NOTE-252: On the Cohomology of Contextuality

Contribution

AMB 2011 (LIT-277) showed that the cohomological witness misses contextuality in two cases: the Hardy model, and some sections of one lopsided Kochen–Specker cover. It conjectured (Conjecture 8.1) that under symmetry and connectedness the witness is complete for strong contextuality. This paper refutes that on the most symmetric non-trivial cover there is. It gives an explicit strongly contextual model on the Bell (2,2,4) scenario on which the obstruction vanishes for every section. So the witness certifies not even logical contextuality there.

It also does three smaller things:

• it generalises the obstruction to odd-degree cohomology;
• it proves that doing so yields nothing for no-signalling models;
• it recasts Ȟ¹ as isomorphism classes of torsors trivialised by the cover.

Key insight

The Hardy mechanism scales up. In Hardy, the formal combination [10] − [11] + [01] over one context has the same marginals as a single section. It therefore closes a “path” that no genuine section can close (Fig 3). Carù builds a model in which every section of every context can be closed this way.

• The design. Two contexts carry “equality-like” supports (b1 = a1; b2 = a2 + 1 mod 4). The other two carry “staircase” supports (b ∈ {a, a − 1}, no wrap-around), which contain enough triples {(i,i), (i,i−1), (i−1,i−1)} for the relaxation to exploit.
• The consequence. Any genuine global assignment has to go round the staircase the wrong way, so none exists. But every section is in a compatible family of ℤ-combinations.
• The lesson. Symmetry of the cover is no protection. What the ℤ-relaxation can glue depends on the support’s affine structure (NOTE-251’s reading of LIT-278), not on the shape of the cover.

Assumptions

• Scenario and model (§2), as in LIT-278.
  • The scenario. ⟨X, M, O⟩, with X finite and M an antichain cover.
  • The event sheaf. ℰ(U) = O
  • Possibilistic models. A model is a subpresheaf S ⊆ ℰ with:
    • (1) S(C) ≠ ∅;
    • (2) flasque beneath the cover (possibilistic no-signalling);
    • (3) compatible families glue.
  • No probabilities. Only possibilistic models are considered, “generated by the support of a probabilistic model”.
  • LC and SC. LC(S, s): s is in no compatible family. SC(S): LC at every s, equivalently S(X) = ∅.
• Coefficients (§3). F is any presheaf of abelian groups “representing” S. It must satisfy (1)–(3) and have an injection i: S ↪ F with i_C(s) ≠ 0. Typically F = F_ℤ S. Footnote 2 allows F_R for any ring R, but every result and example uses ℤ.
• Connected covers throughout. Footnote 3: “From now on, all the covers will be assumed to be connected”.
• The Čech complex (§3).
  • Simplices. q-simplices are ordered tuples (C0, …, Cq) with non-empty intersection. Repeated contexts are allowed: Lemma 5.1 uses (C0, …, C0).
  • Cochains. Cᵠ = ⊕_σ F(|σ|), with the coboundary given correctly, as a sum to q + 1. The face-map range is printed “0 ≤ j ≤ q” for a (q+1)-simplex, the same slip NOTE-250 recorded in AMB.
• The relative sequence (§3).
  • The sequence. 0 ⇒ F_{C̃0} ⇒ F ⇒ F|_{C0}, lifted to 0-cochains. It is right-exact by flasqueness, as in LIT-278.
  • The obstruction. The snake lemma gives γ_{C0}: F(C0) ≅ Ȟ⁰(M, F|{C0}) → Ȟ¹(M, F{C̃0}).
• Cohomological contextuality (§3), quoted from LIT-278.
  • Prop 3.1. γ_{C0}(r0) = 0 iff r0 lies in a compatible family of F.
  • CLC and CSC. CLC(S, s0) means γ(s0) ≠ 0. CSC(S) means CLC at every section.
  • Theorem 3.2. CLC ⇒ LC and CSC ⇒ SC.
• Torsors (§6.2).
  • F-presheaf. A presheaf of sets T with a pointwise left action of F.
  • F-torsor. The action is simply transitive, and some open cover has T(V) ≠ ∅ on each member.
  • Trs(M, F). The torsors trivialised by M, as a pointed set.

Key results

• §4, the counterexample (Fig 4; Table 1, Appendix A). The scenario is (2,2,4): measurements a1, a2 (Alice) and b1, b2 (Bob), with outcomes {0,1,2,3}. Supports, as (a, b) pairs, read from Table 1:

  • S(a1,b1) = {00, 11, 22, 33}, i.e. b1 = a1;
  • S(a1,b2) = {00, 10, 11, 21, 22, 32, 33}, i.e. b2 ∈ {a1, a1 − 1} with no wrap-around;
  • S(a2,b1) = {00, 10, 11, 21, 22, 32, 33}, i.e. b1 ∈ {a2, a2 − 1} with no wrap-around;
  • S(a2,b2) = {01, 12, 23, 30}, i.e. b2 = a2 + 1 mod 4.

  Claim (p. 28). The model is “strongly contextual but not cohomologically contextual (not even cohomologically logically contextual)”, which “essentially” disproves Conjecture 8.1. The support is “by carefully analysing the picture”, i.e. by inspection. For one section, s0 = (a1,b1) ↦ (0,0), an explicit ℤ-family is shown (with the typo noted under corrections).

  My verification.

  • It is a valid model. Every single-measurement marginal of every context is all of {0,1,2,3}, so S is flasque beneath the cover. It is also the support of a genuine no-signalling probability model: my LP found one with every support probability ≥ 0.1 (e.g. p(a1,b1) = 0.3, 0.2, 0.2, 0.3 on 00, 11, 22, 33). The paper does not show this.
  • It is strongly contextual. I enumerated all 4⁴ = 256 assignments: 0 global sections, 0 of the 22 sections extendable. By hand: b1 = a1 and b1 ∈ {a2, a2 − 1} give a2 ∈ {a1, a1 + 1}; b2 = a2 + 1 and b2 ∈ {a1, a1 − 1} give a2 ∈ {a1 − 1, a1 − 2}. The two sets are disjoint mod 4.
  • The obstruction vanishes everywhere. For each of the 22 sections I solved the compatibility system of Prop 3.1 with that section fixed. A family exists over ℤ (exact integer elimination) and over ℤ/2 for all 22. So γ = 0 on every section; ¬CLC, not just ¬CSC.
  • Also over ℚ. Compatible families project onto all of F(C) at every context (rank 4, 7, 7, 4), so γ_C ⊗ ℚ = 0 identically.
  • The corrected family checks. The explicit family, with (a1,b1) corrected to (a1,b2) in the bracket, satisfies all four overlap conditions.
  • Solver calibration. The same code reproduces NOTE-250’s counts: Hardy γ = 0 on 13/13, PR box on 0/8, AMB §8 cover on 9/15.
  • No All-vs-Nothing argument. By LIT-278 Thm 21 read contrapositively, it has none over any ring. Directly over ℤ₄: the affine hulls of S(a1,b2) and S(a2,b1) are all of ℤ₄², since they contain 00, 10 and 11. So the theory is {a1 = b1, b2 = a2 + 1}, which has global solutions.
  • Quantum realisability. Not claimed, and not checked. Whether this support is quantum-realisable is unverified.

  Does it meet Conjecture 8.1’s hypotheses? AMB left them unspecified (“suitable assumptions”), so “counterexample” can only mean “under every condition the example satisfies”.

  • Conditions it meets:
    • a connected nerve;
    • an automorphism group of the cover transitive on measurements and on contexts;
    • no measurement confined to one context (AMB’s diagnosed culprit);
    • Vorob’ev-irreducible.
  • Conditions it fails:
    • pairwise-intersecting contexts, which AMB’s own §8 failure satisfies;
    • model-level symmetry (2 automorphisms).

  So it refutes the conjecture under cover-symmetry, cover-transitivity and “no private measurements”. It does not refute it under a pairwise-intersection or model-symmetry reading. On any reading, it refutes AMB’s diagnosis that private measurements are “crucial”.

• Lemma 5.1. ψ_q: F(C0) → Cᵠ(M, F|{C0}), s0 ↦ (ω ↦ s0|{C0∩|ω|}), is injective. Proof complete; it uses the degenerate simplex (C0, …, C0).

• Lemma 5.2. im ψ_q ⊆ Zᵠ(M, F|{C0}) iff q is even. Proof complete: δ(c^q{s0})(ω) is an alternating sum of q + 2 equal terms. The “only if” holds for s0 ≠ 0, via ω = (C0, …, C0).

• Definition 5.3. The snake lemma on degree 2q gives the q-th obstruction γ^q_{C0}(s0) := γ̃^q(ψ_{2q}(s0)). The text says it lies ”∈ Ȟ^{2q+1}(M, F)”; the diagram puts it in Ȟ^{2q+1}(M, F_{C̃0}), and the text’s F is a slip. CLC_q and CSC_q are defined as for q = 0. γ⁰ = γ (footnote 6).

• Lemma 5.4. γ^q(s0) = 0 iff some s ∈ Z^{2q}(M, F) has p(s(σ)) = s0|{C0∩|σ|} for all σ ∈ N(M){2q}. Proof complete, by exactness.

• Proposition 5.5 (Appendix B). CLC_{q+1}(S, s0) ⇒ CLC_q(S, s0), giving the displayed ladder CSC ⇐ CSC_1 ⇐ … and CLC ⇐ CLC_1 ⇐ ….

  • Proof. f(s)(τ) := s(front (2q+1)-face of τ), shown to be a 2q+2 cocycle with the right projection.
  • My check. The argument is correct. But the face composite is written ”∂{2q+1}∂{2q+2}ν” for ν ∈ N_{2q+3}, where the intended face is the front (2q+2)-tuple, which is ∂{2q+2}∂{2q+3}ν. As printed, it deletes positions 2q+1 and 2q+2, which is not what the relabelling in (6) needs. It is a notational slip.

• Proposition 5.6 (Appendix B). “No-signalling empirical models are cohomologically q-non-contextual for any q ⪈ 0.”

  • Proof. Complete: z(Ci, Cj, Ck) := s_{Cj}|, built from no-signalling lifts, is a 2-cocycle meeting Lemma 5.4 for q = 1. Prop 5.5 then gives all q > 0. I checked the δ² computation.
  • Consequence. Combined with §2’s definition, every model in the paper has CLC_q false for all q ≥ 1. The “hierarchy” has one non-empty level.
  • The paper’s hope. Footnote 8 and p. 30 suggest using it to study signalling models. But the construction as given needs flasqueness for exactness, so for signalling models it is not defined without changes the paper does not make.

• Proposition 6.1. “The model is cohomologically strongly contextual if and only if γ_C is injective for all C ∈ M.”

  • The “if” direction. Correct.
  • The “only if” direction is false. The proof asserts “γ_C(s) ≠ 0 for all sections s ∈ F(C)”, conflating the support sections S(C) with all non-zero elements of F(C) = ℤ[S(C)]. Counterexample: the PR box is CSC (NOTE-250; re-checked), but with s = (a1,b1) ↦ 00 and s′ = (a1,b1) ↦ 11 the family (00+11, 00+11, 00+11, 01+10) over the four contexts is compatible. So γ(s + s′) = 0, and γ_{a1b1} is not injective (checked by solver too).
  • What else is affected. The consequences drawn from the “only if” direction are unsupported: the bound “CSC(S) ⇒ |Ȟ¹(M, F_{C̃0})| ≥ |F(C0)|” and the reading that CSC needs all γ_C injective. The bound’s truth by another route is unverified. Over ℤ both sides are typically countably infinite, so it says little.

• Proposition 6.2. If some γ_{C0} is injective, S is strongly contextual.

  • Proof. Correct.
  • My remark. It is a special case of an immediate fact the paper does not state. If γ_{C0}(s) ≠ 0 for every support section s ∈ S(C0) of a single context, then S is SC, because a global section would restrict to an extendable section at C0. The paper’s point “CSC is stronger than SC” follows from that fact rather than from Props 6.1–6.2.

• Proposition 6.3 (Appendix C). There is a bijection of pointed sets Trs(M, F) ≅ Ȟ¹(M, F), made into a group isomorphism by transport. The paper calls it “a readaptation of a known correspondence”.

  • Proof. Complete in outline. One step, that φ_U is an isomorphism, is “left to the reader”.
  • Minor slips. g([z]) is declared a presheaf into AbGrp, but its sets are affine (the defining equations are inhomogeneous), so it is a presheaf of sets. Also “we now that” appears in the proof.

• Conclusions. Cohomology “cannot provide a full invariant for contextuality (neither logical nor strong)”. The paper proposes obstruction theory, meaning fibrations and Postnikov towers with coefficients in homotopy groups, as future work.

• Footnote 5. It cites Mansfield’s thesis [18] for the conjecture being true for symmetric Kochen–Specker models satisfying a Daykin–Häggkvist condition. This is unverified (not read).

Claims

id	claim	strength	support
C1	The (2,2,4) model of Table 1 is strongly contextual	strong (by inspection in the paper; exhaustive check here)	§4, Fig 4; 0/256 global assignments (my enumeration)
C2	The model is not even cohomologically logically contextual: γ = 0 on every section	moderate in the paper (one section’s family shown, with a typo; the rest “by carefully analysing the picture”); strong after my check	§4; my exact solve, all 22 sections over ℤ and ℤ/2
C3	Hence AMB’s Conjecture 8.1 is false	strong for “symmetry and connectedness of the cover” read as transitivity, a connected nerve, no private measurements; not established for pairwise-intersecting covers or symmetric models	§4 and footnote 5 (“essentially disproving”)
C4	The scenario “verifies any reasonably strong form of symmetry and connectedness”	weak (assertion; false for pairwise intersection, which AMB’s §8 cover has)	p. 28
C5	ψ_q is injective, and lands in cocycles iff q is even	strong (proof)	Lemmas 5.1, 5.2
C6	CLC_{q+1} ⇒ CLC_q	strong (proof; a face-map notational slip)	Prop 5.5, Appendix B
C7	No-signalling models have CLC_q false for all q > 0	strong (proof)	Prop 5.6, Appendix B
C8	Higher obstructions “could potentially be used to study signalling properties”	weak (speculation; the construction assumes flasqueness)	p. 30, footnote 8
C9	CSC ⇔ every γ_C is injective	false as stated (“if” holds; “only if” refuted by the PR box, γ(s + s′) = 0)	Prop 6.1; my computation
C10	Some γ_{C0} injective ⇒ SC	strong (proof)	Prop 6.2
C11	Ȟ¹(M, F) ≅ Trs(M, F), the M-trivialised F-torsors	strong (proof; a known correspondence)	Prop 6.3, Appendix C
C12	The torsor description yields “a new interpretation” of the obstructions	weak (programmatic; nothing derived)	abstract, §6.2, Conclusions
C13	Conjecture 8.1 holds for symmetric Kochen–Specker models under the Daykin–Häggkvist condition	assertion by citation (Mansfield thesis)	footnote 5; unverified

Method

• Construct and inspect. The counterexample is found and argued from bundle diagrams: a section is extendable iff it lies on a closed path through one section per context. The ℤ-relaxation permits paths that cross a context through a signed combination of three sections.
• Generalise the snake. Lift the relative short exact sequence to degree-2q cochains, pre-compose with the diagonal-type map ψ_{2q}, and take the connecting map into Ȟ
• Collapse it. Build explicit cocycles from no-signalling lifts (Prop 5.6) and from front-face truncation (Prop 5.5).
• Torsors. A standard Čech-cocycle ↔ torsor dictionary, written out for the relative presheaves.

Concepts

• Possibilistic empirical model, LC, SC — as in LIT-278 (NOTE-251).
• Bundle diagram — the measurement complex as the base, outcome fibres over its vertices, sections drawn as edges. A global section is a closed path through every fibre once.
• CLC, CSC — non-vanishing γ at some or at every section (LIT-278’s CLC_ℤ, CSC_ℤ).
• False positive — the paper’s (and AMB’s) term for a contextual model or section with vanishing obstruction; it is a false negative of the witness.
• q-th cohomological obstruction γ^q, CLC_q, CSC_q — Definition 5.3. They exist only in odd degree 2q + 1.
• F-torsor; Trs(M, F) — §6.2.

Connections

• LIT-277 (AMB 2011, read in NOTE-250). This paper answers its Conjecture 8.1 in the negative.
  • A third failure, stronger than the other two. AMB’s §8 cover still has γ ≠ 0 on 6 of 15 sections, so it is still certified logically contextual. Carù’s model is invisible to cohomology on all 22 sections.
  • What is not re-examined. Hardy is re-explained (Fig 3; the family checks), but it is not re-analysed. AMB’s §8 cover is not re-analysed at all.
• LIT-278 (Abramsky, Barbosa, Kishida, Lal & Mansfield 2015, read in NOTE-251). Carù takes its definitions (the connecting-homomorphism γ, CLC and CSC, Prop 16 as Prop 3.1, Prop 19 as Thm 3.2). He cites its AvN result as one of the positive cases.
  • What he does not do. He does not place his model relative to AvN.
  • Where it sits. By LIT-278’s Thm 21 it has no AvN argument over any ring, which I confirmed directly over ℤ₄. So it sits where NOTE-251 located AMB’s §8 cover: strongly contextual, outside AvN. It is consistent with Thm 21, and it confirms that SC ⊋ CSC_ℤ even on the most regular small cover.
• LIT-016 (Abramsky & Brandenburger). The sheaf framework, and the identification of flasqueness with no-signalling on which Prop 5.6 turns.
• Carù, “Towards a complete cohomology invariant for non-locality and contextuality”, arXiv 1807.04203. Existence verified from the arXiv abstract page: v1 only, 11 Jul 2018, 733 KB. Content unverified beyond the title. The Conclusions here point to obstruction theory and Postnikov towers as the next step; whether 1807.04203 does that is unverified.
• Mansfield, D.Phil. thesis (2013) [18]. Cited for a special case where the conjecture holds. Not read and not in the record.
• Anthology. A text search of the Anthology of the SOTA’s record finds nothing on cohomology or contextuality, so no ANTH- citation is warranted.

Bearing on the record

• THEORY-012 (Active). It supports the claim and sharpens its cohomology line. It contradicts nothing.
  • What the line says now. The witness is sufficient only, complete on All-vs-Nothing models (LIT-278), and LIT-277’s two failures lie outside that class.
  • What this paper adds.
    • (a) AMB’s conjecture, that symmetry and connectedness of the cover restore completeness, is false. The two-party, two-setting cover with four outcomes carries a strongly contextual model with γ = 0 on every section.
    • (b) The witness can therefore miss strong contextuality entirely, not only on some sections.
    • (c) The new failure also lies outside AvN, so the bullet’s “complete on AvN” framing survives intact.
  • What it does not close. Neither the Hardy gap nor the §8 cover.
  • Proposed wording. See the report line. It needs this paper filed as a LIT first. Until then it can only be named in prose, since no bare code exists.
• THEORY-014 (Proposed). It strengthens the analogy NOTE-250 and NOTE-251 drew; the inference is mine.
  • The analogy. The ℤ-relaxation of the support, which uses signed combinations of possible sections, can succeed where every nonnegative (genuine) gluing fails, and here it succeeds everywhere. Over ℚ, compatible families project onto the whole of F(C) at every context.
  • A side computation. Over ℚ, AMB’s §8 cover has γ = 0 on 13 of 15 sections, against 9 over ℤ, so the choice of coefficients visibly matters.
  • Status change. None warranted.
• LIT-265 (contextual fraction; Deferred). Strong contextuality is non-contextual fraction 0 (LIT-016 Prop 6.3), i.e. CF = 1. So Carù’s model is maximally contextual, since every probabilistic model on this support has CF = 1, yet it is cohomologically invisible. That is the cleanest statement that the cohomological and fractional measures are not monotone in each other. It is my inference.
• THEORY-013, THEORY-015, THEORY-016. No bearing. No-signalling is built in, and the notion is sheaf/Kochen–Specker contextuality only.
• ML practice. It carries nothing. This is algebraic topology for quantum foundations; the author’s DeepMind-funded scholarship is not a connection.
• For filing.
  • Tags. contextuality, quantum-foundations, mathematics, mirroring LIT-277.
  • Why not logic. LIT-278 carries it for its Liar-cycle material. Here there is no logic beyond the cs.LO cross-list and the term “logical contextuality”.
  • Lineage. It extends LIT-277 (answers Conjecture 8.1) and LIT-278 (adopts its framework).

Limitations

• The counterexample is argued from a picture. The paper exhibits one section’s family, with a typo in it, and asks the reader to check the other 21 sections, and strong contextuality, from Fig 4. Both claims are true (my check), but the paper does not prove them.
• “Any reasonably strong form” of symmetry and connectedness is overstated.
  • Connectedness. The cover is not pairwise-intersecting, and AMB’s own failure is.
  • Symmetry. The model’s own symmetry group has order 2. The narrowing of the conjecture to the cover is Carù’s (the added words “of the cover”).
• §5 refutes itself for its stated setting. The hierarchy is empty above level 0 for every no-signalling model. The signalling application is a hope, with a construction that assumes no-signalling.
• Prop 6.1 is false in one direction. The CSC ⇒ |Ȟ¹| ≥ |F(C0)| bound derived from it is unsupported.
• §6.2 is a standard dictionary. No new property of any obstruction follows from it.
• No positive completeness result. The paper gives no characterisation of what the witness detects. The one positive case it mentions (footnote 5, Mansfield’s thesis) is cited, not shown.
• Only ℤ coefficients. Other rings are mentioned (footnote 2) but never used. By LIT-278 Prop 20, any ring receiving a map from ℤ can only make the obstruction vanish more often, so no ring change rescues this model.
• Slips.
  • the face-range typo in §3;
  • Ȟ^{2q+1}(M, F) for Ȟ^{2q+1}(M, F_{C̃0}) in Def 5.3;
  • the face composite in Prop 5.5’s proof;
  • the (a1,b1)/(a1,b2) typo in the displayed family;
  • “(a,b1,a2,b2)” for “(a1,b1,a2,b2)” in the Hardy global section (p. 24);
  • g([z]) typed as AbGrp-valued;
  • “presehaf”, “yelds”, “homomoprhism”, “te sum”.

Open questions

• A sharper conjecture. Is γ complete for strong contextuality on covers whose contexts pairwise intersect, or on models (not just covers) with transitive symmetry? Carù’s example does not touch either reading. AMB’s §8 cover already refutes the first for partial completeness (6/15 detected, SC undetected on 9), though not for “CSC fails entirely”.
• The minimal outcome count. On the (2,2,2) scenario the witness is complete. I enumerated every flasque support: 8 are strongly contextual (the PR-box relabellings), and γ ≠ 0 on all their sections. Is (2,2,3) enough for a complete false negative, or is 4 outcomes the minimum? This is my question; the paper does not address it.
• Quantum realisability. Is Carù’s support, or any complete false negative, quantum-realisable? LIT-278’s uncited claim that all known quantum strong contextuality is AvN would make any quantum example here a counterexample to it.
• What the ℤ-relaxation detects. Does CSC_ℤ imply AvN_R for some R (NOTE-251’s open converse)? Carù’s model is consistent with either answer, since it is neither.
• The follow-up. Does arXiv 1807.04203 give a complete invariant, and via the obstruction theory promised here? Unverified.

Corrections to the seeded skim

• none (there was no seed or dossier)
• Identification note for filing. EPTCS 236, 2017, pp. 21–39, Proceedings QPL 2016, eds. R. Duncan & C. Heunen. I verified DOI 10.4204/EPTCS.236.2 via Crossref (no contact address sent): the title, the sole author Giovanni Carù, pp. 21–39 and volume 236 match, issued 2017. The arXiv categories are quant-ph (primary) and cs.LO. Affiliation: Quantum Group, Department of Computer Science, University of Oxford. arXiv lists no v2.
• The abstract’s “hierarchy of logical implications” is empty above its first level. The abstract says higher obstructions “give rise to a refinement … different ‘levels’ of contextuality are organised in a hierarchy”. The body then proves (Prop 5.6) that every no-signalling model is cohomologically q-non-contextual for all q > 0. The paper’s own empirical models are no-signalling by definition (flasque beneath the cover, §2 condition 2), and the construction of the higher obstruction itself uses that flasqueness (the right-exactness of eq. 3). So for every model the paper defines, the hierarchy collapses to the single level q = 0 that AMB already had. The abstract does not say so; the introduction and Conclusions do.
• The abstract’s “even under symmetry and connectedness restrictions” holds for some such restrictions, not for “any reasonably strong form” (the body’s phrase, p. 28).
  • What holds. The Bell cover {a1b1, a1b2, a2b1, a2b2} has an automorphism group (8 permutations) transitive on measurements and on contexts. Every measurement lies in exactly two contexts, and it is irreducible in Vorob’ev’s sense. That answers the feature AMB called “crucial”, measurements lying in a single context.
  • What does not. Contexts a1b1 and a2b2 are disjoint, so the cover is not pairwise-intersecting. AMB’s own §8 cover {A,B,C},{B,D,E},{C,D,E},{A,D,F},{A,E,G} is pairwise-intersecting, so on that measure of connectedness the new counterexample is less connected than the old one.
  • The model itself. It is not symmetric. It has 2 automorphisms (my count: the identity and the party swap a↔b with outcomes x ↦ 3 − x), against 16 for the PR box on the same cover.
  • Carù’s own qualification. The body says “essentially disproving”, and footnote 5 concedes that the conjecture’s “open-endedness” leaves room for special cases where it holds.
• The conjecture is quoted with words AMB did not write. Carù’s Conjecture 4.1 reads “suitable assumptions of symmetry and connectedness of the cover”. AMB’s Conjecture 8.1 reads “Under suitable assumptions of symmetry and connectedness, the cohomology obstruction is a complete invariant for strong contextuality”. AMB’s preceding sentence says “this model does not satisfy any reasonable criterion for symmetry”. The narrowing to the cover is Carù’s, and it matters, because the new model’s symmetry is small (previous bullet).
• The explicit compatible family has a typo (p. 28). It is printed as {s0, (a2,b1) ↦ (0,0), (a2,b2) ↦ (0,1), [(a1,b1) ↦ (1,1)] − [(a1,b1) ↦ (1,0)] + [(a1,b1) ↦ (0,0)]}.
  • Why it cannot be right as printed. The printed family has two entries over (a1,b1), none over (a1,b2), and (1,0) is not in the support over (a1,b1).
  • The corrected family. With the bracketed combination placed over (a1,b2), it is a compatible ℤ-family through s0 (checked).
• The abstract’s “new interpretation” via torsors is a restatement. Prop 6.3 (Trs(M,F) ≅ Ȟ¹(M,F)) is, in the paper’s words, “a readaptation of a known correspondence”. No property of any obstruction is derived from it. The interpretation is promised for future work.