XXII-C: Freeze one quantum process class and the extraction obligations Stage 2 of a limited kinematic emergence programme

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.

XXII-A fixes the operational Lorentz-cone selection target and its acceptance conditions CheungXXIIA (target:selection, conditions (T1)–(T5), (A1)–(A6)). XXII-B supplies the candidate-data catalogue, hypothesis-ledger syntax, and failure tests CheungXXIIB. The present note selects the process-matrix branch and replaces an unspecified “family of experiments” by the concrete declaration in Section sec:class.

The claim under construction is not

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.

meanings of extraction must be separated. The order-and-feasibility rule ${\mathfrak{E}}_m$ defined in Section 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 $\mathsf{H}$. A later theorem must state its actual finite hypotheses and its actual cone-construction rule before applying them.

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.

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.

The party set is fixed once and for all:

Each party has one qubit input and one qubit output:

Let

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

where ${}^{\mathsf{T}}$ denotes the full matrix transpose. This is the convention used in the original process-matrix probability formula OCB.

For each party set

The first set represents channels. The second represents completely positive trace-nonincreasing instrument elements in the convention (eq:choi) Watrous (Chapter 2).

A matrix $W\in L(\mathcal{H})$ belongs to ${\mathsf{W}}_3$ when

A local finite-outcome instrument is a list $\{{M^a}_{x_a|s_a}{\}}_{x_a}$ of positive matrices whose sum belongs to ${\mathsf{C}}_a$. Its complete joint outcome law is

No definite party ordering or causal-separability decomposition is required in this definition.

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

The set ${\mathsf{W}}_3$ is nonempty and compact. Every $W\in {\mathsf{W}}_3$ satisfies $\mathrm{Tr}W=8$.

Proof. The completely depolarizing channel has Choi matrix $I_{a_I{a}_O}/2$. Substituting these three channels into (eq:normalization) gives

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

is positive. Since every $M_a\in {\mathsf{C}}_a$ has trace two,

Thus $W_0\in {\mathsf{W}}_3$.

The access model is:

1. every local finite-outcome qubit-input/qubit-output instrument is admitted;

2. each party uses its process input/output interface once in a selected experiment;

3. local auxiliary implementations are allowed insofar as they induce the declared instrument maps on that interface;

4. no extra shared ancillary resource is inserted into the signalling tests; correlations already present in $W$ remain allowed;

5. settings are chosen independently of the process resource; the tests use no interparty classical feed-forward outside the declared process;

6. the receiver output used for extraction is its local classical instrument record; sender and other-party records are summed;

7. every outcome, including any declared failure, belongs to the complete outcome alphabet;

8. no postselection on another party's record is used to define influence.

There is no spatial location, distance, speed, or spacetime causal relation in ${\mathsf{A}}_0$.

Let $I=\mathbb{N}=\{1,2,\dots \}$ be a set of experiment identifiers. Define

An object is a family

Experiment $e$ is tested separately using ${\mathsf{A}}_0$. 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.

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 JiaSakharwade. Any later composition theorem must specify the admissible interfaces and prove validity.

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.

The assignment

returns

where the probability functionals and signalling strengths are defined below.

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

Static reduced spectra, mutual information, or other correlation diagnostics may be appended as annotations, but ${\mathfrak{E}}_m$ 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 CheungXXIIB (prop:static) CheungOai265A (thm:separation).

For distinct $a,b\in \mathsf{J}$, let $c$ be the remaining party. For $S,T\in {\mathsf{C}}_a$, a receiver instrument $\{B_z{\}}_z$ at $b$, and $M_c\in {\mathsf{C}}_c$, compare

Factors are placed in the fixed tensor slots. The settings at $b,c$ are identical within the compared pair.

Define

This is the classical-record branch of the unconditional signalling test of XXII-B CheungXXIIB (def:signalling).

For every $W_e\in {\mathsf{W}}_3$,

The maximum exists and belongs to $[0,1]$.

Proof. For finite probability laws,

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 ${\mathsf{B}}_b$. Thus every witness in (eq:strength-definition) is bounded by the right side of (eq:strength-maximum).

Conversely, every $B\in {\mathsf{B}}_b$ can be completed to a binary instrument. Put

Then $B^{\mathrm{c}}\ge 0$, and

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 (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.

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.

For $W,W^{\prime }\in {\mathsf{W}}_3$,

Nevertheless, trace-norm convergence of processes does not generally stabilize the exact predicate ${\Delta }_W(a,b)>0$.

Proof. Each positive tester element has operator norm at most its trace, hence at most two. Moreover $0\le S,T\le 2I$ implies $\Vert S-T{\Vert }_{\infty }\le 2$. Consequently

Trace duality bounds the change of each objective in (eq:strength-maximum) by $8\Vert W-W^{\prime }{\Vert }_1$. Taking maxima gives (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 $|0⟩$ or $|1⟩$ at $A$ and measuring at $B$ shows that

Such definite-order circuits are valid processes OCB. The process $W_0$ in (eq:base-process) has zero signalling strengths, as also checked in Proposition prop:empty-query.

For $0<\delta \le 1$, convexity of the defining conditions gives

The $W_0$-contribution to every signalling difference vanishes, so (eq:strength-maximum) gives

Thus $W_{\delta }\to W_0$ in trace norm as $\delta \downarrow 0$, but the exact edge $A{R}_{\inf }B$ is present for every $\delta >0$ and absent at the limit.

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

That channel construction and its retained systems, records, and reference extension must be stated separately.

Set

The direct relation is

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 $p{⇝}_{D}q$ when $p=q$, or when a finite directed path in ${R_{\inf }}_D$ runs from $p$ to $q$. Every nontrivial path stays within one experiment identifier.

Define

Let

The represented objects are strongly connected components of experiment-use labels. They are not automatically the original individual operation events.

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.

For every $D\in \mathcal{D}({\mathcal{P}}_{3q})$:

1. ${\sim }_D$ is an equivalence relation;

2. (eq:component-order) is well defined and is a partial order;

3. each experiment contributes between one and three components;

4. $E_D$ is countably infinite, and every chain has at most three distinct elements.

Proof. Reflexivity and transitivity of reachability show reflexivity and transitivity of ${\sim }_D$; symmetry is built into its definition.

Suppose $p{\sim }_D{p}^{\prime }$, $q{\sim }_D{q}^{\prime }$, and $p{⇝}_{D}q$. Concatenating paths gives

Thus the order is independent of representatives. It is reflexive and transitive. If both $[p]{⪯}_D[q]$ and $[q]{⪯}_D[p]$, 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.

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.

Suppose distinct raw labels $p,q$ satisfy $p{⇝}_{D}q$ and $q{⇝}_{D}p$. There is no injective map $\iota$ into a vector space with a pointed cone $K$ faithfully representing this raw reachability relation by cone differences.

Proof. Faithfulness would give both

Pointedness forces their common difference to be zero, contradicting injectivity.

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

The $A,B$ outcome law is

Hence

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 CheungXXIIB (prop:signalling-cycle). Neither branch is a physical closed causal loop.

Two data families are equivalent for this freeze when there is:

1. a bijection of experiment identifiers;

2. a party relabelling within each experiment;

3. bijections of the admitted local instrument descriptions and outcome labels preserving deterministic insertions and coarse-graining;

4. equality of the matched complete probability tables.

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.

A matched operational equivalence preserves signalling strengths, direct edges, reachability, and strongly connected components. It therefore induces an order isomorphism

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 (eq:scc) then preserves equivalence classes, and (eq:component-order) preserves their order.

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

Fix a target dimension $m\ge 3$ 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.

For a map $\iota :E_D\to {\mathbb{R}}^m$, define

The closure is the ordinary finite-dimensional vector-space closure.

Let ${\mathcal{A}}_m(D)$ be the set of maps $\iota :E_D\to {\mathbb{R}}^m$ satisfying:

Together with (eq:generated), these are the faithful, spanning, and generated representation requirements of XXII-A CheungXXIIA (def:representation).

The frozen rule is

Its construction is given by (eq:strength-maximum), (eq:direct-relation), (eq:scc), (eq:component-order), and Definitions def:generated–def:feasible.

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 def:extraction neither supplies a solver nor asserts a canonical choice.

A later cone-construction rule must specify a single-valued map

on its claimed admissible domain. Then

would be its candidate representation. No map ${\mathfrak{F}}_m$ is constructed in XXII-C.

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

If $\overline{f}:E_D\to E_{D^{\prime }}$ is the order isomorphism of Proposition prop:covariance, then

is a bijection ${\mathcal{A}}_m(D)\to {\mathcal{A}}_m(D^{\prime })$. It preserves the generated cone as a subset of ${\mathbb{R}}^m$.

Proof. The bijection of carriers preserves injectivity. Because it preserves comparable and arbitrary pairs, it preserves both sets of differences in (eq:generated) and (eq:span). Thus the generated cone, properness, faithfulness, and spanning are unchanged. Applying the inverse order isomorphism gives the inverse correspondence.

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

The constant family $W_e=W_0$ belongs to ${\mathcal{P}}_{3q}$. For its datum $D_0$, every signalling strength vanishes, $E_{D_0}$ is an infinite antichain, and

Proof. Membership follows from Lemma lem:base. For a sender channel $S$, receiver element $B_z$, and other-party channel $M_c$,

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 (eq:generated) is empty. Hence $K_{\iota }=\{0\}$, which has empty interior in ${\mathbb{R}}^m$. It cannot satisfy (eq:proper-injective).

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.

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

Proof. The finite carrier gives finitely many nonzero comparable differences $G=\{g_1,\dots ,g_s\}$. If it generates a proper cone, $G\ne \varnothing$, and pointedness implies $0\notin \mathrm{conv}G$. 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 $z\ne 0$ be the closest point of the compact set $\mathrm{conv}G$ to zero. Minimization gives

For any convergent sequence ${\sum }_j{\lambda }_{n,j}{g}_j$, ${\lambda }_{n,j}\ge 0$, these inequalities bound all coefficients. A convergent coefficient subsequence shows that $\mathrm{cone}G$ is closed. Thus the generated closed cone is $\mathrm{cone}G$.

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 $(1,u)$, $\Vert u{\Vert }_2=1$, generates an extreme ray of ${\mathcal{L}}_m$, as follows from equality in the triangle inequality in any cone decomposition of that vector. There are infinitely many such rays for $m\ge 3$. Invertible linear maps preserve extreme rays.

This is the finite-generation check of XXII-A and XXII-B, with its argument included here CheungXXIIA CheungXXIIB (prop:finite-generation). The auxiliary inner product is only a proof device.

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

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

A later selection theorem on ${\mathcal{P}}_{3q}$ must address:

1. Order theorem and carrier interpretation. Theorem 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.

2. Faithful generated representation. Construct ${\mathfrak{F}}_m$ in (eq:selector) on the claimed admissible domain and prove (eq:proper-injective)–(eq:span), with (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.

3. 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.

4. 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.

5. 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.

6. 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.

The scheduler package $\mathsf{S}$ 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 CheungXXIIA,CheungEn08A.

It is not assigned automatically to every $D\in \mathcal{D}({\mathcal{P}}_{3q})$. A general process matrix need not come with the complete dependency graph required to interpret those statements.

For a later proposed construction ${\mathfrak{F}}_m$, let

be its actual finite additional hypothesis list, including class restrictions where they have substantive force. Its admissible domain would be

The obligations include

and a justified additional-content comparison with $\mathsf{S}$ 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 CheungXXIIB (def:entry, def:exclusion).

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

For a constructed proper cone $K_D\subset {\mathbb{R}}^m$, homogeneity means

Cone strict convexity means

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

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

The supplied standard orthant realization has

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 CheungEn08A (prop:orthant, lem:diagnostic).

The supplied lattice-ball model has a standard three-dimensional generated diamond cone

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 CheungEn08A (prop:ball, lem:diagnostic).

These calculations concern the named predecessor representations. They do not identify either model with a member of ${\mathcal{P}}_{3q}$ or with a future rule ${\mathfrak{F}}_m$. In particular, any order identification must respect the chain-height restriction in Theorem thm:order. The required translation is part of O4, not inherited by citation.

Finally,

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 CheungXXIIA (rem:ball).

For two processes on the same frozen interface define

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 ${\mathbf{W}}^{(n)}\in {\mathcal{P}}_{3q}$ could be compared by, for example,

or by the stronger uniform statement

A future argument must name its topology and prove the convergence it uses. Neither statement is built into Definition 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 prop:strength-stability. Component splitting or merging therefore requires its own exact support or compatibility theorem.

If a later construction replaces $E_D$ by an extended carrier ${\hat{E}}_D$, it must provide a declared map

and state whether it is injective. If the old comparisons are claimed to survive faithfully, it must prove

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.

Even with a fixed closed Lorentz cone, vectors outside it may converge to its boundary:

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.

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 CheungEn08A CheungXXIIB (Failure test F3). Attained null transport is a separate claim, not an implicit requirement of the cone-classification target.

If a later construction also claims a continuous-time channel evolution, it must first construct:

1. a refinement family and its scale or tick prescription;

2. complete channels ${\mathcal{C}}_n(t)$;

3. encoders ${\mathcal{E}}_n$ and decoders ${\mathcal{D}}_n$ on common identified interfaces;

4. a limiting family ${\mathcal{T}}_t$, with its asserted composition law and continuity;

5. the claimed convergence theorem.

For example, one possible complete-channel criterion is

Retained quantum systems, classical histories, and untouched references must be included in the actual comparison interfaces CheungXIXC (def:limit).

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 CheungXIXC (prop:two-limits). 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 CheungXIXC (prop:interpolation). A uniqueness claim needs a specific theorem under specific extra conditions. Naming “projective consistency” or “interpolation” does not prove uniqueness.

The present edge guard is exactly

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 ${\mathfrak{E}}_m$ without an explicit revision.

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

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 CheungOai265B (ass:statistics, thm:main). Its certificate is not an exact process-matrix zero test or an event-cone selection theorem.

{}1.14

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.

{}1.14

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.

On ${\mathcal{P}}_{3q}$ with data assignment ${\mathcal{D}}_{3q}$, supply a predeclared finite additional hypothesis package $\mathsf{H}$ and a single-valued construction ${\mathfrak{F}}_m$ such that:

1. ${\mathcal{D}}_{\mathsf{H}}\ne \varnothing$;

2. for every $D\in {\mathcal{D}}_{\mathsf{H}}$, ${\mathfrak{F}}_m(D)\in {\mathcal{A}}_m(D)$, or in the exactly specified feasibility set of a justified revised carrier;

3. the relevant countermodel audit and genuinely additional-content comparison are proved;

4. the constructed cone-selecting properties are derived from the operational premises, or their additional assumed status is explicit;

5. the representation invariance, limits, and evidence statements required by the claim are proved;

6. 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.

This target has no proof in XXII-C.

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 CheungVIIJ,CheungXXIIA.

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–Lourenço 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 CheungXIXE.

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.

CheungXXIIA

N. Cheung. XXII-A: Freeze the precise claim and the gap. Stage 0 of a limited kinematic emergence programme. Supplied manuscript XXII-A.tex, n.d. Relevant items: target:selection, def:representation, thm:classification, rem:ball, and conditions (T1)–(T5), (A1)–(A6).

CheungXXIIB

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

OCB

O. Oreshkov, F. Costa, and Č. Brukner. Quantum correlations with no causal order. Nature Communications 3 (2012), 1092. doi:\ 10.1038/ncomms2076. Preprint: https://arxiv.org/abs/1105.4464.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Author's text and chapters: https://cs.uwaterloo.ca/ watrous/TQI/.

JiaSakharwade

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

ItoLourenco

M. Ito and B. F. Lourenço. The $p$-cones in dimension $n\ge 3$ are not homogeneous when $p\ne 2$. Linear Algebra and its Applications 533 (2017), 326–335. doi:\ 10.1016/j.laa.2017.07.029. Preprint: https://arxiv.org/abs/1701.05801. Theorem 7 is cited for the classification boundary, not applied to an extracted cone in this note.

CheungEn08A

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

CheungVIIJ

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

CheungXIXC

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

CheungEn08B

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

CheungOai265A

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

CheungOai265B

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

CheungXIXE

N. Cheung. XIX-E, version 1, manuscript XIX-E.tex. Enter Calculus Companion, n.d. Relevant supplied passage: “What remains outside the release.” Documentary programme boundaries, not a universal impossibility claim.