Contribution
The paper gives one framework for contextuality under three kinds of causal structure: a causal order in the physical background, an order in which an experiment’s measurements are made, and adaptive choice of measurement in measurement-based computation. Earlier causal refinements (Mansfield’s ordered measurements, Gogioso and Pinzani’s ordered Bell sites, LIT-813) handled one order at a time over Bell or Leggett–Garg scenarios. Here the enabling relation can depend on outcomes, the scenario need not be Bell-type, and the roles of Nature (causal background) and Experimenter (adaptivity) are separated. It also shows that once causality enters, the deterministic layer is no longer a sheaf.
Key insight
Change only the sheaf of events. Replace “a section assigns each measurement an outcome” with “a strategy assigns each measurement an outcome given the history in which it is performed”. Every later definition (empirical model as a compatible family of distributions, noncontextuality as a global section, the classical polytope) is then the old one applied to the new presheaf. Signalling from the causal past becomes part of what a classical model may do, so it no longer counts as contextuality.
Assumptions
- Finite sets of measurements and outcomes for the polytope and the fixed-point description of histories.
- Each measurement occurs at most once in a history (histories are consistent event sets).
- Nature’s strategies are deterministic and total; probabilities enter only by composing with the distribution monad D_R over a semiring R (non-negative reals for probability).
- The cover still decides which measurements can be performed together; causal structure decides in what order and on what condition (§6a).
Key results
- Histories H(M): the least set of consistent event sets containing ∅ and closed under s ⊳ x (x not yet performed, some t ⊆ s with t ⊢ x), for every outcome of x. For finite X it is reached at a finite stage.
- Proposition 4.1 (monotonicity). A strategy’s outcome for x is fixed at the minimal histories enabling x, and so the same along any chain above them. It may still differ between incomparable causal pasts.
- Proposition 4.2. Strategies are maximal: σ ⊆ τ implies σ = τ.
- Propositions 4.3–4.5. Restriction σ|_U = σ ∩ H(M_U) is a strategy, and strategies form a presheaf Γ on P(X).
- §5. An empirical model is a compatible family in D_R Γ over a cover; it is causally noncontextual when it extends to a global distribution on Γ(X). Noncontextual models are the convex hull of deterministic strategies, so classicality is a linear program, and a causal contextual fraction is suggested, not defined.
- §6a. If ∅ enables everything, Γ is in bijection with the event sheaf E and the theory is LIT-016’s.
- §6b. For the two-site chain ω₁ < ω₂ with the Bell cover, a strategy is a pair of functions I_A → O_A and I_A × I_B → O_B, matching Gogioso and Pinzani’s sheaf of sections. The general GP case is asserted, with proof deferred.
- Proposition 7.1. The union of a compatible family of strategies is deterministic.
- Example 7.2 (no gluing). X = {x, y, z}, ∅ ⊢ x, ∅ ⊢ y, {(x,0)} ⊢ z, {(y,0)} ⊢ z, cover {x, z}, {y, z}. The strategies answering z = 0 after x = 0 and z = 1 after y = 0 are compatible (z is never enabled within {z}) and each is total, but any global strategy must contain {(x,0),(y,0)} and then answer z both ways.
- Example 7.3 (non-unique gluing). With {(x,0),(y,0)} ⊢ z, z is enabled in neither member of the cover, and the gluing leaves its value free. The authors call this pathological.
- Note added in proof. For covers of causally secured sets the sheaf property holds; details are deferred to a sequel.
- Proposition 8.1 and E-strategies. Maximal histories are those with no accessible measurement. An Experimenter strategy is down-closed and co-total, accepting every outcome of each measurement it allows, and ⟨σ | τ⟩ = σ ∩ τ plays it against a Nature strategy. The Anders–Browne OR gate is the E-strategy that measures C_{a⊕b} after A_a and B_b on GHZ; the GHZ parities XXX = +1 and XYY = YXY = YYX = −1 make the XOR of the three outcomes equal a OR b. Chaining two gates needs a choice of measurement that depends on outcomes, which is full adaptivity.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Replacing the event sheaf by the presheaf of Nature’s strategies extends sheaf-theoretic contextuality to causal scenarios without changing its definitions | strong (construction) | §§4–5, Propositions 4.3–4.5 |
| C2 | The flat theory is the case where everything is initially enabled | strong (proof sketch) | §6a |
| C3 | Gogioso and Pinzani’s causal Bell theory is a special case | moderate | §6b: two-party case worked; the general case deferred |
| C4 | Compatible deterministic strategies need not glue, and need not glue uniquely | strong (counterexamples) | Examples 7.2, 7.3, checked here |
| C5 | On causally secured covers the strategy presheaf is a sheaf | not supported here | note added in proof; deferred to a sequel |
| C6 | Adaptive MBQC is an Experimenter strategy played against a flat model | strong for the one-gate case | §§3b, 8a |
| C7 | The framework also covers Leggett–Garg and classical causal networks | not supported here | §9: deferred to an extended version |
Concepts
- causal measurement scenario M = (X, O, ⊢): measurements, outcomes, and an enabling relation from consistent event sets to measurements.
- history: a consistent set of events reachable from ∅ by performing enabled measurements; a play of the game.
- N-strategy: a down-closed, deterministic and total set of histories; a hidden variable that may depend on the causal past.
- E-strategy: a down-closed, co-total set of histories; an adaptive measurement protocol.
- causally secured context: one that can occur on its own, its measurements’ causal prerequisites included (used informally in §3a and the note added in proof).
- causal contextuality: an empirical model on D_R Γ with no global section.
Connections
The base is Abramsky and Brandenburger (LIT-016). The Bell-scenario special case is Gogioso and Pinzani 2021 (LIT-813), cited as [24]; that paper was withdrawn by its authors in April 2024, after this one appeared, and replaced by their trilogy (LIT-800, LIT-808, LIT-788). This paper’s claimed equivalence with “the sheaf of sections” of [24] stands on the two-party example only, and the withdrawal concerns [24]‘s locale of inputs, not that example. The trilogy’s corrected treatment (LIT-808) also finds that causal functions are not always a sheaf. The contextual fraction it proposes to generalise is LIT-265; the cohomology it lists among the framework’s tools is LIT-277 and LIT-278. Contextuality-by-Default (LIT-777) is contrasted, not used: CbD treats signalling as uncharacterised, while this paper declares the causal background and allows signalling only from the causal past.
Bearing on the record
- THEORY-171 (context-dependent causal constraints make deterministic causal assignments fail to glue). Example 7.2 is a second, independent instance in a different formalism: the measurement z has two incomparable causal pasts, as event B does in LIT-808’s Θ₃, and two locally causal answers cannot be joined. It also adds a case THEORY-171 does not state, non-unique gluing (Example 7.3), which LIT-808’s causal functions cannot show, being separated. The two remedies differ: Gogioso and Pinzani restrict the space (tightness), this paper the cover (causally secured sets, unproved here). THEORY-171 could cite it as further support.
- CLAIM-038 (“a further kind of non-gluing”). The same point: here the obstruction is in the causal structure of the contexts, with no probabilities involved.
- CLAIM-037 (formal contextuality needs Contextuality-by-Default when marginals shift with context). This paper is a second route. When the shift can be attributed to a declared causal order, signalling from the causal past is part of the classical model, and the sheaf criterion applies to the relaxed no-signalling conditions. CbD is needed when the signalling cannot be so characterised, which is the paper’s own description of the difference. For language, the subject-to-verb influence that makes Wang et al.’s corpus models signalling (LIT-842) is the kind of thing a declared order could absorb. Wang and Sadrzadeh’s causal analysis of ambiguous phrases (EPTCS 394, not held) takes the Gogioso–Pinzani route.
- CLAIM-100 (extending sheaf contextuality to directed transport
between scenarios whose covers change is the open problem). Partial
prior art, like LIT-808’s: an E-strategy makes which context is
measured depend on earlier outcomes, and playing it against a model
yields a model. But both stay inside one scenario; the paper builds no
map between empirical models on different scenarios. The works it
lists on simulations between contextual systems (Karvonen 2019,
Abramsky, Barbosa, Karvonen and Mansfield 2019, Barbosa, Karvonen and
Mansfield 2021) define maps between empirical models on different
scenarios. They are the nearer test of CLAIM-100’s
defeated_if, and are not held. - No instruction for machine-learning practice; nothing for the anthology.
Limitations
- A short paper: the general equivalence with Gogioso and Pinzani’s theory, the Leggett–Garg and causal-network cases, the sheaf property for good covers and a Vorob’ev-type theorem are all deferred.
- No new quantum example of causal contextuality is computed; the examples illustrate the definitions.
- The causal contextual fraction and a logical characterisation of the polytope’s facets are posed, not given.
- “Causally secured” is used without a formal definition.
- Its cited Bell-scenario predecessor (LIT-813) was later withdrawn, so the special-case claim of §6b now points at a superseded paper.
Open questions
- Question 7.3: which covers make Γ a sheaf? The authors report a positive answer for causally secured covers, unpublished here.
- How the cover condition relates to Gogioso and Pinzani’s tightness (LIT-800, LIT-808): whether one implies the other.
- Whether Experimenter adaptivity can ever create contextuality that the flat model lacks. In the Anders–Browne case the model Nature plays is the ordinary GHZ model.