Source
Abramsky & Brandenburger (2011), LIT-016, §§2.2–2.5, Prop 3.1, Props 4.2–4.4, Thms 5.4 and 5.9, Props 6.1 and 6.3, Thm 8.1 (NOTE-016). The graded version: Abramsky, Barbosa & Mansfield’s contextual fraction (LIT-265, Eq. 3 and Theorem 1, read in NOTE-236). Budroni et al.’s review (LIT-263, §IV.A.1, read in NOTE-844, which calls local consistency “the sheaf condition” and lists the sheaf, marginal-problem, polytope and graph formulations as substantially equivalent; it asserts that equivalence with citations and does not show it).
What was actually shown
An empirical model is a family of distributions, one per maximal context of jointly performable measurements, that agree on overlaps; the agreement is no-signalling, and it is part of the definition (§2.5). The model is noncontextual when there is a global section: one distribution on joint outcome assignments to all measurements whose marginals are the context distributions. Thm 8.1 proves this holds iff there is a factorizable hidden-variable model, and Prop 3.1 that a global section is a deterministic hidden-variable model. Contextuality comes in three nested strengths — probabilistic, possibilistic, strong — witnessed by Bell, Hardy and GHZ (Props 4.2–4.4, 6.1), and strong contextuality is a non-contextual fraction of zero (Prop 6.3).
Over the reals the answer is always yes: noncontextual and no-signalling models span the same subspace (Thm 5.4), so every no-signalling model, the PR box included, has a signed global section (Thm 5.9). Negative probabilities therefore cannot single out quantum theory; what contextuality marks is that no nonnegative section exists.
What this does not say
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Anything about data whose marginals depend on context (signalling, or imperfectly compatible measurements); no-signalling is assumed. Contextuality-by-Default handles that case (see the behavioural claim in this group).
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Anything about generalized (Spekkens) contextuality; how the two notions relate is a separate claim that extends this one.
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Anything infinite; the scenarios are finite.
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That contextuality can always be certified cohomologically. Abramsky, Mansfield & Barbosa’s Čech obstruction (LIT-277, NOTE-250) is a sufficient witness only. It vanishes on every section of the Hardy model, and on 9 of 15 sections of a strongly contextual Kochen–Specker cover (§8), because it tests extension to a compatible family of ℤ-combinations, not to a global section of the support. The criterion here is the exact one; the cohomology is a computable relaxation of it. It is, however, complete on one large class. Every model whose support admits an All-vs-Nothing argument has a non-vanishing obstruction on every section (Abramsky, Barbosa, Kishida, Lal & Mansfield, LIT-278, Thm 21, for a connected cover). An All-vs-Nothing argument is a set of R-linear equations, over any commutative ring R, that each context satisfies but that have no global solution. The class includes:
- GHZ-type n-qubit stabiliser states (Thm 4);
- Peres–Mermin;
- the PR box;
- Kochen–Specker covers failing LIT-277’s GCD condition. This last is the reader’s derivation in NOTE-251, not the paper’s.
Both of LIT-277’s documented failures lie outside it. Hardy is not strongly contextual. The §8 cover is strongly contextual but, by the reader’s application of Thm 21 (NOTE-251), admits no All-vs-Nothing argument over any ring.
Symmetry of the cover does not rescue completeness. Abramsky, Mansfield & Barbosa conjectured that under suitable symmetry and connectedness it does (LIT-277, Conjecture 8.1). Carù (LIT-279, §4 and Appendix A) refutes this for symmetry of the cover.
- The counterexample. It sits on the two-party, two-setting, four-outcome Bell cover, where every measurement lies in two contexts and the cover’s symmetries act transitively. It is strongly contextual, yet the obstruction vanishes on all 22 of its sections, over ℤ and over ℤ/2 (NOTE-252).
- Consequences. On that cover the witness certifies not even logical contextuality. This third failure also lies outside All-vs-Nothing (Thm 21, contrapositive).
- A partial repair. Carù (LIT-280, Thm 7.7; republished unchanged as Thm IV.32 of his
2019 thesis, LIT-281) treats cyclic covers,
where the contexts’ overlaps form a single chordless N-cycle, as in the
Bell (2,2,d) and N-cycle scenarios. On those, the same ℤ/2 obstruction
computed on the (N−1)-th iterated “joint model” is exact for logical
and strong contextuality.
- It removes the Hardy failure and the (2,2,4) counterexample.
- Its cost grows exponentially with the iteration; neither the paper nor the thesis analyses it.
- Beyond cycles nothing general is established. The paper’s extension (Prop 8.2) is false on its own example, and LIT-277’s §8 cover stays undetected at every level under its definitions, which refutes its Conjecture 9.1. The failure of Prop 8.2 and the refutation are the reader’s computations (NOTE-253). The thesis restates the extension and the conjecture unchanged (Prop IV.34, Conjecture IV.41) and adds no result beyond cycles, so these findings apply to it as well (NOTE-254).
- Still open. Whether a complete and computable cohomological test exists off cyclic covers. Whether completeness holds for covers whose contexts pairwise intersect, or for symmetric models.
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The strictness of the hierarchy at the possibilistic step, which leans on a cited result, or GHZ for n other than 4k (NOTE-016). This claim covers the equivalence and Thm 5.9, not those details.