Source
Carù (2018), “Towards a complete cohomology invariant for non-locality and contextuality”, arXiv 1807.04203, LIT-280, read in NOTE-253. Carù (2019), DPhil thesis, Oxford, LIT-281, read in NOTE-254; its Chapters III–IV republish LIT-279 and LIT-280, with the joint model renamed the line model.
What was actually shown
The construction. The joint model S^(1) pulls back adjacent contexts; S^(k) iterates it. S^(k) is a well-defined empirical model, and it is strongly (or logically, at a section) contextual exactly when S is (Props 4.2, 4.6, 5.1). A non-vanishing obstruction on it is sound (Thm 7.2).
Why it helps. On joint contexts, which are unions of full rectangles, the signed “Z-shaped” combinations that fool the ordinary obstruction can no longer sit inside one context (Lemmas 7.4–7.5, Thm 7.6).
The theorem (LIT-280 Thm 7.7; LIT-281 Thm IV.32). On a cyclic scenario with |M| contexts, LC ⇔ CLC^(|M|−1) and SC ⇔ CSC(S^(|M|−1)). The readers confirmed it on Hardy, on Carù’s 2017 (2,2,4) counterexample (detected on all 22 sections at level 3 and on none at levels 1–2), and on 659 random contextual cycle models. Hardy needs level 3, so the bound is tight. This covers the Bell (2,2,d) and N-cycle scenarios, and removes two of the three failures in THEORY-526.
Beyond cycles (the readers’ findings).
- Prop 8.2 (= IV.34) is false as stated. The paper’s own Table 5 model fails at level 2 for section s₁₀ with n = 3; NOTE-253 gives an explicit family.
- The §8 Kochen–Specker cover of LIT-277 is never fully detected. Under the paper’s definitions, a lifted ℤ/2 family through (s_{ABC,A}, s_{AEG,A}) survives at every level, which refutes the closing conjecture that some level is complete for every model (Conjecture 9.1; IV.41 in the thesis). The thesis’s claim that the first line model removes all its false negatives rests on a wrong equation in the level-0 system.
- Cost. For that cover the joint models have 5, 10, 30, 150 and 1,350 contexts at levels 0–4.
What could have come out otherwise: some random cycle model could have escaped detection at level |M| − 1. None of 659 did.
What this does not say
- That a complete cohomological invariant exists for every scenario. Off cycles nothing general is established, and the stated conjecture is false.
- That the invariant is cheap. “Without compromising the practical computability” (§1) is asserted; the objects grow like iterated line graphs.
- That cohomology is the only exact route. The thesis’s Chapter V decides logical and strong contextuality on any scenario by local computation on join trees, at a cost exponential in treewidth; the thesis never connects the two (NOTE-254). That chapter’s reformulation is close to a restatement of the definitions, and its speed comparison uses an unnatural baseline, so it is not filed here.
- Anything probabilistic. As with the ordinary obstruction, only supports are used. The thesis’s probabilistic line model preserves marginals and reflects contextuality one way (Thm IV.40); it is not a completeness result.
Connections
- THEORY-526 is the ordinary obstruction this repairs on cycles, and THEORY-012 the exact criterion both relax.
- The thesis’s Chapters VI–VII (All-vs-Nothing stabiliser groups reduce to three-qubit GHZ; no two-qubit or W-class state is strongly non-local) were not checked by the reading and are not filed.