NOTE-650: Closing Bell

Contribution

Karvonen (LIT-846) and Abramsky, Barbosa, Karvonen and Mansfield (LIT-847) defined classical simulations between empirical models. This chapter asks the converse question: given an arbitrary map from the models of one scenario to the models of another, is it a classical simulation? It answers it for non-adaptive procedures, by turning the map into an empirical model on a new scenario and asking whether that model is non-contextual. The relative question about maps becomes the familiar question about objects. The chapter is also a long exposition of the sheaf framework from the resource side, and it shows that non-local games and Bell functionals are procedures.

Key insight

A transformation between two scenarios is itself a behaviour. For each context of the target, a procedure says which source context to measure and how to read the result; a family of such local instructions, consistent on overlaps, is an empirical model on the hom scenario [S, T]. It is a classical procedure exactly when those local instructions come from one global instruction set, which is to say when that model is non-contextual. A transport can therefore be contextual in the same sense that data can.

Assumptions

• Finite scenarios ⟨X, Σ, O⟩ with Σ a simplicial complex, and no-signalling models (Definition 7, compatibility).
• Single-use boxes, non-adaptive procedures. Each target measurement calls a fixed set of source measurements; adaptivity is set aside because Lemma 39 fails for it (§6.2).
• Shared classical randomness as convex mixtures of deterministic procedures (Definition 21), which lets both π and α be random.
• Convex maps. Only maps preserving convex combinations are candidates, a necessary condition (Lemma 24).

Key results

• Theorem 29. e is contextual iff there is no morphism z → e in Emp (z the model on the empty scenario); logically contextual iff none in Emp_B; strongly contextual iff none in the weak category Emp≤_B.
• Example 27. The procedure of Example 20 simulates the PR box from the triangle’s strongly contextual model (Example 8). No simulation goes back: any simplicial relation from the triangle into the CHSH scenario lands in one context, whose restriction is non-contextual.
• §3.5. A non-local game (input distribution and winning rule) is a probabilistic procedure g : S → [2]; a strategy’s winning probability is EMP(g)e. CHSH: classical value 3/4, the CHSH model 13/16, Tsirelson (2 + √2)/4, the PR box 1 (Example 30).
• Proposition 34. Every satisfiable possibilistic predicate is equivalent to one induced by a possibilistic model.
• Theorem 37. A convex map EMP(S) → EMP(T) is determined by its values on deterministic models, since every no-signalling model is an affine combination of deterministic ones (Theorem 35, from LIT-016) and convex maps preserve existing affine combinations (Lemma 36).
• Lemma 39. A function of a global assignment has a least set of arguments through which it factors, and these least sets are additive over tuples.
• Theorem 40. A map preserving convex combinations and deterministic models is induced by a deterministic procedure iff, for each target context σ, the composite E_S(X_S) → E_T(X_T) → E_T(σ) factors through E_S(τ) for a single source context τ.
• Proposition 42 and Corollary 43. Deterministic procedures S → T are the deterministic models of [S, T] satisfying g_{S,T}; probabilistic procedures give exactly the non-contextual models of ⟨[S, T], g_{S,T}⟩.
• Theorem 44 (v2). A convex map F is induced by a probabilistic procedure iff F = F_e for some e on [S, T] that is non-contextual and satisfies g_{S,T}. Membership in the image of a polytope under a linear map: a linear program.
• Theorem 46, Corollary 47. [−, −] makes Scen^g_Det, Scen^g and Scen^g_B closed categories; Scen^g_B is isomorphic to the category of possibilistic models and weak simulations (Remark 48).

Claims

id	claim	strength	support
C1	Contextuality at each strength is non-simulability from the empty model in the matching category	strong (proof)	Theorem 29
C2	Bell functionals and non-local games are procedures into a one-bit scenario	strong (definitional, with worked CHSH)	§3.5, Example 30
C3	A convex map between model sets is a classical non-adaptive procedure iff some non-contextual model of the hom scenario induces it	strong (proof, as corrected in v2)	Theorem 44 with Corollary 43, Theorem 37
C4	Compatible local procedures glue into a global one exactly when their hom-scenario model is non-contextual	moderate (argued in prose, not stated as a theorem)	§4.5
C5	Scenarios with predicates form a closed category	moderate (proof in outline)	Theorem 46, Corollary 47
C6	The characterisation extends to adaptive procedures	not supported	§6.2: Lemma 39 fails; only the analogue of Theorem 44 is claimed

Method

The relative point of view: properties of objects are recast as properties of morphisms, and then the morphisms are internalised as objects. The technical core is linear: Theorem 37 uses signed global sections to determine convex maps from deterministic models; Lemma 39 supplies canonical least supports for deterministic experiments.

Concepts

• deterministic procedure S → T: a simplicial relation π from X_T to subsets of X_S and outcome maps α_x : E_S(π(x)) → O_T,x.
• probabilistic procedure: a convex mixture of deterministic ones.
• experiment, predicate: a procedure into [n], and into [2].
• simulation e → d: a procedure f with EMP(f)e = d; possibilistic and weak simulations relax “equal” to equal supports and to inclusion.
• hom scenario [S, T]: T’s measurements and contexts, with outcomes ⟨U, α : E_S(U) → O_T,x⟩; the predicate g_{S,T} requires each context’s outcomes to use a context of S jointly.
• contextual simulation: a contextual model of [S, T] (§6.7).

Connections

Revises the morphisms of LIT-846 (stochastic outcome maps only) and of LIT-847 (simplicial maps with adaptive protocols) into mixtures of deterministic procedures over simplicial relations (Remark 28). Uses Abramsky and Brandenburger (LIT-016) for signed global sections (Theorem 35). The CHSH game, Specker’s triangle, and Boole’s conditions of possible experience frame the exposition. The prologue cites Wang, Sadrzadeh, Abramsky and Cervantes (LIT-842) as natural language among the domains where the same structure occurs.

Bearing on the record

• CLAIM-100 (directed transport between scenarios with changing covers is the open problem). This is the closest existing answer. A94 asks “under what conditions can a directed stochastic transport preserve … global compatibility while changing its observational cover”; this chapter characterises, for no-signalling models and non-adaptive procedures, exactly which maps between the models of two arbitrary scenarios are classical transports, and every classical transport preserves non-contextuality (Theorem 29 composed with any procedure). CLAIM-100’s defeated_if asks for a framework that “characterizes stochastic maps between empirical models with different covers, including when they preserve decision-relevant information and global compatibility”. The global-compatibility part is met here; the decision-relevant-information part is not, and signalling data are excluded. My reading: CLAIM-100 should stay Proposed but narrow its open problem to those two parts, and its “Partial prior art” should name this chapter and its two predecessors before Gogioso and Pinzani.
• The manuscript’s §6 gap (“cover-changing or incompatible maps demand separate analysis”). §4.5 gives the analysis for the case the manuscript calls incompatible: a family of local transport kernels compatible on overlaps that does not come from one global kernel is a contextual model of [S, T]. That is CLAIM-070’s limit (“false if one assumes only local kernels with no globally compatible realization”) with a name and a test.
• CLAIM-105 (chains may become more or less contextual). A contextual simulation is the only kind of transport that could add contextuality; the chapter names the notion and develops none of it.
• CLAIM-001 (the owner’s answer: conditional diffusion already covers directed transport). This chapter is prior art from the sheaf side, closer to A94’s question than the diffusion literature, because it decides gluing exactly rather than combining conditional scores.
• Produces THEORY-174, with LIT-846 and LIT-847.
• No instruction for machine-learning practice; nothing for the anthology.

Limitations

• No-signalling models only; nothing on signalling or Contextuality-by-Default data.
• Non-adaptive procedures only for the sharp result (Theorem 40); the adaptive analogue of Theorem 44 is asserted in §6.2 without proof.
• The published chapter states the uncorrected Theorem 44; v2 is the version of reference by the authors’ own note.
• The gluing statement of §4.5 is argued in prose.
• Contextual simulations are named, not studied.

Open questions

• Adaptive versions of Theorem 40 and of the closed structure; a monoidal closed structure, which seems to need directed adaptive tensors (§6.3).
• What contextual simulations can do, and whether they match maps d ⊗ c → e with c contextual (§6.7).
• A Stone-type duality with generalised partial Boolean algebras, with [2] as dualising object (§6.4).