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  pdftitle={XXII-C: Freeze one quantum process class and the extraction obligations},
  pdfauthor={Ngo Cheung},
  pdfsubject={Stage 2 of a limited kinematic emergence programme}
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\title{\textbf{XXII-C: Freeze one quantum process class\\
and the extraction obligations}\\[4pt]
\large Stage 2 of a limited kinematic emergence programme}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
This note fixes a concrete class of countably indexed, separately
tested tripartite process matrices with qubit input and output
systems. It fixes the local instrument access, the exact operational
data assignment, and a strongly connected component quotient of
unconditional signalling. The resulting component order is proved
to be a partial order, and its operational relabelling covariance
is established. An infinite experiment family is distinguished
from infinitely many settings of a single finite-party process;
neither is silently treated as a continuum spacetime. The frozen
extraction returns the component order and a precisely defined
affine-representation feasibility problem, not a constructed
representation. Base-class nonemptiness is proved, while
nonemptiness of a cone-selecting admissible subclass remains open.
Six obligations specify the missing representation, additional
hypotheses, exclusion audit, and limit discipline. No extracted cone
is shown homogeneous or strictly convex, and the Lorentz-cone
classification is not applied to operational data.
\end{abstract}

\section{Purpose, claim, and release boundary}
\label{sec:purpose}

XXII-A fixes the operational Lorentz-cone selection target and its
acceptance conditions
\cite[\loc{target:selection}, conditions (T1)--(T5), (A1)--(A6)]
{CheungXXIIA}.
XXII-B supplies the candidate-data catalogue, hypothesis-ledger syntax,
and failure tests
\cite{CheungXXIIB}.
The present note selects the process-matrix branch and replaces an
unspecified ``family of experiments'' by the concrete declaration in
Section~\ref{sec:class}.

The claim under construction is not
\[
 \text{process-matrix validity}
 \quad\Longrightarrow\quad
 \text{Lorentz-cone selection}.
\]
It is the prospective existence of a nonempty additional operational
package on the frozen class, together with a construction of a faithful
generated affine representation and a justified cone-selection step.
That existence claim is not proved here.

\paragraph{Two meanings of extraction must be separated.}
The order-and-feasibility rule \(\cE_m\) defined in
Section~\ref{sec:representation} is fully specified as a mathematical
rule. Its output includes a representation problem.
A single-valued rule constructing one admissible affine embedding is
not specified or constructed. Calling the former a frozen rule must
not be used to claim completion of the latter.

Similarly, a list of obligations is not a successful additional
hypothesis package \(\HP\). A later theorem must state its actual
finite hypotheses and its actual cone-construction rule before
applying them.

\paragraph{Scope of Stage 2 in this release.}
This is a completed process-class and extraction-interface freeze.
It does not complete the representation construction assigned to
Stage 2 in XXII-A. Those mathematical obligations are explicitly
carried forward rather than marked as discharged by a change of
stage label.

\paragraph{Source discipline.}
Published articles supply the process-matrix and channel frameworks.
Cheung's references identify supplied manuscripts by filename and
internal label. A documentary comparison is not a theorem premise.
The proofs printed here concern the explicitly defined objects of
this note; they do not verify the full predecessor manuscripts.

\section{The frozen process class}
\label{sec:class}

\subsection{Fixed parties, dimensions, and Choi convention}

The party set is fixed once and for all:
\[
 \Party=\{A,B,C\}.
\]
Each party has one qubit input and one qubit output:
\[
 H_{a_I}=H_{a_O}=\C^2,
 \qquad a\in\Party.
\]
Let
\[
 \mathcal H
 =
 \bigotimes_{a\in\Party}
       (H_{a_I}\otimes H_{a_O}),
 \qquad \dim\mathcal H=64.
\]
A written tensor-factor ordering is bookkeeping only. It is not an
operational or temporal order of the parties.

Fix bases and use the transposed, unnormalized Choi convention
\[
 J(\mathcal M)
 =
 \sum_{i,j=0}^{1}
       \kb{i}{j}\otimes\mathcal M(\kb{i}{j}),
 \qquad
 M=J(\mathcal M)^{\mathsf T},
 \label{eq:choi}
\]
where \({}^{\mathsf T}\) denotes the full matrix transpose.
This is the convention used in the original process-matrix
probability formula \cite{OCB}.

For each party set
\[
 \ChannelSet_a
 =
 \{M\geq0:\Tr_{a_O}M=I_{a_I}\},
 \qquad
 \EffectSet_a
 =
 \{B\geq0:\Tr_{a_O}B\leq I_{a_I}\}.
 \label{eq:local-sets}
\]
The first set represents channels. The second represents completely
positive trace-nonincreasing instrument elements in the convention
\eqref{eq:choi}
\cite[Chapter 2]{Watrous}.

\begin{definition}[Valid tripartite qubit process]
\label{def:process}
A matrix \(W\in L(\mathcal H)\) belongs to \(\ProcessSet\) when
\begin{align}
 W&\geq0,
 \label{eq:positive}\\
 \Tr[W(M_A\otimes M_B\otimes M_C)]&=1
 \quad
 \text{for every }M_a\in\ChannelSet_a.
 \label{eq:normalization}
\end{align}
A local finite-outcome instrument is a list
\(\{M^a_{x_a|s_a}\}_{x_a}\) of positive matrices whose sum belongs
to \(\ChannelSet_a\).
Its complete joint outcome law is
\[
 P_W(\boldsymbol x|\boldsymbol s)
 =
 \Tr\!\left[
 W\bigotimes_{a\in\Party}M^a_{x_a|s_a}
 \right].
 \label{eq:probability}
\]
No definite party ordering or causal-separability decomposition is
required in this definition.
\end{definition}

Positivity and \eqref{eq:normalization} imply that
\eqref{eq:probability} is nonnegative and sums to one.
The process-matrix framework does not assume an externally supplied
global causal order \cite{OCB}. This is not a statement that every
valid process lacks a definite-order realization.

\begin{lemma}[Compactness and a base witness]
\label{lem:base}
The set \(\ProcessSet\) is nonempty and compact.
Every \(W\in\ProcessSet\) satisfies \(\Tr W=8\).
\end{lemma}

\begin{proof}
The completely depolarizing channel has Choi matrix
\(I_{a_Ia_O}/2\).
Substituting these three channels into \eqref{eq:normalization}
gives
\[
 1=\Tr(WI_{\mathcal H}/8)=\Tr W/8.
\]
The normalization equations and positivity define a closed subset
of the finite-dimensional positive matrices with trace eight,
which is bounded and hence compact.

The matrix
\[
 W_0=I_{\mathcal H}/8
 \label{eq:base-process}
\]
is positive. Since every \(M_a\in\ChannelSet_a\) has trace two,
\[
 \Tr[W_0(M_A\otimes M_B\otimes M_C)]
 =
 \frac18\prod_{a\in\Party}\Tr M_a=1.
\]
Thus \(W_0\in\ProcessSet\).
\end{proof}

\subsection{Frozen access and experiment family}

\begin{specification}[Access declaration \(\Access\)]
\label{spec:access}
The access model is:
\begin{enumerate}[label=\textup{(P\arabic*)}]
\item every local finite-outcome qubit-input/qubit-output instrument
      is admitted;
\item each party uses its process input/output interface once in a
      selected experiment;
\item local auxiliary implementations are allowed insofar as they
      induce the declared instrument maps on that interface;
\item no extra shared ancillary resource is inserted into the
      signalling tests; correlations already present in \(W\)
      remain allowed;
\item settings are chosen independently of the process resource;
      the tests use no interparty classical feed-forward outside
      the declared process;
\item the receiver output used for extraction is its local classical
      instrument record; sender and other-party records are summed;
\item every outcome, including any declared failure, belongs to the
      complete outcome alphabet;
\item no postselection on another party's record is used to define
      influence.
\end{enumerate}
There is no spatial location, distance, speed, or spacetime causal
relation in \(\Access\).
\end{specification}

\begin{definition}[Frozen family class]
\label{def:class}
Let \(I=\N=\{1,2,\ldots\}\) be a set of experiment identifiers.
Define
\[
 \ProcessClass=(\ProcessSet)^I.
 \label{eq:class}
\]
An object is a family
\[
 \boldsymbol W=(W_e)_{e\in I},
 \qquad W_e\in\ProcessSet.
\]
Experiment \(e\) is tested separately using \(\Access\).
The identifier \(e\) is not a time coordinate, refinement scale,
or causal precedence relation.

No composition of the \(W_e\), no inter-experiment influence test,
and no joint process on the entire family is included in this
definition.
\end{definition}

This is one concrete class, not a menu of operational languages.
The number of parties and their wire dimensions are fixed; the
number of separately addressable experiments is countably infinite.

The family notation does not assert that an unrestricted tensor
product of its members is a valid process. Process-matrix tensor
products can violate normalization when unrestricted joint local
operations are permitted \cite{JiaSakharwade}.
Any later composition theorem must specify the admissible interfaces
and prove validity.

\begin{remark}[Settings are not additional event occurrences]
\label{rem:settings}
A single \(W_e\) can admit infinitely many instrument choices while
still possessing only the three operation-use labels \(A,B,C\).
Changing an instrument setting does not by itself create a new
operation occurrence or a causal comparison between different trials.

The countable raw carrier in this note comes from the explicitly
declared experiment-use tokens \((e,a)\), not from counting the
continuum of instrument settings as events.
\end{remark}

\section{Exact data and unconditional influence}
\label{sec:data}

\subsection{The named data assignment}

\begin{definition}[Exact data assignment]
\label{def:data}
The assignment
\[
 \mathcal D_{\mathrm{3q}}:
 \ProcessClass\longrightarrow\mathcal D(\ProcessClass)
\]
returns
\[
 D(\boldsymbol W)
 =
 (\boldsymbol W,\Access,
        \{P_{W_e}\}_{e\in I},
        \{\Delta_e(a,b)\}_{e\in I,\ a\ne b}),
 \label{eq:data}
\]
where the probability functionals and signalling strengths are
defined below.

The probability and strength entries are derived from
\(\boldsymbol W\); they are not additional independently adjustable
numbers. The datum is exact mathematical data, not a finite
statistical estimate.
\end{definition}

Static reduced spectra, mutual information, or other correlation
diagnostics may be appended as annotations, but \(\cE_m\) below does
not use them as event comparisons. XXII-B's static-state witness and
Oai265-A's documentary separation explain the missing operational
access in such an identification
\cite[\loc{prop:static}]{CheungXXIIB}
\cite[\loc{thm:separation}]{CheungOai265A}.

\subsection{A compact formula for signalling strengths}

For distinct \(a,b\in\Party\), let \(c\) be the remaining party.
For \(S,T\in\ChannelSet_a\), a receiver instrument
\(\{B_z\}_z\) at \(b\), and \(M_c\in\ChannelSet_c\), compare
\[
 P_{b|S}(z)
 =
 \Tr[W_e(S_a\otimes B_z^b\otimes M_c)],
 \qquad
 P_{b|T}(z)
 =
 \Tr[W_e(T_a\otimes B_z^b\otimes M_c)].
\]
Factors are placed in the fixed tensor slots.
The settings at \(b,c\) are identical within the compared pair.

Define
\[
 \Delta_e(a,b)
 =
 \sup_{S,T,\{B_z\},M_c}
 \dtv(P_{b|S},P_{b|T}),
 \qquad
 \dtv(P,Q)=\frac12\sum_z|P(z)-Q(z)|.
 \label{eq:strength-definition}
\]
This is the classical-record branch of the unconditional signalling
test of XXII-B \cite[\loc{def:signalling}]{CheungXXIIB}.

\begin{proposition}[Binary reduction and attainment]
\label{prop:binary}
For every \(W_e\in\ProcessSet\),
\[
 \Delta_e(a,b)
 =
 \max_{\substack{S,T\in\ChannelSet_a\\
                  B\in\EffectSet_b,\ M_c\in\ChannelSet_c}}
 \left|
 \Tr[W_e((S-T)_a\otimes B_b\otimes M_c)]
 \right|.
 \label{eq:strength-maximum}
\]
The maximum exists and belongs to \([0,1]\).
\end{proposition}

\begin{proof}
For finite probability laws,
\[
 \dtv(P,Q)=\max_{Z}|P(Z)-Q(Z)|,
\]
where \(Z\) ranges over subsets of the outcome alphabet.
For any such subset, the coarse-grained receiver element
\(B=\sum_{z\in Z}B_z\) belongs to \(\EffectSet_b\).
Thus every witness in \eqref{eq:strength-definition} is bounded
by the right side of \eqref{eq:strength-maximum}.

Conversely, every \(B\in\EffectSet_b\) can be completed to a binary
instrument. Put
\[
 B^{\mathrm c}
 =
 (I_{b_I}-\Tr_{b_O}B)\otimes I_{b_O}/2.
\]
Then \(B^{\mathrm c}\geq0\), and
\[
 \Tr_{b_O}(B+B^{\mathrm c})=I_{b_I}.
\]
The binary law for \(\{B,B^{\mathrm c}\}\) has total-variation
distance equal to the absolute difference of the \(B\)-outcome
probabilities. This gives the reverse inequality.

The sets in \eqref{eq:local-sets} are closed and bounded:
their elements are positive with trace at most two.
Their finite product is compact, and the objective is continuous.
It therefore attains its supremum.
Finally, each compared binary outcome probability lies in
\([0,1]\), so the resulting distance is at most one.
\end{proof}

The maximization is a finite-dimensional mathematical specification.
No numerical optimization is reported, and no claim is made that
the displayed multilinear problem is one semidefinite program.

\begin{proposition}[Continuity of strengths, not of exact edge membership]
\label{prop:strength-stability}
For \(W,W'\in\ProcessSet\),
\[
 |\Delta_W(a,b)-\Delta_{W'}(a,b)|
 \leq8\|W-W'\|_1.
 \label{eq:strength-stability}
\]
Nevertheless, trace-norm convergence of processes does not generally
stabilize the exact predicate \(\Delta_W(a,b)>0\).
\end{proposition}

\begin{proof}
Each positive tester element has operator norm at most its trace,
hence at most two.
Moreover \(0\leq S,T\leq2I\) implies
\(\|S-T\|_\infty\leq2\).
Consequently
\[
 \|(S-T)_a\otimes B_b\otimes M_c\|_\infty\leq8.
\]
Trace duality bounds the change of each objective in
\eqref{eq:strength-maximum} by \(8\|W-W'\|_1\).
Taking maxima gives \eqref{eq:strength-stability}.

For the last assertion, let \(W_{\to}\) be the definite-order
process that initializes a qubit at \(A\), sends \(A\)'s output
through an identity channel to \(B\), and discards \(B\)'s output.
Party \(C\) receives an independent initialized qubit and has its
output discarded.
Preparing \(\ket0\) or \(\ket1\) at \(A\) and measuring at \(B\)
shows that
\[
 \Delta_{W_{\to}}(A,B)=1.
\]
Such definite-order circuits are valid processes \cite{OCB}.
The process \(W_0\) in \eqref{eq:base-process} has zero signalling
strengths, as also checked in Proposition~\ref{prop:empty-query}.

For \(0<\delta\leq1\), convexity of the defining conditions gives
\[
 W_\delta=(1-\delta)W_0+\delta W_{\to}\in\ProcessSet.
\]
The \(W_0\)-contribution to every signalling difference vanishes,
so \eqref{eq:strength-maximum} gives
\[
 \Delta_{W_\delta}(A,B)=\delta.
\]
Thus \(W_\delta\to W_0\) in trace norm as \(\delta\downarrow0\),
but the exact edge \(A\Infl B\) is present for every \(\delta>0\)
and absent at the limit.
\end{proof}

\paragraph{No unspecified complete-channel norm.}
The frozen extraction uses unconditional classical receiver laws.
A process matrix is not silently treated as a channel between
arbitrarily selected endpoints.
If a later reduction constructs compatible complete channels, their
comparison may use
\[
 \ddiamond(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond
 \cite[Chapter 3]{Watrous}.
\]
That channel construction and its retained systems, records, and
reference extension must be stated separately.

\section{The frozen order-forming operation}
\label{sec:order}

\subsection{Raw experiment-use relation}

Set
\[
 \widetilde E=I\times\Party.
\]
The direct relation is
\[
 (e,a)\Infl_D(f,b)
 \quad\Longleftrightarrow\quad
 e=f,\ a\ne b,\ \Delta_e(a,b)>0.
 \label{eq:direct-relation}
\]
There are no cross-identifier edges in this definition.

The absence of cross-identifier edges records the scope of the
separate-experiment extraction. It is not a measured statement of
geometric separation or of non-signalling for an unspecified joint
experiment.

Write \(p\rightsquigarrow_D q\) when \(p=q\), or when a finite
directed path in \(\Infl_D\) runs from \(p\) to \(q\).
Every nontrivial path stays within one experiment identifier.

\begin{definition}[Component quotient]
\label{def:quotient}
Define
\[
 p\sim_D q
 \quad\Longleftrightarrow\quad
 p\rightsquigarrow_D q
 \ \text{and}\
 q\rightsquigarrow_D p.
 \label{eq:scc}
\]
Let
\[
 E_D=\widetilde E/{\sim_D},
 \qquad
 [p]\preceq_D[q]
 \quad\Longleftrightarrow\quad
 p\rightsquigarrow_D q.
 \label{eq:component-order}
\]
The represented objects are strongly connected components of
experiment-use labels. They are not automatically the original
individual operation events.
\end{definition}

This quotient is the only order-forming operation adopted here.
The rule does not switch between an acyclic restriction and a quotient
after seeing which gives a desired cone.

\begin{theorem}[Order extraction]
\label{thm:order}
For every \(D\in\mathcal D(\ProcessClass)\):
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\sim_D\) is an equivalence relation;
\item \eqref{eq:component-order} is well defined and is a partial order;
\item each experiment contributes between one and three components;
\item \(E_D\) is countably infinite, and every chain has at most
      three distinct elements.
\end{enumerate}
\end{theorem}

\begin{proof}
Reflexivity and transitivity of reachability show reflexivity and
transitivity of \(\sim_D\); symmetry is built into its definition.

Suppose \(p\sim_D p'\), \(q\sim_D q'\), and
\(p\rightsquigarrow_D q\).
Concatenating paths gives
\[
 p'\rightsquigarrow_D p
 \rightsquigarrow_D q
 \rightsquigarrow_D q'.
\]
Thus the order is independent of representatives.
It is reflexive and transitive.
If both \([p]\preceq_D[q]\) and \([q]\preceq_D[p]\), then
\(p\sim_D q\), proving antisymmetry.

A graph on three vertices has between one and three strongly
connected components. No component contains vertices from different
identifiers, since a nontrivial path cannot change identifier.
There are countably many nonempty experiment blocks, so their
disjoint union is countably infinite.
Every chain lies within one block and hence contains at most
three distinct elements.
\end{proof}

This theorem discharges the abstract partial-order part of obligation
O1 for the frozen quotient. It does not prove that the component order
is a spacetime order, that its reachability comparisons are all direct
signalling witnesses, or that the process has a definite-order circuit
realization.

\begin{lemma}[Why cycles cannot be represented injectively without a quotient]
\label{lem:cycle}
Suppose distinct raw labels \(p,q\) satisfy
\(p\rightsquigarrow_D q\) and \(q\rightsquigarrow_D p\).
There is no injective map \(\iota\) into a vector space with a pointed
cone \(K\) faithfully representing this raw reachability relation by
cone differences.
\end{lemma}

\begin{proof}
Faithfulness would give both
\[
 \iota(q)-\iota(p)\in K,
 \qquad
 \iota(p)-\iota(q)\in K.
\]
Pointedness forces their common difference to be zero, contradicting
injectivity.
\end{proof}

\begin{example}[A quotient is a substantive change of carrier]
\label{ex:cycle}
Mix, with probabilities \(0<\lambda<1\) and \(1-\lambda\),
the qubit circuits \(A\to B\) and \(B\to A\), and keep \(C\)
independent.
At \(A,B\), use measure-and-prepare instruments
\[
 \mathcal M_{u|x}(X)=\bra uX\ket u\,\kb{x}{x}.
\]
The \(A,B\) outcome law is
\[
 P(u,v|x,y)
 =
 \lambda\,\delta_{u0}\delta_{vx}
 +(1-\lambda)\,\delta_{v0}\delta_{uy}.
\]
Hence
\[
 P(v=1|x,y)=\lambda x,
 \qquad
 P(u=1|x,y)=(1-\lambda)y.
\]
Both direct signalling edges are present, even though the process
is a mixture of definite-order circuits.
The quotient identifies \(A\) and \(B\), leaving that component
and the isolated \(C\)-component incomparable.

This is the phenomenon recorded in XXII-B
\cite[\loc{prop:signalling-cycle}]{CheungXXIIB}.
Neither branch is a physical closed causal loop.
\end{example}

\subsection{Operational equivalence and order covariance}

\begin{definition}[Matched operational equivalence]
\label{def:equivalence}
Two data families are equivalent for this freeze when there is:
\begin{enumerate}
\item a bijection of experiment identifiers;
\item a party relabelling within each experiment;
\item bijections of the admitted local instrument descriptions and
      outcome labels preserving deterministic insertions and
      coarse-graining;
\item equality of the matched complete probability tables.
\end{enumerate}
The resulting raw-label bijection is denoted by \(f\).
Changes of basis are allowed only with the corresponding changes
of the process and tester descriptions.
\end{definition}

\begin{proposition}[Covariance of the extracted order]
\label{prop:covariance}
A matched operational equivalence preserves signalling strengths,
direct edges, reachability, and strongly connected components.
It therefore induces an order isomorphism
\[
 \overline f:(E_D,\preceq_D)
       \longrightarrow(E_{D'},\preceq_{D'}).
\]
\end{proposition}

\begin{proof}
The matched tables give the same total-variation distances for
every matched intervention pair and context.
The bijection of admissible tests gives equal suprema.
Thus \(f\) preserves the direct relation in both directions.

Applying \(f\) and \(f^{-1}\) to finite paths preserves reachability.
The definition \eqref{eq:scc} then preserves equivalence classes,
and \eqref{eq:component-order} preserves their order.
\end{proof}

This establishes order covariance.
Affine covariance of a future selected representation remains a
separate obligation.

\section{The frozen representation query}
\label{sec:representation}

Fix a target dimension \(m\geq3\) before any proposed selection
argument. It is a desired output dimension, not a spatial metric
or a dimension already derived from the three laboratories.

A cone is proper here when it is closed, convex, pointed, and has
nonempty interior in its ambient finite-dimensional real vector
space. The conical hull consists of finite nonnegative linear
combinations, including zero.

\begin{definition}[Generated cone associated with an embedding]
\label{def:generated}
For a map \(\iota:E_D\to\R^m\), define
\[
 K_\iota
 =
 \overline{\Cone
 \{\iota(y)-\iota(x):
                  x\preceq_D y,\ x\ne y\}}.
 \label{eq:generated}
\]
The closure is the ordinary finite-dimensional vector-space closure.
\end{definition}

\begin{definition}[Representation-feasibility set]
\label{def:feasible}
Let \(\Feasible_m(D)\) be the set of maps
\(\iota:E_D\to\R^m\) satisfying:
\begin{align}
 &\iota\text{ is injective and }K_\iota\text{ is proper},
 \label{eq:proper-injective}\\
 &x\preceq_D y
 \quad\Longleftrightarrow\quad
 \iota(y)-\iota(x)\in K_\iota,
 \label{eq:faithful}\\
 &\R^m
 =
 \Span\{\iota(y)-\iota(x):x,y\in E_D\}.
 \label{eq:span}
\end{align}
Together with \eqref{eq:generated}, these are the faithful,
spanning, and generated representation requirements of XXII-A
\cite[\loc{def:representation}]{CheungXXIIA}.
\end{definition}

\begin{definition}[Predeclared order-and-feasibility extraction]
\label{def:extraction}
The frozen rule is
\[
 \cE_m(D)
 =
 \left(
 \{\Delta_e(a,b)\},
 \Infl_D,
 E_D,
 \preceq_D,
 \Feasible_m(D)
 \right).
 \label{eq:extraction}
\]
Its construction is given by
\eqref{eq:strength-maximum},
\eqref{eq:direct-relation},
\eqref{eq:scc},
\eqref{eq:component-order}, and
Definitions~\ref{def:generated}--\ref{def:feasible}.
\end{definition}

The last output is a feasibility set, not a chosen affine embedding.
It may be empty, and nonempty outputs may contain affinely
inequivalent representations.
Definition~\ref{def:extraction} neither supplies a solver nor
asserts a canonical choice.

A later cone-construction rule must specify a single-valued map
\[
 \cF_m:D\longmapsto\iota_D\in\Feasible_m(D)
 \label{eq:selector}
\]
on its claimed admissible domain.
Then
\[
 V_D=\R^m,\qquad K_D=K_{\iota_D}
\]
would be its candidate representation.
No map \(\cF_m\) is constructed in XXII-C.

No Lorentz cone is independently appended to
\eqref{eq:generated}. Conversely, allowing all maps in the
feasibility set is not an operational derivation of whichever
geometry a later argument prefers.

\begin{proposition}[Covariance of the feasibility problem]
\label{prop:query-covariance}
If \(\overline f:E_D\to E_{D'}\) is the order isomorphism of
Proposition~\ref{prop:covariance}, then
\[
 \iota\longmapsto\iota\circ\overline f^{-1}
\]
is a bijection
\(\Feasible_m(D)\to\Feasible_m(D')\).
It preserves the generated cone as a subset of \(\R^m\).
\end{proposition}

\begin{proof}
The bijection of carriers preserves injectivity.
Because it preserves comparable and arbitrary pairs, it preserves
both sets of differences in
\eqref{eq:generated} and \eqref{eq:span}.
Thus the generated cone, properness, faithfulness, and spanning
are unchanged. Applying the inverse order isomorphism gives the
inverse correspondence.
\end{proof}

This covariance of a set-valued problem does not choose an invariant
member of that set.

\subsection{Nonempty process class versus admissible representation data}

\begin{proposition}[An infinite family can still have an empty query]
\label{prop:empty-query}
The constant family \(W_e=W_0\) belongs to \(\ProcessClass\).
For its datum \(D_0\), every signalling strength vanishes,
\(E_{D_0}\) is an infinite antichain, and
\[
 \Feasible_m(D_0)=\varnothing
 \qquad(m\geq3).
\]
\end{proposition}

\begin{proof}
Membership follows from Lemma~\ref{lem:base}.
For a sender channel \(S\), receiver element \(B_z\), and
other-party channel \(M_c\),
\[
 \Tr[W_0(S\otimes B_z\otimes M_c)]
 =
 \frac18(2)(\Tr B_z)(2)
 =
 \frac12\Tr B_z.
\]
This does not depend on the sender channel.
Thus every strength is zero.

There are no raw edges, so each raw token is a singleton component
and the quotient has no distinct comparable elements.
For every map \(\iota\), the generating set in
\eqref{eq:generated} is empty. Hence \(K_\iota=\{0\}\), which
has empty interior in \(\R^m\).
It cannot satisfy \eqref{eq:proper-injective}.
\end{proof}

The proposition proves non-vacuity of the base process class and
failure of universal representation on that entire class.
It does not provide a witness satisfying a future cone-selecting
hypothesis package.

\subsection{Finite subfamilies retain the finite-generation obstruction}

\begin{proposition}[No finite-event Lorentz generation]
\label{prop:finite}
For a finite set of experiment identifiers, the resulting quotient
carrier is finite.
Any faithful generated proper cone representing that finite carrier
has finitely many extreme rays and cannot be linearly isomorphic
to
\[
 \Lor_m=\{(t,x)\in\R\times\R^{m-1}:t\geq\|x\|_2\}
 \qquad(m\geq3).
\]
\end{proposition}

\begin{proof}
The finite carrier gives finitely many nonzero comparable
differences \(G=\{g_1,\ldots,g_s\}\).
If it generates a proper cone, \(G\ne\varnothing\), and pointedness
implies \(0\notin\operatorname{conv}G\).
Otherwise a nontrivial nonnegative combination summing to zero
would put the negative of a nonzero generator in the same cone.

Choose an auxiliary Euclidean inner product and let \(z\ne0\)
be the closest point of the compact set \(\operatorname{conv}G\)
to zero. Minimization gives
\[
 \langle z,g_j\rangle\geq\|z\|^2>0.
\]
For any convergent sequence
\(\sum_j\lambda_{n,j}g_j\), \(\lambda_{n,j}\geq0\),
these inequalities bound all coefficients.
A convergent coefficient subsequence shows that \(\Cone G\) is
closed. Thus the generated closed cone is \(\Cone G\).

Every extreme ray contains one of the finitely many generators:
a generating decomposition of a nonzero vector on an extreme ray
forces each nonzero summand onto that ray.
There are therefore finitely many extreme rays.

By contrast, every \((1,u)\), \(\|u\|_2=1\), generates an extreme
ray of \(\Lor_m\), as follows from equality in the triangle
inequality in any cone decomposition of that vector.
There are infinitely many such rays for \(m\geq3\).
Invertible linear maps preserve extreme rays.
\end{proof}

This is the finite-generation check of XXII-A and XXII-B, with its
argument included here
\cite{CheungXXIIA}
\cite[\loc{prop:finite-generation}]{CheungXXIIB}.
The auxiliary inner product is only a proof device.

The countable family in Definition~\ref{def:class} removes this
particular finiteness premise. Proposition~\ref{prop:empty-query}
shows why that removal alone is not a solution.

\section{The six extraction obligations}
\label{sec:obligations}

The following are proof obligations, not sufficient hypotheses already
satisfied by the frozen class.

\begin{specification}[Obligations O1--O6]
\label{spec:obligations}
A later selection theorem on \(\ProcessClass\) must address:
\begin{enumerate}[label=\textup{(O\arabic*)}]
\item \textbf{Order theorem and carrier interpretation.}
      Theorem~\ref{thm:order} supplies a partial order for the
      frozen component quotient.
      A later theorem must retain that carrier or explicitly state
      and justify a revised carrier construction.
      It must not identify component reachability with direct
      influence or individual events without a further theorem.
      Acyclicity alone does not establish a complete circuit
      realization or the scheduler package.

\item \textbf{Faithful generated representation.}
      Construct \(\cF_m\) in \eqref{eq:selector} on the claimed
      admissible domain and prove
      \eqref{eq:proper-injective}--\eqref{eq:span}, with
      \eqref{eq:generated}.
      In particular, prove that convexification and closure introduce
      no spurious comparisons.
      Nonemptiness of a feasibility set, if shown nonconstructively,
      must not be reported as an implemented extraction algorithm.

\item \textbf{Infinite-family, extension, or limit compatibility.}
      Explain how the separately tested data family supports the
      claimed common affine representation.
      If no completion or limit is used, say so.
      If one is used, supply its event maps, operational comparisons,
      topology, compatibility, and convergence theorem.
      The experiment identifiers must not become a time or spatial
      coordinate by numerical relabelling.
      Existence and uniqueness are separate claims.

\item \textbf{Relevant countermodel audit.}
      Compare each relevant En08-A countermodel with the actual
      frozen access, component carrier, and proposed cone rule.
      Supply either a justified interpretation and a failed
      additional premise, or a proof that the required interpretation
      is unavailable.
      An unchanged interpreted countermodel satisfying all premises
      and falsifying the conclusion defeats the selection claim.
      Excluding the named examples is not a sufficient classification
      of all remaining cones.

\item \textbf{Honest cone-selection status.}
      Prove homogeneity and strict convexity from the stated
      operational premises, or mark them as additional assumptions
      about the constructed cone.
      In the latter case the theorem remains conditional.
      No classification theorem proves its hypotheses for the
      frozen quantum data merely by being cited.

\item \textbf{Layer separation and invariance.}
      The present relation is formed from operational influence,
      not scheduled ancestry.
      A future schedule comparison requires a certified schedule;
      a geometric comparison requires a separate supplied causal
      carrier and embedding.
      Prove affine invariance of the selected representation under
      the matched operational equivalences.
      State all implementation and statistical allowances actually
      used.
\end{enumerate}
\end{specification}

\subsection{Hypotheses are not obligations renamed}

The scheduler package \(\SP\) retained from XXII-A comprises acyclic
dependency, finite-experiment termination, commutation of independent
operations, schedule-independent complete records, sensitivity equal
to strict reachability in its designated models, and confluence
\cite{CheungXXIIA,CheungEn08A}.

It is not assigned automatically to every
\(D\in\mathcal D(\ProcessClass)\).
A general process matrix need not come with the complete dependency
graph required to interpret those statements.

For a later proposed construction \(\cF_m\), let
\[
 \HP=(H_1,\ldots,H_r)
\]
be its actual finite additional hypothesis list, including class
restrictions where they have substantive force.
Its admissible domain would be
\[
 \mathcal D_{\HP}
 =
 \{D\in\mathcal D(\ProcessClass):
                  H_1(D),\ldots,H_r(D)\}.
 \label{eq:admissible}
\]
The obligations include
\[
 \mathcal D_{\HP}\ne\varnothing
 \label{eq:nonvacuity}
\]
and a justified additional-content comparison with \(\SP\) in a
common operational model universe.

One may not prove the latter merely by observing that process
matrices and scheduler networks have different names.
A dictionary between the compared models, access, and extracted
relations must be stated.
XXII-B fixes the corresponding audit discipline
\cite[\loc{def:entry}, \loc{def:exclusion}]{CheungXXIIB}.

No successful list \(\HP\) and no witness for
\eqref{eq:nonvacuity} are supplied here.
This is compatible with the independently proved nonemptiness of
the unrestricted base class.

\subsection{Classification properties: definitions, not applications}

For a constructed proper cone \(K_D\subset\R^m\), homogeneity means
\[
 \Aut(K_D)=\{A\in\GL(\R^m):A(K_D)=K_D\},
 \qquad
 \forall u,v\in\Int K_D\
 \exists A\in\Aut(K_D):Au=v.
 \label{eq:homogeneity}
\]
Cone strict convexity means
\[
 u,v\in K_D\setminus\{0\},
 \quad u,v\text{ non-proportional}
 \quad\Longrightarrow\quad
 u+v\in\Int K_D.
 \label{eq:strict}
\]

Ito and Louren\c{c}o classify strictly convex homogeneous cones
\cite[Theorem 7]{ItoLourenco}.
XXII-A records the resulting conditional Lorentz classification in
dimension at least three
\cite[\loc{thm:classification}]{CheungXXIIA}.
This note states no application of that result to a cone obtained
from \(\cE_m\), because no such selected cone has been constructed.

Complete positivity is an operator-cone property.
Its relation to the event-difference cone
\eqref{eq:generated} is not an identity.
Likewise, a scheduler interleaving is not automatically an element
of \(\Aut(K_D)\).

\subsection{Exact scope of the En08-A comparisons}

The supplied standard orthant realization has
\[
 K_{\mathrm{orth}}=\R_{\geq0}^{m},\qquad m\geq3.
\]
It is homogeneous: positive diagonal matrices map any interior point
to any other. It is not strictly convex, since
\(e_1,e_2\) are non-proportional and
\(e_1+e_2\in\partial K_{\mathrm{orth}}\).
Thus homogeneity alone does not exclude it
\cite[\loc{prop:orthant}, \loc{lem:diagnostic}]{CheungEn08A}.

The supplied lattice-ball model has a standard three-dimensional
generated diamond cone
\[
 K_{\diamond}=\{(t,x,y):t\geq|x|+|y|\}.
\]
For example, \((1,1,0)\) and \((1,0,1)\) are non-proportional
boundary vectors whose sum is again on the boundary.
This standard cone therefore fails strict convexity
\cite[\loc{prop:ball}, \loc{lem:diagnostic}]{CheungEn08A}.

These calculations concern the named predecessor representations.
They do not identify either model with a member of
\(\ProcessClass\) or with a future rule \(\cF_m\).
In particular, any order identification must respect the chain-height
restriction in Theorem~\ref{thm:order}.
The required translation is part of O4, not inherited by citation.

Finally,
\[
 \Cone\{(1,x):\|x\|_2\leq1\}=\Lor_m.
\]
A Euclidean-ball section is not by itself a non-Lorentzian continuum
cone. The discrete ball-section model and this full continuum
homogenization must remain distinct
\cite[\loc{rem:ball}]{CheungXXIIA}.

\section{Continuum, refinement, and evidence discipline}
\label{sec:limits}

\subsection{What convergence of process data can mean}

For two processes on the same frozen interface define
\[
 \PMdist(W,W')
 =
 \sup_{\text{admitted local instruments}}
 \dtv(P_W,P_{W'}).
 \label{eq:process-distance}
\]
This is an operational probability-comparison quantity for the
specified one-use tests. It is not a diamond norm on an unspecified
global channel.

A family of approximating data
\(\boldsymbol W^{(n)}\in\ProcessClass\) could be compared by, for
example,
\[
 \max_{e\in F}\PMdist(W_e^{(n)},W_e)\longrightarrow0
 \quad
 \text{for every finite }F\subset I,
 \label{eq:finite-family-convergence}
\]
or by the stronger uniform statement
\[
 \sup_{e\in I}\PMdist(W_e^{(n)},W_e)\longrightarrow0.
 \label{eq:uniform-family-convergence}
\]
A future argument must name its topology and prove the convergence
it uses. Neither statement is built into
Definition~\ref{def:class}.

Even convergence of all strengths does not automatically yield
well-defined maps between the component quotients.
A positive limiting strength is eventually witnessed by positive
approximating strengths, but a zero limiting strength may have
positive approximating strengths at every scale, as in
Proposition~\ref{prop:strength-stability}.
Component splitting or merging therefore requires its own exact
support or compatibility theorem.

\subsection{Event and cone extensions}

If a later construction replaces \(E_D\) by an extended carrier
\(\widehat E_D\), it must provide a declared map
\[
 j_D:E_D\longrightarrow\widehat E_D
\]
and state whether it is injective.
If the old comparisons are claimed to survive faithfully, it must
prove
\[
 x\preceq_D y
 \quad\Longleftrightarrow\quad
 j_D(x)\widehat{\preceq}_D j_D(y).
 \label{eq:extension-faithfulness}
\]
Every new event and comparison must have its declared operational
or limiting meaning.
A completion is not licensed merely by drawing a continuum around
the countable experiment family.

For a cone limit, finite-dimensional comparison spaces and maps
must be supplied before statements such as
\(K_n\to K\) have a meaning.
The chosen cone topology must be stated, and the limiting
faithfulness and generation equations must be proved.
An arbitrary norm used to formulate a mathematical convergence
criterion is not a supplied spacetime metric, but neither does it
provide an operational identification by itself.

\begin{remark}[A closed-cone limit can introduce a new comparison]
\label{rem:closure}
Even with a fixed closed Lorentz cone, vectors outside it may
converge to its boundary:
\[
 v_n=(1,1+1/n,0,\ldots,0)\notin\Lor_m,
 \qquad
 v_n\longrightarrow(1,1,0,\ldots,0)\in\Lor_m.
\]
Thus convergence of coordinates alone does not prove the
absence of spurious order comparisons at the limit.

This is a mathematical diagnostic, not a Lorentz cone installed
as the output of the frozen extraction.
\end{remark}

Boundary generation must also be distinguished from an attained
operational event pair. A boundary ray produced by closure need not
be occupied by any actual protocol
\cite{CheungEn08A}
\cite[Failure test F3]{CheungXXIIB}.
Attained null transport is a separate claim, not an implicit
requirement of the cone-classification target.

\subsection{Complete-channel evolution limits are a separate interface}

If a later construction also claims a continuous-time channel
evolution, it must first construct:
\begin{enumerate}
\item a refinement family and its scale or tick prescription;
\item complete channels \(\mathcal C_n(t)\);
\item encoders \(\mathcal E_n\) and decoders \(\mathcal D_n\) on
      common identified interfaces;
\item a limiting family \(\mathcal T_t\), with its asserted
      composition law and continuity;
\item the claimed convergence theorem.
\end{enumerate}
For example, one possible complete-channel criterion is
\[
 \sup_{0\leq t\leq T}
 \ddiamond\bigl(
 \mathcal D_n\mathcal C_n(t)\mathcal E_n,\mathcal T_t
 \bigr)\longrightarrow0
 \quad\text{for every finite }T.
 \label{eq:channel-limit}
\]
Retained quantum systems, classical histories, and untouched
references must be included in the actual comparison interfaces
\cite[\loc{def:limit}]{CheungXIXC}.

The parameter \(t\), the channels, and their composition are not
present in the frozen datum merely because its experiments have
integer identifiers.

XIX-C gives two distinct refinement families sharing a coarse
complete channel and a one-site support ceiling but converging to
different semigroups
\cite[\loc{prop:two-limits}]{CheungXIXC}.
Those families choose different missing refinement data; they do
not satisfy an unstated consistency condition with the coarse tick.
The result is a failure of inference from the coarse data, not a
universal prohibition of unique limits with additional data.

Finite sampling can likewise leave interpolation ambiguity
\cite[\loc{prop:interpolation}]{CheungXIXC}.
A uniqueness claim needs a specific theorem under specific extra
conditions. Naming ``projective consistency'' or ``interpolation''
does not prove uniqueness.

\subsection{Exact guards and finite evidence}

The present edge guard is exactly
\[
 \Delta_e(a,b)>0.
\]
A tolerance rule, such as declaring an edge only when
\(\Delta_e(a,b)>\eta\), defines a different graph and hence potentially
a different quotient order.
It is not substituted into \(\cE_m\) without an explicit revision.

If an implemented receiver law is within \(\varepsilon\) in total
variation of its comparison law for each intervention, the triangle
inequality bounds the change of an intervention-pair witness by
\(2\varepsilon\).
A positive allowance does not establish exact absence of signalling
\cite[\loc{prop:tolerance}]{CheungXXIIB}.

Finite records require coverage, preparation and calibration
assumptions, complete attempted-trial scoring, and any transfer
allowances actually used.
Cheung's Oai265-B supplies a documentary example of keeping such
assumptions separate from its channel conclusion
\cite[\loc{ass:statistics}, \loc{thm:main}]{CheungOai265B}.
Its certificate is not an exact process-matrix zero test or an
event-cone selection theorem.

\section{Component ledger and release status}
\label{sec:ledger}

\begingroup
\small
\renewcommand{\arraystretch}{1.14}
\begin{longtable}{@{}p{.26\linewidth}p{.66\linewidth}@{}}
\toprule
Component & Role and boundary\\
\midrule
\endfirsthead
\toprule
Component & Role and boundary\\
\midrule
\endhead
\bottomrule
\endfoot

XXII-A &
Target, faithful-generation equations, and non-circularity audit.
No positive selection theorem is inherited
\cite{CheungXXIIA}.\\[5pt]

XXII-B &
Catalogue, ledger syntax, signalling-cycle witness,
finite-generation check, and failure tests F1--F5
\cite{CheungXXIIB}.\\[5pt]

OCB process matrices &
Published probability framework for the frozen one-use class;
not an event-cone selection theorem
\cite{OCB}.\\[5pt]

Jia--Sakharwade &
Published warning that unrestricted process tensor products
require a validity analysis. No global composition is inferred
from the separately tested family
\cite{JiaSakharwade}.\\[5pt]

En08-A &
Scheduler obstruction and conditional representation record.
The standard orthant and diamond comparisons are not transferred
to a new extraction without a model dictionary
\cite{CheungEn08A}.\\[5pt]

VII-J &
Separation of scheduled support, operational influence, and
supplied geometric causality
\cite{CheungVIIJ}.\\[5pt]

XIX-C &
Refinement and interpolation non-determination, and the
complete operational-limit interface
\cite{CheungXIXC}.\\[5pt]

En08-B &
Possible later endpoint-transport comparison. Its metric,
speed, and null targets are inputs; they are not derived by
its approximation theorem
\cite{CheungEn08B}.\\[5pt]

Family 265 /
Oai265-A &
Documentary negative control for static area-law records;
not a causal carrier or metric extraction
\cite{CheungOai265A}.\\[5pt]

VIII-A/B/C,
Em02-E,
XIX/XX clock records &
Documentary geometric, lapse, clock, and admission comparisons
as recorded in XXII-A and XIX-E. None supplies the missing
construction \(\cF_m\)
\cite{CheungXXIIA,CheungXIXE}.\\
\end{longtable}
\endgroup

No unrelated field-theoretic admission result, Ward identity, recovery
benchmark, or catalogue claim is made a premise of the process-class
freeze by sharing terminology with it.

\subsection{What is proved and what remains open}

\begingroup
\small
\renewcommand{\arraystretch}{1.14}
\begin{longtable}{@{}p{.39\linewidth}p{.16\linewidth}p{.35\linewidth}@{}}
\toprule
Item & Status & Exact scope\\
\midrule
\endfirsthead
\toprule
Item & Status & Exact scope\\
\midrule
\endhead
\bottomrule
\endfoot

Base class and exact access &
Defined &
Tripartite qubit processes, separately tested countable family\\[4pt]

Base-class nonemptiness &
Proved &
Lemma~\ref{lem:base}; not admissible-subclass non-vacuity\\[4pt]

Exact signalling query &
Proved &
Binary reduction and attained finite-dimensional maximum\\[4pt]

Component partial order &
Proved &
Theorem~\ref{thm:order}; components, not raw cyclic events\\[4pt]

Order and feasibility covariance &
Proved &
Matched operational equivalence; no invariant selected embedding\\[4pt]

Single-valued affine construction &
Open &
No \(\cF_m\) in \eqref{eq:selector}\\[4pt]

Successful finite hypothesis package &
Open &
No witness for \(\mathcal D_{\HP}\ne\varnothing\)\\[4pt]

Faithful generated representation &
Open &
No member of \(\Feasible_m(D)\) selected for admissible data\\[4pt]

Complete countermodel audit &
Open &
No full interpretation and exclusion package\\[4pt]

Homogeneity and strict convexity &
Open &
Neither operationally derived nor satisfied by a constructed cone\\[4pt]

Lorentz classification application &
Not performed &
No extracted cone meeting its hypotheses\\[4pt]

Continuum or event extension &
Not constructed &
Interfaces and proof obligations only\\[4pt]

Geometric causal embedding &
Separate &
Requires an independently supplied carrier and certificate\\
\end{longtable}
\endgroup

The statement that ``none of O1--O6 has been discharged'' would be
incorrect for this revision: the abstract quotient order theorem and
several interface properties are proved.
The cone-selecting conjunction of obligations remains unfulfilled.

\section{Stage 3 handoff and unchanged boundaries}
\label{sec:handoff}

\begin{target}[Selection on the frozen process family]
\label{target:selection}
On \(\ProcessClass\) with data assignment
\(\mathcal D_{\mathrm{3q}}\), supply a predeclared finite additional
hypothesis package \(\HP\) and a single-valued construction
\(\cF_m\) such that:
\begin{enumerate}[label=\textup{(C\arabic*)}]
\item \(\mathcal D_{\HP}\ne\varnothing\);
\item for every \(D\in\mathcal D_{\HP}\),
      \(\cF_m(D)\in\Feasible_m(D)\), or in the exactly specified
      feasibility set of a justified revised carrier;
\item the relevant countermodel audit and genuinely additional-content
      comparison are proved;
\item the constructed cone-selecting properties are derived from the
      operational premises, or their additional assumed status is
      explicit;
\item the representation invariance, limits, and evidence statements
      required by the claim are proved;
\item the resulting conclusion meets XXII-A's conditions (T1)--(T5)
      without installing a spacetime metric, geometric causal relation,
      physical light cone, or invariant speed as input.
\end{enumerate}
This target has no proof in XXII-C.
\end{target}

If homogeneity and strict convexity are used only as assumptions on
the constructed cone, any eventual classification conclusion must be
described as conditional on those assumptions.
The existence and construction obligations are not discharged by
that conditional implication.

A supplied VII-J-type causal relation is still a subsequent embedding
problem. On a definite-order channel realization, certified containment
of every dependency edge is an additional embedding hypothesis; it
does not derive the supplied metric or causal relation
\cite{CheungVIIJ,CheungXXIIA}.

\paragraph{Unchanged boundaries.}
XXII-C constructs no faithful generated affine representation for a
cone-selecting admissible family. It does not derive homogeneity or
strict convexity from the process matrices, apply Ito--Louren\c{c}o
to an operationally extracted cone, or complete the En08-A exclusion
audit.

It proves no Einstein dynamics, holographic reconstruction, Page curve,
gravity-generated entanglement, or quantum evolution without an
external time. No result here changes O.1, XV-B, XVI-N, Route B,
H2, I3, or J2. No quantum coefficient is admitted.
No successful compilation, native programme execution, numerical
optimization, or proof-assistant verification is asserted
\cite{CheungXIXE}.

\paragraph{Final gap statement.}
The remaining gap is the construction, on the frozen family and access
model, of one nonempty additional operational package and one
predeclared affine-construction rule linking its component order to a
faithful generated cone with justified selecting properties.
The exact order query is now defined and its partial-order character
proved. The representation and selection remain open.
Neither the countable experiment family nor the classification
literature supplies those missing constructions by itself.

\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}, \loc{rem:ball},
and conditions (T1)--(T5), (A1)--(A6).

\bibitem{CheungXXIIB}
N.~Cheung.
\emph{XXII-B: Catalogue the additional quantum data and the
hypothesis ledger.
Stage 1 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-B.tex}, n.d.
Relevant labels:
\loc{def:signalling}, \loc{def:entry},
\loc{def:exclusion}, \loc{prop:signalling-cycle},
\loc{prop:static}, \loc{prop:finite-generation},
\loc{prop:tolerance}, and \loc{prop:two-limits}.
Failure tests F1--F5.

\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{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{JiaSakharwade}
D.~Jia and N.~Sakharwade.
Tensor products of process matrices with indefinite causal structure.
\emph{Physical Review A}
\textbf{97} (2018), 032110.
\doi{10.1103/PhysRevA.97.032110}.
Preprint, under the title
\emph{Process Products Need Qualification}:
\url{https://arxiv.org/abs/1706.05532}.

\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 cited for the classification boundary, not applied
to an extracted cone in this note.

\bibitem{CheungEn08A}
N.~Cheung.
\emph{En08-A}, version 1, manuscript \loc{En08-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:main}, \loc{thm:network},
\loc{prop:ball}, \loc{prop:orthant},
\loc{lem:diagnostic}, \loc{thm:mainpositive},
\loc{thm:representation}, and \loc{thm:weaker}.
Registry: \url{https://enter.grok.me/}.
The supplied scope and comparison passages are not represented
as independent verification of the full manuscript.

\bibitem{CheungVIIJ}
N.~Cheung.
\emph{VII-J}, version 1, manuscript \loc{VII-J.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:schedule}, \loc{thm:nosignal},
\loc{thm:physical}, and \loc{thm:release}.
Scheduled ancestry, operational influence, and supplied geometric
causality are distinct interfaces.

\bibitem{CheungXIXC}
N.~Cheung.
\emph{XIX-C}, version 1, manuscript \loc{XIX-C.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{def:limit}, \loc{prop:two-limits},
\loc{prop:interpolation}, and \loc{thm:main}.
Finite certification, complete operational limits, and
refinement non-determination.

\bibitem{CheungEn08B}
N.~Cheung.
\emph{En08-B}, version 1, manuscript \loc{En08-B.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:main}, \loc{def:comparison},
and \loc{cor:operational-limit}.
Reference-stable endpoint approximation on prescribed geometric
and speed inputs; documentary comparison only.

\bibitem{CheungOai265A}
N.~Cheung.
\emph{Oai265-A}, manuscript \loc{Oai265-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied label: \loc{thm:separation}.
Static area-law records, recovery, and radiation histories
are not interchangeable claims.

\bibitem{CheungOai265B}
N.~Cheung.
\emph{Oai265-B}, version 2, manuscript \loc{Oai265-B-v2.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{ass:statistics}, \loc{thm:main},
\loc{eq:coverage}, and \loc{eq:falseaccept}.
Documentary complete-record finite-confidence channel certificate;
not a premise of event-cone selection.

\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 programme boundaries, not a universal impossibility
claim.

\end{thebibliography}

\end{document}