NOTE-016: The sheaf-theoretic structure of non-locality and contextuality

Contribution

The paper gives a single, Hilbert-space-free formalism for Bell non-locality and Kochen–Specker contextuality. Measurements, contexts (a “measurement cover”) and outcome sections form a sheaf ℰ. Empirical models are compatible families of the presheaf D_R ℰ of R-distributions, with R a commutative semiring. Within it the paper proves:

• non-locality/contextuality is exactly the non-existence of a global section;
• the question is linear-algebraic, via an incidence matrix;
• there are three strictly nested strengths of contextuality;
• over the reals every no-signalling model has a global section;
• strong contextuality coincides with maximal contextuality (non-contextual fraction 0) and reduces to CSP/SAT;
• Kochen–Specker results are generic strong contextuality, which Bell-type covers can never exhibit;
• quantum mechanics satisfies no-signalling for arbitrary commuting families (a generalized no-signalling theorem).

Key insight

No-signalling is the sheaf compatibility condition. Locality and non-contextuality are the gluing (global-section) condition. Contextuality is the gap between the two: data that agree on every overlap of contexts but do not glue to one joint distribution over all measurements. The “incompatibility” of measurements is then derived, as the provable non-existence of a joint distribution, rather than postulated (§10). Changing the semiring R (booleans, ℝ≥0, ℝ) changes the strength of the question, and semiring homomorphisms carry solutions only one way (ℝ≥0 → 𝔹), which is what makes the hierarchy.

Assumptions

• A finite set of measurements X and a finite outcome set O, the same O for every measurement (§2.2; footnote § says the generality is not needed).
• A measurement cover 𝓜: a family of subsets of X with ⋃𝓜 = X that is an anti-chain (maximal contexts only) (§2.4).
• Empirical models are no-signalling by definition: a family {e_C}{C∈𝓜}, e_C ∈ D_R ℰ(C), with e_C|{C∩C′} = e_{C′}|_{C∩C′} (§2.5). “We shall only consider no-signalling models in this paper.”
• Distributions: finite-support functions d: X → R with Σd = 1, over a commutative semiring R. The three used are ℝ≥0 (probabilistic), 𝔹 = ({0,1},∨,∧) (possibilistic) and ℝ (signed or “negative” probabilities) (§2.3).
• Hidden-variable models (§8) assume λ-independence (h_Λ does not depend on the context) and parameter independence (compatibility of {h^λ_C} for each λ). Factorizability is h^λ_C(s) = ∏{m∈C} h^λ_C|{m}(s|_m).
• Quantum representations (§9): finite-dimensional Hilbert space, dichotomic observables for simplicity, and a context is a maximal pairwise-commuting subset.
• The notion of contextuality is Kochen–Specker / measurement-contextuality. It is not Spekkens’ generalized (preparation/transformation) noncontextuality, and the paper does not address that notion.

Key results

• Prop 3.1 — A global section of e gives a local/non-contextual deterministic hidden-variable model realizing e (a distribution on the global assignments O
• Prop 4.1 — Solutions over R of the augmented system M′X = V′ (the incidence matrix plus a row of 1s) are in bijection with the global sections of e. For Bell-type (n,k,l) covers the augmentation is unnecessary (stated, not proved).
• Incidence matrix size (§4.2) — for (n,k,l) Bell scenarios the matrix is (kl)^n × l^{kn}, with l^{(k−1)n} ones per row. The (2,2,2) matrix is 16×16 of rank 9.
• Prop 4.2 (Bell) — The model with rows (a,b): ½,0,0,½; (a′,b), (a,b′): ⅜,⅛,⅛,⅜; (a′,b′): ⅛,⅜,⅜,⅛ has no global section over ℝ≥0. Proof: four of the 16 equations give LHS ≥ ½ against 3/8.
• Prop 4.3 (Hardy) — The possibilistic Hardy model (zeros at (a′,b)(0,0), (a,b′)(0,0) and (a′,b′)(1,1)) has no boolean global section. Proof: 4 clauses.
• Prop 4.4 — If MX = V has a solution over ℝ≥0, then MX = V̂ (the support) has one over 𝔹, because ℝ≥0 → 𝔹 is a semiring homomorphism. The converse fails: the Bell model’s support is extendable, by [25] (Mansfield–Fritz: Hardy models are complete for (2,2,2), and at least 3 zeros are needed). Hence possibilistic non-extendability is strictly stronger than probabilistic.
• Prop 5.1 — Every probabilistic model is linearly determined by e^{(0)}, its values on sections with no outcome equal to 1, over partial contexts 𝒰 = {U : ∃C∈𝓜, U ⊆ C}.
• Prop 5.2 — The span of the no-signalling models has dimension ≤ D := Σ_{U∈𝒰} (l−1)^{|U|}. Prop 5.3 — The span of the non-contextual (global-section) models has dimension ≥ D, via explicit linearly independent columns v_{U,s}.
• Theorem 5.4 — For any cover, the linear spans of the non-contextual models and of the no-signalling models coincide, with dimension D. Thm 5.5 — For every (no-signalling) probabilistic model, MX = V has a real solution. Prop 5.6 — rank M = D.
• Theorem 5.9 — “Probability models have local hidden-variable realizations with negative probabilities if and only if they satisfy no-signalling.” (⇐ is Thm 5.5; ⇒ is Prop 5.8, since restrictions of a global section are always compatible.) Corollary drawn: negative probabilities alone cannot characterize QM, because they give the whole no-signalling set, including PR boxes (explicit signed solution in §5.2).
• Prop 5.7 — For a homogeneous cover (all contexts of size n, each j-subset in N_j contexts, p = |𝓜|): D = Σ_{j=0}^{n} C(n,j)·p(l−1)^j / N_j. Worked values:
  • (n,k,l) Bell scenarios: D = (k(l−1)+1)^n. This matches Pironio’s D−1; the difference is marginalization over ∅.
  • 18-vector KS cover: D = 118, against ambient dimension 144.
  • Peres–Mermin square: D = 34, against ambient 48.
  • (n,2,2) scenarios: rank 3
• Strong contextuality (§6) — Define S_e := {s ∈ O^X : ∀C, s|_C ∈ supp(e_C)}. The model e is strongly contextual iff S_e = ∅. Strong contextuality implies possibilistic non-extendability, because extendability implies S_e ≠ ∅. Hardy is not strongly contextual (witness {a↦1, a′↦0, b↦1, b′↦0}), and neither is Bell.
• Prop 6.1 — The GHZ models of type (n,2,2) are strongly contextual for all n ≥ 3. Full proof for n = 4k; n = 4k+2 and odd n ≥ 5 “proceed similarly”; n = 3 via Mermin’s instruction-set argument, sketched. This gives the strict hierarchy Bell < Hardy < GHZ.
• Prop 6.2 (Lal, private communication) — The only strongly contextual no-signalling (2,2,2) models are the PR boxes. Stated without proof.
• Prop 6.3 — Strongly contextual ⇔ maximally contextual, i.e. the non-contextual fraction (the sup of λ over decompositions e = λL + (1−λ)q, with L local and q no-signalling) is 0. Props 6.4–6.5 — Maximal contextuality (and maximal non-locality in Bell scenarios) ⇔ the CSP (X, O, {supp e_C}) has no solution. For dichotomic outcomes this is ⇔ φ_e = ⋀C ⋁{s∈supp e_C} ψ_s is unsatisfiable.
• Monotonicity (§7) — supp e ⊆ supp e′ and e′ strongly contextual ⇒ e strongly contextual. This is the basis of generic (model-independent) results.
• Prop 7.1 (parity condition) — If φ_𝓜 = ⋀_C ONE(C) has a global section, then every common divisor of {|𝓜(m)|} divides |𝓜|. For example, each measurement in an even number of contexts with |𝓜| odd means no section. This generalizes “parity proofs” (e.g. the 18-vector cover: 9 contexts, each vector in 2).
• Prop 7.2 — For the maximal-clique cover 𝓜_G of a graph G, φ_𝓜 has a global section ⇔ G has a stable transversal. Kochen–Specker graph: a faithful orthogonal co-representation in ℝ^d, all maximal cliques of size d, and no stable transversal. Thm 7.3 is cited from Lovász–Saks–Schrijver, not proved here: a graph on n nodes whose complement is (n−d)-connected has a faithful orthogonal co-representation in ℝ
• Theorem 8.1 — For any D_R, e has a realization by a factorizable hidden-variable model ⇔ e has a global section. Proof: the product of single-measurement marginals is a distribution on O^X for each λ; averaging over h_Λ gives the section.
• Prop 9.2 (generalized no-signalling) — For any quantum state ρ and any contexts C, C′ of commuting projective observables, ρ_C|{C∩C′} = ρ{C′}|_{C∩C′}. The paper also notes a reduction to the standard theorem via Tsirelson’s Thm 9.1, which it credits to a referee.
• Prop 9.3 — A cover is of Bell type iff it is the maximal-clique family of the complement of an equivalence relation (so incompatibility is transitive). Prop 9.4 — For any quantum representation of a Bell-type cover and any section s, some product state has s in its support. So there is no Kochen–Specker-type (state-independent) strong contextuality theorem for Bell-type scenarios. Only model-specific ones exist, such as GHZ.

Claims

id	claim	strength	support
C1	An empirical model has a factorizable hidden-variable realization iff it has a global section of D_R ℰ over the cover	strong	Thm 8.1 + Prop 3.1, full proofs
C2	For any finite cover, the non-contextual and no-signalling models span the same real subspace, of dimension D = Σ_{U∈𝒰}(l−1)^{	U	}
C3	”Negative-probability” local HV realizations exist iff the model is no-signalling, so negative probabilities cannot single out QM	strong	Thm 5.9 (Thm 5.5 + Prop 5.8); PR-box signed solution in §5.2
C4	Possibilistic non-extendability is strictly stronger than probabilistic non-extendability	moderate	Prop 4.4 proved; strictness rests on the Bell model’s support being extendable, which is asserted as “easy to show directly” and via [25]
C5	Strong contextuality is strictly stronger than possibilistic non-extendability	strong	§6 argument + explicit Hardy witness assignment
C6	GHZ models are strongly contextual for all n ≥ 3, completing the hierarchy Bell < Hardy < GHZ	moderate	Prop 6.1: full proof for n = 4k; other cases “similarly” or sketched (n = 3 via Mermin)
C7	Strongly contextual ⇔ maximally contextual (non-contextual fraction 0) ⇔ the associated CSP/SAT instance is unsatisfiable	strong	Props 6.3–6.5, proofs (6.3 relies on Thm 8.1)
C8	The only strongly contextual (2,2,2) no-signalling models are the PR boxes	weak	Prop 6.2, private communication from R. Lal, no proof given
C9	A common divisor of the |𝓜(m)| not dividing |𝓜| rules out a Kochen–Specker assignment (generalized parity proof)	strong	Prop 7.1, proof (sufficient condition only)
C10	Kochen–Specker graphs realized by quantum observables give state-independent strong contextuality	moderate	Props 7.2, 9.2 and §9.2 argument; realizability condition via Thm 7.3, cited not proved
C11	Quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just tensor factors	strong	Prop 9.2, direct trace computation (finite-dimensional, projective)
C12	No Kochen–Specker-type (generic) theorem exists for Bell-type covers	strong	Props 9.3–9.4, proofs
C13	The sheaf formalism “opens the door” to powerful sheaf-theoretic (e.g. cohomological) methods	weak	assertion + pointer to preliminary cohomology work [53] (§10)

Method

Mathematical. The paper formalizes the operational scenario as a sheaf of events ℰ: P(X)^op → Set, ℰ(U) = O^U, which is trivially a sheaf. It composes ℰ with the distribution functor D_R to get the presheaf of distributions, whose restriction maps are marginalization. The question “does e glue?” becomes the linear system MX = V, where M is the 0/1 incidence matrix of the restriction map O^X → ∐_C O^C. The system is read over ℝ≥0 (a linear programme), over 𝔹 (SAT/CSP) or over ℝ (unconstrained). Dimension counting (Props 5.1–5.3) settles the real case in general. Explicit small sub-systems give the Bell and Hardy no-go proofs, and parity/graph combinatorics give the Kochen–Specker ones.

Concepts

• measurement cover 𝓜 — an anti-chain of subsets of X covering X. Its elements are the maximal contexts of jointly performable measurements.
• section / event — s: U → O. ℰ is the sheaf of events.
• empirical model — a compatible family {e_C ∈ D_R ℰ(C)}_{C∈𝓜}. Compatibility is no-signalling here, so the term always means a no-signalling model.
• global section — d ∈ D_R ℰ(X) with d|_C = e_C for all C. It is the same thing as a joint distribution over all measurements that marginalizes correctly (Fine-style extendability).
• probabilistically / possibilistically non-extendable — no global section over ℝ≥0 / the support has none over 𝔹. The paper also says “probabilistic / possibilistic non-locality”.
• strongly contextual — S_e = ∅: no global assignment is consistent with the support in every context.
• non-contextual (local) fraction; maximally contextual — the sup of λ in e = λL + (1−λ)q; maximally contextual means that sup is 0.
• factorizable hidden-variable model — λ-independent, parameter-independent (compatible) and product-form per λ. It subsumes Bell locality and “non-contextuality at the level of distributions”.
• Kochen–Specker graph — a graph with a faithful orthogonal co-representation in ℝ^d, all maximal cliques of size d, and no stable transversal.
• generalized no-signalling — marginals of a quantum state on a commuting family do not depend on which compatible family it is measured with.

Connections

Builds on: Fine 1982 (joint distributions ⇔ Bell inequalities), which it puts into sheaf form. It also builds on Abramsky’s “Relational hidden variables and non-locality” [18] (the possibilistic case), Brandenburger–Keisler and Brandenburger–Yanofsky on hidden-variable properties, and Pironio’s / Basoalto–Percival’s transfer matrices and dimension count, which it generalizes from Bell scenarios to arbitrary covers. It credits Isham–Butterfield [57] for the insight that Kochen–Specker is non-existence of global sections of a presheaf, and it separates itself from the topos approach (Isham, Döring, Heunen–Landsman–Spitters). The topos approach uses the operator-algebraic spectral presheaf, has no distribution functor or measurement covers, and does not treat locality or extendability. The paper compares itself with Cabello–Severini–Winter [8] (the graph-theoretic approach, events rather than measurements). Descendants it points to: the Čech-cohomology obstruction of Abramsky–Barbosa–Mansfield [53]. Its notion is not Spekkens’ generalized noncontextuality (dossier items 14, 17, 71). The two should be kept apart.

ML descendants: checked, and much weaker than the dossier implies.

• Belfiore & Bennequin, “Topos and Stacks of Deep Neural Networks” (2106.14587v3; ANTH-LIT-698). Its bibliography contains this paper as [AB11], “Samson Abramsky and Adam Brandenburger … New Journal of Physics, 13(11):113036, 2011”. The single in-text citation (Introduction, p. 4) is in a list of prior uses of sheaves and cosheaves for information networks: Ghrist–Hiraoka, Curry, Robinson–Joslyn, “and Abramsky et al. specially for Quantum Information [AB11]”. A text search of the whole 152-page monograph for “Abramsky”, “contextuality”, “non-local”, “Kochen” and “Isham” finds no other occurrence. The monograph’s machinery is Grothendieck topoi, fibrations/stacks over a DNN’s site, Martin-Löf type theory and homotopy. It uses nothing specific to this paper: no measurement covers, no distribution presheaf, no global-section obstruction. So it cites this paper as related work; it does not build on it.
• Sheaf neural networks. Hansen & Gebhart, “Sheaf Neural Networks” (2012.06333), was read in full (6 pp.). Bodnar et al., “Neural Sheaf Diffusion” (2202.04579, 29 pp.), was text-searched. Neither mentions Abramsky or contextuality. Both descend from cellular sheaves and sheaf Laplacians (Hansen–Ghrist, Curry), not from this paper. Other sheaf-NN papers were not checked; any claim of a lineage there is unverified.
• What is genuinely shared is the abstract “local consistency without global consistency” pattern (compatible families vs. global sections). That is common to all sheaf theory; it did not originate here.

Bearing on the record

This is a quantum-foundations / applied-category-theory paper, and it carries no instruction for ML practice. It should not be cited from any anthology practice or theory. The one existing cross-link is the anthology’s reading of Belfiore–Bennequin (ANTH-LIT-698), which lists “Abramsky-Brandenburger (item 75)” among related work. That is accurate as a description of B&B’s related-work paragraph, and it should not be read as a technical dependency. If a record document or curation entry calls this paper “the bridge” between the quantum cluster and topos/sheaf ML, that overstates a single related-work citation and should be softened. The substantive content (Thms 5.4/5.9, 8.1; the hierarchy; Prop 9.4) bears on the quantum-foundations cluster in this reading record (nucleation LIT-016 / NOTE-016). A reading of the Spekkens items should note that the two senses of “contextuality” differ.

Limitations

• Finite only: finite X and O, one outcome set for all measurements. The measure-theoretic case is flagged as future work (§10.1).
• No-signalling is built into the definition of an empirical model, so the framework says nothing about signalling data, e.g. real experimental data with disturbance. (Handling that is the aim of later work such as Contextuality-by-Default; this paper does not address it.)
• Several headline results rest on less than a full proof:
  • Lal’s PR-box characterization (Prop 6.2) is unproved here.
  • The Bell model’s support being possibilistically extendable, which is needed for the strictness of C4, is asserted and referred to [25].
  • GHZ strong contextuality is fully proved only for n = 4k; the other cases are sketched.
  • The realizability of KS graphs leans on a cited graph-theory theorem.
• The abstract’s “sheaf theory” is mostly vocabulary: ℰ is trivially a sheaf, D_R ℰ is a presheaf, and all proofs are elementary linear algebra and combinatorics. The “powerful methods of sheaf theory” (cohomology) are promised, not used (C13).
• The quantum side (§9) is restricted to finite-dimensional, projective, mostly dichotomic observables.
• The hierarchy is between three qualitative levels. The paper gives no quantitative measure beyond the non-contextual fraction, which it uses only at its extreme value 0.

Open questions

• A measure-theoretic (infinite-outcome) generalization of Thms 5.4, 5.9 and 8.1 (flagged in §10.1).
• Whether cohomological invariants can detect strong contextuality completely. [53] gives witnesses for salient examples only, and this paper does not ask whether the obstruction is also necessary. What later work found is unverified here.
• A necessary-and-sufficient combinatorial condition for KS-type covers. Prop 7.1 is only sufficient for non-existence.
• A quantitative theory interpolating between the levels (the contextual fraction as a graded measure), and a characterization of which covers admit quantum strongly contextual models.
• A published proof of Lal’s result, and its extension beyond (2,2,2).

Corrections to the seeded skim

• None of substance: the dossier’s outline matches the text. Refinements: (a) the paper’s own term is the “non-contextual fraction”, and it appears here only as a tool to prove strong ⇔ maximal contextuality (Prop 6.3). The quantitative “contextual fraction” programme is later work, not this paper. (b) The hierarchy is stated in the Introduction as holding between properties (“strong contextuality implies possibilistic non-locality, which implies probabilistic non-locality”) and between models (Bell < Hardy < GHZ). The dossier gives only the model-level version.
• The dossier says Kochen–Specker graphs give generic strong contextuality ”(§6–7)”. The quantum realizability that makes them quantum witnesses is in §9.2, and only for vector sets in which every vector lies in an orthonormal basis drawn from the set. §7 alone gives only the combinatorial half.
• The dossier’s “ML link: item 69 cites this paper” (item 69 = Belfiore–Bennequin, ANTH-LIT-698) is verified, but it is thinner than “bridge” suggests. The paper is cited exactly once, in a related-work list (p. 4), and nothing in that monograph uses it technically. Neither of the two main sheaf-neural-network papers checked cites it at all (see Connections).