% XXII-B.tex % Standalone pdfLaTeX source. No external .tex, .bib, or .aux files required. % Suggested build: % pdflatex -interaction=nonstopmode -halt-on-error XXII-B.tex % pdflatex -interaction=nonstopmode -halt-on-error XXII-B.tex % % No successful compilation, numerical execution, or machine verification % is asserted. % % Corrections incorporated: % 1. Existence of a successful ledger remains a research objective. % 2. Process-matrix signalling is not automatically a partial order. % 3. Causal separability does not imply acyclicity of the aggregate % pairwise signalling graph. % 4. Standard combs and process tensors have declared temporal slots. % 5. Reference extension is distinguished from preservation of all % input-reference correlations. % 6. An acyclic signalling graph alone does not establish the complete % scheduler package or a circuit realization. % 7. The orthant is homogeneous; homogeneity alone does not exclude it. % 8. A Euclidean-ball section is not, by itself, a non-Lorentzian cone. % 9. Countermodel exclusion and non-vacuity are separate proof obligations. % 10. Boundary attainment and continuum uniqueness are not silently added % to the Lorentz-cone selection target. % 11. All references to XXII-A are bibliographic, not external LaTeX refs. \documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{enumitem} \usepackage{booktabs,array,longtable} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{ pdftitle={XXII-B: Catalogue the additional quantum data and the hypothesis ledger}, pdfauthor={Ngo Cheung}, pdfsubject={Stage 1 of a limited kinematic emergence programme} } \setlength{\emergencystretch}{2em} \setlist[enumerate]{leftmargin=*,itemsep=3pt,topsep=5pt} \setlist[itemize]{leftmargin=*,itemsep=3pt,topsep=5pt} \setlist[description]{leftmargin=0pt,labelsep=.6em,itemsep=5pt} \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}[proposition]{Theorem} \theoremstyle{definition} \newtheorem{definition}[proposition]{Definition} \newtheorem{example}[proposition]{Example} \newtheorem{specification}[proposition]{Specification} \theoremstyle{remark} \newtheorem{remark}[proposition]{Remark} \newcommand{\R}{\mathbb R} \newcommand{\Q}{\mathbb Q} \newcommand{\C}{\mathbb C} \newcommand{\id}{\operatorname{id}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Span}{\operatorname{span}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\Int}{\operatorname{int}} \newcommand{\Cone}{\operatorname{cone}} \newcommand{\Spec}{\operatorname{spec}} \newcommand{\Lor}{\mathcal L} \newcommand{\SP}{\mathsf S} \newcommand{\HP}{\mathsf H} \newcommand{\Sched}{R_{\mathrm{sch}}} \newcommand{\Infl}{R_{\mathrm{inf}}} \newcommand{\Supplied}{R_{\mathrm{sup}}} \newcommand{\ddiamond}{d_{\diamond}} \newcommand{\dtv}{d_{\mathrm{TV}}} \newcommand{\cE}{\mathfrak E} \newcommand{\loc}[1]{\texttt{#1}} \newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|} \newcommand{\kb}[2]{|#1\rangle\langle#2|} \newcommand{\doi}[1]{doi:\ \href{https://doi.org/#1}{\nolinkurl{#1}}} \title{\textbf{XXII-B: Catalogue the additional quantum data\\ and the hypothesis ledger}\\[4pt] \large Stage 1 of a limited kinematic emergence programme} \author{Ngo Cheung} \date{} \begin{document} \maketitle \begin{abstract} This note supplies a catalogue and a hypothesis-ledger syntax for the operational Lorentz-cone selection target fixed in XXII-A. Process matrices, ordered quantum combs and process tensors, static correlation data, and reference-stable channel tables are distinguished. In particular, pairwise signalling need not be a partial order, and a standard process tensor does not derive its declared temporal slot ordering. The ledger separates operational inputs, additional assumptions, derived properties, and undischarged proof obligations. Faithful generation, homogeneity, strict convexity, countermodel exclusion, continuum compatibility, and error allowances receive explicit acceptance conditions. Elementary witnesses make the five failure tests precise. No candidate is shown to satisfy the complete selection target, and no nonempty joint cone-selecting data class is constructed. The Stage 1 release is the completed catalogue and audit interface, not a positive emergence theorem. \end{abstract} \section{Purpose and exact logical status} \label{sec:purpose} XXII-A fixes a research target: operational data, on one named nonempty process class and under a finite predeclared hypothesis package, must yield a faithful generated affine cone linearly isomorphic to a Lorentz cone \cite[\loc{target:selection}, conditions (T1)--(T5)]{CheungXXIIA}. XXII-B catalogues possible data and the obligations of such a package. It does not assert that a successful package exists. The proposed sentence \begin{quote} There exists a finite additional ledger, a named quantum data class, and a nonempty admissible data set excluding the relevant scheduler countermodels \end{quote} is therefore an \emph{objective to be established}, not the principal theorem of this note. Moreover, even establishing that sentence would not alone establish Lorentz-cone selection: excluding named countermodels is a necessary audit, not a sufficient classification of every remaining cone. The baseline scheduler package is retained from XXII-A and En08-A: \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} Their conjunction is denoted by \(\SP\). En08-A supplies the obstruction to selection from this package \cite[\loc{thm:main}, \loc{prop:ball}, \loc{prop:orthant}] {CheungEn08A}. Condition (S5) is a special property of those models, not a general identification of influence with ancestry. \paragraph{What is frozen here.} The finite catalogue of operational languages, the meaning of the candidate data, the ledger-entry syntax, and the failure tests are frozen. A particular process class, extraction algorithm, and successful hypothesis instantiation are \emph{not} selected here. Calling this a completed catalogue does not discharge the substantive non-vacuity and construction obligations of the programme. \paragraph{Source discipline.} Cheung's manuscripts are cited by identifier, supplied version where available, and internal label. Documentary comparisons are not mathematical premises. The operational literature provides definitions and frameworks, not an unspecified cone-selection theorem. The elementary arguments printed below establish their own stated claims; they do not verify the full proofs of the cited manuscripts. \section{Common operational interface} \label{sec:interface} \begin{definition}[Operational declaration] \label{def:declaration} An operational declaration specifies: \begin{enumerate}[label=\textup{(D\arabic*)}] \item a language tag, a finite party or slot set for each experiment, and finite-dimensional input and output Hilbert spaces; \item admissible preparations and local instruments, including any restrictions on interventions; \item the accessible output systems and records, with every attempted outcome and failure retained in the declared interface; \item which auxiliary systems and classical controls participate, and which references are untouched; \item the operational equivalences, relabellings, and comparison maps; \item the event labels, their interpretation, and the proposed relation to be extracted; \item whether the supplied data are exact mathematical maps or probabilities, or finite statistical observations; \item any family of experiments, refinements, or extensions used to enlarge the event carrier. \end{enumerate} A finite party set in one experiment does not require the full experiment family or the event carrier to be finite. \end{definition} Tensor factors and laboratory labels in this declaration are operational interface data. They are not spatial coordinates or a spacetime metric. Conversely, an externally assigned speed, spatial distance, or geometric causal relation must not be hidden inside an ``operational'' label. A local instrument consists of completely positive maps \(\{\mathcal M_{a|x}\}_a\), with \(\sum_a\mathcal M_{a|x}\) trace preserving for each setting \(x\). The setting is chosen independently of the process except where a specified adaptive protocol permits otherwise. Unconditional signalling compares complete instruments with the sender's outcome summed, not normalized postselected branches \cite{OCB,Beckman}. \subsection{Relations and signalling strengths} \label{sec:relations} Three relations remain distinct: \begin{description} \item[Scheduled ancestry \(\Sched\).] Reachability in a complete dependency graph that includes the actual quantum inputs, classical controls, guards, and routing dependencies. \item[Operational influence \(\Infl\).] A change in an accessible receiver output under an admissible intervention, with the rest of the operational context held fixed. \item[Supplied causal relation \(\Supplied\).] An independently declared geometric comparison relation, requiring its own carrier, calibration, and embedding of operational events. \end{description} This separation is the discipline of VII-J and XXII-A \cite{CheungVIIJ,CheungXXIIA}. \begin{definition}[Unconditional signalling test] \label{def:signalling} Fix distinct event labels \(p,q\), a declared class of contexts \(\Gamma\), and admissible deterministic interventions \(\mathcal I,\mathcal J\) at \(p\). All operations other than this replacement are the same within a compared pair. For classical receiver records, let \(P_{q|\mathcal I,\Gamma}\) and \(P_{q|\mathcal J,\Gamma}\) be their unconditional laws and define \[ \Delta_D(p,q) = \sup_{\mathcal I,\mathcal J,\Gamma} \dtv\bigl(P_{q|\mathcal I,\Gamma}, P_{q|\mathcal J,\Gamma}\bigr). \label{eq:classical-strength} \] For complete channels on common input and output comparison spaces, use instead \[ \Delta_D(p,q) = \sup_{\mathcal I,\mathcal J,\Gamma} \ddiamond\bigl(\Phi_{q|\mathcal I,\Gamma}, \Phi_{q|\mathcal J,\Gamma}\bigr), \qquad \ddiamond(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond. \label{eq:channel-strength} \] Write \(p\Infl q\) when \(\Delta_D(p,q)>0\). The receiver interface must not acquire the sender's record through an undeclared readout or side channel. \end{definition} The contexts included in the supremum are part of the declaration. An influence relation for one fixed context and the union of influence witnesses over several contexts are different objects. A diamond norm is used only after a complete channel reduction has been defined. It is not assigned without justification to an arbitrary multi-time probability functional \cite[Chapters 2--3]{Watrous}. Direct influence is not assumed to be transitive or acyclic. A future order extraction must explicitly: \begin{enumerate} \item prove the needed order properties; or \item specify an order-forming operation and prove what it represents. \end{enumerate} For an acyclic directed relation, reflexive transitive closure gives a partial order. For a relation with cycles, quotienting strongly connected components gives an order of \emph{components}, not an injective representation of the original events. Such a quotient must not be left implicit. In complete definite-order channel networks, influence is bounded by scheduled ancestry. Equality requires more than the existence of a wire: a replacement channel may ignore that wire entirely \cite{CheungVIIJ,CheungXXIIA}. \section{Operational languages and candidate data} \label{sec:candidates} The following are alternative tagged interfaces. Moving from one to another requires a declared translation preserving the relevant instruments, observations, and relation. Similar operator notation does not supply such a translation. \subsection{Candidate A: process-matrix signalling data} \label{sec:process} Fix bases and use the unnormalized Choi operator \[ J(\mathcal M) = \sum_{i,j}\kb{i}{j}\otimes\mathcal M(\kb{i}{j}), \qquad M=J(\mathcal M)^{\mathsf T}, \] where \({}^{\mathsf T}\) is the full transpose in those bases. In this convention a process matrix \(W\) assigns \[ P_W(\boldsymbol a|\boldsymbol x) = \Tr\!\left[ W\bigotimes_{k=1}^{N}M^{(k)}_{a_k|x_k} \right]. \label{eq:process-probabilities} \] Positivity and normalization for all admitted local instruments, including the required ancillary extensions, are part of process validity \cite{OCB}. Candidate A consists of \(W\), the instrument-access declaration, and its unconditional signalling tests. The framework does not impose one global party order at the outset. This does \emph{not} mean that every \(W\) lacks definite order, nor that causal nonseparability selects a Lorentz cone. The extracted object initially is a signalling relation. The phrase ``the signalling partial order of \(W\)'' is permissible only after an order theorem or a declared order-forming construction. \begin{proposition}[Causal separability does not ensure an acyclic signalling graph] \label{prop:signalling-cycle} There is a causally separable bipartite process with both \(A\Infl B\) and \(B\Infl A\). Thus its aggregate pairwise signalling relation is not a strict partial order. \end{proposition} \begin{proof} Choose \(0<\lambda<1\). With probability \(\lambda\), initialize a qubit in \(\ket0\), send it through \(A\), then through \(B\), and discard the final output. With probability \(1-\lambda\), use the reversed circuit. The classical choice is independent of the laboratory settings. This is a convex mixture of definite-order processes, hence a causally separable process in the bipartite sense \cite{OCB}. At both laboratories use measure-and-prepare instruments: \[ \mathcal M_{a|x}(X) = \bra aX\ket a\,\kb{x}{x}, \qquad a,x\in\{0,1\}. \] Let \(x,y\) be the settings at \(A,B\), respectively. The resulting complete probability table is \[ P(a,b|x,y) = \lambda\,\delta_{a0}\delta_{bx} +(1-\lambda)\,\delta_{b0}\delta_{ay}. \] Consequently, \[ P(b=1|x,y)=\lambda x, \qquad P(a=1|x,y)=(1-\lambda)y. \] Changing \(x\) changes \(B\)'s unconditional record, and changing \(y\) changes \(A\)'s unconditional record. Both influences are nonzero. The two edges describe alternative order branches; they do not constitute a physical closed loop within either branch. \end{proof} Acyclicity of a particular signalling graph, when it holds, still does not by itself prove a complete circuit realization or every condition in \(\SP\). The En08-A obstruction applies to a proposed construction that uses only that scheduler package, or to an appropriately identified realization of its countermodels. It is not a universal theorem about every process matrix possessing an acyclic graph. \subsection{Candidate B: combs and process tensors on declared slots} \label{sec:combs} A deterministic comb is a positive Choi operator on declared, ordered input/output slots, satisfying recursive trace conditions of the form \[ \Tr_{\mathrm{out},j} C^{(j)} = I_{\mathrm{in},j}\otimes C^{(j-1)}, \qquad C^{(0)}=1, \label{eq:comb-normalization} \] with tensor-factor order fixed by the chosen convention \cite[Theorem 5]{CDP}. A standard multi-time process tensor similarly describes interventions at a declared sequence of time steps, with its specified boundary systems and containment conditions \cite{Pollock}. Accordingly, the expression ``a standard process tensor with non-preinstalled slot order'' is not adopted here. Candidate B may extract \emph{which permitted interventions influence which later outputs}, or a finer operational relation within the declared temporal interface. It cannot report the declared slot ordering as a conclusion derived from its outcome probabilities. An unordered or coherently controlled-order higher-order object requires a different explicitly defined interface. It must not be obtained by dropping the comb normalization conditions while keeping the same name. Memory, temporal correlations, or process-tensor tomography do not supply a unique continuous-time interpolation from one coarse datum. XIX-C's counterexample in Section~\ref{sec:test-refinement} illustrates the missing refinement information \cite[\loc{prop:two-limits}]{CheungXIXC}. It is a warning about that inference, not an assertion that every fully specified process tensor has ambiguous refinement. \subsection{Candidate C: static spectra and mutual information} \label{sec:static} For a declared tensor-product output state \(\rho\), static candidate data may include \[ \Spec(\rho_A),\qquad S(\rho_A),\qquad I(A:B)_\rho = S(\rho_A)+S(\rho_B)-S(\rho_{AB}), \] for a named collection of subsystem splits. For mixed states, a reduced-state spectrum is not by itself an entanglement measure. These quantities are correlation data; their interpretation requires the declared partitions \cite[Chapters 5--6]{Watrous}. \begin{proposition}[Static state data do not determine the operational realization] \label{prop:static} Even an exact labelled bipartite output state, and hence all its static reduced spectra and mutual-information values, can have realizations both with and without a signalling transmission. Such state data alone do not specify an intervention-based event order. \end{proposition} \begin{proof} Consider the labelled Bell output \[ \rho_{AB}=\kb{\Phi^+}{\Phi^+}, \qquad \ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt2}. \] One realization distributes this state to two laboratories with no communication. For any trace-preserving local intervention \(\mathcal I_A\), \[ \Tr_A[(\mathcal I_A\otimes\id_B)(\rho_{AB})] = \Tr_A\rho_{AB}. \] Indeed, pairing with a \(B\)-observable and using \(\mathcal I_A^*(I_A)=I_A\) proves the equality. There is no unconditional signalling from that local intervention. Another realization prepares the Bell pair at a source, retains the \(A\)-register, and transmits its \(B\)-register to a receiver. The baseline labelled output is exactly the same \(\rho_{AB}\). If the source is allowed to replace its outgoing register by \(\kb0 0\) or \(\kb1 1\), the receiver can distinguish the alternatives perfectly. The two realizations have the same static output datum but different operational access and transmission structure. Precisely those missing declarations are not encoded by the state or its spectra. The first equality is the standard distinction between local correlations and signalling \cite{Beckman}. \end{proof} This is an underdetermination statement over unspecified realizations. It does not exclude reconstruction after a sufficiently restrictive dynamical and intervention model is independently declared. Static area-law records provide the corresponding documentary negative control in Oai265-A: an entropy record is not a recovery channel, a radiation history, or an operational metric extraction \cite[\loc{thm:separation}]{CheungOai265A}. A geometric partition or an area appearing in a supplied formula must not be treated as geometry derived from that formula. \subsection{Candidate D: reference-stable complete channel tables} \label{sec:reference} Candidate D consists of indexed complete channels on declared comparison interfaces, optionally supplemented by preparations, intervention settings, and complete correlation records. \emph{Reference stability} means that a channel acts as \(\Phi\otimes\id_R\) on an untouched reference and that a comparison is made completely, including such extensions. For every channel, \[ \Tr_{\mathrm{out}} [(\Phi\otimes\id_R)(\rho_{\mathrm{in},R})] = \rho_R. \label{eq:reference-marginal} \] This leaves the unconditional reference marginal unchanged. It does not imply preservation of all input-reference correlations: a replacement channel satisfies \eqref{eq:reference-marginal} while destroying those correlations. Exact logical identity transport is the stronger statement \[ (\Phi\otimes\id_R)(X)=X \] on identified input and output spaces \cite{Watrous,CheungEn08B}. Thus the term ``reference-preserving table'' must distinguish unchanged reference marginals from successful reference-preserving transmission. Selective conditioning may also change the conditional reference state without signalling to the unconditional reference \cite[\loc{thm:main}]{CheungEn09A}. An intervention-indexed complete table may determine influence within its declared access model. A transport-only table need not. Neither observation supplies a cone-selection theorem here. En08-B supplies a later endpoint-approximation benchmark, but its metric and speed are inputs \cite[\loc{thm:main}, \loc{cor:operational-limit}]{CheungEn08B}. \subsection{Status of the catalogue} All four candidates are admitted for \emph{investigation}, not certified as cone-selecting data classes. None has a proved nonempty joint admissible set satisfying the complete XXII-A target in this note. Finite-dimensional tomography concerns identification of an already declared process interface from informationally complete mathematical data. Exact probabilities are distinguished from their finite-sample estimates. Neither notion supplies an infinite event carrier merely by identifying a finite-dimensional operator \cite{CDP,Pollock,CheungXXIIA}. \section{Cone interface and hypothesis-ledger syntax} \label{sec:ledger} For a finite-dimensional real vector space \(V\), a cone is \emph{proper} here if it is closed, convex, pointed, and has nonempty interior in \(V\). The conical hull of a set \(G\) means the set of all finite nonnegative linear combinations of elements of \(G\), including zero. For \(m\geq3\), set \[ \Lor_m = \{(t,x)\in\R\times\R^{m-1}:t\geq\|x\|_2\}. \] These are the closed-cone conventions of XXII-A \cite{CheungXXIIA}. \begin{definition}[Faithful generated representation obligation] \label{def:representation} For datum \(D\), a proposed extraction must construct an event carrier \(E_D\), a declared partial order \(\preceq_D\), and \((V_D,\iota_D,K_D)\), with \(\iota_D\) injective and \(K_D\) proper, such that \begin{align} p\preceq_D q &\ \Longleftrightarrow\ \iota_D(q)-\iota_D(p)\in K_D, \label{eq:faithfulness}\\ V_D &=\Span\{\iota_D(q)-\iota_D(p):p,q\in E_D\}, \label{eq:spanning}\\ K_D &=\overline{\Cone \{\iota_D(q)-\iota_D(p): p\preceq_D q,\ p\ne q\}}. \label{eq:generation} \end{align} Any quotient, completion, or enlargement defining \(E_D\) is part of the extraction and must be justified. \end{definition} The closure in \eqref{eq:generation} is the finite-dimensional vector-space closure. A continuum construction supplying that vector space requires its own compatibility theorem. An open chronological relation instead uses \(\Int K_D\); it is not substituted for the closed order in \eqref{eq:faithfulness}. \begin{definition}[Ledger entry and status] \label{def:entry} Each ledger entry records: \begin{enumerate} \item its mathematical statement and domain of quantification; \item the operational data or extracted object on which it depends; \item whether it is a declaration, an additional assumption, a derived property, an open obligation, or a documentary comparison; \item its proof or source, where a derivation is claimed; \item the exact conclusion for which it is used. \end{enumerate} The status labels are \[ \mathsf{DECL},\quad \mathsf{ASSUMED},\quad \mathsf{DERIVED},\quad \mathsf{OPEN},\quad \mathsf{DOC}. \] Only \(\mathsf{ASSUMED}\) entries belong to the additional hypothesis list as assumptions. A \(\mathsf{DERIVED}\) entry must name its actual premises and proof. An \(\mathsf{OPEN}\) entry cannot be used as a proved premise. \end{definition} A later paper must fix one actual finite list \(\HP=(H_1,\ldots,H_r)\) before invoking its extraction theorem. The template below is not an assertion that eight hypotheses have been instantiated or satisfied. \begin{specification}[Finite ledger template] \label{spec:ledger} A proposed selection package must address the following entries. \begin{enumerate}[label=\textup{(L\arabic*)}] \item \textbf{Operational class and access.} Name the process class, data assignment, admissible instruments, observations, and equivalences. State every class restriction. Default status here: \(\mathsf{DECL}\) as a required interface, with actual selection \(\mathsf{OPEN}\). \item \textbf{Order formation.} Name the relation extracted and prove that \(\preceq_D\) is a partial order on the claimed carrier. If it is obtained from influence by closure, restriction, or quotient, justify that operation. Default status: \(\mathsf{OPEN}\). \item \textbf{H-faithful-generation.} Construct the representation and prove \eqref{eq:faithfulness}--\eqref{eq:generation}, with properness and a declared fixed dimension \(m\geq3\). A conditional theorem may explicitly assume that the proposed extraction returns such a representation. It must then state that construction and admission remain conditional. Default status: \(\mathsf{OPEN}\). \item \textbf{H-homogeneity.} For \[ \Aut(K_D)=\{A\in\GL(V_D):A(K_D)=K_D\}, \] require \[ \forall u,v\in\Int K_D\ \exists A\in\Aut(K_D): Au=v. \label{eq:homogeneity} \] Either prove this from the declared operational premises or mark it \(\mathsf{ASSUMED}\). Default status: \(\mathsf{OPEN}\). \item \textbf{H-strict-convexity.} Require \[ u,v\in K_D\setminus\{0\},\ u,v\ \text{non-proportional} \quad\Longrightarrow\quad u+v\in\Int K_D. \label{eq:strict-convexity} \] Either prove it or mark it \(\mathsf{ASSUMED}\). Default status: \(\mathsf{OPEN}\). \item \textbf{H-continuum or extension.} If used, specify the refinement or completion, comparison maps, topology, compatibility conditions, and convergence theorem. State separately whether uniqueness is claimed. Default status: \(\mathsf{OPEN}\), or explicitly not applicable to a construction using no such step. \item \textbf{Operational equivalence and extraction invariance.} Prove that equivalent data and permitted relabellings produce affinely isomorphic representations, with an invertible linear map on the cone differences. Do not hide an independently supplied Lorentz cone in the extraction choices. Default status: \(\mathsf{OPEN}\). \item \textbf{Exactness, tolerance, and evidence.} Distinguish exact signalling guards from tolerance guards. If statistical or implemented data are used, give the applicable coverage, calibration, and complete-channel allowances. Default status: \(\mathsf{OPEN}\), with exact mathematical-data branches identified separately. \end{enumerate} \end{specification} The phrases H-faithful-generation, H-homogeneity, and H-strict-convexity are permissible \emph{forms} of entries. They are not automatically operationally derived facts. A predicate on the output of a declared extraction rule is legitimate in a conditional statement; it is not legitimate to omit that rule and supply an arbitrary cone as its answer. A scheduler interleaving is not a linear cone automorphism. Commutation of independent operations therefore does not establish \eqref{eq:homogeneity}. Likewise, complete positivity concerns cones of operators; identifying an operator-positivity cone with a cone of event differences requires a separate representation theorem \cite{CheungEn08A,CheungXXIIA}. \subsection{The conditional classification route} \begin{proposition}[Conditional assembly, not operational derivation] \label{prop:classification} Suppose that, for a particular datum \(D\), the proposed extraction returns the representation of Definition~\ref{def:representation}, with \(\dim V_D=m\geq3\). If \eqref{eq:homogeneity} and \eqref{eq:strict-convexity} hold, then there exists \(T_D\in\GL(V_D,\R^m)\) such that \[ T_D(K_D)=\Lor_m. \] In particular, \[ p\preceq_D q \quad\Longleftrightarrow\quad T_D(\iota_D(q)-\iota_D(p))\in\Lor_m. \] \end{proposition} \begin{proof} The classification of strictly convex homogeneous cones gives the linear isomorphism in dimension at least three \cite[Theorem 7]{ItoLourenco}. Apply \eqref{eq:faithfulness} and the invertibility of \(T_D\). This is the conditional route recorded in En08-A \cite[\loc{thm:weaker}]{CheungEn08A} and XXII-A \cite[\loc{thm:classification}]{CheungXXIIA}. \end{proof} The proposition does not provide its operational representation, derive its cone properties, or establish a nonempty admissible data set. No self-duality or irreducibility assumption is required for this particular classification route. \begin{example}[The orthant already satisfies homogeneity] \label{ex:orthant} For \(m\geq3\), \(K=\R_{\geq0}^{m}\) is proper and homogeneous: for \(u,v\in\Int K\), the diagonal map \[ A=\operatorname{diag}(v_1/u_1,\ldots,v_m/u_m) \] is an automorphism and sends \(u\) to \(v\). It fails strict convexity because \(e_1,e_2\) are non-proportional elements of \(K\), while \(e_1+e_2\in\partial K\). Thus H-homogeneity alone does not exclude the orthant countermodel. \end{example} The scheduler package does not furnish the combined classification hypotheses. Individual claims of independence must be proved in their declared common model universe; failure of the combined implication does not by itself prove independence of each conjunct. \section{Non-vacuity, genuine additional content, and exclusion} \label{sec:acceptance} \subsection{Two separate existence obligations} On a scheduler-comparable branch, let \(\mathfrak U\) be a declared common operational model universe, let \(C(D)\) express the class restrictions, and let \[ H(D)=\bigwedge_{j=1}^{r}H_j(D;\cE) \] express the actual additional assumptions for a fixed proposed extraction rule \(\cE\). The desired joint admissible set is \[ \mathcal A = \{D\in\mathfrak U: \SP(D)\wedge C(D)\wedge H(D)\}. \label{eq:admissible} \] The two distinct requirements are \begin{align} &\mathcal A\ne\varnothing, \label{eq:nonvacuity}\\ &\SP\not\models_{\mathfrak U}(C\wedge H). \label{eq:additional-content} \end{align} Equation~\eqref{eq:nonvacuity} needs an admissible witness or an existence theorem. Equation~\eqref{eq:additional-content} needs a scheduler model failing a stated additional restriction or assumption. Neither is established for a cone-selecting candidate here. This formulation includes class restrictions in the additional-content audit. It does not require every \(H_j\) separately to be independent of \(\SP\). A repeated scheduler condition may be restated for readability, but must be marked as baseline rather than credited as new cone-selecting content. For a general process matrix with no declared complete dependency schedule, \(\SP(D)\) is not silently assigned a meaning. A corresponding branch must state its own operational premises and how the scheduler obstruction is compared with them. Rejecting all scheduler models merely by changing language is not a construction of a successful operational extraction. \subsection{H-exclusion is an audit obligation} \begin{definition}[Relevant countermodel and exclusion certificate] \label{def:exclusion} A relevant countermodel is a specified model, together with a justified interpretation in the proposed operational class, that satisfies the claimed baseline premises and falsifies the claimed conclusion for the proposed extraction. An exclusion certificate identifies an actual class restriction or ledger assumption that fails for that interpretation and proves the failure. If the interpretation itself is impossible, that impossibility requires a proof. \end{definition} The countermodel's process, interventions, event carrier, and extraction must be identified before comparing cones. One cannot transfer the geometry of a predecessor model to a different extraction by sharing its name. If an unchanged relevant countermodel satisfies all the proposed premises, the selection claim is false. If every named countermodel has been excluded, the selection claim is merely \emph{not refuted by those countermodels}. A positive theorem still requires its representation and cone-selection argument \cite{CheungEn08A,CheungXXIIA}. \begin{remark}[The ball section must not be excluded by its shape alone] \label{rem:ball} En08-A's ball-section and orthant examples witness different nonimplications \cite[\loc{prop:ball}, \loc{prop:orthant}] {CheungEn08A}. In the continuum, \[ \Cone\{(1,x):\|x\|_2\leq1\}=\Lor_m. \] Indeed, a nonzero Lorentz-cone vector has \(t>0\) and equals \(t(1,x/t)\), and the reverse inclusion follows from the triangle inequality. A Euclidean-ball section is therefore not, by itself, a falsifier of Lorentz-cone selection. An exclusion audit for En08-A's discrete ball model must examine its complete event representation, not declare every Euclidean ball inadmissible. \end{remark} \subsection{Acceptance mapping to XXII-A} The ledger is to be assessed against XXII-A's conditions (A1)--(A6) \cite{CheungXXIIA}: \begin{enumerate}[label=\textup{(A\arabic*)}] \item one model and one explicitly named extracted relation; \item genuinely additional content, including class restrictions; \item non-vacuity and justified relevant-countermodel exclusion; \item honest assumed or derived status of classification hypotheses; \item faithfulness, generation, and extraction invariance; \item declared limits, evidence, and comparison allowances. \end{enumerate} No candidate in Section~\ref{sec:candidates} is marked as having passed this full audit. \section{Five failure tests} \label{sec:tests} These are rejection and scope tests, not additional sufficient premises for a positive theorem. A later paper must prove the required repairs or explicitly retain the corresponding limitation. \subsection{Test F1: scheduler consistency alone} \label{sec:test-scheduler} Any argument using only \(\SP\) retains En08-A's obstruction \cite[\loc{thm:main}]{CheungEn08A}. A process-matrix, comb, or channel-table description does not remove the obstruction just by being quantum notation. For a new class, however, an actual realization or interpretation of the countermodel must be established; it is not inferred solely from an acyclic signalling graph. \textbf{Pass condition.} Supply the genuinely additional package and its relevant exclusion certificates. If cone properties are assumed rather than derived, the conclusion must remain explicitly conditional. \subsection{Test F2: finite event data and finite generation} \label{sec:test-finite} \begin{proposition}[Finite event differences cannot generate a Lorentz cone] \label{prop:finite-generation} Suppose that \(E_D\) is finite and that Definition~\ref{def:representation} holds. Then \(K_D\) cannot be linearly isomorphic to \(\Lor_m\) for \(m\geq3\). \end{proposition} \begin{proof} Let \(G=\{g_1,\ldots,g_s\}\) be the nonzero comparable differences appearing in \eqref{eq:generation}. Properness excludes the case \(G=\varnothing\) in the stated dimension. Because \(G\) lies in a pointed cone, zero is not in \(\operatorname{conv}G\). Indeed, a convex combination equal to zero would put the negative of a nonzero generator in the same cone. Choose any auxiliary Euclidean inner product on \(V_D\) and let \(z\ne0\) be the closest point of the compact set \(\operatorname{conv}G\) to zero. The minimizing property gives \[ \langle z,g_j\rangle\geq\|z\|^2>0 \qquad(1\leq j\leq s). \] If a sequence \(\sum_j\lambda_{n,j}g_j\), \(\lambda_{n,j}\geq0\), converges, these inequalities bound all its coefficients. A convergent coefficient subsequence shows that \(\Cone G\) is closed. Thus \(K_D=\Cone G\). Every extreme ray of \(\Cone G\) contains one of the finitely many generators: decomposing a nonzero vector on an extreme ray into its generating summands forces every nonzero summand onto that ray. Hence \(K_D\) has finitely many extreme rays. The Lorentz cone has an extreme ray \(\R_{\geq0}(1,u)\) for every \(\|u\|_2=1\). Extremality follows from equality in the triangle inequality in a decomposition of \((1,u)\) into two cone vectors. There are infinitely many such rays when \(m\geq3\). Invertible linear maps preserve extreme rays, giving a contradiction. \end{proof} The auxiliary inner product in this proof is a mathematical proof device, not an installed spacetime metric. \textbf{Pass condition.} Declare an infinite operational family, a justified carrier extension, or a limiting construction, and prove that it produces \eqref{eq:faithfulness}--\eqref{eq:generation}. A finite party set is compatible with such a family. Finite tomography of a declared operator is not this carrier construction. The separate metric warning is also retained: a finite ordered event sample does not determine an unrestricted spacetime metric or unique continuum realization \cite{CheungXXIIA,CheungEn08A}. This does not deny conformal reconstruction from a full continuum chronological structure under additional hypotheses. \subsection{Test F3: a generated boundary is not an attained null protocol} \label{sec:test-boundary} The distinction already recorded in En08-A and XXII-A is a distinction between cone generation and operational attainment \cite{CheungEn08A,CheungXXIIA}. For an elementary cone check, consider \[ G= \{(1,x):x\in\Q^{m-1},\ \|x\|_2<1\}. \] Every generator is strictly inside \(\Lor_m\), as is every nonzero finite conical combination of these generators. Nevertheless, \[ \overline{\Cone G}=\Lor_m. \label{eq:boundary-generated} \] To verify this, approximate every point of the closed unit ball by rational points in its open ball and then use conical scaling. Thus the generated closed cone has nonzero null rays, although none of the generators, or their nonzero finite combinations, is null. This is a generator example, not a constructed quantum event representation. \textbf{Pass condition.} If an attained null protocol is claimed, exhibit its actual event pair and complete operational witness. If only cone selection or boundary approximation is claimed, retain the non-attainment boundary explicitly. Attainment is not an additional necessary condition for the cone-selection target itself. \subsection{Test F4: tolerance is not exact causal containment} \label{sec:test-tolerance} \begin{proposition}[Allowance bound and an exact small-signal witness] \label{prop:tolerance} Suppose that complete comparison channels for every intervention at \(p\) equal one channel \(\Theta_q\), and that \[ \ddiamond(\Phi_{q|\mathcal I},\Theta_q)\leq\varepsilon \] uniformly over the declared interventions and contexts. Then the actual signalling strength obeys \[ \Delta_D(p,q)\leq\min\{1,2\varepsilon\}. \] A positive allowance does not imply \(\Delta_D(p,q)=0\). \end{proposition} \begin{proof} The triangle inequality bounds the difference between two actual intervention channels by \(2\varepsilon\). The half-diamond distance between channels is at most one \cite[Chapter 3]{Watrous}. For the failure of exactness, on a qubit set \[ \mathcal R_0(X)=\Tr(X)\kb0 0,\qquad \mathcal F_\delta=(1-\delta)\mathcal R_0+\delta\id, \qquad 0<\delta\leq1. \] Both maps are channels. Homogeneity of the norm and the input \(\kb1 1\) give \[ \ddiamond(\mathcal F_\delta,\mathcal R_0) = \delta\,\ddiamond(\id,\mathcal R_0)=\delta. \] The upper bound is the maximum channel distance one; the input \(\kb1 1\) gives the matching lower bound for \(\ddiamond(\id,\mathcal R_0)\). The two bit preparations satisfy \[ \frac12\| \mathcal F_\delta(\kb1 1)-\mathcal F_\delta(\kb0 0) \|_1=\delta>0. \] The channel is arbitrarily close to a replacement channel while still transmitting an unconditional signal. \end{proof} \textbf{Pass condition.} Declare whether absence of an edge means the exact guard \(\Delta_D(p,q)=0\), or a tolerance guard such as \(\Delta_D(p,q)\leq\eta\). Prove the transfer from the implemented comparisons to that guard. A tolerance graph must not be called an exact causal order without a separate theorem. Finite statistical channel certificates require their actual calibration, coverage, and trial assumptions. Oai265-B's complete-record certificate is a model for keeping those assumptions explicit; its error interval is not exact non-signalling or cone selection \cite[\loc{ass:statistics}, \loc{thm:main}]{CheungOai265B}. \subsection{Test F5: a common coarse channel does not select a unique limit} \label{sec:test-refinement} \begin{proposition}[Explicit refinement non-determination] \label{prop:two-limits} Fix \(00\), \[ \mathcal T_t^c(\kb1 1) = e^{-ct}\kb1 1+(1-e^{-ct})\kb0 0. \] The trace distance between the \(a\)- and \(b\)-outputs is \(|e^{-at}-e^{-bt}|>0\). \end{proof} These families do not satisfy an unasserted exact subdivision or projective-consistency relation with their common coarse tick. They choose different missing refinement data. They therefore obstruct inference from a coarse member and support certificate, not every continuum construction with additional consistency data. A continuum ledger must specify complete comparison interfaces and, for a channel-evolution claim, a convergence statement such as \[ \sup_{0\leq t\leq T} \ddiamond\bigl( \mathcal D_n\mathcal C_n(t)\mathcal E_n,\mathcal T_t \bigr)\longrightarrow0 \quad\text{for every finite }T, \label{eq:operational-limit} \] together with the claimed composition law and continuity \cite[\loc{def:limit}]{CheungXIXC}. A spatial or event-cone limit needs its additional carrier and order-compatibility statements. \textbf{Pass condition.} Provide the refinement data and prove the claimed existence and, if asserted, uniqueness. Merely naming ``projective consistency'' or ``interpolation'' is not a uniqueness proof. XIX-C's interpolation warning likewise records that finite sampling constraints may leave untested intermediate behaviour \cite[\loc{prop:interpolation}]{CheungXIXC}. Uniqueness may instead be explicitly left outside the claim. \section{Documentary comparisons and negative controls} \label{sec:documentary} The supplied catalogue family identifiers are documentary records, not independently verified external theorem inputs. Their role is to prevent an unrelated result from being substituted for the missing operational representation. \begingroup \small \renewcommand{\arraystretch}{1.15} \begin{longtable}{@{}p{.24\linewidth}p{.69\linewidth}@{}} \toprule Component & Role and boundary\\ \midrule \endfirsthead \toprule Component & Role and boundary\\ \midrule \endhead \bottomrule \endfoot En08-A & Baseline scheduler obstruction and conditional cone-representation language. The ball and orthant must be interpreted with their actual model scopes \cite{CheungEn08A}.\\[5pt] VII-J & Three-relation separation. A supplied physical causal embedding does not derive the relation to which it embeds the protocol \cite{CheungVIIJ}.\\[5pt] XIX-C & Refinement and interpolation warnings, including the complete operational-limit interface. No native continuum limit is imported into this note \cite{CheungXIXC}.\\[5pt] En08-B & Later endpoint approximation by reference-stable finite transport. The distance, speed, and null targets are prescribed there; approximation does not select them \cite{CheungEn08B}.\\[5pt] Family 265 / Oai265-A--B & Static area-law records are not an operational causal cone. A certificate for one implemented recovery channel is not a metric or a classification of event orders \cite{CheungOai265A,CheungOai265B}.\\[5pt] Families 211, 280, 069, 282 & The supplied records concern random-tree limits, VOA-net claims, quantum geometric Langlands, and Ward-related claims. No operational event-carrier extraction and Lorentz-cone selection argument is supplied by those records \cite{CatalogueRecord}.\\[5pt] Oai282-A--B; Oai069-A & The supplied companion results have independent finite recovery or local Ward scopes. They do not verify the catalogue claims or construct the representation required here \cite{CheungOai282,CheungOai069A}.\\[5pt] VIII-A/B/C; Em02-E; XIX/XX clock records & Declared conical geometries, classical lapse prescriptions, finite registers, and clock/admission boundaries remain documentary comparisons. No unsupplied proof or operational selection package is imported \cite{CheungXXIIA,CheungXIXE}.\\ \end{longtable} \endgroup No claim here says that these subjects could never contribute to a future construction. The narrower statement is that the supplied records do not discharge the specified selection obligations. A documentary separation, inherited interface, or source dependency does not automatically establish a proof-transfer premise. \section{Completed release and Stage 2 handoff} \label{sec:release} The completed catalogue release consists of: \begin{enumerate} \item the operational declaration and three-relation discipline; \item the four candidate data classes with their corrected meanings; \item the finite ledger template and its status syntax; \item the non-vacuity, additional-content, and exclusion audits; \item the five explicit failure tests; \item the conditional classification reduction, with its missing operational premises left visible. \end{enumerate} Propositions~\ref{prop:signalling-cycle}, \ref{prop:static}, \ref{prop:finite-generation}, \ref{prop:tolerance}, and \ref{prop:two-limits} are scoped mathematical failure witnesses. They do not jointly construct a quantum cone-selection model. Proposition~\ref{prop:classification} is conditional and supplies no non-vacuity theorem. \paragraph{The precise Stage 2 handoff.} A later construction must provide: \begin{enumerate}[label=\textup{(B\arabic*)}] \item one frozen quantum process class and one named data assignment, with all access and comparison interfaces stated; \item one predeclared extraction rule and one actual finite additional hypothesis list, including every operative class restriction; \item a proof of non-vacuity and of genuinely additional content relative to the properly compared scheduler baseline; \item an order theorem and a faithful generated affine representation, including any required carrier extension or continuum step; \item the relevant countermodel exclusion certificates; \item either an operational derivation of the cone-selecting properties or an explicitly conditional theorem assuming them; \item the extraction-invariance and evidence/error statements required by the claim. \end{enumerate} Together these must discharge XXII-A's target conditions (T1)--(T5) and acceptance conditions (A1)--(A6) \cite{CheungXXIIA}. Any item deferred to Stage 3 remains open; it is not discharged by calling the catalogue complete. A subsequent comparison with a supplied geometric causal relation is still a separate embedding problem. \paragraph{Unchanged boundaries.} XXII-B does not exhibit a nonempty \emph{joint cone-selecting} process class; derive homogeneity or strict convexity from a quantum observable; construct the required operational affine representation; or establish that any candidate passes the complete exclusion audit. It does not identify scheduled ancestry with supplied geometric causality. No Einstein dynamics, holographic reconstruction, Page curve, gravity-generated entanglement, or quantum evolution without an external time is established. The note supplies no result changing 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 successful programme execution or pdfLaTeX build is asserted \cite{CheungXIXE}. \paragraph{Final gap statement.} The remaining gap is the construction and proof of one nonempty operational package linking additional quantum data to the faithful generated event cone and its selecting properties. Until that package is supplied, homogeneity plus strict convexity is a conditional classification route, and scheduler consistency alone remains obstructed as a Lorentz-cone selection principle. \begin{thebibliography}{99} \bibitem{CheungXXIIA} N.~Cheung. \emph{XXII-A: Freeze the precise claim and the gap. Stage 0 of a limited kinematic emergence programme}. Supplied manuscript \loc{XXII-A.tex}, n.d. Relevant items: \loc{target:selection}, \loc{def:representation}, \loc{thm:classification}, and conditions (T1)--(T5), (A1)--(A6). No external LaTeX reference file is required by this note. \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{prop:ball}, \loc{prop:orthant}, \loc{thm:mainpositive}, \loc{thm:representation}, and \loc{thm:weaker}. Supplied manuscript passages and scope records. Registry: \url{https://enter.grok.me/}. \bibitem{CheungVIIJ} N.~Cheung. \emph{VII-J}, manuscript \loc{VII-J.tex}. Enter Calculus Companion, n.d. Relevant supplied scope: scheduled support, operational influence, supplied causal relation, and \loc{thm:release}. Documentary identification is not independent proof verification. \bibitem{OCB} O.~Oreshkov, F.~Costa, and \v{C}.~Brukner. Quantum correlations with no causal order. \emph{Nature Communications} \textbf{3} (2012), 1092. \doi{10.1038/ncomms2076}. Preprint: \url{https://arxiv.org/abs/1105.4464}. \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{10.1103/PhysRevA.80.022339}. Preprint: \url{https://arxiv.org/abs/0904.4483}. \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{10.1103/PhysRevA.97.012127}. Preprint: \url{https://arxiv.org/abs/1512.00589}. \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{10.1103/PhysRevA.64.052309}. Preprint: \url{https://arxiv.org/abs/quant-ph/0102043}. \bibitem{Watrous} J.~Watrous. \emph{The Theory of Quantum Information}. Cambridge University Press, 2018. Author's text and chapters: \url{https://cs.uwaterloo.ca/~watrous/TQI/}. \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{10.1016/j.laa.2017.07.029}. Preprint: \url{https://arxiv.org/abs/1701.05801}. Theorem 7 is the classification used here. \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}. The explicit refinement witness is supplied in the source materials. \bibitem{CheungEn08B} N.~Cheung. \emph{En08-B}, version 1, manuscript \loc{En08-B.tex}. Enter Calculus Companion, n.d. Relevant labels: \loc{lem:identity}, \loc{thm:main}, \loc{def:comparison}, and \loc{cor:operational-limit}. Reference-stable endpoint approximation on prescribed geometric inputs. \bibitem{CheungEn09A} N.~Cheung. \emph{En09-A}, version 1, manuscript \loc{En09-A.tex}. Enter Calculus Companion, n.d. Relevant label: \loc{thm:main}. Supplied complete clock-readout and reference-extension statements; not a premise of Lorentz-cone selection. \bibitem{CheungOai265A} N.~Cheung. \emph{Oai265-A}, manuscript \loc{Oai265-A.tex}. Enter Calculus Companion, n.d. Relevant supplied label: \loc{thm:separation}. Documentary distinction between static area-law records, recovery, and radiation histories. \bibitem{CheungOai265B} N.~Cheung. \emph{Oai265-B}, version 2, manuscript \loc{Oai265-B-v2.tex}. Enter Calculus Companion, n.d. Relevant labels: \loc{ass:statistics}, \loc{thm:main}, \loc{eq:coverage}, and \loc{eq:falseaccept}. Supplied complete-record finite-confidence recovery certificate. \bibitem{CheungOai282} N.~Cheung. \emph{Oai282-A} and \emph{Oai282-B}, Enter Calculus Companion manuscripts, n.d. Relevant supplied labels: Oai282-A \loc{thm:main}; Oai282-B \loc{thm:main}, \loc{thm:coverage}, and \loc{def:wardmeaning}. Documentary local Ward scopes, not operational metric extraction. \bibitem{CheungOai069A} N.~Cheung. \emph{Oai069-A}, version 1, manuscript \loc{Oai069-A.tex}. Enter Calculus Companion, n.d. Relevant label: \loc{thm:main}. Supplied finite recovery and independent free-scalar consistency benchmark, not verification of catalogue Family 069. \bibitem{CheungXIXE} N.~Cheung. \emph{XIX-E}, version 1, manuscript \loc{XIX-E.tex}. Enter Calculus Companion, n.d. Relevant supplied passage: ``What remains outside the release.'' Documentary boundary record. \bibitem{CatalogueRecord} OpenAI. \emph{Supplied mathematics-catalogue record}, n.d. Documentary family identifiers 265, 211, 280, 069, and 282. Publication metadata and independently verified proofs are not supplied by this citation. The record is not a theorem premise. \end{thebibliography} \end{document}