% XXII-A.tex % Single-file source for pdfLaTeX; no BibTeX or external files required. % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XXII-A.tex % pdflatex -interaction=nonstopmode -halt-on-error XXII-A.tex % % No successful compilation or machine verification is asserted. % % Mathematical corrections incorporated: % 1. A finite event order is distinguished from an ambient cone order. % 2. A Euclidean-ball section is not itself a non-Lorentzian cone. % 3. Direct operational influence is not assumed to be transitive. % 4. Closed causal order is distinguished from open chronology. % 5. Chronology may determine conformal structure under additional % continuum hypotheses, but does not determine the full metric. % 6. The future selection statement is a research target, not a theorem. % 7. A finitely generated event-difference cone cannot be Lorentzian % in dimension at least three. % \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{enumitem} \usepackage{booktabs,tabularx,array} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={XXII-A: Freeze the precise claim and the gap}, pdfsubject={Stage 0 of a limited kinematic emergence programme} } \setlength{\emergencystretch}{2em} \setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt} \setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt} \newtheorem{theorem}{Theorem} \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{target}{Research target} \newtheorem{remark}{Remark}[section] \newcommand{\R}{\mathbb R} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Int}{\operatorname{int}} \newcommand{\Span}{\operatorname{span}} \newcommand{\Cone}{\operatorname{cone}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\Lor}{\mathcal L} \newcommand{\Sched}{R_{\mathrm{sch}}} \newcommand{\Infl}{R_{\mathrm{inf}}} \newcommand{\Supplied}{R_{\mathrm{sup}}} \newcommand{\SP}{\mathsf S} \newcommand{\HP}{\mathsf H} \newcommand{\ddiamond}{d_{\diamond}} \newcommand{\cE}{\mathfrak E} \newcommand{\loc}[1]{\texttt{#1}} \newcolumntype{Y}{>{\raggedright\arraybackslash}X} \title{\textbf{XXII-A: Freeze the precise claim and the gap}\\[3pt] \large Stage 0 of a limited kinematic emergence programme} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} This note fixes the meaning and evidentiary boundary of a prospective quantum-operational Lorentz-cone selection theorem. It separates scheduled ancestry, intervention-based operational influence, and an independently supplied geometric causal relation. It records the scheduler-consistency obstruction of Cheung's En08-A and the conditional homogeneous strictly convex cone classification used there. The latter is a classification theorem, not a derivation of its hypotheses from quantum data. A precise future target is formulated in terms of a nonempty, named process class, predeclared additional hypotheses, and a faithful generated affine cone representation. Finite event orders, continuum completions, cone boundaries, and conformal metric information are distinguished. No process class and additional hypothesis package satisfying the target are constructed here. The completed deliverable is the Stage 0 specification and obstruction audit, not a positive selection theorem. \end{abstract} \section{Claim, conventions, and source boundary} \label{sec:claim} The claim under construction is \emph{not} \[ \text{scheduler consistency} \quad\Longrightarrow\quad \text{Lorentz-cone selection}. \] Cheung's En08-A explicitly obstructs that implication \cite[\loc{thm:main}]{CheungEn08A}. The narrower claim is that \emph{there exists a named class of quantum operational data and a finite, predeclared list of genuinely additional hypotheses under which an operationally extracted cone is linearly isomorphic to a Lorentz cone}. This is an open target. XXII-A neither establishes its existential claim nor supplies the missing process class and hypotheses. In particular, a target statement must not be promoted to a theorem merely because its conclusion has been written precisely. \begin{definition}[Cone and cone order] \label{def:cone} Let \(V\) be a finite-dimensional real vector space. A cone \(K\subset V\) is \emph{proper} if it is closed, convex, pointed, \(K\cap(-K)=\{0\}\), and has nonempty interior in \(V\). Its translation-invariant order is \[ u\leq_K v \quad\Longleftrightarrow\quad v-u\in K. \] For \(m\geq3\), the closed Lorentz cone is \[ \Lor_m = \bigl\{(t,x)\in\R\times\R^{m-1}:t\geq\|x\|_2\bigr\}. \] The statement that \(K\) is \emph{linearly isomorphic to a Lorentz cone} means that an invertible linear map \(T:V\to\R^m\) satisfies \(T(K)=\Lor_m\). \end{definition} A finite event poset is not itself a vector-space cone. Accordingly, ``the extracted order is Lorentzian'' will mean the precise representation statement in Section~\ref{sec:target}, not a linear bijection from finitely many events onto \(\Lor_m\). The present convention uses a \emph{closed} cone order. An open chronological relation would instead use \(\Int K\). Passing between these two relations requires an explicitly stated closure or boundary argument; it is not a notational identification. \paragraph{Possible operational languages.} Process matrices, quantum combs, and multi-time process tensors provide different operational frameworks \cite{OCB,CDP,Pollock}. They are possible languages for Stage 1, not interchangeable models and not hypotheses of a selection theorem in this note. A future choice must specify: \begin{enumerate} \item the finite party or slot set and the input/output Hilbert spaces; \item the admissible preparations, instruments, interventions, and retained outputs; \item whether the datum consists of exact complete maps, exact intervention probabilities, or finite statistical records; \item the operational equivalence and relabelling conventions; \item which relation is being extracted. \end{enumerate} Finite parties do not necessarily mean finitely many admissible interventions. Conversely, finitely many observed event comparisons do not supply a continuum carrier. \paragraph{Source convention.} The Cheung references below identify unpublished manuscripts by their supplied identifiers, versions where available, and internal labels. A cited restatement is not an independent verification of the predecessor proof. En08-A is used for the obstruction; the cone classification is attributed directly to Ito and Louren\c{c}o. Other series references are documentary comparisons unless expressly identified otherwise. \section{The three relations} \label{sec:relations} The separation adopted from VII-J is a distinction between mathematical objects and their inputs, not a claim that they can never coincide in a particular certified model \cite{CheungVIIJ}. \begin{definition}[Scheduled ancestry] A complete dependency graph includes every quantum input, classical control, guard, routing decision, and retained record actually used by the protocol. For distinct operation events \(p,q\), write \(p\Sched q\) if there is a directed dependency path from \(p\) to \(q\). Its reflexive closure is the scheduled event order. An edge records an operation that the protocol calls. It does not assert that the operation is sensitive to every input it receives. \end{definition} \begin{definition}[Operational influence] \label{def:influence} Fix an operational process and an accessible output at \(q\). Two admissible interventions \(\mathcal I,\mathcal J\) at \(p\) must have common comparison interfaces; all other operations and preparations are held fixed. Let \(\Phi_{q\mid\mathcal I}\) and \(\Phi_{q\mid\mathcal J}\) be the resulting complete output channels. Set \[ \ddiamond(\Phi,\Psi) =\frac12\|\Phi-\Psi\|_{\diamond}, \qquad \Delta_{p\to q} = \sup_{\mathcal I,\mathcal J} \ddiamond\bigl(\Phi_{q\mid\mathcal I}, \Phi_{q\mid\mathcal J}\bigr). \] Write \(p\Infl q\) when \(\Delta_{p\to q}>0\). For an interface containing only classical records, the corresponding definition is a change in the unconditional output marginal under the intervention. In a process-matrix formulation the admissible instrument insertions and signalling comparison must be specified within that formulation. \end{definition} ``Complete'' means that retained quantum systems and classical histories are included and that equality is stable under an untouched reference. Instrument outcomes are either retained or summed according to a declared deterministic interface. Normalized postselected branches are not substituted for complete channels. Shared entanglement and conditional steering do not by themselves constitute unconditional signalling \cite{Beckman,CheungVIIJ}. Direct operational influence is a directed relation, but it is not assumed here to be transitive. If a later extraction uses its transitive closure, an acyclic restriction, or a quotient, that operation must be named. The relation being represented must then be the resulting relation, not an unannounced replacement of \(\Infl\). \begin{definition}[Supplied causal relation] Let \(X\) be a separately declared spatial carrier with distance \(d_{C^*}\), including any stipulated metric-length completion. On \(\R\times X\), define \[ (s,x)\Supplied(t,y) \quad\Longleftrightarrow\quad t-s\geq d_{C^*}(x,y). \label{eq:supplied} \] A comparison with operational events additionally requires a declared location map \(\ell:E\to\R\times X\). The carrier, distance, calibration, and completion are inputs to this comparison. \end{definition} \begin{theorem}[Three-relation separation] \label{thm:separation} Consider a finite acyclic network of completely positive trace-preserving operations, with complete dependency bookkeeping and intervention-independent initial preparation. Then \[ \Infl\subseteq\Sched \] for distinct events, and the inclusion can be strict. Neither relation is automatically contained in the pullback of \(\Supplied\) along an independently chosen location map. If every dependency edge is certified to respect \(\Supplied\), then scheduled ancestry, and hence operational influence, is contained in that pullback. These statements express the separation adopted from VII-J; they do not identify the three relations. \end{theorem} \begin{proof} Pull an output observable backwards through the circuit. Operations outside its dependency ancestors act on factors carrying the identity observable and disappear because the adjoint of a trace-preserving map preserves the identity. Thus an intervention outside the ancestor set cannot change the output channel. The argument remains valid after tensoring with an untouched reference. This is the finite-channel content of VII-J's scheduled-support and no-signalling statements \cite[\loc{thm:schedule}, \loc{thm:nosignal}]{CheungVIIJ}. For strictness, let a scheduled wire from \(p\) enter an operation at \(q\) whose observed output is the replacement channel \[ \mathcal R_\sigma(X)=\Tr(X)\sigma. \] Every trace-preserving intervention upstream gives \(\mathcal R_\sigma\circ\mathcal I=\mathcal R_\sigma\). There is scheduled ancestry but no influence on that output. An independent location assignment need not respect the schedule. For example, assign the endpoints of a scheduled identity transmission the locations \((1,x)\) and \((0,x)\), in that order. Its operational transmission is unchanged, while \eqref{eq:supplied} fails. Conversely, certified causal containment of every edge propagates along a path by transitivity of the supplied relation. This is the separate embedding hypothesis of VII-J \cite[\loc{thm:physical}]{CheungVIIJ}. \end{proof} The theorem applies to the stated definite-order channel networks. It is not an assertion that every process matrix possesses such a dependency graph; the process-matrix framework does not generally assume a predefined global causal order \cite{OCB}. \section{The scheduler-consistency obstruction} \label{sec:obstruction} Write \(\SP\) for the package consisting of: \begin{enumerate}[label=\textup{(S\arabic*)}] \item acyclic dependency; \item termination of every finite experiment; \item commutation of independent operations; \item schedule-independent complete joint records; \item exact operational sensitivity equal to strict reachability; \item confluence of legal finite executions. \end{enumerate} Condition (S5) is an extra property of the countermodels. It must not be read as a general identity between scheduled ancestry and influence. \begin{theorem}[Recorded obstruction of En08-A] \label{thm:obstruction} The package \(\SP\) does not force a generated reachability cone to be quadratic or linearly isomorphic to a Lorentz cone. En08-A provides local record networks with these consistency properties and, in the statement of its \loc{thm:main}, additionally: bounded degree, finite support, finitely many local reduction shapes, no intervention-induced change outside the dependency future, local point distinguishability, and finite pasts for finite collections of records. Its countermodels permit a Euclidean metric ball as a one-step spatial section on a discrete slice and permit a generated positive-orthant cone. The consistency assumptions also do not imply interpolation, a manifold topology, or causal availability rules for an otherwise unspecified type system. \end{theorem} \begin{proof}[Justification by the cited obstruction] This is the obstruction recorded in En08-A \cite[\loc{thm:main}]{CheungEn08A}. The source construction is \loc{thm:network}; the two geometric realizations are \loc{prop:ball} and \loc{prop:orthant}. The source also uses a two-event chain to witness failure of interpolation and separately discusses the missing typing rules. XXII-A does not reconstruct those networks or claim an independent verification of their proofs. \end{proof} \begin{remark}[What the ball countermodel does and does not say] \label{rem:ball} The two countermodels witness different failures of inference. A Euclidean-ball section is compatible with scheduler consistency. It is not, merely by being a ball, a non-Lorentzian continuum cone: indeed, \[ \Cone\bigl\{(1,x):\|x\|_2\leq1\bigr\}=\Lor_m \quad\text{in }\R\times\R^{m-1}. \] A discrete one-step section and its generated reachability structure must therefore be distinguished from this full continuum homogenization. The orthant supplies the direct non-Lorentzian cone obstruction in dimension at least three. The statement ``ball and orthant are countermodels'' is shorthand for the separately specified nonimplications in En08-A, not a claim that every cone over a Euclidean ball is non-Lorentzian \cite[\loc{prop:ball}, \loc{prop:orthant}]{CheungEn08A}. \end{remark} The obstruction is relative to its hypotheses. A future quantum process class may exclude a countermodel by a proved class restriction or by a genuinely additional hypothesis. Such an exclusion must be shown; it does not follow from the word ``quantum.'' Conversely, a countermodel outside a newly restricted class does not refute a theorem about that class. En08-A's positive representation statements are conditional on a faithful generated affine cone representation and further cone properties \cite[\loc{thm:mainpositive}, \loc{thm:representation}] {CheungEn08A}. They do not construct that representation from a raw event order. \subsection*{Four warnings retained with their exact scope} The following distinctions are retained from En08-A; their predecessor arguments are not re-proved here \cite{CheungEn08A}. \begin{enumerate} \item A boundary ray of a generated, closed cone need not contain the difference of any attained operational event pair. Boundary generation or limiting approximation is not an attained null protocol. \item A small diamond distance from a non-signalling channel is not exact non-signalling, and hence is not exact causal containment. A positive error allowance permits a nonzero influence. \item A finite ordered event sample does not determine an unrestricted spacetime metric or a unique continuum realization. \item Chronology does not determine the full metric, including its conformal scale. \end{enumerate} The last warning is not a denial of causal reconstruction theorems. With an entire continuum spacetime and suitable causal hypotheses, chronological information can determine conformal structure \cite{HKM,Malament}. Those continuum assumptions are not furnished by a finite operational event sample. In particular, ``chronology alone does not determine the metric'' must not be strengthened to ``chronology can never determine a conformal class.'' \section{The available conditional classification} \label{sec:classification} \begin{definition}[Homogeneity and cone strict convexity] For a proper cone \(K\subset V\), set \[ \Aut(K)=\{A\in\GL(V):A(K)=K\}. \] The cone is \emph{homogeneous} if \(\Aut(K)\) acts transitively on \(\Int K\). It is \emph{strictly convex in the cone sense} if, for nonzero, non-proportional \(u,v\in K\), \[ u+v\in\Int K. \] Equivalently, its boundary has no line segment joining points on different nonzero rays. \end{definition} \begin{theorem}[Conditional Lorentz classification] \label{thm:classification} Let \(K\subset V\) be a homogeneous, strictly convex proper cone, with \(\dim V=m\geq3\). Then there is an invertible linear map \(T:V\to\R^m\) such that \[ T(K)=\Lor_m. \] This is the classification consequence recorded as En08-A \loc{thm:weaker}. Homogeneity and strict convexity are additional hypotheses, not consequences of \(\SP\). \end{theorem} \begin{proof} Apply Ito and Louren\c{c}o's classification of strictly convex homogeneous cones \cite[Theorem 7]{ItoLourenco}. Their lower-dimensional alternatives are excluded by \(\dim V\geq3\). The remaining alternative is a Lorentz cone. The classification is not re-proved here; its operational use is the conditional statement recorded in En08-A \cite[\loc{thm:weaker}]{CheungEn08A}. \end{proof} Self-duality and irreducibility need not be added to the hypotheses of Theorem~\ref{thm:classification}. Their appearance in another representation theorem does not make them part of this weaker classification route. \begin{remark}[Only conditional conformal information] \label{rem:conformal} If a later extraction supplies the hypotheses of Theorem~\ref{thm:classification}, the selected cone determines the Lorentzian quadratic form \[ q_T(v)=(Tv)_0^2-\sum_{j=1}^{m-1}(Tv)_j^2 \] up to positive multiplication. This concerns the full cone, not merely the comparisons among a sparse set of embedded events \cite[\loc{prop:quadraticunique}]{CheungEn08A}. For completeness, the elementary uniqueness calculation is as follows. In Lorentz coordinates write another quadratic form with the same null boundary and positive sign on the cone interior as \[ \widetilde q(t,x)=a t^2+2t\,b\cdot x+x^{\mathsf T}Cx, \qquad C=C^{\mathsf T}. \] Its vanishing on \((1,u)\), \(\|u\|_2=1\), and on \((1,-u)\) gives \(b=0\) and \(u^{\mathsf T}Cu=-a\) for every unit \(u\). Thus \(C=-aI\), and positivity on \((1,0)\) gives \(a>0\). The ambiguity here is a positive constant on one affine carrier. A future field of cones on a manifold would require separate arguments and could leave a position-dependent conformal factor. No metric field or manifold is constructed in XXII-A. \end{remark} A scheduler change is not automatically an element of \(\Aut(K)\). Commutation of independent operations concerns legal execution interleavings; homogeneity concerns transitivity of a linear group on the interior of an extracted cone. Renaming the former does not prove the latter. \section{The frozen target and its acceptance conditions} \label{sec:target} \begin{definition}[Faithful generated affine representation] \label{def:representation} Let \(E_D\) be the declared event carrier associated with operational datum \(D\), and let \(\preceq_D\) be a declared partial order extracted from scheduled ancestry or operational influence. A faithful generated affine cone representation consists of \((V_D,\iota_D,K_D)\), where \(V_D\) is a finite-dimensional real vector space, \(\iota_D:E_D\to V_D\) is injective, and \(K_D\) is a proper cone satisfying \begin{align} p\preceq_D q &\quad\Longleftrightarrow\quad \iota_D(q)-\iota_D(p)\in K_D, \label{eq:faithful}\\ V_D &=\Span\{\iota_D(q)-\iota_D(p):p,q\in E_D\}, \label{eq:span}\\ K_D &=\overline{\Cone \{\iota_D(q)-\iota_D(p): p\preceq_D q,\ p\ne q\}}. \label{eq:generated} \end{align} If the carrier is obtained by an operational quotient or completion, that construction and the relation on the resulting carrier are part of the extraction. \end{definition} Equation~\eqref{eq:faithful} prevents an arbitrary supercone from being called an extracted causal cone. Equation~\eqref{eq:generated} prevents an independently appended Lorentz cone from supplying the desired answer. Neither equation follows merely from drawing a dependency graph. \paragraph{Finite-generation check.} If \(E_D\) is finite, the set of generators in \eqref{eq:generated} is finite. Its conical hull is a closed polyhedral cone. By contrast, \(\Lor_m\), \(m\geq3\), has the infinitely many extreme rays \[ \R_{\geq0}(1,u), \qquad u\in S^{m-2}. \] Invertible linear maps preserve extreme rays. Therefore a finite event carrier with exactly the generation rule \eqref{eq:generated} cannot supply the proposed Lorentz cone. A later theorem must consequently declare an infinite operational family, an operationally justified extension, or a limiting construction. This is not a requirement of infinitely many parties. It is a requirement that the claimed continuum cone not be substituted for the finite event-difference cone without a connecting construction. A finite tomographically complete list can determine a declared finite-dimensional process; that fact is different from determining a continuum event carrier \cite{CDP,Pollock}. \begin{target}[Operational Lorentz-cone selection] \label{target:selection} Find one named quantum process class \(\mathcal P\), one named data assignment \(\mathcal D(\mathcal P)\), a finite list \[ \HP=(H_1,\ldots,H_r), \] and one predeclared extraction rule \(\cE\), such that: \begin{enumerate}[label=\textup{(T\arabic*)}] \item The admissible data set \[ \mathcal D_{\HP} = \{D\in\mathcal D(\mathcal P): H_1(D),\ldots,H_r(D)\} \] is nonempty. \item For every \(D\in\mathcal D_{\HP}\), the rule \(\cE\) constructs a specified order \(\preceq_D\) and a faithful generated affine representation \((V_D,\iota_D,K_D)\) as in Definition~\ref{def:representation}. \item On a declared fixed dimension \(m\geq3\), an invertible linear map \(T_D:V_D\to\R^m\) exists with \[ T_D(K_D)=\Lor_m. \] Consequently the represented event comparisons satisfy \[ p\preceq_D q \quad\Longleftrightarrow\quad T_D\bigl(\iota_D(q)-\iota_D(p)\bigr)\in\Lor_m. \] \item The extraction uses only the specified operational data, their declared non-geometric interface, and \(\HP\). No spacetime metric, geometric causal relation, physical light cone, or invariant speed is installed as input. \item The additional class restrictions and hypotheses are not supplied merely by \(\SP\). Their status is stated before the extraction is applied. \end{enumerate} A comparison with a supplied VII-J-type causal relation is a subsequent embedding problem, not part of the selection. \end{target} Research target~\ref{target:selection} is a specification of the result to be established in later stages. It has no proof in this note. \subsection*{Hypothesis and non-circularity audit} A proposed solution must satisfy the following acceptance conditions. \begin{enumerate}[label=\textup{(A\arabic*)}] \item \textbf{One model and one relation.} Freeze the process class, admissible data, intervention access, equivalences, and extraction rule. State whether the represented relation is scheduled ancestry, direct influence, or a declared order constructed from influence. A state-positivity cone is not an operational event cone without an order-identification theorem. \item \textbf{Genuine additional content.} Include class restrictions in the hypothesis ledger. In a common operational language, their conjunction with \(\HP\) must contain content not forced by \(\SP\). This does not require every individual \(H_j\) to be independent of \(\SP\), but it excludes a package that merely renames it. \item \textbf{Non-vacuity and countermodel exclusion.} Exhibit admissible data satisfying the complete package. For each relevant En08-A countermodel, identify and justify the failed class restriction or hypothesis. An unchanged countermodel satisfying all proposed premises defeats the proposed conclusion. \item \textbf{Honest use of classification.} Homogeneity and strict convexity may be proved from operational data, or they may be explicitly assumed as properties of the extracted cone. In the latter case, the result is conditional on those assumptions. The classification theorem does not discharge their operational derivation. \item \textbf{Generation, faithfulness, and invariance.} Prove \eqref{eq:faithful}--\eqref{eq:generated}, including the absence of spurious comparisons introduced by convexification or closure. The same declared data, up to the permitted relabellings, must give affinely isomorphic representations. The input must not prescribe the cone whose extraction is claimed. \item \textbf{Limits and evidence.} If a completion or refinement is used, give its comparison maps, topology, compatibility, and convergence theorem. A finite list of universally quantified structural hypotheses is not automatically a finite empirical certificate. Statistical records require their own coverage and implementation allowances. \end{enumerate} Externally supplied correlations are permissible operational inputs, but their preparation and access model must be declared. A static entanglement table does not by itself provide intervention access or a temporal influence relation. Likewise, a quantum comb's declared slot ordering must not be misreported as an ordering derived from its outcome probabilities \cite{CDP,OCB,Beckman}. \section{Continuum and comparison ledger} \label{sec:ledger} XIX-C supplies two specific cautions for any later continuum step \cite[\loc{prop:two-limits}, \loc{prop:interpolation}]{CheungXIXC}: \begin{itemize} \item two refinement families can share one coarse complete channel, obey one-site scheduled locality, and converge to different semigroups; \item finite sampling constraints can leave interpolation ambiguity. \end{itemize} The first counterexample chooses different missing refinement data; it does not assume an additional projective-consistency condition. Such consistency would itself be an extra hypothesis. These warnings do not exclude a unique continuum limit after suitable additional data and assumptions are supplied. The documentary roles of the other named components are fixed below. None substitutes for Research target~\ref{target:selection}. \begin{center} \small \renewcommand{\arraystretch}{1.16} \begin{tabularx}{\linewidth} {@{}p{.20\linewidth}p{.22\linewidth}Y@{}} \toprule Component & Role in XXII-A & Boundary\\ \midrule En08-A & Obstruction and conditional representation record & Scheduler consistency does not select the cone; faithful affine representation and cone structure are additional \cite{CheungEn08A}.\\[3pt] VII-J & Three-relation discipline & Scheduled support and operational influence are distinguished; physical containment needs a causal embedding \cite{CheungVIIJ}.\\[3pt] XIX-C & Continuum warning & One finite channel or finite sampling record does not select the missing refinement and continuum evolution \cite{CheungXIXC}.\\[3pt] En08-B & Possible later approximation layer & Finite SWAP protocols approximate a prescribed null boundary. The metric and speed are inputs; endpoint approximation is not cone selection or a global continuum evolution \cite[\loc{thm:main}, \loc{cor:operational-limit}]{CheungEn08B}.\\[3pt] VIII-A/B/C & Documentary geometric comparison & Declared conical families and fixed-deficit packets are not an operational derivation of their cone or deficit \cite{CheungVIII}.\\[3pt] Em02-E & Documentary clock comparison & The reported relational lapse uses an installed classical background; its quantum audit is not supplied here \cite{CheungEm02E}.\\[3pt] XIX-A--E; XX-A--F & Clock and admission boundaries & Finite registers, scheduled support, and classical lapse prescriptions are not silently promoted to native continuum evolution or a derived metric \cite{CheungXIXA,CheungXIXE}.\\[3pt] OpenAI catalogue & Negative-control record & Area laws, random-tree limits, VOA-net claims, and local Ward records do not supply the missing operational cone-selection argument \cite{CatalogueRecord}.\\ \bottomrule \end{tabularx} \end{center} This ledger records scopes, not proof-transfer paths. In particular, null-boundary approximation in En08-B cannot be used to derive the metric or speed with which its targets were defined. Similarly, an independent geometric construction does not become an operational extraction by changing the name of its input. \section{Exact release and remaining gap} \label{sec:release} The completed Stage 0 release consists of: \begin{enumerate} \item the separation in Theorem~\ref{thm:separation}; \item the cited obstruction in Theorem~\ref{thm:obstruction}; \item the conditional classification in Theorem~\ref{thm:classification}; \item the unproved but precise Research target~\ref{target:selection} and its acceptance audit. \end{enumerate} There is no fourth theorem asserting positive selection. For the later programme, the intended obligations are: \begin{description}[leftmargin=0pt,labelsep=.6em] \item[Stage 1.] Name and freeze the operational process class, its additional data, the relation to be extracted, and the finite hypothesis ledger. Establish non-vacuity and explain the relevant countermodel exclusions. \item[Stage 2.] Construct the faithful generated carrier and order representation, including any necessary quotient, extension, or continuum limit. Prove the needed compatibility and convergence statements. \item[Stage 3.] Prove the cone-selecting properties and apply an appropriate classification, or state a explicitly conditional selection theorem. Any comparison with a supplied geometric causal relation follows as a separately certified embedding. \end{description} XXII-A does not derive homogeneity or strict convexity from quantum data; construct an attained null protocol; identify scheduled ancestry with geometric causality; or construct a metric field. It proves no Einstein dynamics, gravity-generated entanglement, holographic reconstruction, Page curve, or quantum evolution without an external time. The note supplies no result changing the separate series obligations O.1, XV-B, XVI-N, Route B, H2, I3, or J2. No quantum coefficient is admitted. No predecessor proof is represented as machine-checked, and no native programme execution or successful pdfLaTeX build is asserted. These are scope boundaries, not universal impossibility claims \cite{CheungXIXE}. \paragraph{The precise remaining gap.} Stage 1 must supply \emph{the additional quantum operational data and the finite hypothesis package, on one nonempty frozen process class, that go beyond the scheduler package \(\SP\) and are sufficient for a faithful generated Lorentz-cone selection}. Until that package and its connecting proofs are supplied, scheduler consistency remains obstructed as a selection principle, and homogeneity-plus-strict-convexity remains a conditional classification route rather than an operational emergence theorem. \begin{thebibliography}{99} \bibitem{CheungEn08A} N.~Cheung. \emph{En08-A}, version 1, manuscript \loc{En08-A.tex}. Enter Calculus Companion, n.d. Relevant labels: \loc{thm:main}, \loc{thm:network}, \loc{prop:ball}, \loc{prop:orthant}, \loc{thm:mainpositive}, \loc{thm:representation}, \loc{thm:weaker}, and \loc{prop:quadraticunique}. Manuscript identification and passages supplied with this note. Registry: \url{https://enter.grok.me/}. \bibitem{CheungVIIJ} N.~Cheung. \emph{VII-J}, version 1, manuscript \loc{VII-J.tex}. Enter Calculus Companion, n.d. Relevant labels: \loc{thm:schedule}, \loc{thm:nosignal}, \loc{thm:physical}, and \loc{thm:release}. Scheduled ancestry, operational influence, and supplied causal relation are distinct interfaces. \bibitem{ItoLourenco} M.~Ito and B.~F.~Louren\c{c}o. The \(p\)-cones in dimension \(n\geq3\) are not homogeneous when \(p\ne2\). \emph{Linear Algebra and its Applications} \textbf{533} (2017), 326--335. doi:\ \href{https://doi.org/10.1016/j.laa.2017.07.029} {\nolinkurl{10.1016/j.laa.2017.07.029}}. Preprint: \url{https://arxiv.org/abs/1701.05801}. Theorem 7 is the classification used here. \bibitem{OCB} O.~Oreshkov, F.~Costa, and \v{C}.~Brukner. Quantum correlations with no causal order. \emph{Nature Communications} \textbf{3} (2012), 1092. doi:\ \href{https://doi.org/10.1038/ncomms2076} {\nolinkurl{10.1038/ncomms2076}}. \bibitem{CDP} G.~Chiribella, G.~M.~D'Ariano, and P.~Perinotti. Theoretical framework for quantum networks. \emph{Physical Review A} \textbf{80} (2009), 022339. doi:\ \href{https://doi.org/10.1103/PhysRevA.80.022339} {\nolinkurl{10.1103/PhysRevA.80.022339}}. \bibitem{Pollock} F.~A.~Pollock, C.~Rodr\'iguez-Rosario, T.~Frauenheim, M.~Paternostro, and K.~Modi. Non-Markovian quantum processes: Complete framework and efficient characterization. \emph{Physical Review A} \textbf{97} (2018), 012127. doi:\ \href{https://doi.org/10.1103/PhysRevA.97.012127} {\nolinkurl{10.1103/PhysRevA.97.012127}}. \bibitem{Beckman} D.~Beckman, D.~Gottesman, M.~A.~Nielsen, and J.~Preskill. Causal and localizable quantum operations. \emph{Physical Review A} \textbf{64} (2001), 052309. doi:\ \href{https://doi.org/10.1103/PhysRevA.64.052309} {\nolinkurl{10.1103/PhysRevA.64.052309}}. \bibitem{HKM} S.~W.~Hawking, A.~R.~King, and P.~J.~McCarthy. A new topology for curved space-time which incorporates the causal, differential, and conformal structures. \emph{Journal of Mathematical Physics} \textbf{17} (1976), 174--181. doi:\ \href{https://doi.org/10.1063/1.522874} {\nolinkurl{10.1063/1.522874}}. \bibitem{Malament} D.~B.~Malament. The class of continuous timelike curves determines the topology of spacetime. \emph{Journal of Mathematical Physics} \textbf{18} (1977), 1399--1404. doi:\ \href{https://doi.org/10.1063/1.523436} {\nolinkurl{10.1063/1.523436}}. \bibitem{CheungXIXC} N.~Cheung. \emph{XIX-C}, version 1, manuscript \loc{XIX-C.tex}. Enter Calculus Companion, n.d. Relevant labels: \loc{def:limit}, \loc{prop:two-limits}, \loc{prop:interpolation}, and \loc{thm:main}. Supplied passages concern finite certification and continuum non-determination. \bibitem{CheungEn08B} N.~Cheung. \emph{En08-B}, version 1, manuscript \loc{En08-B.tex}. Enter Calculus Companion, n.d. Relevant labels: \loc{thm:main}, \loc{cor:operational-limit}, and \loc{def:comparison}. Supplied passages concern resource-controlled approximation of a prescribed null boundary. \bibitem{CheungVIII} N.~Cheung. \emph{VIII-A}, version 3; \emph{VIII-B}, version 3; and \emph{VIII-C}. Enter Calculus Companion manuscripts, n.d. Documentary conical-family and fixed-deficit comparison records supplied with the XXII-A materials; not mathematical premises of the selection target. \bibitem{CheungEm02E} N.~Cheung. \emph{Em02-E}. Enter Calculus Companion manuscript, n.d. Classical relational-lapse and incomplete quantum-audit scope record as supplied with the XXII-A materials. No unsupplied theorem or proof is imported here. \bibitem{CheungXIXA} N.~Cheung. \emph{XIX-A}, version 1, manuscript \loc{XIX-A.tex}. Enter Calculus Companion, n.d. Relevant label: \loc{thm:main}. Supplied statement concerns clock separation before coordinate identification. \bibitem{CheungXIXE} N.~Cheung. \emph{XIX-E}, version 1, manuscript \loc{XIX-E.tex}. Enter Calculus Companion, n.d. Relevant passage: ``What remains outside the release.'' Documentary boundary record; no claim of independent proof verification. \bibitem{CatalogueRecord} OpenAI. \emph{OpenAI mathematics catalogue}, n.d. Supplied documentary record, including family identifiers 265, 211, 280, 069, and 282. The catalogue claims are neither verified nor used as theorem premises in XXII-A. \end{thebibliography} \end{document}