% XIX-E.tex % Self-contained pdfLaTeX source. % No external bibliography, figures, code, or input files are required. % % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XIX-E.tex % pdflatex -interaction=nonstopmode -halt-on-error XIX-E.tex % % No successful compilation or proof-assistant execution is asserted. % Anonymous submission: authorized authorship metadata and stable copies % of cited unpublished versions must be supplied before signed submission. % % Principal scope corrections: % 1. XIX-D is a quantum audit, not a completed quantum construction. % 2. XIX-B compares first coordinate coefficients, not the complete % harmonic relational scalar with a coordinate candidate. % 3. XVI-H reports a general-test kernel under root inputs; its partial % contraction, not the kernel itself, is the partial witness datum. % 4. XIV-H's historical H2 stop is not allowed to erase XIV-I's I2 advance. % 5. Finite quantum updates are not denied; their native continuum, % physical, and external-time-free interpretations remain separate. % 6. Oai comparison-sheet authorship is distinguished from catalogue % attribution to OpenAI. % 7. Release boundaries are not universal nonexistence theorems. \documentclass[11pt,a4paper]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=27mm]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,tabularx,array,longtable} \usepackage{enumitem} \usepackage{microtype} \usepackage{xurl} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={XIX-E: Boundary of the Clock-Separation Programme}, pdfauthor={}, pdfsubject={Nontransfer, physical interpretation, and unchanged construction obligations}, pdfkeywords={relational clock, boundary inventory, continuum evolution, scheduled support, quantum certificate, characteristic kernel, spacetime emergence} } \setlength{\emergencystretch}{3em} \setlength{\parskip}{3pt} \setlength{\parindent}{15pt} \numberwithin{equation}{section} \setlist[enumerate]{leftmargin=*,itemsep=3pt} \setlist[itemize]{leftmargin=*,itemsep=3pt} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{convention}[theorem]{Convention} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\dd}{\mathrm d} \newcommand{\id}{\operatorname{id}} \newcommand{\im}{\operatorname{im}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\Lie}{\mathcal L} \newcommand{\Rec}{\mathsf{Rec}} \newcommand{\CV}{\mathsf C_{\mathrm{VI}}} \newcommand{\CE}{\mathsf C_{\mathrm E}} \newcommand{\CI}{\mathsf C_{\mathrm{VIII}}} \newcommand{\RX}{\mathsf R_{\mathrm X}} \newcommand{\Rconv}{\mathsf R_{\mathrm{conv}}} \newcommand{\NN}{\mathcal N_{\mathrm E}} \newcommand{\Alg}{\mathcal A} \newcommand{\ket}[1]{\lvert #1\rangle} \newcommand{\bra}[1]{\langle #1\rvert} \newcommand{\kb}[2]{\lvert #1\rangle\langle #2\rvert} \newcommand{\pair}[2]{\left\langle #1,#2\right\rangle} \newcommand{\norm}[1]{\left\lVert #1\right\rVert} \newcommand{\abs}[1]{\left\lvert #1\right\rvert} \newcommand{\sid}[1]{\texttt{\detokenize{#1}}} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \title{\textbf{XIX-E}\\[3pt] Boundary of the Clock-Separation Programme\\ \large Nontransfer, Physical Interpretation, and Unchanged Obligations} \author{} \date{} \begin{document} \maketitle \begin{abstract} We record the exact boundary of a programme separating a harmonic relational scalar, an original-rod scalar-clock lapse, a prescribed incidence register, and a coordinate-density ramp. The first three stages concern classical prescriptions and finite interfaces; the fourth inventories unclosed quantum inputs. This final note neither identifies their targets nor completes a quantum or continuum construction. The boundary is supported by self-contained comparison arguments. Fixed density multiplication and fixed convolution are different operations. A density ramp can fail the first harmonic pure-gauge response test in its transition region without contradicting a plateau-restricted identity. Pure-trace evaluations have a smooth kernel blind sector. Agreement away from an extension locus does not select an extension, and finite normal order does not imply a finite-dimensional ambiguity. Scheduled channel support does not select a metric, clock calibration, or continuous-time generator. Even exactly consistent subdivisions of one finite channel can have different continuous-time limits. Finally, localization and smoothing coefficients do not transfer without a specified comparison map and parameter identification. The remaining claims are version-qualified documentary boundaries: VI's RG-open certificate, the five quantum inputs of the original-rod lapse, the native incidence continuum realization, the complete characteristic map, insertion and route obligations, and physical or holographic interpretation. External catalogue entries are not theorem premises. No inherited characteristic quantum coefficient, quantum evolution without external time, compilation, independent predecessor verification, or full spacetime emergence is released. \end{abstract} \noindent\textbf{Keywords:} clock separation; release boundary; relational observable; scheduled support; continuum non-determination; kernel recovery; quantum certificate; spacetime emergence. \tableofcontents %===================================================================== \section{Question and exact kind of conclusion} \label{sec:scope} %===================================================================== The question is what remains outside the clock-separation programme after XIX-A, XIX-B, XIX-C, and XIX-D \cite{XIXA,XIXB,XIXC,XIXD}. The answer is a boundary inventory, supplemented by explicit non-determination arguments. It is not a new theorem that every future clock identification, quantum construction, or spacetime realization is impossible. The four entries remain \begin{equation} \boxed{\CV,\qquad\CE,\qquad\CI,\qquad\RX.} \label{eq:labels} \end{equation} They denote, respectively: \begin{enumerate}[label=\textup{(\roman*)}] \item a scalar dressed by a specified harmonic-coordinate map; \item a matter-clock lapse pulled back using the original clock-slice harmonic spatial rods; \item a prescribed incidence parameter together with finite reading and update interfaces; \item a prescribed profile placed in a coordinate density. \end{enumerate} These are four time-related entries, not four certified physical or quantum clocks. The fourth is the named non-clock. The motivating manuscript records are Cheung's VI, VIII-A/B/C, X-A, X-A-conv, Em02-E, and the analytic and operational comparisons specified below \cite{CheungVI,CheungVIIIA,CheungVIIIB,CheungVIIIC, CheungXA,CheungXAconv,CheungEm02E}. Their equations, domains, gauges, auxiliary inputs, and releases are not identified by their names. Relational-coordinate constructions provide established context \cite{BFHPR}. Quantum fields on a classical background and their states, composites, and products provide another context \cite{HW,BF}. Neither context automatically supplies the particular quantum inputs or comparisons required by this programme. \subsection{Three different statements} \begin{definition}[Three boundary statements] \label{def:boundary} We distinguish: \begin{enumerate}[label=\textup{(\roman*)}] \item \textbf{Nonidentity on a declared comparison.} Two specified prescriptions differ on a common domain, or have incompatible declared types without an additional map. \item \textbf{Non-determination by specified data.} There are two realizations agreeing on the named input data and disagreeing on the requested target. \item \textbf{Not constructed in this release.} The manuscript contains no realization and proof of the requested object. \end{enumerate} Only a separately proved incompatibility over an exhaustive candidate class warrants a universal obstruction within that class. \end{definition} Statements of type (iii) are release statements, not mathematical proofs of nonexistence. Theorems in this note concern type (ii) or the explicitly displayed prescriptions of type (i). Source-status inventories are identified as documentary. \subsection{Source discipline} All proofs below use displayed definitions and elementary arguments. The unprinted proofs, computational artifacts, catalogue claims, and reported checks of predecessor manuscripts are not theorem premises. A citation records an attributed source and its scope; it does not independently verify that source. Likewise, a source dependency, a continuation label, or a proposed comparison is not itself a proved identification. The convention \[ \boxed{\text{Declared path: null}} \] means that this note supplies no cross-programme identification or proof-transfer path. It does not prohibit a future explicitly defined comparison, and does not describe a mathematical map as identically zero. %===================================================================== \section{The preceding stages and the necessary scope corrections} \label{sec:stages} %===================================================================== \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.12\linewidth}YY@{}} \toprule Stage & Supplied result or inventory & Conclusion not inherited\\ \midrule XIX-A & Four labelled objects and classical detecting nonidentities & A common physical clock, common functional, or quantum upgrade\\[4pt] XIX-B & Mismatch of one direct first-coordinate comparison & A common coordinate field through that route, or a quantum anomaly\\[4pt] XIX-C & Finite incidence interfaces and continuum non-determination & Native continuum evolution, dynamical cone, or invariant physical speed\\[4pt] XIX-D & Quantum input inventory and nontransfer criteria & VI certificate closure, lapse gauge closure, or native quantum evolution\\ \bottomrule \end{tabularx} \end{center} This table summarizes supplied manuscripts \cite{XIXA,XIXB,XIXC,XIXD}. Its entries are not premises of the independent comparison proofs below. \paragraph{The coordinate mismatch has a precise target.} XIX-B compares VI's \emph{first coordinate coefficient} \(w^\mu[h]\) with the prepared coefficient \(X_{\mathrm{cand}}^\mu[R]\). After the specified signature transport and equal ordinary zero Cauchy data, its detecting forcing difference is \[ \mathfrak r_\lambda^\mu =\partial_t\bigl((\sigma-1)V_\lambda^{0\mu}\bigr). \] That paper does not compare the complete matter scalar \(T\circ H_q^{-1}\) directly with a coordinate coefficient. Nor does its coordinate mismatch establish nonidentity of every possible complete observable, whose scalar legs would also have to be matched \cite{XIXB}. The compact scalar-switch defect is another classical identity. It is not an obstruction to smooth harmonic-coordinate existence, and it is not a loop anomaly. \paragraph{The fourth stage is an audit.} XIX-A/B/C are classical or finite-interface stages. XIX-D is a quantum-certificate inventory, not a completed quantum theory. Its elementary comparison channels do not become the native incidence dynamics \cite{XIXD}. \paragraph{Finite quantum updates are not excluded.} A finite register can have a complete quantum channel, instrument, or unitary implementation. The missing conclusion is a specified native continuum evolution or a clock-defined quantum evolution without external time, not the existence of every finite quantum update \cite{XIXC,XIXD}. \paragraph{A constructed kernel is not erased by a partial witness.} The supplied XVI-H record reports a compact general-test three-slot kernel under its stated root inputs. The partial object is its selected contraction to witness data. We do not relabel the reported kernel itself as merely a partial kernel \cite{CheungXVIH}. Its full-kernel admission and quantum-product obligations remain separate. \paragraph{Historical and later analytic gates are not conflated.} XIV-H's H2 amplitude gap is the stop \emph{within that sheet}. XIV-I subsequently reports a constructed internal I2 amplitude module and a remaining I3 joint-map gap. XIV-J reports a supported-sector reference and a remaining full-map gap at J2 \cite{CheungXIVH,CheungXIVI,CheungXIVJ}. The shorthand ``H2/I3/J2 unresolved'' must not erase that narrower I2 advance. %===================================================================== \section{Prescribed profiles, convolution, and clocks} \label{sec:profiles} %===================================================================== \subsection{External inputs are not matter-defined clocks} \begin{definition}[Clock criterion used for this comparison] \label{def:clock} A dynamical scalar-clock prescription specifies a scalar functional of the dynamical configuration, a region on which its gradient is nonzero and timelike, an orientation, and a calibration. A fixed external profile or a fixed smoothing operation is not such a prescription merely because coordinate time occurs in its definition. \end{definition} For example, let \[ \sigma_\epsilon(t) =\chi\!\left(\frac{t}{\epsilon}\right), \qquad \chi(u)=0\ (u\le0),\quad \chi(u)=1\ (u\ge1), \] with \(\chi\) smooth and nonconstant. Held fixed, \(\sigma_\epsilon\) has zero variation with respect to the metric and matter fields. Its differential vanishes on both plateaux. It therefore does not satisfy Definition~\ref{def:clock} as a dynamical clock on the full slab. A strictly monotone transition segment may still reparametrize an interval. This is an externally prescribed parametrization, not the construction of a matter-defined clock. For X-A-conv, the corresponding non-clock entry \(\Rconv\) means the \emph{fixed convolution prescription}: kernel, cutoff, width, and placement. It is an operation, not a scalar clock field. The field dependence of an observable formed with that operation is not asserted to vanish. These distinctions do not prove that no separately constructed smoothed scalar could be a clock. They prevent identification of the external regulator data with that scalar. \subsection{Density multiplication and convolution do not coincide} Let \[ (M_\sigma f)(t)=\sigma(t)f(t), \qquad (S_\rho f)(t)=\int_\R \rho(u)f(t-u)\,\dd u, \] where \(\rho\in C_c^\infty(\R)\) is nonnegative, has integral one, and is strictly positive near zero. \begin{proposition}[Different placement operations] \label{prop:placement} If \(\sigma\) is smooth and nonconstant, the commutator \(M_\sigma S_\rho-S_\rho M_\sigma\) is nonzero on \(C_c^\infty(\R)\). More explicitly, \begin{equation} \bigl(M_\sigma S_\rho-S_\rho M_\sigma\bigr)f(t) =\int\rho(u)\bigl(\sigma(t)-\sigma(t-u)\bigr)f(t-u)\,\dd u. \label{eq:commutator} \end{equation} \end{proposition} \begin{proof} The formula follows by subtraction. There is \(t_c\) with \(\sigma'(t_c)\ne0\). Choose a sufficiently small \(u_0>0\) such that \(\rho(u_0)>0\) and \(\sigma(t_c)-\sigma(t_c-u_0)\ne0\). The latter difference has a fixed nonzero sign on a small interval about \(u_0\). Choose a nonnegative nonzero \(f\) supported in the corresponding interval about \(t_c-u_0\). The integral at \(t_c\) has that strict sign. \end{proof} This proposition is an elementary placement comparison, not a reconstruction of X-A-conv's complete native equation. The supplied X-A and X-A-conv records have different defining prescriptions and different gauge conclusions \cite{CheungXA,CheungXAconv}. They are not glued into one regulator or one defect. \subsection{A fixed density ramp can fail the harmonic response test} Use the flat background \[ \eta=\operatorname{diag}(-1,1,1,1), \qquad \Box_\eta=-\partial_t^2+\Delta \] on \(M_T=[0,T]\times\R^3\). For a smooth symmetric perturbation \(h\), set \[ A_1^{\alpha\mu}[h] =\frac12(\eta^{\rho\sigma}h_{\rho\sigma})\eta^{\alpha\mu} -h^{\alpha\mu}. \] The first density-ramped harmonic correction is the smooth zero-data solution of \begin{equation} \Box_\eta W_\sigma^\mu[h] =-\partial_\alpha\bigl(\sigma A_1^{\alpha\mu}[h]\bigr). \label{eq:ramp-equation} \end{equation} Ordinary coordinate-time position and velocity vanish at \(t=0\). The zero-data solution operator for a smooth spatially compact forcing \(F\) can be written componentwise as \[ G_-F(t,\mathbf x) =-\frac1{4\pi}\int_0^t(t-s) \int_{S^2}F(s,\mathbf x+(t-s)\omega)\,\dd\omega\,\dd s. \] Differentiation of the spherical means verifies \(\Box_\eta G_-F=F\) and the data. For a compact smooth vector field \(\xi\) vanishing in an initial collar, let \[ (K\xi)_{\mu\nu} =\partial_\mu\xi_\nu+\partial_\nu\xi_\mu. \] \begin{proposition}[Transition defect and plateau qualification] \label{prop:ramp} For the displayed prescription, \begin{equation} \Box_\eta\bigl(W_\sigma^\mu[K\xi]-\xi^\mu\bigr) =(\sigma-1)\Box_\eta\xi^\mu -(\partial_\alpha\sigma)A_1^{\alpha\mu}[K\xi]. \label{eq:ramp-defect} \end{equation} There is a compact \(\xi\) in the transition region for which \(W_\sigma[K\xi]\ne\xi\). If \(\supp\xi\) lies strictly inside a region where \(\sigma=1\), the displayed defect vanishes and the two fields agree. \end{proposition} \begin{proof} Direct differentiation gives \[ A_1^{\alpha\mu}[K\xi] =(\partial_\rho\xi^\rho)\eta^{\alpha\mu} -\partial^\alpha\xi^\mu-\partial^\mu\xi^\alpha, \qquad \partial_\alpha A_1^{\alpha\mu}[K\xi] =-\Box_\eta\xi^\mu. \] Substitute into \eqref{eq:ramp-equation} and subtract \(\Box_\eta\xi^\mu\). Choose \(t_c\in(0,\epsilon)\) with \(0<\sigma(t_c)<1\). Let a compact spatial bump \(\psi\) equal one near \(\mathbf x_c\), and choose a compact time bump \(a\) in the transition region with \[ a'(t_c)=0,\qquad a''(t_c)\ne0. \] Set \(\xi^0=a\psi\), \(\xi^i=0\). Near the detecting point, \[ A_1^{00}=a'\psi,\qquad \Box_\eta\xi^0=-a''\psi. \] The right side of \eqref{eq:ramp-defect} there is \[ (1-\sigma(t_c))a''(t_c)\ne0. \] The difference field cannot vanish identically. For plateau-supported \(\xi\), the right side is zero and the difference has zero data. Smooth uniqueness is local: the wave energy on an inward-moving ball has nonpositive boundary flux, and starts at zero. Enlarging the ball treats any chosen point. Thus the homogeneous difference vanishes without an assumption of finite total energy. \end{proof} The transition witness does not contradict X-A's reported plateau-supported classical intertwiner through second order \cite{CheungXA}. It also does not reproduce the distinct convolution obstruction of X-A-conv. A correction beginning at second perturbative order cannot change the first coefficient in \eqref{eq:ramp-defect}. Likewise, in \(E[[\hbar]]\), a nonzero \(D_0\in E\) cannot be cancelled by an element of \(\hbar E[[\hbar]]\), because the constant coefficient remains \(D_0\). Neither statement is a quantum Ward calculation. %===================================================================== \section{Witness recovery, extension, and analytic admission} \label{sec:kernel} %===================================================================== \subsection{Pure-trace data have a blind sector} For this subsection use \(\eta_+=\operatorname{diag}(1,-1,-1,-1)\). Consider compact smooth kernels with two symmetric tensor slots and one scalar slot on \((\R^4)^3\). \begin{proposition}[A smooth kernel invisible to all pure-trace pairs] \label{prop:blind} All pure-trace evaluations fail to determine an unrestricted compact smooth three-slot kernel, even when symmetry under exchange of the two tensor slots is required. They also fail to determine its functional on a specified free solution family. \end{proposition} \begin{proof} Let \(A^{12}=A^{21}=1\), all other components zero. Then \(A^{\mu\nu}\eta_{+,\mu\nu}=0\). For nonnegative nonzero \(b,c\in C_c^\infty(\R^4)\), define \[ K^{\mu\nu,\rho\sigma}(x_1,x_2,x_3) =A^{\mu\nu}A^{\rho\sigma}b(x_1)b(x_2)c(x_3). \] This is nonzero and tensor-slot symmetric. Every contraction with \[ (\eta_{+,\mu\nu}\psi_1) \otimes(\eta_{+,\rho\sigma}\psi_2)\otimes\phi \] vanishes. For a free comparison, choose both metric inputs with constant \(h_{12}=h_{21}=1\) and all other components zero. They satisfy the componentwise free wave equation. Let \(\phi(t)=\cos(mt)\), \(m>0\), which satisfies \((\partial_t^2-\Delta+m^2)\phi=0\). Choose the support of \(c\) where \(\cos(mt)>0\). The kernel evaluation is \[ 4\left(\int b\,\dd^4x\right)^2 \int c(x)\cos(mt)\,\dd^4x>0. \] Thus zero and \(K\) agree on all pure-trace tests but differ on that free solution family. \end{proof} The free family in this proof is not identified with VI's solution jets or an inherited gravitational free-shell class. The argument does not show nonuniqueness of a kernel constructed from complete root data. It shows non-determination from the selected witness evaluations. This is the comparison shape in XVIII-A \cite{CheungXVIIIA}. XVI-H's reported general-test construction, its selected contraction, and its full-kernel wavefront and admission questions remain separate \cite{CheungXVIH}. The witness sources and responses of XVI-B/XVI-D are smooth fields on a spacetime slab. As regular distributions they live on that slab; they do not automatically become compact three-slot kernels on a configuration product \cite{CheungXVIB,CheungXVID}. A map changing the number and meaning of the slots must be specified. \subsection{An off-locus reference does not select its extension} \begin{proposition}[Restriction and finite normal order leave ambiguity] \label{prop:extension} Let \(U=(-1,1)\times\R^d\), \(d\ge1\), with coordinates \((q,x)\), and let \(S=\{q=0\}\). If a distribution \(t^\circ\) on \(U\setminus S\) has an extension \(t\) to \(U\), then \[ t+f(x)\delta(q),\qquad f\in C_c^\infty(\R^d), \] has the same restriction. The space of the displayed ambiguities is infinite-dimensional, although every one has normal order zero. \end{proposition} \begin{proof} The added distribution is supported on \(S\), so its restriction to the complement vanishes. The map \(f\mapsto f(x)\delta(q)\) is injective: pairing against \(\chi(q)\psi(x)\), where \(\chi(0)=1\), recovers \(\int f\psi\). The domain \(C_c^\infty(\R^d)\) is infinite-dimensional, proving the last assertion. \end{proof} Specified scaling, wavefront, normalization, or admissible-contact conditions may reduce this ambiguity. They are additional hypotheses. The proposition assumes extension existence and does not establish it for a characteristic target. Microlocal extension frameworks require their actual distribution and estimates \cite{BF}; citing such a framework is not the application proof. \begin{lemma}[Reconstruction compatibility] \label{lem:descent} For a surjective linear reconstruction \(\Rec:P\to K\), a linear \(F:P\to V\) descends to \(K\) if and only if \[ \ker\Rec\subseteq\ker F. \] The induced map is unique. \end{lemma} \begin{proof} Necessity follows from \(F=M\Rec\). For sufficiency, define \(M(\Rec p)=F(p)\). Equal reconstructions have a difference in \(\ker\Rec\), so the definition is independent of the representative. Surjectivity proves uniqueness. \end{proof} For example, \[ q\delta'(q)+\delta(q)=0, \] as follows by pairing with a test and differentiating \(q\psi(q)\) at zero. Its two summands are individually nonzero. A presentation-degree projection that breaks this relation does not automatically descend. Lemma~\ref{lem:descent} and Proposition~\ref{prop:extension} explain why an off-locus reference, finite normal-jet array, or descent criterion is not the missing full characteristic operation. They do not prove that every possible operation fails. \subsection{The version-qualified analytic boundary} The supplied records state \cite{CheungXIVH,CheungXIVI,CheungXIVJ,CheungXVB}: \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.17\linewidth}YY@{}} \toprule Record & Reported advance & Remaining target\\ \midrule XIV-H, v1 & Exact internal maps and regular charts & Uniform internal amplitude at H2 within that sheet\\[3pt] XIV-I, v1 & Internal amplitude module constructed at I2 & Complete joint characteristic map at I3\\[3pt] XIV-J, v1 & Reconstruction identities and supported \(K_6+K_8+K_9\) reference off the characteristic locus & Full retained map at J2\\[3pt] XV-B, v2 & Technical descent, relative lifting, normal-jet and extension interfaces & Graph-specific complete characteristic realization and subsequent admission\\ \bottomrule \end{tabularx} \end{center} The full retained sum still includes \[ K_{47}+K_5+K_6+K_8+K_9, \] where \(K_{47}\) regroups two branches and \(K_5\) retains both metric-slot placements. A reference for the supported sector is not that complete sum after cross-line composition. The remaining same-target obligations are: \[ \begin{gathered} \text{full characteristic operation},\\ \text{joint section or stated prescription and complete remainder},\\ \text{uniform transverse estimates},\\ \text{contact compatibility},\\ \text{output wavefront and normalization},\\ \text{quantum admission in the stated output space}. \end{gathered} \] No step is undertaken here. In particular, the documentary target \[ Q_HR_1^{(1)}\in\mathscr E_H \] is not asserted. No differential or Ward defect is applied to an undefined inherited coefficient. The existence of XV-B as a technical manuscript is acknowledged. Its downstream quantum-admission exit is not opened by the technical interfaces alone. %===================================================================== \section{Operational support does not determine spacetime or evolution} \label{sec:operational} %===================================================================== We use complete finite channels: completely positive, trace-preserving maps, including all outcomes and failure branches. Their adjoints are unital. These conventions, reference extensions, and the diamond norm are standard finite-dimensional channel notions \cite{Watrous}. \subsection{What scheduled support proves} Let \(X\) be a finite site set, \[ \mathcal H_X=\bigotimes_{x\in X}\mathcal H_x, \qquad \Alg(Y)=\mathcal B(\mathcal H_Y)\otimes I_{X\setminus Y}. \] At tick \(k\), let \[ \Phi_k=\bigotimes_{B\in\mathscr B_k}\Phi_{k,B}, \] where \(\mathscr B_k\) is a partition and each block map is a channel. Define backward ancestor sets by \[ N_{t,t}(Y)=Y, \] \[ N_{s,t}(Y) =\bigcup\{B\in\mathscr B_{s+1}: B\cap N_{s+1,t}(Y)\ne\varnothing\}. \] \begin{proposition}[Scheduled support and conditional physical containment] \label{prop:support} For \(\Phi_{s,t}=\Phi_t\circ\cdots\circ\Phi_{s+1}\), \[ \Phi_{s,t}^*(\Alg(Y))\subseteq\Alg(N_{s,t}(Y)). \] Equal input reductions on \(N_{s,t}(Y)\) therefore give equal output reductions on \(Y\). The statement is stable under an untouched reference when the joint ancestor--reference reductions agree. Physical causal containment additionally follows if every dependency edge, including control and guard inputs, has a location in a separately specified transitive physical causal relation. \end{proposition} \begin{proof} For one layer, an observable is the identity on every disjoint block. The corresponding adjoints preserve that identity. An intersecting block can enlarge support only to its own input block. Iterate backward to obtain the inclusion. Pairing with input states proves snapshot determinacy. Tensoring the linear maps with a reference identity gives the joint referenced statement. Under the additional embedding hypothesis, each dependency path maps to a chain in the physical causal relation. Transitivity places its initial event in the physical past of its final event. \end{proof} The support inclusion can be strict: a scheduled input may be followed by a constant-output channel and have no actual influence. A hidden distant guard is still an input dependency. This is the finite support shape of VII-E, VII-J, and XIX-C \cite{CheungVIIE,CheungVIIJ,XIXC}. It proves no causal embedding when the embedding is not supplied. If a metric \(d\) and block-diameter bound \(\Delta\) are supplied, each backward step adds at most \(\Delta\), so \[ N_{s,t}(Y)\subseteq \{x:d(x,Y)\le(t-s)\Delta\}. \] With a supplied tick duration \(\tau\), the scalar envelope is \(\Delta/\tau\). Changing the external calibration changes that report. This is not a proof against Lorentz-invariant propagation in another constructed model; it shows that neither calibration nor a frame action is furnished by the support proof. \subsection{Commuting channels do not furnish an observable algebra} \begin{proposition}[Superoperator commutation is insufficient] \label{prop:channel-algebra} A unital completely positive map need not be multiplicative. Commuting channel superoperators, without additional regional algebra hypotheses, do not imply commutation of their images as observables. \end{proposition} \begin{proof} On \(M_2(\C)\), let \[ \mathcal D(A)=P_0AP_0+P_1AP_1. \] For the Pauli matrix \(X\), \[ \mathcal D(X)=0,\qquad \mathcal D(X^2)=\mathcal D(I)=I \ne\mathcal D(X)^2. \] For the second assertion, two identity superoperators commute. Their images of the Pauli matrices \(X\) and \(Z\) do not commute: \([X,Z]\ne0\). \end{proof} The example does not challenge locality proved from actual commuting regional algebras. It excludes an inference from superoperator commutation alone. En05-C's tensor-free snapshot--channel--adjoint equivalence is an algebraic dependence statement, not a metric reconstruction. En05-B and En06-B use their own prescribed geometries, routes, and speed calibration. En06-A distinguishes operational limits, observable-product hypotheses, and frame covariance \cite{CheungEn05B,CheungEn05C,CheungEn06A,CheungEn06B}. Those positive scopes are not denied and are not transferred to the incidence register. \subsection{Even complete causal order does not select the metric scale} \begin{proposition}[Conformal ambiguity of causal data and clock lapse] \label{prop:conformal} Let \(q\) be a time-oriented Lorentzian metric and \(\widetilde q=\Omega^2q\), where \(\Omega>0\) is smooth. The future causal curves, and hence their pointwise causal relation, are the same for both metrics. For a fixed scalar \(T\) with timelike gradient, \[ N[q,T]=\bigl(-q^{-1}(\dd T,\dd T)\bigr)^{-1/2} \] satisfies \begin{equation} N[\widetilde q,T]=\Omega N[q,T]. \label{eq:conformal-lapse} \end{equation} Thus causal order alone does not determine this calibrated raw lapse or proper-time scale. \end{proposition} \begin{proof} For every tangent \(v\), \(\widetilde q(v,v)=\Omega^2q(v,v)\). Its causal character and the chosen future cone are unchanged. The same curves are therefore future causal. Since \(\widetilde q^{-1}=\Omega^{-2}q^{-1}\), substitution gives \eqref{eq:conformal-lapse}. For the proper-time statement it suffices to take \(\Omega=\lambda>0\) constant: proper times and lengths are multiplied by \(\lambda\), with causal order unchanged. \end{proof} The proposition deliberately retains only causal data and the fixed scalar. If calibrated lengths or a particular conical metric are declared as input, they are extra data. We do not vary a native installed cone while calling its complete packet unchanged. The native construction already supplies that cone; supplying it is not deriving it from a support certificate. A relational lapse additionally requires its rods and pullback. No identification of those data follows from causal order. \subsection{A finite channel does not select its continuous-time completion} \begin{proposition}[Exactly consistent subdivisions with different limits] \label{prop:continuum} One finite qubit channel admits two exactly coarse-consistent subdivision families with different continuous-time limits. Each family is complete and reference stable. \end{proposition} \begin{proof} Take the common coarse channel to be \(\id\), and let \[ H_0=0,\qquad H_1=2\pi\kb{1}{1}, \qquad \mathcal T_t^a(\rho) =e^{-itH_a}\rho e^{itH_a},\quad a=0,1. \] Both are continuous channel groups with \(\mathcal T_1^a=\id\). At tick duration \(1/n\), put \[ \Phi_n^a=\mathcal T_{1/n}^a,\qquad \mathcal C_n^a(t)=(\Phi_n^a)^{\lfloor nt\rfloor}. \] Then \((\Phi_n^a)^n=\id\) exactly. The bound \[ \norm{e^{-itH_a}-e^{-isH_a}}\le\norm{H_a}\abs{t-s} \] and expansion of the difference of two unitary conjugations imply \[ \norm{\mathcal C_n^a(t)-\mathcal T_t^a}_\diamond \le 2\norm{H_a}/n. \] It holds with arbitrary references and uniformly in \(t\). At \(t=1/2\), the first limit fixes \(\ket{+}\), whereas the second sends it to \(\ket{-}\). The corresponding output states are orthogonal. The limiting evolutions differ. \end{proof} This is an elementary comparison, not the native incidence continuum limit. Extra spectral, refinement, calibration, or dynamical conditions might select one completion. They are not consequences of the common finite channel. Likewise, a finite stencil need not be isotropic. Cardinal unit steps with waiting on \(\mathbb Z^2\) reach exactly \(\abs{x_1}+\abs{x_2}\le n\) after \(n\) ticks. Rescaling by \(n\) yields an \(\ell^1\) ball, not a Euclidean disk. Periodic-graph velocity polytopes provide related context \cite{Fritz}; that fixed-graph conclusion is not imported as a no-go for growing directional families such as those reported in En05-B. %===================================================================== \section{Asymptotics, states, and quantum certificate boundaries} \label{sec:quantum} %===================================================================== \subsection{A source coefficient needs its actual comparison map} \begin{lemma}[Asymptotic transfer requires a map] \label{lem:asymptotic} Let \(E,F\) be topological vector spaces and \(B:E\to F\) continuous and linear. Suppose \[ t_\delta=\delta^{-r}C+R_\delta,\qquad \delta^rR_\delta\longrightarrow0. \] Then \[ \delta^rBt_\delta\longrightarrow BC. \] Nonvanishing of \(C\) does not imply nonvanishing of \(BC\). \end{lemma} \begin{proof} Apply \(B\) to \(\delta^rt_\delta=C+\delta^rR_\delta\). Continuity proves the limit. A nonzero \(C\) may lie in \(\ker B\), even when \(B\ne0\). \end{proof} The supplied Em02-A record calculates localization powers in \(\epsilon\) for its displayed scalar seeds \cite{CheungEm02A}. Em02-D calculates an inverse-cubic smoothing coefficient for its fixed distribution model \cite{CheungEm02D}. Neither calculation defines a relational-lapse quantum contraction or an incidence-register kernel. The parameters also remain different: \[ F(\epsilon,\delta)=\epsilon^2\delta^{-3} \] tends to zero along \(\epsilon=\delta^2\), is constant along \(\epsilon=\delta^{3/2}\), and diverges along \(\epsilon=\delta\). A joint limit needs its own regime. An \(O(\delta^{-2})\) remainder after a leading \(\delta^{-3}\) term is neither a finite part nor an evaluated remainder divergence. Both zero and \(\delta^{-2}C\) satisfy such an upper-bound shape. No lower-order subtraction for another observable is licensed by it. \subsection{The two clock quantum slots remain unclosed here} The supplied VI record leaves its quantum finite-renormalization certificate RG-open: the quantum products, complete counterterm catalogue, radiative vector, response matrix, and quantum switch identity are not supplied there \cite{CheungVI}. The supplied Em02-E record fixes the classical lapse using the original inflaton-slice harmonic rods, while leaving the quantum package incomplete \cite{CheungEm02E}. Its five audit slots are: \begin{enumerate}[label=\textup{(\roman*)}] \item gauge fixing, the complete coupled operator, Green distributions, and normalized covariance; \item a state or appropriate parametrix with the required singular and tail properties; \item the required quantum insertion products and admissible counterterm class, including their observable images; \item a controlled complete remainder and any requested regulator-removal theorem; \item the retained-order quantum gauge identity and its admissible repair. \end{enumerate} None is constructed in this note. No limit toward \(t=0\) is taken. A parametrix is not a selected physical state. A classical Hessian is not its quantum contraction. A product prescription for local insertions is not automatically one for an independently specified nonlocal coordinate kernel \cite{HW,BF}. XIX-D supplies the relevant audit comparisons, not their native realization \cite{XIXD}. \subsection{The scoped Q11 record is neither erased nor enlarged} The supplied XVI-Q Q11 record reports closure of O.1 only for its one-loop, order-\(\kappa^2\) bulk two-point sector of \(S_{\mathrm{inv}}\), at zero mass, gauge held fixed, with QdS external legs and XVI-P's enlarged prescription \cite{CheungXVIQ,CheungXVIP}. It reports: \[ \begin{gathered} 11\text{ four-dimensional directions} +10\text{ evanescent operators} =21\text{ counterterm directions},\\ 23\text{ bulk-identity rows}. \end{gathered} \] A row count is not a counterterm dimension. The reported Q02/Q03 rebuild confirmation retains its same-computer and independent-Feynman-rule qualifications. No calculation or artifact is rerun here. The Q11 report does not change XVI-O's different prescription \cite{CheungXVIO}. QdS is an external-leg scheme, not a de Sitter state, not \(\NN\), and not horizon entropy. The scoped bulk check does not supply a relational insertion certificate, full bulk BV restoration, three-point closure, higher loops, or ultraviolet completeness. Outside that scope, \sid{O.1 = NOT_CLOSED} remains the supplied standing record \cite{CheungXVIQ}. \subsection{The insertion obstruction and Route B stay separate} XVI-N's supplied conditional theorem concerns one massless special-family instance and a specified regular, seed-preserving, antifield-linear completion class \cite{CheungXVIN}. It is not a theorem excluding every massive clock construction. Conversely, choosing \(m>0\) for the free stress benchmark of XVIII-B does not discharge that massless insertion theorem. Changing the family changes the quantified problem, not its answer \cite{CheungXVIIIB}. The spatially compact Route B is not discharged in the supplied r12 record. Its unbounded-switch support difficulty cannot be repaired by merely invoking compact Cauchy cutoffs. No such repair, insertion completion, or prepared interacting BV complex is constructed here. %===================================================================== \section{External catalogue and operational negative controls} \label{sec:external} %===================================================================== \subsection{The catalogue is a source record, not a proof premise} The following descriptions are the contents and attributed scopes of the supplied catalogue record dated 7 October 2026 \cite{Catalogue}. They are not independently verified theorems of this note. \begin{center} \small \begin{tabularx}{\linewidth}{@{}p{.12\linewidth}YY@{}} \toprule Family & Supplied recorded topic & Nontransfer boundary\\ \midrule 260 & Spacetime Penrose inequalities in the described asymptotically flat or asymptotically hyperbolic/anti-de Sitter settings, under energy, decay, and trapping hypotheses & Not a de Sitter state, an original-rod lapse, or the QdS external-leg scheme\\[4pt] 362 & Large-data global smoothness for a stated relativistic Vlasov--Maxwell system and admissible initial-data class & Not a metric evolution theorem, Einstein dynamics, or spacetime emergence\\[4pt] 376 & Universal computation in a stated forced Navier--Stokes flow construction & Not a derivation of a Lorentzian metric or of gravitational dynamics\\[4pt] 271 & Heisenberg-ferromagnet/Bloch-type claims and a reported Lean scope naming 271-04 & Not a clock, a VI proof, or a quantum lapse premise\\ \bottomrule \end{tabularx} \end{center} No Penrose formula, global regularity argument, forced-flow simulation, or Lean proof is reconstructed here. The catalogue attributes these external records to OpenAI. That attribution does not authenticate their proofs or establish a comparison map into the clock programme. Even a valid inequality for geometric initial data would not select a quantum state by itself. Smoothness of a matter PDE on a supplied background is not derivation of that background. Encoding computation into a supplied fluid forcing is not a matter--geometry equation. VI is not the identifier 271-04. The reported Lean scope is not a proof-assistant run on VI, Em02-E, or this manuscript. \subsection{Ward comparisons are not gravitational dictionaries} The supplied Oai282-B and Oai069-A comparison manuscripts are attributed to N.~Cheung, whereas the separate family catalogue is attributed to OpenAI \cite{CheungOai282B,CheungOai069A,Catalogue}. These authorship records must not be merged. Oai282-B reports a free local conformal Ward action and finite coverage in its declared homogeneous stress class, with a two-dimensional residual family. Local action is not global integration. Its finite coverage does not cover unrestricted conserved tensors. Oai069-A supplies its separate free-scalar finite defect and local-system benchmark. Neither is a gravitational fixed point or an inherited counterterm map. As a direct kinematic check, \(T_{\mu\nu}=0\) is symmetric, conserved, and trace free on any Lorentzian metric. Those identities impose no equation on the freely chosen metric. They cannot alone require a specified Einstein equation. A matter--geometry law is an additional equation, not a consequence of the tensor adjectives. VII-H's free massless susceptibility and XVIII-B's massive stress response retain their stated vacuum, insertion, retarded, and normalization data \cite{CheungVIIH,CheungXVIIIB}. They are useful benchmarks, not clock responses or gravitational dictionaries. \subsection{Recovery, representation, and emergence are different targets} The finite measurement and implementation interfaces associated with Cheung's Enter calculus concern their stated operational class \cite{CheungI,CheungEnter}. They do not select a field state, supply an unknown observable, or execute a native compilation. For an elementary recovery distinction, let \[ V:\mathcal H_M\to\mathcal H_A\otimes\mathcal H_R,\qquad V\ket\psi=\ket0_A\otimes\ket\psi_R. \] The complete output preserves the input isometrically. The accessible \(A\) output is constant. No decoder from \(A\) alone can recover every input, because it receives the same state for all inputs. Thus complete preservation and recovery through declared access are different. This is the comparison shape of the supplied Hawk04 records \cite{CheungHawk04A,CheungHawk04B}. It is not a black-hole model. Selected Psy, Hawk, and En certificates retain their own preparation, residual, access, schedule, and approximation hypotheses. No corpus-wide theorem about those series is claimed. Operator reconstruction, a supplied metric dictionary, derivation of Einstein dynamics, and holographic reconstruction are distinct projects. AdS/CFT error-correction and entanglement-wedge reconstruction have their own duality and algebraic hypotheses \cite{ADH,DHW}. A resting conical collar or a finite decoder does not supply that dictionary. Likewise, the IX partial-product and causal-interface records retain their own sharp-insertion domains and order boundaries. The relative IX-Free-2 selected-word construction through \(\hbar^3\) is not erased by an earlier IX-Free stop, but does not supply IX-D's missing sharp time-ordered or normalized retarded maps \cite{CheungIX,CheungIXFreeTwo,CheungIXD}. %===================================================================== \section{Boundary theorem and exact outside list} \label{sec:release} %===================================================================== \begin{theorem}[Non-determination underlying the release boundary] \label{thm:boundary} On the explicit comparison domains of this note: \begin{enumerate}[label=\textup{(\roman*)}] \item density multiplication and fixed convolution are not the same operation, and the displayed fixed density ramp fails a transition-region harmonic pure-gauge test; \item all pure-trace evaluations do not determine an unrestricted compact smooth three-slot kernel or its functional on the displayed free solution family; \item off-locus agreement does not select an extension, and a finite normal-order bound does not imply a finite-dimensional ambiguity; \item scheduled channel support does not determine a physical causal embedding, metric scale, clock lapse, or continuous-time completion; \item an asymptotic source coefficient need not survive a comparison operation, and independent localization and smoothing parameters cannot be identified by notation. \end{enumerate} These are failures of determination by the named data. They are not universal obstructions to constructions supplied with additional comparison, normalization, dynamical, or physical hypotheses. \end{theorem} \begin{proof} Propositions~\ref{prop:placement} and~\ref{prop:ramp} prove (i). Proposition~\ref{prop:blind} proves (ii). Proposition~\ref{prop:extension} proves (iii). For (iv), Proposition~\ref{prop:support} isolates the additional physical embedding hypothesis. Proposition~\ref{prop:channel-algebra} separates channel statements from observable products. Proposition~\ref{prop:conformal} supplies equal causal data with different clock normalization, and Proposition~\ref{prop:continuum} supplies equal coarse channel data with different limits. Lemma~\ref{lem:asymptotic} and the explicit two-parameter example prove (v). No predecessor theorem is used in these proofs. \end{proof} \subsection{What remains outside the release} The exact outside list is: \begin{enumerate}[label=\textbf{E\arabic*.}] \item \textbf{Identification.} No identification of the four labelled prescriptions is supplied. The direct first-coordinate comparison excluded in XIX-B is not reopened or replaced. Other complete-observable comparisons remain separate questions. \item \textbf{Native continuum evolution.} No continuum realization of the VIII incidence package or autonomous matter-driven cone evolution is constructed. Finite channels and illustrative continuous-time comparison groups are not denied or renamed as that realization. \item \textbf{Clock quantum certificates.} VI's RG-open certificate is not closed. Em02-E's five audit slots and retained-order quantum gauge identity are not supplied or verified. \item \textbf{External-time-free quantum evolution.} No quantum evolution without an external time is constructed. A prescribed controller, coordinate time, or reparametrization does not establish that claim. \item \textbf{Inherited coefficient and analytic exit.} No quantum coefficient of the inherited characteristic graph is admitted. The full joint characteristic map, its subsequent section, remainder, transverse, contact, output, and normalization obligations remain outside this note. The downstream XV-B quantum-admission exit is not opened. \item \textbf{Insertion and route obligations.} XVI-N and spatially compact Route B are not discharged. The historical H2 record, the later I2 advance, and remaining I3/J2 full-target gaps retain their separate scopes. \item \textbf{Regulator, witness, and catalogue transfers.} Neither the external density ramp nor the fixed convolution prescription is promoted to a dynamical clock. A partial contraction is not promoted to a full kernel. The catalogue families are not promoted to a lapse, state, Einstein equation, or spacetime-emergence theorem. \item \textbf{Physical and holographic interpretation.} No invariant physical speed, derived metric, Einstein dynamics, de Sitter state, horizon entropy, Page curve, entanglement wedge, or full spacetime emergence is constructed. \item \textbf{Verification and implementation.} No predecessor proof is independently verified. No numerical or symbolic artifact is executed. No native programme compilation, proof-assistant execution, or successful pdfLaTeX build is asserted. \end{enumerate} This list is an explicit scope statement. It does not claim that no unexamined manuscript or future construction can meet any item. \subsection{Unchanged standing record} \[ \boxed{ \begin{gathered} \text{Declared path: null},\\ \text{VI quantum certificate: RG-open in the supplied record},\\ \text{Em02-E quantum gauge identity: not verified here},\\ \sid{O.1 = NOT_CLOSED} \quad\text{outside the separately scoped Q11 report},\\ \text{XVI-N and spatially compact Route B: not discharged},\\ \text{I3/J2 full characteristic target: not completed here},\\ \text{XV-B downstream quantum-admission exit: not opened},\\ Q_HR_1^{(1)}\in\mathscr E_H:\ \text{not asserted}. \end{gathered}} \] The documentary qualifications are those of the cited versions \cite{CheungVI,CheungEm02E,CheungXVIQ, CheungXVIN,CheungXIVI,CheungXIVJ,CheungXVB}. The H2 label continues to describe XIV-H's own stop, without withdrawing XIV-I's narrower I2 construction. %===================================================================== \section{Conclusion and successor boundary} \label{sec:conclusion} %===================================================================== The programme has separated prescriptions and identified additional constructions that their comparison does not supply. Its finite and classical results are useful within their stated domains. They are not evidence that their quantum, geometric, or continuum successors have already been built. A future connecting paper must specify the common target, domains, input maps, operators, boundary conditions, normalizations, allowed changes, and identity or limit to be proved. A future physical interpretation must additionally supply the relevant calibration, causal embedding, frame action, or matter--geometry law. A quantum paper must construct its quantum objects before calculating their gauge defects. No such successor is opened here. The final statement is a boundary, not a universal prohibition: \[ \boxed{ \begin{gathered} \text{prescription comparison}\ne\text{quantum certification},\\ \text{scheduled support}\ne\text{derived spacetime geometry},\\ \text{witness recovery}\ne\text{full-kernel admission},\\ \text{documentary scope}\ne\text{proof transfer},\\ \text{outside this release}\ne\text{impossible in every future model}. \end{gathered}} \] Here \(\ne\) denotes non-identification of the declared objects. %===================================================================== \section*{Declarations} \addcontentsline{toc}{section}{Declarations} %===================================================================== \paragraph{Mathematical scope.} The proofs concern displayed classical prescriptions, distributional comparisons, finite complete channels, and non-determination witnesses. They construct no native clock quantum coefficient or incidence continuum realization. \paragraph{Documentary scope.} Unpublished records are cited by supplied version and filename. Their full proofs and artifacts are not independently authenticated. The catalogue descriptions are attributed records, not premises. No exhaustive scan of the manuscript series is claimed. \paragraph{Bibliographic checks.} Bibliographic consultation of published primary sources is not verification of an unpublished predecessor proof. The Enter-calculus citation retains the accepted-article status and date; its final article metadata should be checked before signed submission. No approval or acceptance of the present manuscript is inferred. \paragraph{Static debugging.} The source has one internal bibliography, defined notation, unique local theorem labels, and no external inputs. Scope statements have been separated from mathematical non-determination claims. Historical gate labels, later partial advances, complete kernels, and selected contractions are not collapsed. \paragraph{Execution.} No successful pdfLaTeX build, proof-assistant run, independent numerical or symbolic execution, native programme compilation, hardware realization, or referee certification is asserted. \paragraph{Authorship and availability.} The manuscript is anonymous. Citation of N.~Cheung does not establish authorship of this note. A signed submission must supply authorized authorship metadata and stable copies or archival locations for the cited unpublished versions. %===================================================================== \begin{thebibliography}{99} \addcontentsline{toc}{section}{References} %===================================================================== \bibitem{XIXA} \emph{XIX-A: Clock Separation Before Coordinate Identification: Four Labelled Objects and Five Non-Identifications}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-A.tex}, n.d. Classical comparison, not quantum certification. \bibitem{XIXB} \emph{XIX-B: A Witness Mismatch Between Two Coordinate Cauchy Prescriptions: Unswitched Harmonic Coordinates and Prepared Coordinate Completion}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-B.tex}, n.d. Specified direct first-coordinate comparison only. \bibitem{XIXC} \emph{XIX-C: Finite Incidence Registers and the Missing Continuum Evolution: Matching, Records, Promised Tokens, and Scheduled Support}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-C.tex}, n.d. Finite-interface and continuum non-determination comparison. \bibitem{XIXD} \emph{XIX-D: Quantum-Certificate Inventory for Separated Clock Prescriptions: Non-Determination, Counterterm Images, and Unclosed Quantum Gates}, anonymous draft supplied in the present manuscript exchange, \sid{XIX-D.tex}, n.d. Quantum audit and self-contained comparison arguments; no native quantum certificate is inferred. \bibitem{CheungVI} N.~Cheung, \emph{Relational Einstein--scalar observables at second order: classical descent and a compact-switch obstruction}, Landmark VI, version 1, supplied unpublished manuscript, \sid{VI.tex}, n.d. Enter Calculus Companion. Classical descent, separate switch defect, and RG-open quantum finite-renormalization audit. \bibitem{CheungVIIIA} N.~Cheung, \emph{Landmark VIII-A}, version 3, supplied unpublished manuscript, \sid{VIII-A-v3.tex}, n.d. Enter Calculus Companion. Finite matching and record--residual interfaces; relevant supplied locator: \sid{A:thm:residual}. \bibitem{CheungVIIIB} N.~Cheung, \emph{Landmark VIII-B}, version 3, supplied unpublished manuscript, \sid{VIII-B-v3.tex}, n.d. Enter Calculus Companion. Packet dictionary, finite source and promised tokens, and parameter--lapse separation. \bibitem{CheungVIIIC} N.~Cheung, \emph{Landmark VIII-C}, version 3, supplied unpublished manuscript, \sid{VIII-C-v3.tex}, n.d. Enter Calculus Companion. Frozen-packet incidence, guarded composition, and the separate mixed-germ, acquisition, embedding, and dynamics interfaces. \bibitem{CheungXA} N.~Cheung, \emph{X-A}, version 1, supplied unpublished manuscript, \sid{X-A.tex}, n.d. Enter Calculus Companion. Fixed density ramp, transition response test, and the plateau-supported classical identity. \bibitem{CheungXAconv} N.~Cheung, \emph{X-A-conv}, version 1, supplied unpublished manuscript, \sid{X-A-conv.tex}, n.d. Enter Calculus Companion. Distinct fixed convolution prescription and its stated first-order covariance obstruction. \bibitem{CheungEm02E} N.~Cheung, \emph{Em02-E}, version 1, supplied unpublished manuscript, \sid{Em02-E-v1.tex}, n.d. Enter Calculus Companion. Original inflaton-slice rods, classical relational lapse, and incomplete quantum audit. Relevant supplied locators: \sid{thm:lapse}, \sid{thm:release}. \bibitem{CheungXVIIIA} N.~Cheung, \emph{Landmark XVIII-A: Prepared Relational Observables: Non-Determination, Normalization Compatibility, and Conditional Ultraviolet Matching}, version 2, supplied unpublished manuscript, \sid{XVIII-A-v2.tex}, n.d. Enter Calculus Companion. Witness, complete kernel, class, normalization, and quantum admission remain distinct. \bibitem{CheungXVIIIB} N.~Cheung, \emph{Landmark XVIII-B}, version 2, supplied unpublished manuscript, \sid{XVIII-B-v2.tex}, n.d. Enter Calculus Companion. Free massive scalar stress response on a classical Minkowski background; not a native clock or insertion certificate. \bibitem{CheungXVIB} N.~Cheung, \emph{Landmark XVI-B: Prepared-source diagnostics and the complete-insertion boundary}, version 1, supplied unpublished manuscript, \sid{XVI-B.tex}, n.d. Enter Calculus Companion. Source, response, root pairing, and class-kernel interfaces. \bibitem{CheungXVID} N.~Cheung, \emph{Landmark XVI-D: Prepared witness response and completion domains}, version 1, supplied unpublished manuscript, \sid{XVI-D.tex}, n.d. Enter Calculus Companion. Smooth response, coordinate candidate, and conditional inherited-coordinate compatibility. \bibitem{CheungXVIH} N.~Cheung, \emph{Landmark XVI-H: Compact polarized kernel and witness recovery}, version 1, supplied unpublished manuscript, \sid{XVI-H.tex}, n.d. Enter Calculus Companion. General-test compact kernel under root inputs; selected witness contraction and admission remain distinct. \bibitem{CheungXIVH} N.~Cheung, \emph{Landmark XIV-H: Internal kernel relations on the operator route}, version 1, supplied unpublished manuscript, \sid{XIV-H.tex}, n.d. Relevant supplied locators: \sid{prop:K5exact}, \sid{thm:H2}, \sid{thm:release}. H2 is the stop within this sheet. \bibitem{CheungXIVI} N.~Cheung, \emph{Landmark XIV-I}, version 1, supplied unpublished manuscript, \sid{XIV-I.tex}, n.d. Enter Calculus Companion. Relevant supplied locators: \sid{thm:I2}, \sid{thm:I3}, \sid{thm:release}. Internal amplitude construction and remaining complete joint-map gap. \bibitem{CheungXIVJ} N.~Cheung, \emph{Landmark XIV-J}, version 1, supplied unpublished manuscript, \sid{XIV-J.tex}, n.d. Enter Calculus Companion. Relevant supplied locators: \sid{prop:descent}, \sid{lem:null}, \sid{thm:reference}, \sid{thm:J2}, \sid{cor:noadmission}. Supported-sector reference is not full characteristic admission. \bibitem{CheungXVB} N.~Cheung, \emph{XV-B}, version 2, supplied unpublished manuscript, \sid{XV-B.tex}, n.d. Enter Calculus Companion. Technical descent, relative lifting, finite normal jets, and extension interfaces; no complete inherited coefficient or downstream quantum admission is inferred. \bibitem{CheungXVIN} N.~Cheung, \emph{XVI-N}, version r12, supplied unpublished manuscript, \sid{XVI-N.tex}, n.d. Enter Calculus Companion. Relevant supplied locator: \sid{thm:massless-obstruction}. Conditional massless special-family result and separately undischarged spatially compact Route B. \bibitem{CheungXVIP} N.~Cheung, \emph{XVI-P}, version p23, supplied unpublished manuscript, \sid{XVI-P.tex}, n.d. Enter Calculus Companion. Declared two-point prescription, external-leg scheme, and separately scoped identity checks. \bibitem{CheungXVIQ} N.~Cheung, \emph{XVI-Q}, release Q11, supplied unpublished manuscript, \sid{XVI-Q.tex}, n.d. Enter Calculus Companion. Relevant supplied locators: \sid{rem:o1}, \sid{rem:qds}, \sid{rem:p6}, \sid{rem:insertion}, \sid{rem:package}. Scoped two-point closure report, verification qualifications, and unchanged inherited insertion and admission boundaries. \bibitem{CheungXVIO} N.~Cheung, \emph{XVI-O}, version o70-sync, supplied unpublished manuscript, \sid{XVI-O.tex}, n.d. Enter Calculus Companion. Kernel two-point Ward subsector under its own prescription, not the enlarged Q11 prescription or bulk BV restoration. \bibitem{CheungEm02A} N.~Cheung, \emph{Em02-A}, version 6, supplied unpublished manuscript, \sid{Em02-A-v6.tex}, n.d. Enter Calculus Companion. Relevant supplied locator: \sid{thm:transition}. Localization coefficients in \(\epsilon\), not smoothing coefficients in \(\delta\). \bibitem{CheungEm02D} N.~Cheung, \emph{Em02-D}, version 4, supplied unpublished manuscript, \sid{Em02-D-v4.tex}, n.d. Enter Calculus Companion. Relevant supplied locator: \sid{thm:main}. Inverse-cubic leading distribution for one fixed model; no transferred finite part or quantum lapse coefficient. \bibitem{CheungVIIE} N.~Cheung, \emph{Landmark VII-E}, version 1, supplied unpublished manuscript, \sid{VII-E.tex}, n.d. Enter Calculus Companion. Observation interfaces and prescribed scheduled support. \bibitem{CheungVIIJ} N.~Cheung, \emph{Landmark VII-J}, version 1, supplied unpublished manuscript, \sid{VII-J.tex}, n.d. Enter Calculus Companion. Timed dependency, actual influence, physical containment, and fixed-cone transport comparisons. \bibitem{CheungEn05B} N.~Cheung, \emph{En05-B}, version 1, supplied unpublished manuscript, \sid{En05-B.tex}, n.d. Enter Calculus Companion. Isotropic matched operational-limit construction on its declared experiment family and prescribed speed. No incidence continuum identification is imported. \bibitem{CheungEn05C} N.~Cheung, \emph{En05-C}, version 1, supplied unpublished manuscript, \sid{En05-C.tex}, n.d. Enter Calculus Companion. Relevant supplied locator: \sid{thm:comparison}. Tensor-free snapshot--channel--adjoint dependence, not metric reconstruction. \bibitem{CheungEn06A} N.~Cheung, \emph{En06-A}, version 1, supplied unpublished manuscript, \sid{En06-A.tex}, n.d. Enter Calculus Companion. Corrected layer account and distinct operational, observable-product, and frame-covariance interfaces. \bibitem{CheungEn06B} N.~Cheung, \emph{En06-B}, version 1, supplied unpublished manuscript, \sid{En06-B.tex}, n.d. Enter Calculus Companion. Prescribed routes, speed, complete payload transport, and legal scheduling; not derived geometry. \bibitem{CheungVIIH} N.~Cheung, \emph{Landmark VII-H}, version 1, supplied unpublished manuscript, \sid{VII-H.tex}, n.d. Enter Calculus Companion. Free massless scalar susceptibility, retarded prescription, logarithmic extension, and contact ambiguity. Not a gravitational dictionary. \bibitem{CheungIX} N.~Cheung, \emph{IX sharp-insertion and partial-product records}, supplied unpublished manuscripts: \sid{IX-A.tex}, version 1; \sid{IX-B-v2.tex}, version 2; \sid{IX-C.tex}, version 1; \sid{IX-Free.tex}, version 1; \sid{IX-E.tex}, version 1; \sid{IX-E-2.tex}, version 1; \sid{IX-E-3.tex}, version 1; n.d. Enter Calculus Companion. Version-qualified comparison collection, not one theorem or a completed clock-product domain. \bibitem{CheungIXFreeTwo} N.~Cheung, \emph{IX-Free-2}, version 1, supplied unpublished manuscript, \sid{IX-Free-2.tex}, n.d. Enter Calculus Companion. Relative truncated partial product and selected-word coherence through \(\hbar^3\); not time ordering or full quantum algebra. \bibitem{CheungIXD} N.~Cheung, \emph{Landmark IX-D}, version 3, supplied unpublished manuscript, \sid{IX-D-v3.tex}, n.d. Enter Calculus Companion. Separate sharp time-ordered extension and normalized retarded maps remain outside its supplied release. \bibitem{CheungOai282B} N.~Cheung, \emph{Oai282-B}, version 1, supplied unpublished manuscript, \sid{Oai282-B.tex}, n.d. Enter Calculus Companion. Local free-scalar conformal Ward construction within its declared homogeneous class; not OpenAI family verification. \bibitem{CheungOai069A} N.~Cheung, \emph{Oai069-A}, version 1, supplied unpublished manuscript, \sid{Oai069-A.tex}, n.d. Enter Calculus Companion. Finite Ward-defect and finite-monodromy/local-system benchmarks. No gravitational dictionary is imported. \bibitem{CheungI} N.~Cheung, \emph{Landmark I}, version 2, supplied unpublished manuscript, \sid{I.tex}, n.d. Enter Calculus Companion. Finite destructive measurement characterization inside a stated structural class; not a field instrument. \bibitem{CheungEnter} N.~Cheung, ``The Enter Calculus: A Linear Quantum Lambda Calculus with an Operational Characterization of Measurement,'' \emph{Frontiers in Physics} \textbf{14} (2026), article 1954943. DOI: \sid{10.3389/fphy.2026.1954943}. Accepted-article record dated 28 September 2026; article number and DOI as supplied. Finite operational context only. % The publisher volume listing confirms the title, author, % and accepted-record date. This does not verify any predecessor proof. \bibitem{CheungHawk04A} N.~Cheung, \emph{Hawk04-A}, version 1, supplied unpublished manuscript, \sid{Hawk04-A.tex}, n.d. Enter Calculus Companion. Complete preservation and recovery through declared access are separate tasks. \bibitem{CheungHawk04B} N.~Cheung, \emph{Hawk04-B}, version 2, supplied unpublished manuscript, \sid{Hawk04-B.tex}, n.d. Enter Calculus Companion. Classical readout and quantum-return decoding retain different interfaces. \bibitem{Catalogue} \emph{OpenAI mathematics catalogue and companion scope records}, attributed to OpenAI in the supplied documentary record, dated 7 October 2026. Recorded comparison families: 260, 271, 362, 376, 282, and 069. Family 271's reported Lean scope names 271-04 only. Catalogue attribution, source proofs, and Lean execution are not independently authenticated by this manuscript. No catalogue theorem is a premise. \bibitem{BFHPR} R.~Brunetti, K.~Fredenhagen, T.-P.~Hack, N.~Pinamonti, and K.~Rejzner, ``Cosmological perturbation theory and quantum gravity,'' \emph{Journal of High Energy Physics} \textbf{08} (2016), 032. DOI: \sid{10.1007/JHEP08(2016)032}. arXiv:1605.02573. Relational-observable context, not an identification of native clock prescriptions. \bibitem{HW} S.~Hollands and R.~M.~Wald, ``Quantum fields in curved spacetime,'' \emph{Physics Reports} \textbf{574} (2015), 1--35. DOI: \sid{10.1016/j.physrep.2015.02.001}. arXiv:1401.2026. States, composite observables, and products on classical globally hyperbolic backgrounds. \bibitem{BF} R.~Brunetti and K.~Fredenhagen, ``Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds,'' \emph{Communications in Mathematical Physics} \textbf{208} (2000), 623--661. DOI: \sid{10.1007/s002200050004}. arXiv:math-ph/9903028. Distributional products and extension framework; not applied to an undefined inherited graph. \bibitem{Watrous} J.~Watrous, \emph{The Theory of Quantum Information}, Cambridge University Press, Cambridge, 2018. DOI: \sid{10.1017/9781316848142}. Complete channels, references, and diamond-norm comparison. \bibitem{Fritz} T.~Fritz, ``Velocity polytopes of periodic graphs and a no-go theorem for digital physics,'' \emph{Discrete Mathematics} \textbf{313} (2013), 1289--1301. DOI: \sid{10.1016/j.disc.2013.02.010}. arXiv:1109.1963v3. The revised arXiv version includes the historical-attribution corrigendum. Fixed periodic-graph context only; no obstruction is transferred to changing refinement families. \bibitem{ADH} A.~Almheiri, X.~Dong, and D.~Harlow, ``Bulk locality and quantum error correction in AdS/CFT,'' \emph{Journal of High Energy Physics} \textbf{04} (2015), 163. DOI: \sid{10.1007/JHEP04(2015)163}. arXiv:1411.7041. External holographic comparison under its own duality hypotheses. \bibitem{DHW} X.~Dong, D.~Harlow, and A.~C.~Wall, ``Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality,'' \emph{Physical Review Letters} \textbf{117} (2016), 021601. arXiv:1601.05416. External reconstruction context, not an incidence or clock-programme premise. \end{thebibliography} \end{document}