Source
Gogioso and Pinzani, The Topology of Causality (2023), LIT-808, Section 2.3: Definition 2.22, Theorem 2.30, Definition 2.23, Proposition 2.33 and Theorem 2.34, with the worked example on the space Θ₃, as read in NOTE-612. The spaces, tip events and tightness are those of The Combinatorics of Causality, LIT-800 (Definitions 3.13–3.16, Proposition 3.32, Theorem 3.33), as read in NOTE-635.
What was actually shown
In Abramsky and Brandenburger’s framework (LIT-016), the deterministic local assignments always form a sheaf. Contextuality comes only from probabilistic (or possibilistic) data that admits no global section (THEORY-012). Gogioso and Pinzani replace the no-signalling scenario with a space of input histories Θ, which can encode definite, indefinite and input-dependent causal order. They give Θ the lowerset topology. A causal function on an open set λ assigns outputs to the tip events of λ’s histories. Theorem 2.30 shows these form a separated presheaf, in which a compatible family glues iff its compatible join is causal.
The new phenomenon comes from non-tight spaces, where one event can be the tip of two different histories under a common extension. The worked case is Θ₃, the meet of the spaces of total(A, B) ∨ discrete(C) and discrete(A) ∨ total(C, B). Event B has causal past {A, B} in one and {B, C} in the other. Take λ = {A:a, B:b}↓ and λ′ = {B:b, C:c}↓. These are disjoint opens, so any pair of sections is compatible. Put f(A:a, B:b)_B = 1 on λ and f′(B:b, C:c)_B = 0 on λ′. Each is causal in its context, but the join gives B two different outputs on {A:a, B:b, C:c}, which Θ₃’s causal functions forbid. So there is no gluing, and the presheaf is not a sheaf. This example was followed step by step in the reading.
Theorem 2.34, which the record reports and has not checked (QUESTION-143), generalises it. With at least two outputs per event, the presheaf is a sheaf iff Θ has no solipsistic contextuality witness: two tip-equivalent histories whose minimal common extensions are not all histories (Proposition 2.33). Tight spaces are always sheaves. Every space induced by a causal order is tight (LIT-800, Proposition 3.32). The effect therefore needs causal constraints that differ between contexts, which no single order supplies. By Theorem 2.48, a witness yields a deterministic empirical model on the fully solipsistic cover that extends to no standard empirical model. The introduction states that 1767 of the 2644 causally complete spaces on 3 events with binary inputs have this property. That figure was not checked.
What could have come out otherwise: the sheaf property could have survived the passage from no-signalling to arbitrary causal spaces, as the authors’ withdrawn 2021 paper (LIT-813) asserted for definite orders. For order-induced (tight) spaces it does survive, and for non-tight ones it fails.
What this does not say
- Not that quantum or any physical data show it. Every example is a combinatorial space and a deterministic assignment. No experiment, and no quantum process, is shown to produce a solipsistically contextual model. The examples in LIT-788 are all on the standard cover, where a deterministic model always glues (Corollary 2.50).
- Not that it is the same as Kochen–Specker contextuality or non-locality. Those are failures of probabilistic data to come from a global distribution over a sheaf of deterministic sections. This is a failure of the deterministic sections themselves, one level down.
- Not that it arises for a fixed causal order. Order-induced spaces, definite or indefinite, are tight and give sheaves. The effect needs contexts with incompatible causal pasts for one event, as in meets of order-induced spaces with non-nested pasts (LIT-800, Theorem 3.33).
- Not that the general theorem is checked here. The proof of Theorem 2.34 was not read, and the paper is an unrefereed preprint. The characterisation is QUESTION-143.
- Not a statement about language, meaning or translation. The analogy to contexts of interpretation with incompatible dependency structure is the reader’s to make and is not drawn by the paper.
Note of 2026-10-19: the existence claim, re-derived
The promotion audit’s batch 008 re-derived the Θ₃ example (ADR-047 v2). Θ₃ is the meet of the spaces of total(A, B) ∨ discrete(C) and discrete(A) ∨ total(C, B). So event B has causal past {A, B} in one and {B, C} in the other. {B:b} alone is a history in neither, so λ = {A:a, B:b}↓ and λ′ = {B:b, C:c}↓ share no history, and any two sections on them are compatible. Set B’s output to 1 on λ, where its past is {A:a, B:b}, and to 0 on λ′, where its past is {B:b, C:c}. Each is causal on its open. On the extension {A:a, B:b, C:c}, B is the tip of both sub-histories, so a causal function there must give B both outputs. No gluing exists, and the presheaf of causal functions is not a sheaf. LIT-843’s Example 7.2 is an independent second instance of the same kind. That is everything the title asserts. The characterisation, which says exactly which spaces fail, is QUESTION-143.