein subshell \cite{CheungXIG}. Neither a universal no-go nor that physical class obstruction is assumed in the theorem. \end{remark} %===================================================================== \section{Preparation residuals and retarded response} \label{sec:preparation} %===================================================================== \begin{proposition}[Density weighting retains a derivative term] \label{prop:weighted} For $\sigma\in C^\infty(M)$ and a compact smooth vector field $J$, \begin{equation} \int_M\sigma\,\partial_\mu J^\mu\dd^4x = -\int_M(\partial_\mu\sigma)J^\mu\dd^4x. \label{eq:weightedIBP} \end{equation} When $\dd\sigma$ is nonzero, the right side need not vanish. Thus an unweighted total-divergence equivalence does not imply equality after density weighting. \end{proposition} \begin{proof} The integral of $\partial_\mu(\sigma J^\mu)$ vanishes by compact support. The product rule gives the identity. If $\partial_\nu\sigma$ is nonzero on an open set, choose a nonnegative nonzero compact test $\chi$ there. Set $J^\nu=\chi\partial_\nu\sigma$ and other components zero. The residual equals $-\int\chi(\partial_\nu\sigma)^2\dd^4x\neq0$. \end{proof} This elementary residual explains why the preparation derivatives must be retained in the specification. It neither derives the component arrays in \cref{eq:doclayer} nor identifies a source with an observable kernel \cite{CheungXVIB,CheungXIIC}. \begin{proposition}[Late source agreement does not determine the response] \label{prop:lateresponse} There are smooth sources $s_1,s_2$ agreeing for every $t\ge\eps$ whose zero-data retarded solutions \[ u_i''=s_i,\qquad u_i(0)=u_i'(0)=0 \] differ for every $t>\eps$. \end{proposition} \begin{proof} Take nonzero $a\ge0$ in $C_c^\infty((0,\eps))$ and set $s_1=a$, $s_2=0$. For $t\ge\eps$, \begin{equation} u_1(t)-u_2(t)=tM_0-M_1,\qquad M_0=\int_0^\eps a(r)\dd r,\quad M_1=\int_0^\eps ra(r)\dd r. \label{eq:layermoments} \end{equation} Here $M_0>0$ and $M_1<\eps M_0$, so the difference is strictly positive for $t>\eps$. \end{proof} \begin{remark}[The object to match] A gravitational application must compare the complete prepared response and completed insertion, not merely late sources. Source, propagated response, coordinate completion, scalar completion, and compact class kernel are different stages. XVI-D's supplied witness response $U_{\mathrm D}=-\Pc V_\lambda$ retains its separate coordinate-identification hypotheses \cite{CheungXVID}. The proposition supplies no new identification of those stages. \end{remark} Changing $\eps$, initial data, smearing, or state generally changes the preparation or experiment. It is not declared an admissible finite renormalization merely because a subtraction scale is also present. Scheme comparisons below hold at fixed preparation unless an independent equivalence is proved. %===================================================================== \section{Microlocal operations and their boundaries} \label{sec:microlocal} %===================================================================== We use the standard wavefront criteria for products, pullbacks, and proper pushforwards. Their topological implementation must respect the relevant continuity or hypocontinuity hypotheses \cite{BDHintro,BDHops}. These analytical tools are not substitutes for an undefined characteristic graph product. \begin{lemma}[Nonvanishing smooth multiplication preserves local wavefront] \label{lem:switchWF} For $u\in\D'(X)$ and $a\in C^\infty(X)$, let $U=\{a\neq0\}$. Then \[ \WF(au)|_U=\WF(u)|_U,\qquad \WF(au)\subseteq\WF(u). \] \end{lemma} \begin{proof} Smooth multiplication does not enlarge wavefront. On $U$, $a^{-1}$ is smooth and $u=a^{-1}(au)$, giving the reverse local inclusion. \end{proof} A switch can remove a distribution on an open region where it vanishes. The lemma only excludes local regularization by multiplication with a nonvanishing smooth switch. \subsection{Balanced forests} \begin{definition}[Balanced line] A bidistribution $L_e$ on $M^2$ is \emph{balanced} if \[ \WF(L_e)\subseteq \{(x,p;y,-p):p\neq0\} \] in the fixed flat trivialization. \end{definition} \begin{theorem}[Compact balanced-forest composition] \label{thm:forest} Let a finite forest have vertex set $V$ and edge set $E$. Give every edge a balanced bidistribution and pull it back to $M^V$ by its endpoint projection. Multiply by a compact smooth joint weight. Then: \begin{enumerate}[label=\textup{(\roman*)}] \item the finite edge product is defined; \item its pushforward over any set of internal vertices is defined by proper support; \item for $n\ge1$ external slots, the output wavefront is contained in \begin{equation} Z_n= \left\{ (x_1,p_1;\ldots;x_n,p_n): \sum_{a=1}^np_a=0 \right\}\setminus0. \label{eq:Zn} \end{equation} \end{enumerate} Finite differentiations and smooth multiplications preserve this bound. It excludes an all-future or all-past nonzero causal tuple, and therefore implies the corresponding microcausal kernel condition. With no external slots, the compact pushforward is a scalar. \end{theorem} \begin{proof} Endpoint projections are submersions, so their pullbacks exist. A forbidden product cancellation would assign nonzero edge covectors, with opposite signs at their endpoints, whose sum at every vertex is zero. The participating edges form a subforest. A leaf of that subforest has one participating incident edge, so its vertex equation forces that edge covector to vanish. Removing the edge and repeating forces every participating covector to vanish, a contradiction. The product criterion therefore applies. The compact smooth weight makes the product compactly supported. Projection is then proper on its support. For a pushforward wavefront point, the internal covectors are zero. Summing the vertex covectors cancels each edge pair, leaving zero total external covector. This proves \cref{eq:Zn}. Differentiation and smooth multiplication do not enlarge that bound. Finally, future and past causal cones are pointed. A tuple contained entirely in either cone with zero sum must have every covector zero, which is excluded. \end{proof} The same forest mechanism occurs in the supplied finite-width free construction of VII-I \cite{CheungVIII}. Its use here is independently proved and does not transfer that paper's model or quantum algebra. \begin{example}[A cyclic raw product need not converge] \label{ex:cycle} Two parallel delta lines between two vertices lead formally to $\delta(r)^2$. Take a nonzero real $\rho\in C_c^\infty(\R)$ and $\rho_\tau(r)=\tau^{-1}\rho(r/\tau)$. For $\phi(0)>0$, \[ \int\rho_\tau(r)^2\phi(r)\dd r = \tau^{-1}\phi(0)\int\rho(s)^2\dd s+O(1). \] The raw regularization diverges. This does not exclude a separately authorized renormalized extension. It shows why the forest argument alone supplies no loop product. \end{example} \subsection{Weak limits do not preserve admission} \begin{proposition}[Smooth compact approximants can acquire singular wavefront] \label{prop:weaklimit} There are smooth compact $u_\tau$ with $\WF(u_\tau)=\varnothing$ for every $\tau>0$ such that $u_\tau\to\delta_0$ in $\D'(\R^d)$. \end{proposition} \begin{proof} Let $\rho\in C_c^\infty(\R^d)$ with integral one and set $u_\tau(x)=\tau^{-d}\rho(x/\tau)$. For every compact smooth test, \[ \pair{u_\tau}{\phi} = \int\rho(y)\phi(\tau y)\dd y\longrightarrow\phi(0). \] The limit has every nonzero covector in its wavefront at zero. \end{proof} Positive-regulator admission and weak convergence thus do not give admission of the limit. A limit theorem needs uniform microlocal control in a suitable wavefront-restricted topology. The distinction is explicitly retained in XIII-E and Em02-E \cite{CheungXIIIE,CheungEmIIE}. %===================================================================== \section{Descent, class transfer, and formal orders} \label{sec:descent} %===================================================================== \begin{proposition}[Exact reconstruction descent] \label{prop:descent} Let $P,K,Y$ be vector spaces, $\Rec:P\to K$ a surjective linear map, and $\widetilde M:P\to Y$ linear. A unique linear map $M:K\to Y$ with $\widetilde M=M\Rec$ exists if and only if \begin{equation} \Ker\Rec\subseteq\Ker\widetilde M. \label{eq:descent} \end{equation} If $K$ has the quotient topology induced by $\Rec$ and $\widetilde M$ is continuous, the descended map is continuous. \end{proposition} \begin{proof} Necessity follows by applying $M\Rec$ to a null presentation. For sufficiency, define $M(\Rec p)=\widetilde M(p)$. Presentations with equal reconstruction differ by an element of $\Ker\Rec$, so the definition is independent of the presentation. Surjectivity gives uniqueness. The quotient topology gives continuity. \end{proof} The criterion has documentary counterparts in XIV-J and XV-B \cite{CheungXIVJ,CheungXVB}. Agreement on a smaller open reference domain does not prove the global condition: a nonzero distribution supported outside that domain is invisible to restriction there. \begin{example}[Highest presentation degree is not intrinsic] \label{ex:nullpresentation} For a nonzero distribution $R(r)$ in independent variables, \[ \bigl(q\delta'(q)+\delta(q)\bigr)\otimes R(r)=0. \] The two summands separately are nonzero and opposite. Thus a chosen polynomial-degree splitting of a presentation does not automatically descend to reconstructed kernels. This does not rule out a quotient filtration defined from images of filtered presentation spaces. It rules out treating a representative-dependent highest term as an intrinsic joint symbol without the required descent theorem \cite{CheungXIVJ,CheungXVB}. \end{example} \begin{proposition}[Conditional class-obstruction transfer] \label{prop:classtransfer} Suppose $u_0\in\K$ satisfies \[ [u_0]_{\mathrm{sol}}\cap\Kmc=\varnothing. \] If a constructed prepared kernel $u_{\mathrm D}$ obeys $u_{\mathrm D}-u_0\in\Isol$, then \[ [u_{\mathrm D}]_{\mathrm{sol}}=[u_0]_{\mathrm{sol}}, \qquad [u_{\mathrm D}]_{\mathrm{sol}}\cap\Kmc=\varnothing. \] A proof that $u_{\mathrm D}$ belongs to a different class removes this transfer argument; it does not establish that the new class intersects $\Kmc$. Admission of that class requires a separate proof. \end{proposition} \begin{proof} Since $\Isol$ is a subspace, \[ u_{\mathrm D}+\Isol = u_0+(u_{\mathrm D}-u_0)+\Isol = u_0+\Isol. \] The assumed empty intersection therefore transfers. The calculation uses equality of classes, not a criterion for a different class. Merely distinguishing classes contains no wavefront or representative-existence assertion, so it supplies no admission conclusion. \end{proof} The transfer logic is elementary. Its documentary appearance in XVI-A/B does not import XI-G's physical obstruction or decide its applicability to a prepared observable \cite{CheungXVIA,CheungXVIB,CheungXIG}. \begin{proposition}[Formal-order separation] \label{prop:hbar} Let $A$ be a vector space or an $\hbar$-independent quotient. A nonzero $a_0\in A$ cannot be cancelled in $A[[\hbar]]$ by positive formal powers: \[ a_0+\sum_{\ell\ge1}\hbar^\ell a_\ell\neq0. \] If $Q_\ell$ are well-defined linear maps on the declared coefficient domain and \[ Q=Q_0+\hbar Q_1+\cdots,\qquad O=O_0+\hbar O_1+\cdots, \] then $QO=0$ implies $Q_0O_0=0$. \end{proposition} \begin{proof} Formal series are equal coefficientwise. The coefficient at $\hbar^0$ proves both assertions. \end{proof} A nonzero classical defect must be repaired at classical order by a completed classical expression, an explicitly changed model, or a justified quotient. It cannot be cancelled by an asserted positive-$\hbar$ loop. This is a formal-series statement, not a claim about numerical cancellation at one nonzero value of $\hbar$. \begin{remark}[Formal ideals and the BRST boundary] In $\C[[\hbar]]$, $(\hbar)$ contains $\hbar\mathbf1$ but not $\mathbf1$. A commutator proportional to $\hbar\mathbf1$ therefore does not alone prove that a formal quotient is the zero algebra. Inverting $\hbar$ changes the problem. The supplied X-G sheet separately constructs a free causal BRST differential and excludes its specified exact source-preserving variational upgrade. It leaves the interacting quantum BRST construction unresolved \cite{CheungXG}. Neither that upgrade nor an interacting identity is inferred from the formal-order proposition. \end{remark} %===================================================================== \section{The selected quantum coefficient and its inventory} \label{sec:oneloop} %===================================================================== Free products, time ordering, and normalized retarded maps have different domains and normalization obligations \cite{FR,FRBV,CheungIXD}. A name for any of these operations does not define it on a nonlocal insertion. \begin{assumption}[Declared normalized retarded family] \label{ass:retarded} For the arguments used below, normalized multilinear maps $R_n$ have been constructed, with \[ \mathcal R_V(O) = O+R_1(V;O)+\frac12R_2(V,V;O)+\cdots. \] Their distributional domains, causal normalization, graded signs, and required equation-ideal descent are proved. They have nonnegative formal $\hbar$ expansions. The maps have no additional hidden $\kappa$ dependence; any such dependence has first been expanded and included in the coefficient inventory. Interaction arguments are even, and the factorial convention is the one displayed. If \[ R_1(V;O) = \frac{\ii}{\hbar} \bigl(T_2^{\mathrm{ren}}(V,O)-V\star_H O\bigr) \] is used, divisibility of the numerator by $\hbar$ is proved on the declared pair. \end{assumption} Let \[ V(\kappa)=\kappa V_1+\kappa^2V_2+O(\kappa^3), \qquad O_\eps(\kappa)=O_{0,\eps} +\kappa O_{1,\eps}+\kappa^2O_{2,\eps}+O(\kappa^3). \] The entries may have their own nonnegative $\hbar$ expansions. \begin{proposition}[Complete formal extraction at order $\kappa^2\hbar$] \label{prop:inventory} Under \cref{ass:retarded}, \begin{equation} \begin{split} [\kappa^2\hbar]\,\mathcal R_{V(\kappa)}(O_\eps(\kappa)) = [\hbar]\Bigl( &O_{2,\eps} +R_1(V_1;O_{1,\eps}) +R_1(V_2;O_{0,\eps})\\ &+\frac12R_2(V_1,V_1;O_{0,\eps}) \Bigr). \end{split} \label{eq:completecoefficient} \end{equation} \end{proposition} \begin{proof} Multilinearity gives \[ [\kappa^2]R_1(V;O) = R_1(V_1;O_{1,\eps})+R_1(V_2;O_{0,\eps}). \] Two interaction arguments contribute $\frac12R_2(V_1,V_1;O_{0,\eps})$. Three or more begin at $\kappa^3$. The direct insertion contributes $O_{2,\eps}$. Extracting $[\hbar]$ proves the formula. \end{proof} The formula is an inventory identity, not an evaluation or a proof that every listed map exists. Ghost vertices, quartic vertices, preparation derivatives, insertion normalizations, and Wick-ordering corrections must appear in the specified entries. A claimed one-loop interpretation also needs its graph-counting convention; the notation $[\hbar]$ alone does not prove a graph inventory. \begin{remark}[Normalized order costs a numerator order] If a numerator is $N(\hbar)=\hbar N_1+\hbar^2N_2+\cdots$, then $[\hbar](\ii N/\hbar)=\ii N_2$. Thus the first positive-$\hbar$ coefficient of a normalized retarded map requires the numerator through order $\hbar^2$, not only its leading term \cite{CheungIXD}. \end{remark} \begin{proposition}[Applying one completion twice changes its coefficient] \label{prop:doublecompletion} For a formal linear map $\mathcal C_\kappa=\id+\kappa L+O(\kappa^2)$, \[ \mathcal C_\kappa^2=\id+2\kappa L+O(\kappa^2). \] Applying the same completion again to an already completed insertion therefore requires an explicit subtraction or factorization accounting for the extra term. \end{proposition} \begin{proof} Multiply the two series and compare their linear coefficients. \end{proof} X-F's supplied identity is at $\kappa^2\hbar^0$ on its plateau-supported transformations; X-E's one-tree defect belongs to its own truncated expression \cite{CheungXE,CheungXF}. They provide no loop counterterms for \cref{eq:completecoefficient}. XVI-L's complete-source contribution is one specified two-edge graph, conditional on its root inputs. Completing its source is not completing the one-loop inventory, and its scale family is not a physical beta function or a BRST/BV/Ward identity \cite{CheungXVIL}. %===================================================================== \section{Exact normalization and Ward compatibility} \label{sec:normalization} %===================================================================== \subsection{Admissible changes must be complete} At the selected order, let $C$ be the full allowed counterterm vector space after reconstruction relations, exchange symmetry, derivative identities, reality, support, and locked subgraph normalizations are imposed. A bookkeeping decomposition \[ C=C_{\mathrm{bulk}}+C_{\mathrm{ins}} +C_{\mathrm{prep}}+C_{\mathrm{switch}} \] need not be direct. Integration by parts and field redefinitions are used only when authorized in this space. A collision-supported tensor is not automatically the Euler derivative of a local bulk functional. Variational integrability and symmetry compatibility are separate tests \cite{CheungXVIL,CheungXVIM}. \begin{assumption}[Constructed affine Ward problem] \label{ass:ward} All contributions and extensions at the selected order have already been defined. The reference prescription has defect $a\in Y^1$. Specified linear maps are \[ d_1:Y^1\to Y^2,\qquad L:C\to Y^1, \] and the exact counterterm law is \begin{equation} a(c)=a+Lc. \label{eq:affineward} \end{equation} Consistency has been proved: \[ d_1a=0,\qquad d_1L=0. \] The spaces, maps, domains, and exhaustion of allowed changes are part of the application data. \end{assumption} A renormalized BV or BRST interpretation needs the corresponding constructed operations and anomaly identities \cite{FRBV,FrobWard}; those are not supplied merely by naming the affine problem. \begin{theorem}[Exact Ward-normalization compatibility] \label{thm:ward} Under \cref{ass:ward}, let \[ H_{\mathrm{adm}}^1=\Ker d_1/\im L. \] Then: \begin{enumerate}[label=\textup{(\roman*)}] \item a counterterm $c$ with $a(c)=0$ exists exactly when $[a]=0$ in $H_{\mathrm{adm}}^1$; \item if $c_0$ is one solution, all solutions are $c_0+\Ker L$; \item replacing the reference prescription by an admissible change preserves the obstruction class. \end{enumerate} Here $\im L$ is the actual image, not its closure. This is a restricted algebraic obstruction quotient. Identifying it with ambient BRST cohomology requires an identification of $C,L,d_1$, their domains, and their relations with that complex. The theorem does not provide the unresolved interacting quantum BRST construction of X-G. \end{theorem} \begin{proof} Since $d_1L=0$, the quotient is defined. The equation $a+Lc=0$ is equivalent to $a\in\im L$, proving (i). Given $c_0$, the equation for $c$ reduces to $L(c-c_0)=0$, proving (ii). A reference change sends $a$ to $a+Lc_*$, preserving its quotient class, proving (iii). \end{proof} \begin{remark}[Exact versus approximate solvability] If topological spaces are used and $\im L$ is not closed, the quotient need not be Hausdorff. Membership in $\overline{\im L}$ can support approximation but not an exact solution. An approximate Ward theorem requires its own norm, error bound, and admissible family. \end{remark} \subsection{Finite probes require coverage} \begin{proposition}[Finite validation on a declared defect space] \label{prop:finitevalidation} Let $D_0$ be a vector space of possible defects and $P:D_0\to\C^m$ a linear test map. The implication \[ P(a)=0\Longrightarrow a=0 \quad\text{for every }a\in D_0 \] holds exactly when $P$ is injective. No finite test map is injective on an infinite-dimensional $D_0$. \end{proposition} \begin{proof} The first assertion is $\Ker P=\{0\}$. If $D_0$ is infinite-dimensional, choose $m+1$ linearly independent elements. Their images are dependent, giving a nonzero element of $\Ker P$. \end{proof} Injectivity on a vector space containing every possible defect is a sufficient global coverage certificate. It need not be necessary when testing only a smaller affine reachable family: there the exact condition is that its intersection with $\Ker P$ contain no nonzero reachable defect. Thus the proposition does not replace an affine-family analysis by a stronger unstated requirement. \begin{example}[An incomplete Ward test] Let $C=\R$, $Y^1=\R^2$, $Lc=(c,0)$, and $a=(0,1)$. The first-coordinate test vanishes at $c=0$, but the full defect is never zero. The second coordinate separates $a$ from every allowed adjustment. \end{example} When the complete spaces and bases are finite-dimensional, exact cancellation is equivalent to \[ \rank(L\mid a)=\rank L. \] This matrix test is decisive only after coverage and the full allowed-change space are established. It does not define an absent distributional product. The two-probe coverage in Oai282-B concerns one homogeneous free-scalar stress class, with its own improvement scalar $L=-Q/2$ and ordered Wick algebra. Oai282-A's finite Ward tests are likewise restricted to their declared data \cite{CheungOaiIIA,CheungOaiIIB}. Neither supplies the gravitational map $L:C\to Y^1$ here. The identical letter $L$ is not an identification of objects. The supplied Oai069-A summary is a further comparison for finite free-scalar defect bookkeeping, not an imported theorem or a chiral-to-four-dimensional bridge \cite{CheungOai069A}. Family 282's recorded local-Ward conclusion does not establish global conformal integration or Einstein--scalar scale-to-conformal enhancement \cite{Catalogue}. \subsection{What survives compatible finite changes} Suppose a constructed coefficient changes by \[ F(c)=F(0)+Bc,\qquad B:C\to E_{\mathrm{out}} \] with $B$ linear. If $c_0+\Ker L$ is the solution set, define \begin{equation} \mathcal A_{\mathrm{phys}}=B(\Ker L). \label{eq:physicalambiguity} \end{equation} \begin{corollary}[Invariant quotient and tests] \label{cor:scheme} The algebraic class of $F(c)$ in $E_{\mathrm{out}}/\mathcal A_{\mathrm{phys}}$ is independent of the compatible prescription. A linear test $\ell$ is invariant under all compatible changes exactly when \[ \ell\circ B|_{\Ker L}=0. \] \end{corollary} \begin{proof} Two solutions differ by an element of $\Ker L$, and their coefficients differ by its image under $B$. The test is invariant exactly when it annihilates every such difference. \end{proof} If a topological quotient is intended, its topology and any closedness requirement must also be supplied. This result does not assert that the gravitational spaces $C,L,B$ have been evaluated. \subsection{Explicit scale dependence is not total RG dependence} At fixed preparation, a prediction may be written $P(g(\mu),\mu)$. A scale-independence statement concerns the total transport, for example \begin{equation} \left( \mu\partial_\mu+\sum_i\beta_i(g)\partial_{g_i} \right)P=0, \label{eq:RGtotal} \end{equation} with operator mixing and state transport when required. Its explicit derivative at fixed $g$ is not that total derivative \cite{BDF}. \begin{example}[Compensated dependence] If \[ P(g,\mu)=g+\log(\mu/\mu_0),\qquad \frac{\dd g}{\dd\log\mu}=-1, \] then the explicit derivative is one while the total derivative along the running trajectory is zero. \end{example} A subtraction-scale family for one graph is not thereby the physical beta function of a complete quantity. The supplied VI audit remains RG-open; XVI-M's degree-four curvature-span obstruction is graph-specific, and its broader matched-scale release is not established by an unspecified cutoff \cite{CheungVI,CheungXVIM}. %===================================================================== \section{Normal order and independent parameters} \label{sec:parameters} %===================================================================== \begin{proposition}[Finite normal order can retain infinite tangential freedom] \label{prop:normalorder} Let $S=\{r=0\}$ in $Y\times\R^c$, where $c\ge1$ and $Y$ contains a positive-dimensional open coordinate set. The order-zero space \[ \{a(y)\delta(r):a\in C_c^\infty(Y)\} \] is infinite-dimensional. A finite normal derivative-order bound therefore does not imply finite-dimensional ambiguity. \end{proposition} \begin{proof} Choose arbitrarily many independent compact smooth functions $a_j$ on $Y$. If $\sum_jb_ja_j(y)\delta(r)=0$, test with $\psi(y)\chi(r)$ where $\chi(0)=1$. This gives $\sum_jb_ja_j=0$, forcing every $b_j=0$. \end{proof} The positive-dimensional qualification matters: at a single point, jets through a fixed finite order are finite-dimensional. For a general stratum, an expression \[ \sum_{|\alpha|\le N}a_\alpha(y)\partial_r^\alpha\delta(r) \] has finitely many normal labels but may have infinitely many tangential coefficients. Locality, covariance, dimension, field degree, and admissible normalization constraints must be checked before counting parameters. \begin{proposition}[An infinite derivative orbit can share one parameter] \label{prop:parentparameter} The distributions $\delta_0^{(n)}$, $n\ge0$, on $\R$ are linearly independent. Nevertheless, if a parent prescription changes by $u\mapsto u+c\delta_0$ and derivative compatibility is imposed, then \[ \partial^nu\mapsto\partial^nu+c\delta_0^{(n)} \] uses the same parameter $c$ for every $n$. \end{proposition} \begin{proof} In a nontrivial finite relation, let $N$ be the largest index with coefficient $b_N\neq0$. Test with a function equal to $x^N$ near zero. Lower derivatives vanish there while the $N$th derivative is $N!$, a contradiction. The parameter statement follows by differentiating the single parent change. \end{proof} This separates distribution-orbit dimension from the number of independent normalization parameters. It is the parent-versus-descendant distinction documented in IX-B \cite{CheungIXB}, not a parameter count for a gravitational continuum theory. %===================================================================== \section{Conditional fixed-order assembly} \label{sec:assembly} %===================================================================== Extension theorems apply to distributions already defined on their stated reference domains and satisfying their extension hypotheses \cite{BF,Dang}. A theorem about collision extension cannot substitute for an undefined noncoincident characteristic product. Extending one primitive also does not prove compatibility with every singular spectator. \begin{assumption}[Complete analytical package] \label{ass:analytic} At the fixed selected order, the application supplies: \begin{enumerate}[label=\textbf{A\arabic*.}] \item a complete finite contribution inventory for \cref{eq:completecoefficient}; \item actual joint distributions for every graph operation, including tests crossing required noncoincident characteristic loci; \item coherent extensions on collision strata, reference agreement, locked normalization, and reconstruction descent; \item checked wavefront conditions for every multiplication, pullback, singular-spectator composition, and attachment; \item proper support for every projection and a proved admission bound for each final contribution or the complete summed output; \item continuity in a declared topology, with the required bounded-set or hypocontinuity qualifications; \item a constructed and complete affine Ward problem satisfying \cref{ass:ward}, with an exact solution. \end{enumerate} \end{assumption} \begin{theorem}[Conditional fixed-order assembly] \label{thm:assembly} Under \cref{ass:retarded,ass:analytic}, the finite assembled sum defines an admitted coefficient at the selected order and satisfies the selected Ward identity. A continuous state functional on its declared physical output domain gives a well-defined expectation. This conclusion is an observable-level construction. It does not itself construct a measurement instrument or a quantum-return channel. \end{theorem} \begin{proof} A2--A4 make every graph operation an actual distributional operation. A3 guarantees authorized presentation independence. A5 permits the projections and proves admission. A1 identifies the resulting finite sum with the formal inventory, and A6 supplies its claimed continuity. A7 and \cref{thm:ward} give the exact equation $a+Lc=0$, which is the selected identity. A continuous state functional can then be evaluated on a coefficient in its domain. No instrument or channel construction occurs in these steps. \end{proof} These assumptions are substantive. They have not been discharged for the inherited characteristic graph. Compact support, a retarded-pole representation, or an unnormalized extension family does not supply the missing joint map or output theorem. A regulator route additionally needs convergence in an appropriate fixed wavefront-restricted space, not merely weak convergence. Product and pushforward limit passage must use their actual topological hypotheses \cite{BDHops}. %===================================================================== \section{Ultraviolet construction and observable matching} \label{sec:uv} %===================================================================== \subsection{A candidate mechanism remains external} Asymptotic safety is one possible candidate mechanism. The gravitational flow framework and fixed-point programme of Reuter and Reuter--Saueressig are external \cite{ReuterFlow,ReuterSaueressig}. They do not verify an Enter sheet or construct the present relational insertion. An exact candidate theory space must include or justify excluding operators generated by its flow. Keeping Einstein--scalar field content does not mean retaining only the Einstein--Hilbert and scalar quadratic action terms. Higher-curvature, matter, vacuum, and insertion terms require a declared treatment. For an exact essential-coupling flow, write \[ \frac{\dd g}{\dd t}=\beta(g),\qquad t=\log(k/k_0). \] Redundant directions must be identified before essential physical inputs are counted. \begin{definition}[Ultraviolet critical family] For a specified exact flow with fixed point $g_*$, the ultraviolet critical family consists of trajectories existing for all sufficiently large $t$ and approaching $g_*$ as $t\to+\infty$, modulo the stated redundant identifications. A finite-dimensionality claim concerns that exact flow and essential quotient. \end{definition} \begin{proposition}[Projected fixed points do not determine the exact verdict] \label{prop:truncation} For every $N\ge1$, a smooth flow on $\R^{N+1}$ can have a projected fixed point in its first $N$ coordinates but no full fixed point. Two flows can also have the same projection and different dimensions of their ultraviolet-attracted families. Consequently, a critical-family dimension claim requires a theorem about the exact flow and essential quotient, not only an eigenvalue count in a truncation. \end{proposition} \begin{proof} Write $(x,y)\in\R^N\times\R$. The vector field \[ \beta(x,y)=(-x,1) \] has no zero, while its projected flow $-x$ has fixed point zero. Next compare \[ \beta_-(x,y)=(-x,-y),\qquad \beta_+(x,y)=(-x,y). \] Both project to $-x$ and have fixed point $(0,0)$. Every trajectory of the first approaches it, whereas approach for the second requires $y=0$. Their attracted families have dimensions $N+1$ and $N$. \end{proof} The proposition is a logical projection counterexample, not an argument against asymptotic safety. Truncation stability can be evidence, but does not replace control of omitted directions. \subsection{The preparation residual belongs to the match} Let $E_{\mathrm{out}}$ and $Y$ be Hausdorff locally convex spaces. After explicit comparison maps, let $F_\Lambda\in E_{\mathrm{out}}$ be the candidate's complete prepared observable. Preparation width $\eps>0$ is fixed. \begin{assumption}[Observable-level continuum matching] \label{ass:matching} The application supplies: \begin{enumerate}[label=\textbf{U\arabic*.}] \item specified cutoff theories, physical states or representations, and complete prepared observables; \item interpretation- and access-preserving comparison maps placing the observables in one $E_{\mathrm{out}}$; \item convergence $F_\Lambda\to F_\infty$ in that topology as $\Lambda_{\UV}\to\infty$; \item a continuous target symmetry map $Q:E_{\mathrm{out}}\to Y$ and transported defects $A_\Lambda$ with \begin{equation} A_\Lambda\to0,\qquad QF_\Lambda-A_\Lambda\to0; \label{eq:wardmatching} \end{equation} \item a low-energy comparison with an independently constructed prepared EFT observable, including the density-weighting residual, all preparation and switch derivatives, propagated layer data, and both required completion legs, with a controlled truncation remainder; \item a theorem fixing independent physical parameters and any independent observable calibrations. \end{enumerate} If a physical record is compared, the instrument and its observable comparison must additionally be specified. Matching the bulk action alone does not discharge U5. \end{assumption} \begin{theorem}[Conditional symmetry-preserving matching] \label{thm:matching} Under \cref{ass:matching}, \[ QF_\infty=0. \] For every continuous linear test $\ell$ whose constructed low-energy comparison gives \[ \left|\ell(F_\infty)-\ell(F_{\eps,p}^{\EFT})\right| \le r_{\ell,\eps,p}, \] the continuum observable recovers that prepared EFT prediction to the stated error. Exact equality with a finite perturbative truncation does not follow unless its remainder vanishes. Nor does this observable comparison establish equality of complete instruments or quantum channels. \end{theorem} \begin{proof} Continuity gives $QF_\Lambda\to QF_\infty$. The two limits in \cref{eq:wardmatching} give $QF_\Lambda\to0$. Hausdorffness yields $QF_\infty=0$. Continuity of $\ell$ permits evaluation of the limit. The displayed remainder is precisely the controlled comparison supplied by U5; it is not identified with zero. No complete-instrument hypothesis is implied by a linear-observable estimate alone. \end{proof} The preparation residual cannot be dropped by appealing to an unweighted divergence. The missing quantum contraction and remainder questions in Em02-E are a documentary model for the type of error control U5 needs, not that control itself \cite{CheungXVIB,CheungEmIIE}. \begin{proposition}[Positivity on a common physical algebra] \label{prop:state} Let $\mathfrak A$ be one common unital $*$-algebra. If normalized positive functionals $\omega_\Lambda$ converge pointwise on $\mathfrak A$ to $\omega_\infty$, then $\omega_\infty$ is linear, normalized, and positive. The common physical algebra is essential. This is not a positivity theorem for the full gauge-fixed ghost algebra. Hadamard product data alone do not supply the positive functionals in the hypothesis. \end{proposition} \begin{proof} Linearity passes through pointwise limits. Also $\omega_\infty(\mathbf1)=1$, and for every $a\in\mathfrak A$, \[ \omega_\infty(a^*a) =\lim_\Lambda\omega_\Lambda(a^*a)\ge0. \] \end{proof} Changing products, domains, or identifications requires compatible transport before this proposition applies. In gauge-fixed field theory, positivity must be stated on the physical algebra or representation. Formal coefficientwise expectations alone are not an ordinary positive measurement law. \subsection{Parameter control does not follow from action counting} If an exact construction depends only on a parameter in a specified finite-dimensional manifold, and its observable, state, and dynamics are all determined by that same parameter, the limit introduces no new independent parameters. This is conditional dependence, not existence of such a construction. Relational insertion normalizations must be fixed by calibration or controlled within the same exact critical family. Counting bulk couplings alone is insufficient. Conversely, the infinitely many possible measurement test functions are not automatically independent couplings of the theory. None of the supplied Cheung records or catalogue comparisons used here proves a asymptotic-safety fixed point for this prepared Einstein--scalar target. No exhaustive assertion about every manuscript in either corpus is made. Family 282 is not the missing fixed-point theorem. %===================================================================== \section{Documentary constraint ledger} \label{sec:ledger} %===================================================================== Each row records a comparison licensed by the supplied version or excerpt and the conclusion that is \emph{not} imported. The table is not a dependency graph, an exhaustive corpus audit, or verification of its source proofs. Generic titles are retained where complete metadata was not supplied. \begingroup \small \begin{longtable}{@{}P{.47\linewidth}P{.47\linewidth}@{}} \toprule Sheet and comparison used & What is not imported\\ \midrule \endfirsthead \toprule Sheet and comparison used & What is not imported\\ \midrule \endhead \bottomrule \endfoot \textbf{I, v2} \cite{CheungI}. Destructive computational-basis readout and consumption of the measured name. & No field instrument, apparatus limit, record-free uniqueness, or gravitational coefficient. \\[4pt] \textbf{Hawk04-A, v1} \cite{CheungHawk04A}. Complete preservation is distinguished from recovery through declared access. & No subsystem recovery, physical evaporation, or holographic identification follows from global unitarity alone. \\[4pt] \textbf{Hawk04-B, v2} \cite{CheungHawk04B}. Classical-output processing is distinguished from quantum-return decoding. & No decoder is supplied by a classical record, and no recovery task is assigned to the present correlator. \\[4pt] \textbf{XVII-A, v1} \cite{CheungRegisteredXVIIA}. The separate finite-matrix bounded-effect benchmark resolves the identifier collision. & Its reconstruction maps and time band are not this observable, an entanglement wedge, or spacetime emergence. \\[4pt] \textbf{VI, v1} \cite{CheungVI}. Classical relational descent and the RG-open quantum finite-renormalization audit remain distinct. & No complete quantum counterterm space, Ward identity, or RG certificate. \\[4pt] \textbf{VII-C, v1} \cite{CheungVIIC}. One logarithmic obstruction under its specified old-product and continuity requirements. & No universal obstruction to all algebras, topologies, or interacting constructions. \\[4pt] \textbf{VII-D, v1} \cite{CheungVIID}. One strong numerical finite part with a quantitative test-function remainder. & A finite part is not a binary algebra product or the present gravitational characteristic coefficient. \\[4pt] \textbf{VII-H, v1} \cite{CheungVIIH}. A free-field contact-ambiguity and invariant-test benchmark. & No identification with the reflected triangle, the inherited graph, or ultraviolet completion. \\[4pt] \textbf{VII-I, v1} \cite{CheungVIII}. Balanced forests and finite-width free constructions illustrate separate admission questions. & No transfer of sharp-product agreement, interacting BRST, or complete loop composition. \\[4pt] \textbf{IX-B, v2} \cite{CheungIXB}. One parent prescription can control an infinite derivative orbit. & No count of gravitational essential inputs or second-stage sharp multiplication. \\[4pt] \textbf{IX-D, v3} \cite{CheungIXD}. Free wordwise coherence, causal factorization, and normalized retarded orders are distinct. & No sharp $T_2$, normalized $R_1$, or $R_2$ on the unconstructed sector. \\[4pt] \textbf{X-A-conv, v1} \cite{CheungXAconv}. The particular frozen convolution has a covariance obstruction. & No successful ultraviolet-regulator identification and no theorem forbidding every convolution regularization. \\[4pt] \textbf{X-E, v1} \cite{CheungXE}. The one-tree defect belongs to its fixed model and normalization. & No complete one-loop inventory or loop counterterms. \\[4pt] \textbf{X-F, v1} \cite{CheungXF}. The plateau-supported all-tree identity is at $\kappa^2\hbar^0$. & No $\hbar^1$ identity, ghost completion, bulk BV restoration, or identification with the XII--XIV track. \\[4pt] \textbf{X-G, v1} \cite{CheungXG}. Free BRST, the specified variational-upgrade obstruction, formal-order qualifications, and loop-domain gaps. & No interacting quantum BRST upgrade or positive-$\hbar$ cancellation of a nonzero classical coefficient. \\[4pt] \textbf{XI-E, v1} \cite{CheungXIE}. Named polynomial Hadamard admission has its own domain. & No admission of every later response or loop product. \\[4pt] \textbf{XI-G, v1} \cite{CheungXIG}. The detecting free-shell class has a reported microcausal-representative obstruction. & No universal no-go, imported obstruction hypothesis, or decision on the Einstein subshell. \\[4pt] \textbf{XII-C, v1} \cite{CheungXIIC}. The cubic-only density weight retains the actual source derivatives without changing the lower-order freeze. & No derivation here of its component arrays, metric response, or observable-kernel admission. \\[4pt] \textbf{XIII-E, v1} \cite{CheungXIIIE}. Positive-regulator admission and good-set convergence precede the unfinished characteristic-set limit at E1. & No characteristic-limit theorem, uniform transverse estimates, or admitted coefficient linear in $\hbar$. \\[4pt] \textbf{XIV-H, v1} \cite{CheungXIVH}. Exact internal maps and regular charts stop at its uniform-amplitude gate H2. & No joint cross-line product or common characteristic representation from kernel existence alone. \\[4pt] \textbf{XIV-I, v1} \cite{CheungXIVI}. Its internal amplitude is constructed at I2; the joint characteristic map remains unresolved at I3. & Internal frequency estimates are not transverse estimates, a new graph regulator, or quantum admission. \\[4pt] \textbf{XIV-J, v1} \cite{CheungXIVJ}. Reconstruction-null relations and the supported $K_6+K_8+K_9$ reference are precise local constraints. & No full six-branch map across the characteristic locus, joint section, output theorem, or J2 closure. \\[4pt] \textbf{XV-B, v2} \cite{CheungXVB}. Technical relative lifting and finite normal-jet interfaces sharpen descent tests. & These technical theorems do not open the downstream quantum-admission exit or construct the inherited full map. \\[4pt] \textbf{XV-F, supplied v3 record} \cite{CheungXVF}. The source-recovery audit preserves its F0 stop. & No graph completion by documentary patches or canonical-domain equality. \\[4pt] \textbf{XVI-A, v1} \cite{CheungXVIA}. Normalization and retarded-slot obligations are recorded separately from source notation. & No constructed retarded quantum coefficient. \\[4pt] \textbf{XVI-B, v1} \cite{CheungXVIB}. Source, response, root, auxiliary evaluation, and class-kernel pairing are different objects. & No source-to-observable identification, late-response equivalence, or class admission from a source change. \\[4pt] \textbf{XVI-D, v1} \cite{CheungXVID}. A smooth prepared witness response has a separate coordinate-compatibility condition. & No ultraviolet subtraction, full joint kernel, or discharge of I3/J2. \\[4pt] \textbf{XVI-H, supplied v1 record} \cite{CheungXVIH}. A compact general-test kernel and partial witness recovery are reported under root inputs. & No full-kernel wavefront or class comparison is transferred; no claim that its root construction is nonunique. \\[4pt] \textbf{XVI-I, v1} \cite{CheungXVII}. A reduced two-edge sector has an extension family under imported cone hypotheses. & No selected interacting coefficient or ultraviolet completion from that family. \\[4pt] \textbf{XVI-J, v1} \cite{CheungXVIJ}. One narrowed undifferentiated square has a selected extension. & No equality with the entire XVI-I family or full interaction. \\[4pt] \textbf{XVI-K, v1} \cite{CheungXVIK}. A full Hessian is retained inside its specified selected sector. & Constant-slot regression is not the complete one-loop inventory. \\[4pt] \textbf{XVI-L, v1} \cite{CheungXVIL}. The complete source is retained in one two-edge contribution and a subtraction-scale family. & No full one-loop observable, physical beta function, Wilson match, or BRST/BV/Ward conclusion. \\[4pt] \textbf{XVI-M, supplied v1 record} \cite{CheungXVIM}. A degree-four component lies outside its declared curvature-square span. & The graph-specific obstruction and divergence statement are not broad renormalizability or a Ward identity. \\[4pt] \textbf{XVI-N, r12} \cite{CheungXVIN}. Theorem 11 treats one conditional massless special family of antifield-linear completions. & No arbitrary-$m>0$ theorem, Route B discharge, prepared BV complex, or ultraviolet conclusion. \\[4pt] \textbf{XVI-O, o70-sync} \cite{CheungXVIO}. The kernel counterterm has a stated prescription and a separate subsector verdict. & No bulk BV restoration; O.1 remains \NOTCLOSED. \\[4pt] \textbf{XVI-P, supplied p23 record} \cite{CheungXVIP}. A renormalized $\kappa^2$ two-point sector and external-leg scheme are recorded separately. & QdS is not a de Sitter state, the frozen insertion is not completed, and O.1 is not closed. \\[4pt] \textbf{Em01-C, v3} \cite{CheungEmIC}. Characteristic coordinates, source reduction, and conditional fixed-kernel scale criteria. & The physical complete-sum origin coefficient and crossing-locus limit remain unevaluated. \\[4pt] \textbf{Em02-E, v1} \cite{CheungEmIIE}. The classical relational lapse and incomplete quantum audit provide a regulator-removal checklist. & No quantum covariance, state, counterterm classification, remainder, or gauge closure is constructed by this comparison. \\[4pt] \textbf{Oai282-A, v1} \cite{CheungOaiIIA}. Finite local Ward residuals retain contacts and coverage conditions. & No unrestricted identity, Family 282 verification, or gravitational counterterm map. \\[4pt] \textbf{Oai282-B, v1} \cite{CheungOaiIIB}. The ordered free-scalar Wick class has its own $L=-Q/2$ and two-probe stress coverage. & No interacting improvement, unrestricted tensor coverage, global conformal action, or identification with the XIV programme's objects. \\[4pt] \textbf{Oai069-A, supplied summary} \cite{CheungOai069A}. A described finite free-scalar Ward-defect comparison. & No full theorem is imported from the unavailable source; no positive-channel or gravitational interpretation of its finite-monodromy decoder. \\[4pt] \textbf{OpenAI catalogue / Family 282, supplied record} \cite{Catalogue}. A recorded local-Ward scope comparison. & No independent attribution audit, proof verification, global action, gravitational fixed point, or ultraviolet match; no corpus-wide exclusivity claim. \\ \end{longtable} \endgroup \subsection{External analytical frameworks} The cited external articles provide wavefront calculus, extension methods, causal perturbation theory, BV anomaly analysis, relational-correlator examples, and RG frameworks \cite{BDHintro,BDHops,BF,Dang,FR,FRBV,FrobWard,BDF,BFR, FrobCorrelation,ReuterFlow,ReuterSaueressig}. Their hypotheses must be checked at each use. None verifies a companion sheet or supplies the missing characteristic product by analogy. In particular, an extension theorem starts with an actual distribution on its stated reference domain. It cannot manufacture an undefined joint characteristic operation. \subsection{Version and source limitations} XVI-E/F/G are referenced within the supplied documentary material, but their full auditable versions are not supplied as mathematical inputs here. Later reports of witness compatibility do not establish full-kernel admission. Earlier unresolved labels are not reinstated after a later source reports a narrower closure. For Oai069-A, the supplied material gives a scope summary, not a full theorem and proof. Its row therefore records only that requested comparison. For XV-B, ``not opened'' refers to the inherited quantum-admission exit, not nonexistence of the separately supplied technical manuscript or its interface theorems. Likewise, XIV-H's H2 stop is version-local: XIV-I reports its named internal I2 advance while retaining I3 as the different missing operation. No catalogue counts, release-date claims, formalization percentages, or current-status claims are needed by any proof here. %===================================================================== \section{Research gates and their non-identification boundaries} \label{sec:programme} %===================================================================== These are proposed acceptance conditions, not completed constructions. Passing one gate does not pass the others or change a separately inherited admission status. \subsection{Gate 1: specification and source audit} Complete \cref{req:contract}. Select exact versions of every proposed comparison and reproduce or separately assume each analytical input needed in the application. Retain the preparation derivatives exposed by \cref{prop:weighted} and the cubic-only freeze of XII-C. \paragraph{Pass condition.} A reproducible target definition, configuration domain, allowed-change catalogue, and applicability verdict for each imported estimate. \paragraph{Does not release.} Specification and source identities do not define a response, full class kernel, field instrument, or interacting quantum product. \subsection{Gate 2: the full prepared kernel} Construct the kernel on arbitrary joint tests. Prove support, slot symmetry, wavefront or trisolution control, and every all-free-triples comparison used for class transfer. Pure-trace witness recovery is insufficient by \cref{thm:witness,cor:smoothwitness}. \paragraph{Pass condition.} An admitted full kernel in its stated topology, or an exhaustive obstruction for a precisely fixed candidate class. Missing information alone gives an unresolved verdict. \paragraph{Does not release.} Changing a class removes a transfer argument but does not prove microcausal admission. Kernel admission is not observable closure, measurement, or quantum graph composition. \subsection{Gate 3: the complete characteristic operation} Construct the joint map on tests meeting the characteristic locus, then its joint section and controlled remainder, uniform transverse estimates, contact compatibility, and complete output wavefront theorem. The required full internal sum retains \[ K_{47}+K_5+K_6+K_8+K_9, \] with both $K_5$ placements. The supported $K_6+K_8+K_9$ reference is not the full map. Internal pole estimates do not define the subsequent grouped cross-line operation \cite{CheungXIVH,CheungXIVI,CheungXIVJ}. Em01-C's crossing-locus target and XIII-E's alternative characteristic-limit target also retain their own missing hypotheses \cite{CheungEmIC,CheungXIIIE}. VII-C's obstruction and VII-D's finite part are separate comparisons, not substitutes \cite{CheungVIIC,CheungVIID}. \paragraph{Pass condition.} Actual joint distributions, reconstruction descent, coherent extensions, spectator compatibility, and admitted final outputs for the same complete coefficient. \paragraph{Does not release.} A defined internal line, proper projection, finite normal jet, or collision extension theorem does not define an absent characteristic product. A completed mathematical graph still does not by itself establish a Ward identity or open an inherited exit without its exact admission and comparison obligations. \subsection{Gate 4: inventory, assembly, and the selected identity} Evaluate every entry of \cref{eq:completecoefficient}, including ghosts, preparation terms, and insertion normalizations. Count scalar completion once. Construct the full defect and admissible counterterm map before applying \cref{thm:ward}. If the leading classical coefficient fails, its repair must be classical; positive-$\hbar$ cancellation is excluded by \cref{prop:hbar}. \paragraph{Pass condition.} One complete fixed-order interacting correlator or response with one proved symmetry identity on its declared domain. \paragraph{Does not release.} The abstract affine quotient is not an interacting BRST complex. A result for a newly specified model does not admit the inherited companion quantum coefficient without an explicit identification theorem. \subsection{Gate 5: physical normalization and scale transport} Determine $B(\Ker L)$ and a physically calibrated quotient class or invariant test. Keep preparation fixed. Prove the total RG transport, including insertion and state transport, instead of reading a beta function from explicit $\mu$ dependence. \paragraph{Pass condition.} A complete finite-renormalization ledger and a proved statement of what survives all compatible changes. \paragraph{Does not release.} A graph-specific curvature-span obstruction, divergence formula, or subtraction-scale family is not a full renormalizability theorem, physical beta function, or broad matched-scale release. \subsection{Gate 6: a specified ultraviolet candidate} If asymptotic safety is selected, construct the exact flow, essential quotient, omitted-direction control, and critical-family theorem. Include the relational insertion's flow and mixing, not only bulk action flow. State the physical state or representation and its cutoff transport. \paragraph{Pass condition.} A named candidate with a controlled continuum construction and an exact parameter statement. \paragraph{Does not release.} A finite truncation is not the exact flow. No result here discharges XVI-N, repairs Route B, restores bulk BV symmetry, or closes O.1. Family 282 is not the required gravitational fixed point. \subsection{Gate 7: prepared observable-level matching} Supply \cref{ass:matching}. Retain the preparation residual, propagated response data, physical state, symmetry transport, and low-energy remainder. Cutoff removal is separate from $\eps\downarrow0$ and from an adiabatic limit. \paragraph{Pass condition.} A connecting theorem recovering the same prepared Einstein--scalar observable with controlled errors and no unexplained independent calibration inputs. \paragraph{Does not release.} Matching a correlator is not a decoder or complete-instrument comparison. It does not automatically open XV-B, admit the separately inherited coefficient, or construct spacetime emergence, an entanglement wedge, a Page curve, or an island. %===================================================================== \section{Corrections and exact mathematical release} \label{sec:release} %===================================================================== The substantive debugging corrections are: \begin{enumerate} \item The identifier is consistently XVIII-A, version 2; the separate XVII-A benchmark is not overwritten. \item Readout, observable representation, quantum return, and physical recovery are kept distinct. \item Smooth witness non-determination is explicit; the singular example is used for failure of wavefront determination. \item Class difference is not promoted to class admission. \item Exact Ward cancellation uses the actual counterterm image, not its closure. \item Finite-probe injectivity is stated on its declared vector space, with the weaker affine-family qualification kept separate. \item Positive formal quantum orders do not repair a nonzero classical coefficient. \item Normalized retarded extraction accounts for division by $\hbar$ and any explicit coupling dependence of the maps. \item Balanced forests are not treated as loop products, and weak convergence is not wavefront convergence. \item Finite normal-order ambiguity is distinguished from finite-dimensional ambiguity; point support is distinguished from positive-dimensional support. \item Observable matching retains preparation residuals and a nonzero perturbative remainder. \item Documentary stopping rules remain version-local; neither catalogue analogies nor technical lifting theorems change an inherited admission status. \end{enumerate} \begin{theorem}[Exact mathematical release] \label{thm:release} This manuscript establishes: \begin{enumerate}[label=\textup{(\roman*)}] \item pure-trace witness non-determination of compact kernels, microcausal admission, and free-shell classes, including smooth compact non-determination; \item the weighted-divergence residual and the late-source/retarded-response non-implication; \item local wavefront equality under nonvanishing smooth multiplication and compact balanced-forest composition; \item reconstruction descent, conditional class-obstruction transfer, and formal-order separation; \item exact formal coefficient inventory under a declared normalized retarded family; \item exact affine Ward compatibility and the quotient invariant under compatible finite changes; \item finite-probe, bounded-normal-order, and projected-flow non-implications; \item conditional fixed-order assembly and symmetry-preserving continuum observable matching. \end{enumerate} The independence or conditionality stated in each result is retained. No complete inherited gravitational characteristic coefficient, evaluated interacting one-loop observable, bulk BV restoration, or ultraviolet completion is established. \end{theorem} \begin{proof} Item (i) is \cref{thm:witness,cor:smoothwitness}; (ii) is \cref{prop:weighted,prop:lateresponse}; (iii) is \cref{lem:switchWF,thm:forest}; (iv) is \cref{prop:descent,prop:classtransfer,prop:hbar}; (v) is \cref{prop:inventory}; (vi) is \cref{thm:ward,cor:scheme}; (vii) is \cref{prop:finitevalidation,prop:normalorder,prop:truncation}; and (viii) is \cref{thm:assembly,thm:matching}. The final sentence states the boundary of the constructions. The substantive hypotheses of the conditional theorems have not been supplied for the concrete inherited gravitational target. \end{proof} \paragraph{Unchanged documentary statuses.} XV-B's inherited admission exit is not opened. XVI-N and Route B are not discharged. No inherited characteristic quantum coefficient is admitted. O.1 remains \NOTCLOSED. These statements delimit the present work; they are not additional mathematical nonexistence theorems. \section*{Declarations} \addcontentsline{toc}{section}{Declarations} \paragraph{Submission scope.} The source is a standalone analytical interface manuscript. Its submission claim must be the theorem package printed here, not a completed gravitational one-loop or ultraviolet construction. Responsible author metadata and independent technical review remain necessary. \paragraph{Source scope.} Cheung references identify supplied versions, excerpts, or summaries. Their full parent proofs and artifacts are not independently certified. The supplied OpenAI catalogue attribution is not authenticated here and is not a premise. Stable archival sources should accompany actual review. \paragraph{Originality.} Elementary linear-algebra, distributional, and formal-series arguments are not claimed as new foundational theorems. Corresponding supplied arguments are attributed. The contribution is their coherent proof package and application contract. \paragraph{Operational scope.} No field instrument, calibrated apparatus, quantum-return decoder, or physical code-recovery task is constructed. A correlator coefficient is not automatically a probability law; measurement implementation remains separate. \paragraph{Physical scope.} No new metric, gravitational Hamiltonian, de Sitter state, spacetime-emergence theorem, entanglement wedge, island, Page curve, or evaporation model is derived. \paragraph{Verification.} No successful \texttt{pdflatex} build, numerical or symbolic execution, proof-assistant verification, independent referee validation, registration, or journal acceptance is asserted. \paragraph{Authorship.} Citing N.~Cheung does not attribute authorship of this revision to that author. Authorship authorization must be supplied by the submitting author. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} % Documentary references: supplied versions or summaries. % No journal metadata or archival availability is inferred. \bibitem{CheungI} N.~Cheung, \emph{Landmark I}, version 2, unpublished manuscript, n.d. Supplied \sourceid{I.tex}; destructive-readout semantics and scope. Enter Calculus Companion, \url{https://enter.grok.me}. \bibitem{CheungHawk04A} N.~Cheung, \emph{Hawk04-A}, version 1, unpublished manuscript, n.d. Supplied \sourceid{Hawk04-A.tex}; complete preservation versus declared-output recovery. \url{https://enter.grok.me}. \bibitem{CheungHawk04B} N.~Cheung, \emph{Hawk04-B}, version 2, unpublished manuscript, n.d. Supplied \sourceid{Hawk04-B.tex}, \sourceid{thm:classical}; classical-output recovery boundary. \url{https://enter.grok.me}. \bibitem{CheungRegisteredXVIIA} N.~Cheung, \emph{XVII-A: One boundary reconstruction is not holographic emergence}, version 1, unpublished manuscript, n.d. Supplied \sourceid{XVII-A.tex}, \sourceid{lem:timeband}, and \sourceid{thm:release}; separate finite-matrix AdS$_2$ benchmark. \url{https://enter.grok.me/browse/XVII-A}. \bibitem{CheungVI} N.~Cheung, \emph{Landmark VI}, version 1, unpublished manuscript, n.d. Supplied classical relational-descent and quantum finite-renormalization/RG audit. \url{https://enter.grok.me}. \bibitem{CheungVIIC} N.~Cheung, \emph{VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:main}; specified continuity obstruction. \url{https://enter.grok.me}. \bibitem{CheungVIID} N.~Cheung, \emph{VII-D: Finite part of the regulated Cauchy-reflection triangle}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:finitepart}; numerical distributional finite part. \url{https://enter.grok.me}. \bibitem{CheungVIIH} N.~Cheung, \emph{VII-H: Normalization-independent connected quantum responses: complete ambiguity, observable descent, and a nonzero free-field benchmark}, version 1, unpublished manuscript, n.d. Supplied contact-normalization and physical-matching ledger. \url{https://enter.grok.me}. \bibitem{CheungVIII} N.~Cheung, \emph{Landmark VII-I}, version 1, unpublished manuscript, n.d. Supplied \sourceid{lem:tree}, \sourceid{thm:rampadmission}, and \sourceid{thm:release}. \url{https://enter.grok.me}. \bibitem{CheungIXB} N.~Cheung, \emph{Landmark IX-B}, version 2, unpublished manuscript, n.d. Supplied parent finite-part conversion and derivative-orbit distinction; \sourceid{thm:descendants}, \sourceid{cor:conversion}. \url{https://enter.grok.me}. \bibitem{CheungIXD} N.~Cheung, \emph{Landmark IX-D}, version 3, unpublished manuscript, n.d. Supplied \sourceid{thm:release}; free, time-ordered, and normalized retarded interfaces. \url{https://enter.grok.me}. \bibitem{CheungXAconv} N.~Cheung, \emph{X-A-conv}, version 1, unpublished manuscript, n.d. Supplied fixed-convolution covariance comparison. \url{https://enter.grok.me}. \bibitem{CheungXE} N.~Cheung, \emph{Landmark X-E}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:release}; fixed-normalization one-tree construction and truncation defect. \url{https://enter.grok.me}. \bibitem{CheungXF} N.~Cheung, \emph{Landmark X-F}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:release}; plateau-supported $\kappa^2\hbar^0$ tree identity. \url{https://enter.grok.me}. \bibitem{CheungXG} N.~Cheung, \emph{Landmark X-G}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:release}; free BRST, variational-upgrade obstruction, and unresolved interacting interfaces. \url{https://enter.grok.me}. \bibitem{CheungXIE} N.~Cheung, \emph{Landmark XI-E}, version 1, unpublished manuscript, n.d. Supplied \sourceid{def:domain}, \sourceid{thm:star}, and release. \url{https://enter.grok.me}. \bibitem{CheungXIG} N.~Cheung, \emph{Landmark XI-G}, version 1, unpublished manuscript, n.d. Supplied \sourceid{def:class}, \sourceid{thm:criterion}, and \sourceid{thm:obstruction}; detecting free-shell class. \url{https://enter.grok.me}. \bibitem{CheungXIIC} N.~Cheung, \emph{XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question}, version 1, unpublished manuscript, n.d. Supplied cubic-only freeze and unintegrated density-source convention. \url{https://enter.grok.me}. \bibitem{CheungXIIIE} N.~Cheung, \emph{Landmark XIII-E}, version 1, unpublished manuscript, n.d. Supplied \sourceid{prop:finite}, \sourceid{prop:good-limit}, \sourceid{def:E1-missing}, and \sourceid{thm:release}. \url{https://enter.grok.me}. \bibitem{CheungXIVH} N.~Cheung, \emph{XIV-H: Internal kernel relations on the operator route}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:H2}, \sourceid{prop:K5exact}, and \sourceid{thm:release}. \url{https://enter.grok.me}. \bibitem{CheungXIVI} N.~Cheung, \emph{Landmark XIV-I}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:I2}, \sourceid{thm:I3}, \sourceid{prop:canonical}, and \sourceid{thm:release}. \url{https://enter.grok.me}. \bibitem{CheungXIVJ} N.~Cheung, \emph{Landmark XIV-J}, version 1, unpublished manuscript, n.d. Supplied \sourceid{prop:descent}, \sourceid{lem:null}, \sourceid{thm:reference}, \sourceid{thm:J2}, and \sourceid{cor:noadmission}. \url{https://enter.grok.me}. \bibitem{CheungXVB} N.~Cheung, \emph{Landmark XV-B}, version 2, unpublished manuscript, n.d. Supplied \sourceid{thm:lifting}, \sourceid{thm:jetnull}, and \sourceid{thm:release}; technical reconstruction interfaces. \url{https://enter.grok.me}. \bibitem{CheungXVF} N.~Cheung, \emph{Landmark XV-F}, supplied version 3 record, unpublished manuscript, n.d. Source-recovery audit and unresolved F0 package. \url{https://enter.grok.me}. \bibitem{CheungXVIA} N.~Cheung, \emph{Landmark XVI-A}, version 1, unpublished manuscript, n.d. Supplied normalization, class-transfer audit, and retarded-slot obligations. \url{https://enter.grok.me}. \bibitem{CheungXVIB} N.~Cheung, \emph{Landmark XVI-B}, version 1, unpublished manuscript, n.d. Supplied \sourceid{prop:transfer} and fixed-width complete-pairing specification. \url{https://enter.grok.me}. \bibitem{CheungXVID} N.~Cheung, \emph{Landmark XVI-D}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:response-package}; prepared witness response. \url{https://enter.grok.me}. \bibitem{CheungXVIH} N.~Cheung, \emph{XVI-H: Compact polarized kernel and witness recovery}, supplied version 1 record, unpublished manuscript, n.d. Supplied \sourceid{thm:trep}, \sourceid{prop:tmatch}, and full-kernel wavefront/class boundary. \url{https://enter.grok.me/browse/XVI-H}. \bibitem{CheungXVII} N.~Cheung, \emph{Landmark XVI-I}, version 1, unpublished manuscript, n.d. Supplied reduced two-edge sector and unnormalized extension family. \url{https://enter.grok.me}. \bibitem{CheungXVIJ} N.~Cheung, \emph{Landmark XVI-J}, version 1, unpublished manuscript, n.d. Supplied narrowed square and conditional root attachment. \url{https://enter.grok.me}. \bibitem{CheungXVIK} N.~Cheung, \emph{Landmark XVI-K}, version 1, unpublished manuscript, n.d. Supplied selected-sector full Hessian and constant-slot regression. \url{https://enter.grok.me}. \bibitem{CheungXVIL} N.~Cheung, \emph{Landmark XVI-L}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:Lmain}, complete-source two-edge contribution, scale-family scope, and L.8 matching questions. \url{https://enter.grok.me}. \bibitem{CheungXVIM} N.~Cheung, \emph{Landmark XVI-M}, supplied version 1 record, unpublished manuscript, n.d. Supplied \sourceid{thm:wpB-dp}, \sourceid{cor:wpB-MC}, and top-versus-full scale-tensor qualifications. \url{https://enter.grok.me}. \bibitem{CheungXVIN} N.~Cheung, \emph{Landmark XVI-N}, revision r12, unpublished manuscript, n.d. Supplied \sourceid{thm:massless-obstruction}; Theorem 11 and Route A/Route B qualifications. \url{https://enter.grok.me}. \bibitem{CheungXVIO} N.~Cheung, \emph{Landmark XVI-O}, revision o70-sync, unpublished manuscript, n.d. Supplied kernel-prescription verdict and separate unresolved bulk O.1 status. \url{https://enter.grok.me}. \bibitem{CheungXVIP} N.~Cheung, \emph{Landmark XVI-P}, supplied p23 record, unpublished manuscript, n.d. Selected two-point sector, external-leg scheme, and retained bulk-restoration boundary. \url{https://enter.grok.me}. \bibitem{CheungEmIC} N.~Cheung, \emph{Em01-C}, version 3, unpublished manuscript, n.d. Supplied \sourceid{Em01-C-v3.tex}, \sourceid{thm:coordinates}, and \sourceid{thm:release}; physical complete-sum limit remains unevaluated. \url{https://enter.grok.me}. \bibitem{CheungEmIIE} N.~Cheung, \emph{Em02-E}, version 1, unpublished manuscript, n.d. Supplied classical relational-lapse release and incomplete quantum regulator-removal audit. \url{https://enter.grok.me}. \bibitem{CheungOaiIIA} N.~Cheung, \emph{Oai282-A}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:ward}; finite local-Ward insertion contract and coverage conditions. \url{https://enter.grok.me}. \bibitem{CheungOaiIIB} N.~Cheung, \emph{Oai282-B}, version 1, unpublished manuscript, n.d. Supplied \sourceid{thm:coverage} and \sourceid{thm:release}; ordered free-scalar Wick class and homogeneous stress coverage. \url{https://enter.grok.me}. \bibitem{CheungOai069A} N.~Cheung, \emph{Oai069-A}, unpublished manuscript, n.d.; version not supplied. Cited only through the supplied scope summary concerning a finite free-scalar Ward-defect comparison. No full theorem or finite-monodromy implementation is assumed or independently verified here. \url{https://enter.grok.me}. \bibitem{Catalogue} \emph{OpenAI mathematics catalogue / Family 282}, attributed to OpenAI in the supplied companion record. Documentary summary only; attribution and mathematical proofs are not independently authenticated here. 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