XXII-B: Catalogue the additional quantum data and the hypothesis ledger Stage 1 of a limited kinematic emergence programme

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.

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 CheungXXIIA (target:selection, conditions (T1)–(T5)). XXII-B catalogues possible data and the obligations of such a package. It does not assert that a successful package exists.

The proposed sentence

There exists a finite additional ledger, a named quantum data class, and a nonempty admissible data set excluding the relevant scheduler countermodels

is therefore an 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:

1. acyclic dependency;

2. termination of every finite experiment;

3. commutation of independent operations;

4. schedule-independent complete joint records;

5. exact operational sensitivity equal to strict reachability;

6. confluence of legal finite executions.

Their conjunction is denoted by $\mathsf{S}$. En08-A supplies the obstruction to selection from this package CheungEn08A (thm:main, prop:ball, prop:orthant). Condition (S5) is a special property of those models, not a general identification of influence with ancestry.

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 not selected here. Calling this a completed catalogue does not discharge the substantive non-vacuity and construction obligations of the programme.

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.

An operational declaration specifies:

1. a language tag, a finite party or slot set for each experiment, and finite-dimensional input and output Hilbert spaces;

2. admissible preparations and local instruments, including any restrictions on interventions;

3. the accessible output systems and records, with every attempted outcome and failure retained in the declared interface;

4. which auxiliary systems and classical controls participate, and which references are untouched;

5. the operational equivalences, relabellings, and comparison maps;

6. the event labels, their interpretation, and the proposed relation to be extracted;

7. whether the supplied data are exact mathematical maps or probabilities, or finite statistical observations;

8. any family of experiments, refinements, or extensions used to enlarge the event carrier.

A finite party set in one experiment does not require the full experiment family or the event carrier to be finite.

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 OCB,Beckman.

Three relations remain distinct:

Scheduled ancestry $R_{\mathrm{sch}}$.

Reachability in a complete dependency graph that includes the actual quantum inputs, classical controls, guards, and routing dependencies.

Operational influence $R_{\inf }$.

A change in an accessible receiver output under an admissible intervention, with the rest of the operational context held fixed.

Supplied causal relation $R_{\sup }$.

An independently declared geometric comparison relation, requiring its own carrier, calibration, and embedding of operational events.

This separation is the discipline of VII-J and XXII-A CheungVIIJ,CheungXXIIA.

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

For complete channels on common input and output comparison spaces, use instead

Write $p{R}_{\inf }q$ when ${\Delta }_D(p,q)>0$. The receiver interface must not acquire the sender's record through an undeclared readout or side channel.

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 Watrous (Chapters 2–3).

Direct influence is not assumed to be transitive or acyclic. A future order extraction must explicitly:

1. prove the needed order properties; or

2. specify an order-forming operation and prove what it represents.

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 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 CheungVIIJ,CheungXXIIA.

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.

Fix bases and use the unnormalized Choi operator

where ${}^{\mathsf{T}}$ is the full transpose in those bases. In this convention a process matrix $W$ assigns

Positivity and normalization for all admitted local instruments, including the required ancillary extensions, are part of process validity 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 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.

There is a causally separable bipartite process with both $A{R}_{\inf }B$ and $B{R}_{\inf }A$. Thus its aggregate pairwise signalling relation is not a strict partial order.

Proof. Choose $0<\lambda <1$. With probability $\lambda$, initialize a qubit in $|0⟩$, 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 OCB.

At both laboratories use measure-and-prepare instruments:

Let $x,y$ be the settings at $A,B$, respectively. The resulting complete probability table is

Consequently,

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.

Acyclicity of a particular signalling graph, when it holds, still does not by itself prove a complete circuit realization or every condition in $\mathsf{S}$. 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.

A deterministic comb is a positive Choi operator on declared, ordered input/output slots, satisfying recursive trace conditions of the form

with tensor-factor order fixed by the chosen convention CDP (Theorem 5). A standard multi-time process tensor similarly describes interventions at a declared sequence of time steps, with its specified boundary systems and containment conditions Pollock.

Accordingly, the expression “a standard process tensor with non-preinstalled slot order” is not adopted here. Candidate B may extract 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 sec:test-refinement illustrates the missing refinement information CheungXIXC (prop:two-limits). It is a warning about that inference, not an assertion that every fully specified process tensor has ambiguous refinement.

For a declared tensor-product output state $\rho$, static candidate data may include

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 Watrous (Chapters 5–6).

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.

Proof. Consider the labelled Bell output

One realization distributes this state to two laboratories with no communication. For any trace-preserving local intervention ${\mathcal{I}}_A$,

Indeed, pairing with a $B$-observable and using ${\mathcal{I}}_A^{\ast }(I_A)=I_A$ 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 $|0⟩⟨0|$ or $|1⟩⟨1|$, 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 Beckman.

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 CheungOai265A (thm:separation). A geometric partition or an area appearing in a supplied formula must not be treated as geometry derived from that formula.

Candidate D consists of indexed complete channels on declared comparison interfaces, optionally supplemented by preparations, intervention settings, and complete correlation records.

Reference stability means that a channel acts as $\Phi \otimes {\mathrm{id}}_R$ on an untouched reference and that a comparison is made completely, including such extensions. For every channel,

This leaves the unconditional reference marginal unchanged. It does not imply preservation of all input-reference correlations: a replacement channel satisfies (eq:reference-marginal) while destroying those correlations. Exact logical identity transport is the stronger statement

on identified input and output spaces 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 CheungEn09A (thm:main).

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 CheungEn08B (thm:main, cor:operational-limit).

All four candidates are admitted for 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 CDP,Pollock,CheungXXIIA.

For a finite-dimensional real vector space $V$, a cone is 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 $m\ge 3$, set

These are the closed-cone conventions of XXII-A CheungXXIIA.

For datum $D$, a proposed extraction must construct an event carrier $E_D$, a declared partial order ${⪯}_D$, and $(V_D,{\iota }_D,K_D)$, with ${\iota }_D$ injective and $K_D$ proper, such that

Any quotient, completion, or enlargement defining $E_D$ is part of the extraction and must be justified.

The closure in (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 $\mathrm{int}{K}_D$; it is not substituted for the closed order in (eq:faithfulness).

Each ledger entry records:

1. its mathematical statement and domain of quantification;

2. the operational data or extracted object on which it depends;

3. whether it is a declaration, an additional assumption, a derived property, an open obligation, or a documentary comparison;

4. its proof or source, where a derivation is claimed;

5. the exact conclusion for which it is used.

The status labels are

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.

A later paper must fix one actual finite list $\mathsf{H}=(H_1,\dots ,H_r)$ before invoking its extraction theorem. The template below is not an assertion that eight hypotheses have been instantiated or satisfied.

A proposed selection package must address the following entries.

1. 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 $\mathsf{OPEN}$.

2. Order formation. Name the relation extracted and prove that ${⪯}_D$ is a partial order on the claimed carrier. If it is obtained from influence by closure, restriction, or quotient, justify that operation. Default status: $\mathsf{OPEN}$.

3. H-faithful-generation. Construct the representation and prove (eq:faithfulness)–(eq:generation), with properness and a declared fixed dimension $m\ge 3$. 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: $\mathsf{OPEN}$.

4. H-homogeneity. For

   $$\mathrm{Aut}(K_D)=\{A\in \mathrm{GL}(V_D):A(K_D)=K_D\},$$

   require

   $$\forall u,v\in \mathrm{int}{K}_D\exists A\in \mathrm{Aut}(K_D):Au=v.$$

   Either prove this from the declared operational premises or mark it $\mathsf{ASSUMED}$. Default status: $\mathsf{OPEN}$.

5. H-strict-convexity. Require

   $$u,v\in K_D\setminus \{0\},u,v\text{non-proportional}⟹u+v\in \mathrm{int}{K}_D.$$

   Either prove it or mark it $\mathsf{ASSUMED}$. Default status: $\mathsf{OPEN}$.

6. 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.

7. 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: $\mathsf{OPEN}$.

8. 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.

The phrases H-faithful-generation, H-homogeneity, and H-strict-convexity are permissible 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 (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 CheungEn08A,CheungXXIIA.

Suppose that, for a particular datum $D$, the proposed extraction returns the representation of Definition def:representation, with $\dim V_D=m\ge 3$. If (eq:homogeneity) and (eq:strict-convexity) hold, then there exists $T_D\in \mathrm{GL}(V_D,{\mathbb{R}}^m)$ such that

In particular,

Proof. The classification of strictly convex homogeneous cones gives the linear isomorphism in dimension at least three ItoLourenco (Theorem 7). Apply (eq:faithfulness) and the invertibility of $T_D$. This is the conditional route recorded in En08-A CheungEn08A (thm:weaker) and XXII-A CheungXXIIA (thm:classification).

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.

For $m\ge 3$, $K={{\mathbb{R}}_{\ge 0}}^m$ is proper and homogeneous: for $u,v\in \mathrm{int}K$, the diagonal map

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.

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.

On a scheduler-comparable branch, let $\mathfrak{U}$ be a declared common operational model universe, let $C(D)$ express the class restrictions, and let

express the actual additional assumptions for a fixed proposed extraction rule $\mathfrak{E}$. The desired joint admissible set is

The two distinct requirements are

Equation (eq:nonvacuity) needs an admissible witness or an existence theorem. Equation (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 $\mathsf{S}$. 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, $\mathsf{S}(D)$ 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.

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.

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 not refuted by those countermodels. A positive theorem still requires its representation and cone-selection argument CheungEn08A,CheungXXIIA.

En08-A's ball-section and orthant examples witness different nonimplications CheungEn08A (prop:ball, prop:orthant). In the continuum,

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.

The ledger is to be assessed against XXII-A's conditions (A1)–(A6) CheungXXIIA:

1. one model and one explicitly named extracted relation;

2. genuinely additional content, including class restrictions;

3. non-vacuity and justified relevant-countermodel exclusion;

4. honest assumed or derived status of classification hypotheses;

5. faithfulness, generation, and extraction invariance;

6. declared limits, evidence, and comparison allowances.

No candidate in Section sec:candidates is marked as having passed this full audit.

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.

Any argument using only $\mathsf{S}$ retains En08-A's obstruction CheungEn08A (thm:main). 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.

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.

Suppose that $E_D$ is finite and that Definition def:representation holds. Then $K_D$ cannot be linearly isomorphic to ${\mathcal{L}}_m$ for $m\ge 3$.

Proof. Let $G=\{g_1,\dots ,g_s\}$ be the nonzero comparable differences appearing in (eq:generation). Properness excludes the case $G=\varnothing$ in the stated dimension.

Because $G$ lies in a pointed cone, zero is not in $\mathrm{conv}G$. 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 $z\ne 0$ be the closest point of the compact set $\mathrm{conv}G$ to zero. The minimizing property gives

If a sequence ${\sum }_j{\lambda }_{n,j}{g}_j$, ${\lambda }_{n,j}\ge 0$, converges, these inequalities bound all its coefficients. A convergent coefficient subsequence shows that $\mathrm{cone}G$ is closed. Thus $K_D=\mathrm{cone}G$.

Every extreme ray of $\mathrm{cone}G$ 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 ${\mathbb{R}}_{\ge 0}(1,u)$ for every $\Vert u{\Vert }_2=1$. 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 $m\ge 3$. Invertible linear maps preserve extreme rays, giving a contradiction.

The auxiliary inner product in this proof is a mathematical proof device, not an installed spacetime metric.

Pass condition. Declare an infinite operational family, a justified carrier extension, or a limiting construction, and prove that it produces (eq:faithfulness)–(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 CheungXXIIA,CheungEn08A. This does not deny conformal reconstruction from a full continuum chronological structure under additional hypotheses.

The distinction already recorded in En08-A and XXII-A is a distinction between cone generation and operational attainment CheungEn08A,CheungXXIIA.

For an elementary cone check, consider

Every generator is strictly inside ${\mathcal{L}}_m$, as is every nonzero finite conical combination of these generators. Nevertheless,

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.

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.

Suppose that complete comparison channels for every intervention at $p$ equal one channel ${\Theta }_q$, and that

uniformly over the declared interventions and contexts. Then the actual signalling strength obeys

A positive allowance does not imply ${\Delta }_D(p,q)=0$.

Proof. The triangle inequality bounds the difference between two actual intervention channels by $2\epsilon$. The half-diamond distance between channels is at most one Watrous (Chapter 3).

For the failure of exactness, on a qubit set

Both maps are channels. Homogeneity of the norm and the input $|1⟩⟨1|$ give

The upper bound is the maximum channel distance one; the input $|1⟩⟨1|$ gives the matching lower bound for $d_{\diamond }(\mathrm{id},{\mathcal{R}}_0)$. The two bit preparations satisfy

The channel is arbitrarily close to a replacement channel while still transmitting an unconditional signal.

Pass condition. Declare whether absence of an edge means the exact guard ${\Delta }_D(p,q)=0$, or a tolerance guard such as ${\Delta }_D(p,q)\le \eta$. 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 CheungOai265B (ass:statistics, thm:main).

Fix $0<p<1$ and

There are two families of one-site complete channels with the same coarse member ${\Phi }_{\ast }$ and different uniform compact-time diamond-norm limits.

Proof. This is the witness of XIX-C CheungXIXC (prop:two-limits), whose calculation is included to specify the exact extent of the test.

Choose $0<a<b<1$. For $c\in \{a,b\}$, set

and define

Every channel acts on one site and has its complete reference extension. Since ${\mathcal{R}}^2=\mathcal{R}$, for $n\ge 2$,

Set

It is a channel, ${{\mathcal{T}}_0}^c=\mathrm{id}$, and ${{\mathcal{T}}_{t+s}}^c={{\mathcal{T}}_t}^c{{\mathcal{T}}_s}^c$.

For every finite $T$,

show that $(1-c/n{)}^{\lfloor nt\rfloor }\to e^{-ct}$ uniformly on $0\le t\le T$. Because $\Vert \mathrm{id}-\mathcal{R}{\Vert }_{\diamond }\le 2$, this gives

But ${{\Phi }_1}^a={{\Phi }_1}^b={\Phi }_{\ast }$, while for $t>0$,

The trace distance between the $a$- and $b$-outputs is $|e^{-at}-e^{-bt}|>0$.

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

together with the claimed composition law and continuity CheungXIXC (def:limit). A spatial or event-cone limit needs its additional carrier and order-compatibility statements.

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 CheungXIXC (prop:interpolation). Uniqueness may instead be explicitly left outside the claim.

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.

{}1.15

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.

The completed catalogue release consists of:

1. the operational declaration and three-relation discipline;

2. the four candidate data classes with their corrected meanings;

3. the finite ledger template and its status syntax;

4. the non-vacuity, additional-content, and exclusion audits;

5. the five explicit failure tests;

6. the conditional classification reduction, with its missing operational premises left visible.

Propositions prop:signalling-cycle, prop:static, prop:finite-generation, prop:tolerance, and prop:two-limits are scoped mathematical failure witnesses. They do not jointly construct a quantum cone-selection model. Proposition prop:classification is conditional and supplies no non-vacuity theorem.

precise Stage 2 handoff. A later construction must provide:

1. one frozen quantum process class and one named data assignment, with all access and comparison interfaces stated;

2. one predeclared extraction rule and one actual finite additional hypothesis list, including every operative class restriction;

3. a proof of non-vacuity and of genuinely additional content relative to the properly compared scheduler baseline;

4. an order theorem and a faithful generated affine representation, including any required carrier extension or continuum step;

5. the relevant countermodel exclusion certificates;

6. either an operational derivation of the cone-selecting properties or an explicitly conditional theorem assuming them;

7. the extraction-invariance and evidence/error statements required by the claim.

Together these must discharge XXII-A's target conditions (T1)–(T5) and acceptance conditions (A1)–(A6) 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.

boundaries. XXII-B does not exhibit a nonempty 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 CheungXIXE.

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.

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, and conditions (T1)–(T5), (A1)–(A6). No external LaTeX reference file is required by this note.

CheungEn08A

N. Cheung. En08-A, version 1, manuscript En08-A.tex. Enter Calculus Companion, n.d. Relevant labels: thm:main, prop:ball, prop:orthant, thm:mainpositive, thm:representation, and thm:weaker. Supplied manuscript passages and scope records. Registry: https://enter.grok.me/.

CheungVIIJ

N. Cheung. VII-J, manuscript VII-J.tex. Enter Calculus Companion, n.d. Relevant supplied scope: scheduled support, operational influence, supplied causal relation, and thm:release. Documentary identification is not independent proof verification.

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.

CDP

G. Chiribella, G. M. D'Ariano, and P. Perinotti. Theoretical framework for quantum networks. Physical Review A 80 (2009), 022339. doi:\ 10.1103/PhysRevA.80.022339. Preprint: https://arxiv.org/abs/0904.4483.

Pollock

F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi. Non-Markovian quantum processes: Complete framework and efficient characterization. Physical Review A 97 (2018), 012127. doi:\ 10.1103/PhysRevA.97.012127. Preprint: https://arxiv.org/abs/1512.00589.

Beckman

D. Beckman, D. Gottesman, M. A. Nielsen, and J. Preskill. Causal and localizable quantum operations. Physical Review A 64 (2001), 052309. doi:\ 10.1103/PhysRevA.64.052309. Preprint: https://arxiv.org/abs/quant-ph/0102043.

Watrous

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

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 the classification used here.

CheungXIXC

N. Cheung. XIX-C, version 1, manuscript XIX-C.tex. Enter Calculus Companion, n.d. Relevant labels: def:limit, prop:two-limits, prop:interpolation, and thm:main. The explicit refinement witness is supplied in the source materials.

CheungEn08B

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

CheungEn09A

N. Cheung. En09-A, version 1, manuscript En09-A.tex. Enter Calculus Companion, n.d. Relevant label: thm:main. Supplied complete clock-readout and reference-extension statements; not a premise of Lorentz-cone selection.

CheungOai265A

N. Cheung. Oai265-A, manuscript Oai265-A.tex. Enter Calculus Companion, n.d. Relevant supplied label: thm:separation. Documentary distinction between static area-law records, recovery, and radiation histories.

CheungOai265B

N. Cheung. Oai265-B, version 2, manuscript Oai265-B-v2.tex. Enter Calculus Companion, n.d. Relevant labels: ass:statistics, thm:main, eq:coverage, and eq:falseaccept. Supplied complete-record finite-confidence recovery certificate.

CheungOai282

N. Cheung. Oai282-A and Oai282-B, Enter Calculus Companion manuscripts, n.d. Relevant supplied labels: Oai282-A thm:main; Oai282-B thm:main, thm:coverage, and def:wardmeaning. Documentary local Ward scopes, not operational metric extraction.

CheungOai069A

N. Cheung. Oai069-A, version 1, manuscript Oai069-A.tex. Enter Calculus Companion, n.d. Relevant label: thm:main. Supplied finite recovery and independent free-scalar consistency benchmark, not verification of catalogue Family 069.

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 boundary record.

CatalogueRecord

OpenAI. 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.