NOTE-251: Contextuality, Cohomology and Paradox

Contribution

AMB 2011 (LIT-277) introduced a Čech obstruction γ(s) that certifies, when non-zero, that a support section s does not extend. It showed on examples that the obstruction sometimes works and sometimes does not. This paper adds three things.

• A class on which it always works. Suppose the outcomes carry a ring structure R and the model’s support satisfies R-linear equations with no global solution: an “All-vs-Nothing” argument, AvN_R. Then the obstruction, with coefficients in R or in ℤ, is non-zero on every section (Theorem 21).
• A large quantum source of such models. Every state stabilised by the group generated by an “AvN triple” of n-qubit Pauli operators is one (Theorem 4). For independent triples these states form a 2ⁿ⁻³-dimensional subspace.
• A more abstract definition. γ is re-defined as the connecting homomorphism of the snake lemma. AMB had only pointed to this form in a Remark.

The paper also offers a short, informal reading of Liar cycles as contextual models, with the PR box as the length-4 Liar cycle.

Key insight

AvN arguments are linear-algebraic, and so is cohomology with free-module coefficients. Take the free R-module F_R S on the support presheaf.

• The families are affine. A compatible family of formal combinations must have the same coefficient sum in every context, since all of them restrict to the empty context. So if it passes through 1·s₀, it consists of affine combinations.
• The counit evaluates them. The counit of the free–forgetful adjunction, F_R U E → E, turns formal affine combinations into actual affine combinations of outcome vectors. It therefore sends a compatible family in F_R S to a compatible family in the affine closure Aff S.
• The consequence. A vanishing obstruction yields a global section of Aff S. But the R-linear theory of S equals that of Aff S, so a global section of Aff S would satisfy every equation of the theory, which is exactly what an AvN argument forbids.

The Hardy-style “false positives” of AMB are exactly what the affine closure can glue. Cohomology over R sees the model only up to its affine closure.

Assumptions

• Scenario (§3).
  • The measurement cover. ⟨X, M, O⟩, with X a finite set of measurements. M is a cover of X, and an antichain: it lists the maximal contexts.
  • Outcomes. One outcome set O for all measurements.
  • The event sheaf. ℰ(U) = O
• The model is the support presheaf only (Def 1). A model is a subpresheaf S ⊆ ℰ with three conditions.
  • (E1) S(C) ≠ ∅ for every context C.
  • (E2) Flasque beneath the cover. Restrictions S(U′) → S(U) are surjective for U ⊆ U′ ⊆ C. By LIT-016 this “amounts to” no-signalling.
  • (E3) Compatible families induce a (unique) global section in S(X).
  • No probabilities. Probabilities are discarded throughout: “we shall only work with the associated support presheaf” (p. 7).
• Contextuality (Def 2).
  • LC(S, s). s belongs to no compatible family.
  • LC(S). LC(S, s) holds for some s.
  • SC(S). LC(S, s) holds for all s, equivalently S(X) = ∅.
• Rings (§4.2, footnote 3). R is commutative with unit. The AvN_R results need the scenario ⟨X, M, R⟩: the outcomes are identified with R, so each ℰ(U) = R^U is an R-module. For binary models this means embedding {0, 1} in R, as box 25 does with ℤ/3.
• The cover must be connected, although Theorem 21 does not say so.
  • Where the paper says it. Prop 16 (γ(r₀) = 0 iff r₀ extends to a compatible family) is stated for connected M. Footnote 5 says “we are only interested in connected covers”.
  • Where it is used. The coefficient-sum remark at the end of §5.3, which the proof of Theorem 21 uses, needs compatibility over the empty intersection. A Čech cocycle guarantees that only on a connected cover.
• Stabiliser setting (§4.1).
  • The group. The Pauli n-group Pₙ: n-fold tensor products of {I, X, Y, Z} with global phase in {±1, ±i}.
  • Qubits only. Eigenvalues ±1 are relabelled as ℤ₂ = {0, 1}, so the construction is n-qubit only. There are no qudits and no other Pauli dimension anywhere in the paper.
  • Measurements. Each party measures a local Pauli. A joint context is one Pauli per site, and the “measurement h” of the proof is the corresponding product.
• Paradox section (§2). It takes over, as argued “extensively and rather compellingly” in the cited literature (Wen 2001; Cook 2004; Walicki 2009), that a Liar cycle’s content is preserved by replacing reference and truth predicates with Boolean equations. That premise is not argued in this paper.

Key results

• Definition 3 (AvN triple). ⟨e, f, g⟩ in Pₙ, with global phases +1 and pairwise commuting, such that:
  • (A1) at each site at least two of eᵢ, fᵢ, gᵢ are equal;
  • (A2) the number of sites with eᵢ = gᵢ ≠ fᵢ, all distinct from I, is odd.
• Theorem 4. Let S be the subgroup generated by an AvN triple, and V_S its stabilised subspace. Every |ψ⟩ ∈ V_S, under Pauli measurements, “admits an All-vs-Nothing argument”: the parity equations its support satisfies are inconsistent.
  • Proof. Complete. By (A2), efg = −h with h ∈ Pₙ of phase +1, which forces odd parity for h. By (A1), any global assignment gives h even parity.
  • Size of the class. The remark after the theorem says that for linearly independent check vectors, dim V_S = 2ⁿ⁻³ (cited to Nielsen–Chuang and Caves).
  • My check of V_S ≠ ∅. V_S is never empty. −I ∈ S would need efg = −I, but (A2) forces efg to be non-identity at the (A2) sites. The paper does not say this.
  • My numerical check. I sampled 40 commuting AvN triples for n = 3 and 12 for n = 4. For each, a random state in V_S under local X/Y/Z measurements gave a strongly contextual model that is AvN over ℤ₂, with no failures.
• GHZ (§4.1). The four Mermin parity equations, with the corrected right-hand sides of v2, sum to 0 = 1. This is a complete hand proof of strong contextuality for GHZ(3). It fills the n = 3 case that NOTE-016 recorded as only sketched in LIT-016 (LIT-277 had a machine check).
  • My extension. By brute force, the GHZ(n) model with local X/Y measurements is AvN over ℤ₂ for n = 3, 4, 5, 6. The paper does not claim this for general n.
• Box 25 (§4.2). The paper says Pironio–Bancal–Scarani’s ‘box 25’ “admits no parity AvN argument” but satisfies six equations mod 3.
  • My check of the six equations. Each variable’s coefficients sum to 3, and the right-hand sides sum to 8 ≡ 2 (mod 3), so they are jointly inconsistent.
  • Unverified. Whether box 25’s support satisfies them, and whether it has no parity argument. I did not obtain the box.
• Definitions 5–6. An R-linear equation is φ = ⟨C, a, b⟩, satisfied by s when Σ_{m∈C} a(m)s(m) = b. The model’s theory is T_R(S) = ⋃_C T_R(S(C)). AvN_R(S) means T_R(S) has no global solution g: X → R.
• Proposition 7. AvN_R(S) ⇒ SC(S). The proof is complete and one line.
• Affine closure (§4.3).
  • Solution sets are affine. The Galois connection S ⊆ M(Γ) ⟺ T(S) ⊇ Γ gives aff ≤ M∘T, with equality when R is a field.
  • Definition 8. (Aff S)(C) := aff(S(C)). Aff is natural under restriction (eq. 1), so Aff S is flasque beneath the cover. Aff S(X) can be non-empty when S(X) is empty.
• Proposition 9. AvN_R(S) ⇒ SC(Aff S), with the converse when R is a field. Both directions are proved.
  • A slip. The statement writes “AvN(S)” without the subscript.
• Cohomology (§5).
  • Definitions 10–14. The Čech complex as in AMB. The coboundary’s face range is now correctly 0 ≤ j ≤ q + 1, fixing the typo NOTE-250 found.
  • Relative presheaves. F|_U and F_Ū are defined as in AMB, with the exact sequence 0 → F_Ū → F → F|_U.
  • Exactness. The paper argues that the cochain map C⁰(M, F) → C⁰(M, F|_U) is surjective by flasqueness beneath the cover. This answers the exactness question NOTE-250 raised.
  • The snake lemma. It gives a connecting map Ȟ⁰(M, F|{C₀}) ≅ F(C₀) → Ȟ¹(M, F{C̄₀}).
  • A minor overstatement. The text calls both rows “short exact”, but the bottom row’s right-hand surjectivity is not argued. It is also not needed for the exactness used.
• Definition 15. γ(r₀) is the image of r₀ under that connecting homomorphism.
  • My check. Unwinding the snake construction (lift r₀’s restrictions to a 0-cochain of F, apply δ⁰, read the result in Z¹(F_{C̄₀})) gives exactly LIT-277’s concrete γ(s₁) = [δ⁰(s₁, …, sₙ)]. The two definitions agree, which the paper does not state.
• Proposition 16. For connected M, γ(r₀) = 0 iff there is a compatible family {r_C ∈ F(C)} with r_{C₀} = r₀.
  • Where the proof is. v2 says “For a proof … see [2]”. The proof is in v1 (Prop 5.7): exactness at Ȟ⁰(F|{C₀}), so ker γ = im(Ȟ⁰(F) → Ȟ⁰(F|{C₀})).
• Definition 18. CLC_R(S, s₀) means γ_{F_R S}(s₀) ≠ 0. CSC_R means γ ≠ 0 for all sections.
• Proposition 19. CLC_R ⇒ LC and CSC_R ⇒ SC. This is LIT-277’s Prop 4.3, generalised to any ring; the proof is complete.
• Proposition 20. For a ring homomorphism R′ → R, γ_{R′}(s) = 0 ⇒ γ_R(s) = 0, so CSC_R ⇒ CSC_{R′}. Since ℤ maps to every ring, CSC_R ⇒ CSC_ℤ.
  • A slip. The proof’s first line swaps the direction (“If h: R → R′”). The statement is right.
• The coefficient-sum remark (end of §5.3). A compatible family has equal coefficient sums in every context. So a family through 1·s₀ consists of affine combinations.
• Theorem 21 (main). AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S).
  • The proof. The new step is the second. By Prop 16, a vanishing γ gives a compatible family of formal affine combinations, and the counit ε pushes it to a compatible family of Aff S. The proof is complete, given connectedness.
• The logical analogue (p. 16), sketched. AvN_R(e, s₀) ⇒ LC(Aff S, s₀) ⇒ CLC_R(S, s₀) ⇒ CLC_ℤ(S, s₀) ⇒ LC(S, s₀), where AvN_R(e, s₀) means no solution of the theory extends s₀.
  • Its support. “Essentially the same strategy”. My check: each step does follow by the same argument, with (E3) for Aff S giving the global section in the first step.
• Discussion claims. The first implication of Theorem 21 reverses for general rings under “a more abstract notion of equational consistency, in terms of quotient modules”. That is asserted, not shown.
• Liar cycles (§2).
  • The encoding. The cycle S₁: S₂ is true, …, S_N: S₁ is false becomes x₁ = x₂, …, x_{N−1} = x_N, x_N = ¬x₁, each equation fibred over its two variables.
  • The claims. Any N − 1 of the equations are consistent, and all N are not. “Up to rearrangement, the Liar cycle of length 4 corresponds exactly to the PR box” via x₁ ∼ a₂, x₂ ∼ b₁, x₃ ∼ a₁, x₄ ∼ b₂. The usual Liar reasoning “corresponds precisely” to searching for a univocal path in the PR bundle.
  • My check of the PR identification. It is exact at the level of supports: three correlated edges {00, 11} and one anti-correlated edge {01, 10}.
  • My computation. I encoded the Liar cycle as a support model, with contexts {xᵢ, xᵢ₊₁} and support {00, 11}, or {01, 10} on the closing edge. For N = 3, …, 8 it is a valid model (both single-variable marginals are {0, 1}, so it is no-signalling), strongly contextual and AvN over ℤ₂. γ ≠ 0 for all 2N sections over both ℤ and ℤ₂. By Theorem 21 that holds for every N ≥ 3. The paper does not state or apply this: it defers “the cohomology of paradox”.
  • Where the encoding fails. It breaks down at N = 1 and N = 2, including the classic Liar the paper names. For N = 1 the only “context” {x₁} has empty support (x₁ = ¬x₁), violating (E1). For N = 2 both equations sit over the same set {x₁, x₂}, violating the antichain condition, and merging them gives empty support. The paper does not note this.
• The completeness claim (p. 17). “all known quantum examples of strong contextuality are of AvN type, hence all such examples are captured by cohomology”. This is an assertion with no citation or survey.

What it does to LIT-277’s documented gaps (NOTE-250).

• The Hardy gap is not closed or narrowed.
  • Why no AvN argument exists. Hardy is not strongly contextual, so only the logical chain could apply, and it cannot. Every Hardy context’s support contains three affinely independent points of R², so (Aff S)(C) = R² = ℰ(C) at every context, over every ring. Then T_R(S) is trivial, and AvN_R(e, s₀) fails for every R and every s₀.
  • My computation. It agrees: Hardy is not AvN over ℤ₂ or ℤ/3, and γ = 0 on all 13 sections over ℤ and ℤ₂.
  • What the paper does with the case. v1 still named Hardy as a false positive (§5.3). v2 deletes the sentence and does not discuss the case at all.
• The strongly contextual Kochen–Specker cover with γ = 0 on 9/15 sections is explained but not narrowed.
  • The cover. LIT-277’s §8 cover is {A,B,C},{B,D,E},{C,D,E},{A,D,F},{A,E,G}.
  • What Theorem 21 implies. Read contrapositively: γ_ℤ = 0 on some section means not CSC_ℤ, hence not CSC_R, hence not SC(Aff S), hence not AvN_R. That holds for every ring R.
  • What follows. The cover is a strongly contextual model with no All-vs-Nothing argument over any ring. It shows that the class Theorem 21 covers is strictly smaller than strong contextuality.
  • My computation. It agrees: not AvN over ℤ₂ or ℤ/3, and γ ≠ 0 on exactly 6 of the 15 sections over ℤ and ℤ₂.
  • What the paper says. It does not mention the example.
• LIT-277’s positive results are all AvN over ℤ₂, so Theorem 21 recovers them.
  • The examples. PR box (the Liar-4 cycle). GHZ. Peres–Mermin (row parities 1, column parities 0, summing to 3 ≡ 1 against 0). The triangle (x + y = 1 three times). The 18-vector set (Σ_{m∈C} x_m = 1 over 9 contexts, with each vector in 2 contexts, gives 0 ≡ 9).
  • Checks. I confirmed PR, GHZ, Peres–Mermin and the triangle by computation, and the 18-vector case by the parity count.
  • LIT-277’s Prop 6.2 too, on connected covers. A Kochen–Specker model failing the GCD condition is AvN over ℤ/g with g = gcd{d_m}. Summing Σ_{m∈C} x_m = 1 over the contexts gives Σ_m d_m x_m ≡ 0 against |𝒰| ≢ 0 (mod g).
  • What the paper says. It asserts the subsumption (“These results will be subsumed and greatly generalised in §6”, p. 15) but does not show it example by example. The derivations above are mine.

Claims

id	claim	strength	support
C1	An AvN_R model is strongly contextual	strong (proof)	Prop 7
C2	AvN_R(S) ⇒ SC(Aff S); converse when R is a field	strong (proof)	Prop 9
C3	γ is the connecting homomorphism of the snake lemma, and γ(r₀) = 0 iff r₀ lies in a compatible family (connected cover)	strong (proof in v1 only; v2 defers it to “[2]”, i.e. v1)	Def 15, Prop 16 (v1 Prop 5.7); agrees with LIT-277’s concrete γ (my check)
C4	CLC_R ⇒ LC, CSC_R ⇒ SC; a ring map R′ → R transports vanishing, so CSC_R ⇒ CSC_ℤ	strong (proof)	Props 19, 20
C5	AvN_R ⇒ SC(Aff S) ⇒ CSC_R ⇒ CSC_ℤ ⇒ SC: every AvN model’s strong contextuality is witnessed cohomologically on every section	strong (proof; connectedness implicit)	Theorem 21
C6	The analogous chain holds per section for logical contextuality	moderate (sketch: “essentially the same strategy”)	p. 16; steps check out (my reading)
C7	Every state in the stabilised subspace of an AvN triple of n-qubit Paulis admits a parity AvN argument; V_S has dimension 2ⁿ⁻³ for independent triples	strong (proof; the dimension is cited)	Theorem 4 and the remark after it; my random tests n = 3, 4 agree
C8	Mermin’s and “the other All-vs-Nothing arguments … in the literature” come down to exhibiting AvN triples	weak (assertion)	p. 8; Peres–Mermin, for one, is not a local-Pauli stabiliser-state argument, though it is AvN over ℤ₂ in the general sense
C9	Box 25 of Pironio–Bancal–Scarani admits no parity AvN argument but is AvN over ℤ₃	weak (assertion; the equations are shown, the support is not)	§4.2; the six displayed equations are inconsistent mod 3 (my check); the rest is unverified
C10	”all known quantum examples of strong contextuality are of AvN type”	weak (assertion, uncited)	p. 17, v2 only
C11	The length-4 Liar cycle “corresponds exactly” to the PR box	strong (explicit identification)	§2 variable renaming; exact at support level (my check)
C12	Semantic paradoxes such as Liar cycles have the structure of contextuality; the paper’s cohomological tools “can be applied” to them	weak (analogy; application deferred)	§2, Discussion; for N ≥ 3 it does follow from Theorem 21 (my derivation), but not for N = 1, 2
C13	Reduction of reference and truth to Boolean equations preserves a paradox’s content	assertion by citation	§2, citing Wen, Cook, Walicki
C14	No bipartite quantum behaviour “of this kind” is strongly contextual	assertion by citation	p. 4, citing Lal “private communication” and Mansfield QPL 2014 informal proceedings; “of this kind” must mean the 2-party, 2-setting, 2-outcome scenario, since bipartite pseudo-telepathy (the magic-square game) is strongly contextual in larger scenarios (my remark)
C15	The first implication of Theorem 21 reverses over any ring with a “quotient module” notion of consistency	weak (assertion)	Discussion

Method

• Linearise the support.
  • The theory. For each context take all R-linear equations its support satisfies. The union is the model’s theory T_R(S).
  • AvN. The model is AvN_R when that theory has no global solution.
  • Aff S. Equivalently for fields, Aff S is strongly contextual. Aff S is the model whose context supports are the affine hulls.
• Define the obstruction abstractly. Free the support presheaf over R, lift 0 → F_{C̄₀} → F → F|_{C₀} to cochains (right-exact by flasqueness), and take the snake-lemma connecting map.
• Compare the two. The coefficient-sum argument makes any compatible family through 1·s₀ affine. The counit of Free ⊣ Forget evaluates it into Aff S, so vanishing cohomology produces a global section of Aff S, contradicting AvN.
• Produce quantum instances. Stabiliser algebra, conditions (A1) and (A2), turns three commuting Paulis into an odd-parity constraint on their product that no global assignment can meet.

Concepts

• Empirical model (this paper) — the support subpresheaf S ⊆ ℰ satisfying (E1)–(E3), not the probability table. LIT-016’s “empirical model” is the table.
• Logically contextual at s, LC(S, s) — s is in no compatible family. This is LIT-277’s “possibilistic non-extendability” under the later name.
• Strongly contextual, SC — LC at every section, i.e. S(X) = ∅.
• Univocal path — in the bundle diagram, a closed path through every fibre exactly once, i.e. a global assignment.
• AvN triple — Def 3 above.
• R-linear equation, theory T_R(S), AvN_R — Defs 5–6.
• Affine closure Aff S — per-context affine hull, flasque below the cover.
• CLC_R, CSC_R — “cohomologically” logically or strongly contextual: γ_{F_R S} ≠ 0 at some or at every section.
• Cohomological obstruction γ(r₀) — the image of r₀ ∈ F(C₀) ≅ Ȟ⁰(M, F|{C₀}) under the connecting map into Ȟ¹(M, F{C̄₀}).
• Liar cycle of length N — the equations x₁ = x₂, …, x_N = ¬x₁, each fibred over its variables.

Connections

• LIT-277 (AMB 2011, read in NOTE-250). This paper is its sequel and cites it as [5].
  • What it keeps. The Čech machinery and the free-module coefficients (now any ring R rather than ℤ).
  • What it changes. It makes the definition abstract (a connecting homomorphism) and replaces example-by-example computation with a class theorem.
  • What it drops. Every negative result: the Hardy false positive (which v1 still mentioned) and the §8 cover. It also drops LIT-277’s Conjecture 8.1 (completeness under symmetry and connectedness). The conjecture is not mentioned, although the Discussion’s “precise characterisation of what cohomology detects” is its descendant.
  • Verdict. The two are complementary, not successive.
• LIT-016 (Abramsky & Brandenburger). It supplies the sheaf formalism, the equivalence of no-signalling with flasqueness beneath the cover, and the hierarchy LC ⊋ SC. §4.1 completes, by hand, the GHZ(3) case that NOTE-016 recorded as sketched.
• LIT-265 (contextual fraction; Deferred). No overlap in content. Everything here is possibilistic: AvN and the cohomology classes certify CF = 1 (strong contextuality), and they grade nothing below it.
• The stabiliser and resource thread. It cites Howard et al. 2014 (contextuality as the “magic” for quantum computation) and Raussendorf–Briegel. Theorem 4 is stated for n-qubit Pauli stabiliser states only.
• Paradox literature. Wen 2001, Cook 2004 and Walicki 2009 (and in v1 Levchenkov 2000) supply the “paradoxes as equations” reduction. Nothing in the nucleation record covers semantic paradoxes, Liar sentences or self-reference: a text search of the literature and theory folders finds only unrelated uses of “paradox” and “liar” (conspiracy liars, LIT-203). No citation is warranted.
• Penrose, “On the cohomology of impossible figures” (1992). Noted as a conceptual resemblance only (Discussion).
• Later work (existence verified from arXiv abstract pages; not read, content unverified beyond what the abstracts say).
  • Carù, “On the Cohomology of Contextuality”, arXiv 1701.00656 (2017-01-03). Its abstract says it shows the obstruction “is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions … disproving a previous conjecture”, presumably LIT-277’s Conjecture 8.1. That is unverified.
  • Carù, “Towards a complete cohomology invariant for non-locality and contextuality”, arXiv 1807.04203 (2018-07-11).
  • Okay, Roberts, Bartlett & Raussendorf, “Topological proofs of contextuality in quantum mechanics”, arXiv 1701.01888 (2017-01-07). Its abstract says it treats Mermin-type parity proofs cohomologically, in an MBQC-oriented framework.
• Anthology. Nothing there on contextuality or cohomology (as in NOTE-250), so no ANTH- citation.

Bearing on the record

• THEORY-012 (Active). This paper supports it and sharpens its new “does not say” line. It contradicts nothing.
  • What the line gets right. The cohomological witness is sufficient, not necessary, and nothing here claims otherwise.
  • What it now leaves out. The witness is not merely incomplete: it is complete on a named class. Every All-vs-Nothing model, over any coefficient ring, has a non-vanishing obstruction on every section (Theorem 21). The class includes every n-qubit stabiliser state picked out by an AvN triple (Theorem 4), GHZ, Peres–Mermin, the PR box and ¬GCD Kochen–Specker covers.
  • The failures, located. The two known failures lie outside the class. Hardy is not strongly contextual, and its affine closure is the full event sheaf. The §8 cover is strongly contextual but has no AvN argument over any ring, which follows from Theorem 21.
  • Proposed wording. See the report line. It needs this paper filed as a LIT first.
• THEORY-014 (Proposed). There is a sharper parallel than NOTE-250 could draw, and it is my inference; the paper never mentions negativity.
  • The mechanism. Cohomology fails exactly where affine (signed, coefficient-sum-1) combinations of support sections can glue. Those are ring-valued “quasi-sections”, and the obstruction is non-zero exactly when even they cannot glue through s₀ (Prop 16 plus the coefficient-sum remark).
  • How it fits THEORY-014. It is a discrete, support-level analogue of THEORY-014’s “negativity is universal over ℝ; only its unavoidability is non-classical”. Over a field, AvN is exactly strong contextuality of the affine closure (Prop 9).
  • Status change. None warranted; this is an analogy.
• THEORY-015, THEORY-016. No bearing: this is Kochen–Specker/sheaf contextuality only, with no preparations and no Spekkens notion.
• THEORY-013. No bearing beyond the boundary NOTE-250 drew. No-signalling (flasqueness, E2) is built in.
• The logic topic. The paper earns a logic tag, fourth.
  • Why. Its arXiv cross-list is cs.LO. It uses a Galois connection between theories and models, it cites the Robinson joint-consistency theorem (v1 only), and it recasts Liar cycles.
  • The limit. What it proves about logic is only the support-level identification of Liar-4 with the PR box. “Semantic paradoxes have the structure of contextuality” is an analogy with one exact instance, and the cohomology of paradox is promised, not delivered.
  • For a future Liar/paradox LIT. This paper would be the natural anchor on the contextuality side. Kishida’s separate work on the subject is not this paper and was not read.
• ML practice. It carries nothing. This is algebraic topology and stabiliser theory for quantum foundations, and it does not belong in the Anthology.
• For filing.
  • Tags. contextuality, quantum-foundations, mathematics, logic.
  • Lineage. It extends LIT-277 and LIT-016.
  • Not proposed. philosophy-of-language, metaphysics and epistemology: the paradox material is a formal analogy, not an analysis of truth or reference.

Limitations

• Completeness on AvN, not a characterisation. Theorem 21 is one-directional. Whether every cohomologically strongly contextual model is AvN_R for some R is not addressed; the paper lists it as future work.
  • Where the known failures sit. All of LIT-277’s failures lie outside the AvN class. Hardy is not strongly contextual. The §8 cover is strongly contextual but, by Theorem 21, has no AvN argument over any ring.
• Only supports are used. It is blind to probabilistic contextuality, and to anything below CF = 1.
• Qubit stabiliser only. Theorem 4 needs ±1-eigenvalue Paulis relabelled into ℤ₂. No qudit or other-ring quantum family is given. The only non-ℤ₂ example (box 25, mod 3) is a no-signalling box, not a quantum model, and its details are unverified.
• Unsupported headline sentences.
  • “all known quantum examples of strong contextuality are of AvN type” (C10). Uncited.
  • “the other All-vs-Nothing arguments … come down to exhibiting AvN triples” (C8). Unshown.
  • The field-free converse (C15). Asserted only.
• Paradox. It is an analogy with one exact case.
  • What is borrowed. The reduction of truth and reference to Boolean equations is taken from the literature, not argued.
  • Where it breaks. The encoding does not produce a valid model for N = 1 (the Liar itself) or N = 2.
  • What is not given. No general correspondence theorem.
• Proof availability.
  • Prop 16. The published text (v2 = LIPIcs) proves Theorem 21 using Prop 16, whose proof it defers to “[2]”, the arXiv version, which v2 itself now replaces. The proof is in v1 only.
  • Connectedness. The theorem silently assumes a connected cover.
• Slips.
  • Prop 20. The proof’s first line reverses h’s direction.
  • Prop 9. It drops the subscript R.
  • The snake diagrams. The text calls both rows “short exact” without arguing the bottom row’s surjectivity. It is not needed.
  • The abstract. “contexuality”.
  • v1’s GHZ equations. They had the wrong right-hand sides for the stated state; corrected in v2.

Open questions

• The converse. Does CSC_ℤ imply AvN_R for some R? The §8 cover shows that SC does not imply AvN; whether CSC does is the question. A counterexample, or a proof, would settle “what cohomology detects”.
• Conjecture 8.1. LIT-277’s completeness conjecture, under symmetry and connectedness, is dropped here without comment. Per its abstract, Carù 1701.00656 disproves it (unverified).
• Quantum strong contextuality beyond AvN. Is there a quantum-realisable strongly contextual model with no AvN argument over any ring? That would refute C10. The §8 cover is not claimed to be quantum-realisable.
• Qudits. Do stabiliser AvN triples generalise to qudit Paulis with ℤ_d outcomes? The ring-general Theorem 21 would then apply at once. The paper does not say.
• Cohomology of paradox. Can the promised treatment handle the paradoxes that are not cycles (Yablo-type infinite chains, which would need an infinite cover) and the N = 1, 2 cases the support encoding cannot express?

Corrections to the seeded skim

• none (there was no seed or dossier)
• Identification note for filing. The journal-ref is 24th EACSL Annual Conference on Computer Science Logic (CSL 2015), LIPIcs 41, pp. 211–228, editor Stephan Kreutzer. The DOI 10.4230/LIPIcs.CSL.2015.211 resolves at doi.org (a 302 redirect) to the Dagstuhl DROPS landing page, whose citation metadata give this title, the five authors and pp. 211–228. No contact address was sent. The paper was not checked against Crossref, as the brief warned that it is not there. The arXiv categories are quant-ph (primary), cs.LO and math.AT. The affiliations are the Department of Computer Science, Oxford, and, for Lal, the Faculty of Philosophy, Cambridge (v2 only).
• The abstract’s “remarkable connection between contextuality and logical paradoxes” is an analogy plus one exact case, not a result. The body gives no theorem about paradoxes. It gives one page (§2): Liar cycles re-encoded as Boolean equations (following Wen, Cook and Walicki), the observation that any N − 1 of the N equations are consistent, and an explicit renaming showing that the length-4 Liar cycle’s equations are the PR box’s support. The cohomological treatment of paradox is explicitly deferred (“We plan to give an extended treatment … in future work”, §2; “We aim to give a detailed treatment of this ‘cohomology of paradox’ in future work”, Discussion).
• The abstract’s “All-vs-Nothing proofs … are witnessed by cohomological obstructions” is proved, but for strong contextuality and in one direction only. Theorem 21 has a complete proof, and the logical-contextuality analogue is stated with the proof only sketched (“essentially the same strategy”). The converse (does cohomology detect only AvN models?) is not claimed, and “a precise characterisation of what cohomology detects” is listed as future work.
• v1’s GHZ equations are wrong, and v2 fixes them silently. v1 §4.1 gives right-hand sides X₁⊕Y₂⊕Y₃ = 0, Y₁⊕Y₂⊕X₃ = 0, Y₁⊕X₂⊕Y₃ = 0, X₁⊕X₂⊕X₃ = 1. v2 gives 1, 1, 1, 0. I computed the support of (|000⟩+|111⟩)/√2 under X/Y measurements: XXX has even parity and the three XYY-type contexts odd, so v2 is right for the stated state. v1’s equations belong to (|000⟩−|111⟩)/√2. The contradiction (LHS sums to 0, RHS to 1) holds either way.
• Other v1 → v2 changes.
  • Material removed in v2 (the conference cut). §2.1–2.2 (the Bell table, “logical Bell inequalities” Σpᵢ ≤ N − 1 from Abramsky–Hardy, and its violation by 1/4). The Robinson Joint Consistency Theorem paragraph, which notes that binary gluing of theories holds while three pairwise-compatible theories need not glue, with the Specker triangle as the propositional counterexample. The Outline. The proof of Prop 16 (v1 Prop 5.7). The sentence “An example for which a false positive arises is the Hardy model” and the ¬GCD sentence in §5.3. The pointer to circuits with feedback (Berry/Esterel).
  • Material added in v2. Footnote 1: the base space is discrete, but “the interesting twisting occurs in the fibres, not in the base” (the Möbius strip). The Remark on why the obstruction lives in Ȟ¹.
  • Typos corrected in v2. v1 says the cochain map “C⁰(M,F) → C⁰(M,F_Ũ) is surjective”; v2 correctly has F|_U. v1’s second snake diagram ends its bottom row with ”→ 0”, which v2 drops.
  • Framing changes. In v1, “Mermin’s argument, and the other All-vs-Nothing arguments …” is the same, but v1 does not claim “all known quantum examples of strong contextuality are of AvN type”. That sentence is new in v2 (p. 17).