XXII-A: Freeze the precise claim and the gap Stage 0 of a limited kinematic emergence programme

This note fixes the meaning and evidentiary boundary of a prospective quantum-operational Lorentz-cone selection theorem. It separates scheduled ancestry, intervention-based operational influence, and an independently supplied geometric causal relation. It records the scheduler-consistency obstruction of Cheung's En08-A and the conditional homogeneous strictly convex cone classification used there. The latter is a classification theorem, not a derivation of its hypotheses from quantum data. A precise future target is formulated in terms of a nonempty, named process class, predeclared additional hypotheses, and a faithful generated affine cone representation. Finite event orders, continuum completions, cone boundaries, and conformal metric information are distinguished. No process class and additional hypothesis package satisfying the target are constructed here. The completed deliverable is the Stage 0 specification and obstruction audit, not a positive selection theorem.

The claim under construction is not

Cheung's En08-A explicitly obstructs that implication CheungEn08A (thm:main). The narrower claim is that there exists a named class of quantum operational data and a finite, predeclared list of genuinely additional hypotheses under which an operationally extracted cone is linearly isomorphic to a Lorentz cone.

This is an open target. XXII-A neither establishes its existential claim nor supplies the missing process class and hypotheses. In particular, a target statement must not be promoted to a theorem merely because its conclusion has been written precisely.

Let $V$ be a finite-dimensional real vector space. A cone $K\subset V$ is proper if it is closed, convex, pointed, $K\cap (-K)=\{0\}$, and has nonempty interior in $V$. Its translation-invariant order is

For $m\ge 3$, the closed Lorentz cone is

The statement that $K$ is linearly isomorphic to a Lorentz cone means that an invertible linear map $T:V\to {\mathbb{R}}^m$ satisfies $T(K)={\mathcal{L}}_m$.

A finite event poset is not itself a vector-space cone. Accordingly, “the extracted order is Lorentzian” will mean the precise representation statement in Section sec:target, not a linear bijection from finitely many events onto ${\mathcal{L}}_m$.

The present convention uses a closed cone order. An open chronological relation would instead use $\mathrm{int}K$. Passing between these two relations requires an explicitly stated closure or boundary argument; it is not a notational identification.

operational languages. Process matrices, quantum combs, and multi-time process tensors provide different operational frameworks OCB,CDP,Pollock. They are possible languages for Stage 1, not interchangeable models and not hypotheses of a selection theorem in this note. A future choice must specify:

1. the finite party or slot set and the input/output Hilbert spaces;

2. the admissible preparations, instruments, interventions, and retained outputs;

3. whether the datum consists of exact complete maps, exact intervention probabilities, or finite statistical records;

4. the operational equivalence and relabelling conventions;

5. which relation is being extracted.

Finite parties do not necessarily mean finitely many admissible interventions. Conversely, finitely many observed event comparisons do not supply a continuum carrier.

convention. The Cheung references below identify unpublished manuscripts by their supplied identifiers, versions where available, and internal labels. A cited restatement is not an independent verification of the predecessor proof. En08-A is used for the obstruction; the cone classification is attributed directly to Ito and Lourenço. Other series references are documentary comparisons unless expressly identified otherwise.

The separation adopted from VII-J is a distinction between mathematical objects and their inputs, not a claim that they can never coincide in a particular certified model CheungVIIJ.

A complete dependency graph includes every quantum input, classical control, guard, routing decision, and retained record actually used by the protocol. For distinct operation events $p,q$, write $p{R}_{\mathrm{sch}}q$ if there is a directed dependency path from $p$ to $q$. Its reflexive closure is the scheduled event order.

An edge records an operation that the protocol calls. It does not assert that the operation is sensitive to every input it receives.

Fix an operational process and an accessible output at $q$. Two admissible interventions $\mathcal{I},\mathcal{J}$ at $p$ must have common comparison interfaces; all other operations and preparations are held fixed. Let ${\Phi }_{q\mid \mathcal{I}}$ and ${\Phi }_{q\mid \mathcal{J}}$ be the resulting complete output channels. Set

Write $p{R}_{\inf }q$ when ${\Delta }_{p\to q}>0$.

For an interface containing only classical records, the corresponding definition is a change in the unconditional output marginal under the intervention. In a process-matrix formulation the admissible instrument insertions and signalling comparison must be specified within that formulation.

“Complete” means that retained quantum systems and classical histories are included and that equality is stable under an untouched reference. Instrument outcomes are either retained or summed according to a declared deterministic interface. Normalized postselected branches are not substituted for complete channels. Shared entanglement and conditional steering do not by themselves constitute unconditional signalling Beckman,CheungVIIJ.

Direct operational influence is a directed relation, but it is not assumed here to be transitive. If a later extraction uses its transitive closure, an acyclic restriction, or a quotient, that operation must be named. The relation being represented must then be the resulting relation, not an unannounced replacement of $R_{\inf }$.

Let $X$ be a separately declared spatial carrier with distance $d_{C^{\ast }}$, including any stipulated metric-length completion. On $\mathbb{R}\times X$, define

A comparison with operational events additionally requires a declared location map $\ell :E\to \mathbb{R}\times X$. The carrier, distance, calibration, and completion are inputs to this comparison.

Consider a finite acyclic network of completely positive trace-preserving operations, with complete dependency bookkeeping and intervention-independent initial preparation. Then

for distinct events, and the inclusion can be strict.

Neither relation is automatically contained in the pullback of $R_{\sup }$ along an independently chosen location map. If every dependency edge is certified to respect $R_{\sup }$, then scheduled ancestry, and hence operational influence, is contained in that pullback.

These statements express the separation adopted from VII-J; they do not identify the three relations.

Proof. Pull an output observable backwards through the circuit. Operations outside its dependency ancestors act on factors carrying the identity observable and disappear because the adjoint of a trace-preserving map preserves the identity. Thus an intervention outside the ancestor set cannot change the output channel. The argument remains valid after tensoring with an untouched reference. This is the finite-channel content of VII-J's scheduled-support and no-signalling statements CheungVIIJ (thm:schedule, thm:nosignal).

For strictness, let a scheduled wire from $p$ enter an operation at $q$ whose observed output is the replacement channel

Every trace-preserving intervention upstream gives ${\mathcal{R}}_{\sigma }\circ \mathcal{I}={\mathcal{R}}_{\sigma }$. There is scheduled ancestry but no influence on that output.

An independent location assignment need not respect the schedule. For example, assign the endpoints of a scheduled identity transmission the locations $(1,x)$ and $(0,x)$, in that order. Its operational transmission is unchanged, while (eq:supplied) fails. Conversely, certified causal containment of every edge propagates along a path by transitivity of the supplied relation. This is the separate embedding hypothesis of VII-J CheungVIIJ (thm:physical).

The theorem applies to the stated definite-order channel networks. It is not an assertion that every process matrix possesses such a dependency graph; the process-matrix framework does not generally assume a predefined global causal order OCB.

Write $\mathsf{S}$ for the package consisting of:

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.

Condition (S5) is an extra property of the countermodels. It must not be read as a general identity between scheduled ancestry and influence.

The package $\mathsf{S}$ does not force a generated reachability cone to be quadratic or linearly isomorphic to a Lorentz cone.

En08-A provides local record networks with these consistency properties and, in the statement of its thm:main, additionally: bounded degree, finite support, finitely many local reduction shapes, no intervention-induced change outside the dependency future, local point distinguishability, and finite pasts for finite collections of records.

Its countermodels permit a Euclidean metric ball as a one-step spatial section on a discrete slice and permit a generated positive-orthant cone. The consistency assumptions also do not imply interpolation, a manifold topology, or causal availability rules for an otherwise unspecified type system.

Proof. This is the obstruction recorded in En08-A CheungEn08A (thm:main). The source construction is thm:network; the two geometric realizations are prop:ball and prop:orthant. The source also uses a two-event chain to witness failure of interpolation and separately discusses the missing typing rules. XXII-A does not reconstruct those networks or claim an independent verification of their proofs.

The two countermodels witness different failures of inference. A Euclidean-ball section is compatible with scheduler consistency. It is not, merely by being a ball, a non-Lorentzian continuum cone: indeed,

A discrete one-step section and its generated reachability structure must therefore be distinguished from this full continuum homogenization.

The orthant supplies the direct non-Lorentzian cone obstruction in dimension at least three. The statement “ball and orthant are countermodels” is shorthand for the separately specified nonimplications in En08-A, not a claim that every cone over a Euclidean ball is non-Lorentzian CheungEn08A (prop:ball, prop:orthant).

The obstruction is relative to its hypotheses. A future quantum process class may exclude a countermodel by a proved class restriction or by a genuinely additional hypothesis. Such an exclusion must be shown; it does not follow from the word “quantum.” Conversely, a countermodel outside a newly restricted class does not refute a theorem about that class.

En08-A's positive representation statements are conditional on a faithful generated affine cone representation and further cone properties CheungEn08A (thm:mainpositive, thm:representation). They do not construct that representation from a raw event order.

The following distinctions are retained from En08-A; their predecessor arguments are not re-proved here CheungEn08A.

1. A boundary ray of a generated, closed cone need not contain the difference of any attained operational event pair. Boundary generation or limiting approximation is not an attained null protocol.

2. A small diamond distance from a non-signalling channel is not exact non-signalling, and hence is not exact causal containment. A positive error allowance permits a nonzero influence.

3. A finite ordered event sample does not determine an unrestricted spacetime metric or a unique continuum realization.

4. Chronology does not determine the full metric, including its conformal scale.

The last warning is not a denial of causal reconstruction theorems. With an entire continuum spacetime and suitable causal hypotheses, chronological information can determine conformal structure HKM,Malament. Those continuum assumptions are not furnished by a finite operational event sample. In particular, “chronology alone does not determine the metric” must not be strengthened to “chronology can never determine a conformal class.”

For a proper cone $K\subset V$, set

The cone is homogeneous if $\mathrm{Aut}(K)$ acts transitively on $\mathrm{int}K$. It is strictly convex in the cone sense if, for nonzero, non-proportional $u,v\in K$,

Equivalently, its boundary has no line segment joining points on different nonzero rays.

Let $K\subset V$ be a homogeneous, strictly convex proper cone, with $\dim V=m\ge 3$. Then there is an invertible linear map $T:V\to {\mathbb{R}}^m$ such that

This is the classification consequence recorded as En08-A thm:weaker. Homogeneity and strict convexity are additional hypotheses, not consequences of $\mathsf{S}$.

Proof. Apply Ito and Lourenço's classification of strictly convex homogeneous cones ItoLourenco (Theorem 7). Their lower-dimensional alternatives are excluded by $\dim V\ge 3$. The remaining alternative is a Lorentz cone. The classification is not re-proved here; its operational use is the conditional statement recorded in En08-A CheungEn08A (thm:weaker).

Self-duality and irreducibility need not be added to the hypotheses of Theorem thm:classification. Their appearance in another representation theorem does not make them part of this weaker classification route.

If a later extraction supplies the hypotheses of Theorem thm:classification, the selected cone determines the Lorentzian quadratic form

up to positive multiplication. This concerns the full cone, not merely the comparisons among a sparse set of embedded events CheungEn08A (prop:quadraticunique).

For completeness, the elementary uniqueness calculation is as follows. In Lorentz coordinates write another quadratic form with the same null boundary and positive sign on the cone interior as

Its vanishing on $(1,u)$, $\Vert u{\Vert }_2=1$, and on $(1,-u)$ gives $b=0$ and $u^{\mathsf{T}}Cu=-a$ for every unit $u$. Thus $C=-aI$, and positivity on $(1,0)$ gives $a>0$.

The ambiguity here is a positive constant on one affine carrier. A future field of cones on a manifold would require separate arguments and could leave a position-dependent conformal factor. No metric field or manifold is constructed in XXII-A.

A scheduler change is not automatically an element of $\mathrm{Aut}(K)$. Commutation of independent operations concerns legal execution interleavings; homogeneity concerns transitivity of a linear group on the interior of an extracted cone. Renaming the former does not prove the latter.

Let $E_D$ be the declared event carrier associated with operational datum $D$, and let ${⪯}_D$ be a declared partial order extracted from scheduled ancestry or operational influence.

A faithful generated affine cone representation consists of $(V_D,{\iota }_D,K_D)$, where $V_D$ is a finite-dimensional real vector space, ${\iota }_D:E_D\to V_D$ is injective, and $K_D$ is a proper cone satisfying

If the carrier is obtained by an operational quotient or completion, that construction and the relation on the resulting carrier are part of the extraction.

Equation (eq:faithful) prevents an arbitrary supercone from being called an extracted causal cone. Equation (eq:generated) prevents an independently appended Lorentz cone from supplying the desired answer. Neither equation follows merely from drawing a dependency graph.

-generation check. If $E_D$ is finite, the set of generators in (eq:generated) is finite. Its conical hull is a closed polyhedral cone. By contrast, ${\mathcal{L}}_m$, $m\ge 3$, has the infinitely many extreme rays

Invertible linear maps preserve extreme rays. Therefore a finite event carrier with exactly the generation rule (eq:generated) cannot supply the proposed Lorentz cone.

A later theorem must consequently declare an infinite operational family, an operationally justified extension, or a limiting construction. This is not a requirement of infinitely many parties. It is a requirement that the claimed continuum cone not be substituted for the finite event-difference cone without a connecting construction. A finite tomographically complete list can determine a declared finite-dimensional process; that fact is different from determining a continuum event carrier CDP,Pollock.

Find one named quantum process class $\mathcal{P}$, one named data assignment $\mathcal{D}(\mathcal{P})$, a finite list

and one predeclared extraction rule $\mathfrak{E}$, such that:

1. The admissible data set

   $${\mathcal{D}}_{\mathsf{H}}=\{D\in \mathcal{D}(\mathcal{P}):H_1(D),\dots ,H_r(D)\}$$

   is nonempty.

2. For every $D\in {\mathcal{D}}_{\mathsf{H}}$, the rule $\mathfrak{E}$ constructs a specified order ${⪯}_D$ and a faithful generated affine representation $(V_D,{\iota }_D,K_D)$ as in Definition def:representation.

3. On a declared fixed dimension $m\ge 3$, an invertible linear map $T_D:V_D\to {\mathbb{R}}^m$ exists with

   $$T_D(K_D)={\mathcal{L}}_m.$$

   Consequently the represented event comparisons satisfy

   $$p{⪯}_{D}q⟺T_D({\iota }_D(q)-{\iota }_D(p))\in {\mathcal{L}}_m.$$

4. The extraction uses only the specified operational data, their declared non-geometric interface, and $\mathsf{H}$. No spacetime metric, geometric causal relation, physical light cone, or invariant speed is installed as input.

5. The additional class restrictions and hypotheses are not supplied merely by $\mathsf{S}$. Their status is stated before the extraction is applied.

A comparison with a supplied VII-J-type causal relation is a subsequent embedding problem, not part of the selection.

Research target target:selection is a specification of the result to be established in later stages. It has no proof in this note.

A proposed solution must satisfy the following acceptance conditions.

1. One model and one relation. Freeze the process class, admissible data, intervention access, equivalences, and extraction rule. State whether the represented relation is scheduled ancestry, direct influence, or a declared order constructed from influence. A state-positivity cone is not an operational event cone without an order-identification theorem.

2. Genuine additional content. Include class restrictions in the hypothesis ledger. In a common operational language, their conjunction with $\mathsf{H}$ must contain content not forced by $\mathsf{S}$. This does not require every individual $H_j$ to be independent of $\mathsf{S}$, but it excludes a package that merely renames it.

3. Non-vacuity and countermodel exclusion. Exhibit admissible data satisfying the complete package. For each relevant En08-A countermodel, identify and justify the failed class restriction or hypothesis. An unchanged countermodel satisfying all proposed premises defeats the proposed conclusion.

4. Honest use of classification. Homogeneity and strict convexity may be proved from operational data, or they may be explicitly assumed as properties of the extracted cone. In the latter case, the result is conditional on those assumptions. The classification theorem does not discharge their operational derivation.

5. Generation, faithfulness, and invariance. Prove (eq:faithful)–(eq:generated), including the absence of spurious comparisons introduced by convexification or closure. The same declared data, up to the permitted relabellings, must give affinely isomorphic representations. The input must not prescribe the cone whose extraction is claimed.

6. Limits and evidence. If a completion or refinement is used, give its comparison maps, topology, compatibility, and convergence theorem. A finite list of universally quantified structural hypotheses is not automatically a finite empirical certificate. Statistical records require their own coverage and implementation allowances.

Externally supplied correlations are permissible operational inputs, but their preparation and access model must be declared. A static entanglement table does not by itself provide intervention access or a temporal influence relation. Likewise, a quantum comb's declared slot ordering must not be misreported as an ordering derived from its outcome probabilities CDP,OCB,Beckman.

XIX-C supplies two specific cautions for any later continuum step CheungXIXC (prop:two-limits, prop:interpolation):

• two refinement families can share one coarse complete channel, obey one-site scheduled locality, and converge to different semigroups;

• finite sampling constraints can leave interpolation ambiguity.

The first counterexample chooses different missing refinement data; it does not assume an additional projective-consistency condition. Such consistency would itself be an extra hypothesis. These warnings do not exclude a unique continuum limit after suitable additional data and assumptions are supplied.

The documentary roles of the other named components are fixed below. None substitutes for Research target target:selection.

{}1.16

This ledger records scopes, not proof-transfer paths. In particular, null-boundary approximation in En08-B cannot be used to derive the metric or speed with which its targets were defined. Similarly, an independent geometric construction does not become an operational extraction by changing the name of its input.

The completed Stage 0 release consists of:

1. the separation in Theorem thm:separation;

2. the cited obstruction in Theorem thm:obstruction;

3. the conditional classification in Theorem thm:classification;

4. the unproved but precise Research target target:selection and its acceptance audit.

There is no fourth theorem asserting positive selection.

For the later programme, the intended obligations are:

Stage 1.

Name and freeze the operational process class, its additional data, the relation to be extracted, and the finite hypothesis ledger. Establish non-vacuity and explain the relevant countermodel exclusions.

Stage 2.

Construct the faithful generated carrier and order representation, including any necessary quotient, extension, or continuum limit. Prove the needed compatibility and convergence statements.

Stage 3.

Prove the cone-selecting properties and apply an appropriate classification, or state a explicitly conditional selection theorem. Any comparison with a supplied geometric causal relation follows as a separately certified embedding.

XXII-A does not derive homogeneity or strict convexity from quantum data; construct an attained null protocol; identify scheduled ancestry with geometric causality; or construct a metric field. It proves no Einstein dynamics, gravity-generated entanglement, holographic reconstruction, Page curve, or quantum evolution without an external time.

The note supplies no result changing the separate series obligations O.1, XV-B, XVI-N, Route B, H2, I3, or J2. No quantum coefficient is admitted. No predecessor proof is represented as machine-checked, and no native programme execution or successful pdfLaTeX build is asserted. These are scope boundaries, not universal impossibility claims CheungXIXE.

precise remaining gap. Stage 1 must supply {the additional quantum operational data and the finite hypothesis package, on one nonempty frozen process class, that go beyond the scheduler package $\mathsf{S}$ and are sufficient for a faithful generated Lorentz-cone selection}. Until that package and its connecting proofs are supplied, scheduler consistency remains obstructed as a selection principle, and homogeneity-plus-strict-convexity remains a conditional classification route rather than an operational emergence theorem.

CheungEn08A

N. Cheung. En08-A, version 1, manuscript En08-A.tex. Enter Calculus Companion, n.d. Relevant labels: thm:main, thm:network, prop:ball, prop:orthant, thm:mainpositive, thm:representation, thm:weaker, and prop:quadraticunique. Manuscript identification and passages supplied with this note. Registry: https://enter.grok.me/.

CheungVIIJ

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

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.

OCB

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

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.

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.

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.

HKM

S. W. Hawking, A. R. King, and P. J. McCarthy. A new topology for curved space-time which incorporates the causal, differential, and conformal structures. Journal of Mathematical Physics 17 (1976), 174–181. doi:\ 10.1063/1.522874.

Malament

D. B. Malament. The class of continuous timelike curves determines the topology of spacetime. Journal of Mathematical Physics 18 (1977), 1399–1404. doi:\ 10.1063/1.523436.

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. Supplied passages concern finite certification and continuum non-determination.

CheungEn08B

N. Cheung. En08-B, version 1, manuscript En08-B.tex. Enter Calculus Companion, n.d. Relevant labels: thm:main, cor:operational-limit, and def:comparison. Supplied passages concern resource-controlled approximation of a prescribed null boundary.

CheungVIII

N. Cheung. VIII-A, version 3; VIII-B, version 3; and VIII-C. Enter Calculus Companion manuscripts, n.d. Documentary conical-family and fixed-deficit comparison records supplied with the XXII-A materials; not mathematical premises of the selection target.

CheungEm02E

N. Cheung. Em02-E. Enter Calculus Companion manuscript, n.d. Classical relational-lapse and incomplete quantum-audit scope record as supplied with the XXII-A materials. No unsupplied theorem or proof is imported here.

CheungXIXA

N. Cheung. XIX-A, version 1, manuscript XIX-A.tex. Enter Calculus Companion, n.d. Relevant label: thm:main. Supplied statement concerns clock separation before coordinate identification.

CheungXIXE

N. Cheung. XIX-E, version 1, manuscript XIX-E.tex. Enter Calculus Companion, n.d. Relevant passage: “What remains outside the release.” Documentary boundary record; no claim of independent proof verification.

CatalogueRecord

OpenAI. OpenAI mathematics catalogue, n.d. Supplied documentary record, including family identifiers 265, 211, 280, 069, and 282. The catalogue claims are neither verified nor used as theorem premises in XXII-A.