Source
Abramsky, Mansfield & Barbosa (2011), LIT-277, read in full in NOTE-250, which re-ran every example as exact linear algebra. Abramsky, Barbosa, Kishida, Lal & Mansfield (2015), LIT-278, read in full in NOTE-251, both arXiv versions (v2 is the published text; Prop 16’s proof survives only in v1). Carù (2017), LIT-279, read in NOTE-252, with the counterexample verified exhaustively.
What was actually shown
The obstruction (LIT-277, §4). Free the support presheaf of an empirical model over ℤ. For a section s₁ of context C₁, choose no-signalling lifts sᵢ that agree with it on each overlap; δ⁰(s₁, …, sₙ) vanishes on C₁, and its class γ(s₁) in the cohomology of the kernel presheaf relative to C₁ need not vanish, because (s₁, …, sₙ) is not a relative cochain. LIT-278 recovers the same class as the snake lemma’s connecting map, which NOTE-251 checked agrees with the concrete one.
The criterion (LIT-277 Prop 4.2; LIT-278 Prop 16). γ(s₁) = 0 iff s₁ lies in a compatible family of formal ℤ-linear combinations of support sections. So γ(s) ≠ 0 for some s witnesses logical contextuality, and γ ≠ 0 on every section witnesses strong contextuality (Prop 4.3, Prop 19). Neither is necessary.
Where it is complete (LIT-278, Thm 21). Call a model All-vs-Nothing over a commutative ring R when the R-linear equations every context’s support satisfies have no global solution. Then AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S). The step that matters: a compatible family through 1·s₀ has coefficient sum 1 in every context, so the counit evaluates it to a global section of the affine closure, which an AvN argument forbids. The proof assumes a connected cover, which the theorem does not state. Every state stabilised by an AvN triple of n-qubit Paulis gives such a model (Thm 4). GHZ, Peres–Mermin, the PR box and the triangle are AvN over ℤ₂, which is why LIT-277 found γ ≠ 0 on all their sections.
Where it fails.
- Hardy (LIT-277 §5). γ = 0 on all 13 support sections, over ℤ and ℤ/2 (NOTE-250’s count; the paper shows one). Every Hardy context’s affine hull is the whole event set, so no AvN argument exists over any ring.
- A Kochen–Specker cover (LIT-277 §8). {A,B,C},{B,D,E},{C,D,E},{A,D,F}, {A,E,G} is strongly contextual, yet γ = 0 on 9 of its 15 sections.
- The (2,2,4) Bell cover (LIT-279 §4). A strongly contextual model, supported on b₁ = a₁, b₂ = a₂ + 1 mod 4 and two “staircase” contexts, has γ = 0 on all 22 sections over ℤ and ℤ/2, and over ℚ. The cover is connected, its symmetries act transitively, and no measurement is private. That refutes LIT-277’s Conjecture 8.1 for symmetry of the cover, and its diagnosis that private measurements are the cause.
The failures share one mechanism: signed combinations such as [10] − [11] + [01] close a loop inside a context that no actual section closes. Cohomology over R sees a model only up to its affine closure.
What could have come out otherwise: symmetry of the cover could have forced completeness, as LIT-277 conjectured. It does not.
What this does not say
- That cohomology detects only All-vs-Nothing models. Theorem 21 runs one way. Whether every cohomologically strongly contextual model is AvN over some ring is open (NOTE-251).
- That all quantum strong contextuality is All-vs-Nothing. LIT-278 asserts it (p. 17) with no citation or survey.
- Anything probabilistic. Only supports are used, so the witness is blind to Bell–CHSH-type contextuality with full support and cannot grade models below contextual fraction 1.
- That higher-degree obstructions help. For no-signalling models every level above the first is empty (LIT-279, Prop 5.6).
- That strong contextuality is equivalent to every γ_C being injective. LIT-279’s Prop 6.1 is false in that direction; the PR box refutes it (NOTE-252).
- That Liar cycles are contextual models in general. The length-4 cycle is the PR box at the level of supports, and NOTE-251 derives cohomological strong contextuality for every length N ≥ 3; the encoding fails at N = 1 and 2, the classic Liar included.
Connections
- THEORY-012 holds the exact criterion, a global section of the support or of the distributions. This account is about a computable relaxation of it, and refines the cohomology line THEORY-012 carries in its “does not say”.
- THEORY-524 carries it further: on cyclic covers an iterated joint model makes the same obstruction exact.
- THEORY-014’s point that signed global sections always exist over ℝ has a discrete analogue here, noted in NOTE-251 as an inference: the obstruction fails exactly where affine combinations can glue.