Source
Budroni, Cabello, Gühne, Kleinmann & Larsson (2022), LIT-263, §IV.A.2, §V.B.1–2 (Eqs. 78–79) and App. A.6, read in NOTE-844. The theorem is Vorob’ev (1962), with Kellerer (1964) and Malvestuto (1988), none held.
What was actually shown
Represent a marginal scenario as a hypergraph: observables are vertices, contexts are hyperedges. Call it acyclic when its hyperedges can be ordered E_1, …, E_n so that each E_i ∩ (E_1 ∪ … ∪ E_{i−1}) lies inside a single earlier E_j (the running-intersection property, Eq. 78). An equivalent condition is being the clique hypergraph of a chordal graph. The review states Vorob’ev’s theorem: “Any marginal scenario represented by an acyclic hypergraph admits a joint probability distribution”. The surrounding text adds the hypothesis that the given distributions coincide on their intersections.
The mechanism is gluing. For A − B − C with p₁(a,b) and p₂(b,c) agreeing on B,
p(a,b,c) = p₁(a,b)p₂(b,c)/p(b) = p₁(a|b)p₂(c|b)p(b) (Eq. 79),
which is Fine’s construction for CHSH. The running-intersection order lets this be repeated one context at a time.
Consequences drawn in the review:
- For sharp measurements, pairwise compatibility implies joint compatibility (Specker’s principle). So a chordal compatibility graph always admits a noncontextual model.
- The smallest compatibility graphs that can be contextual contain an induced cycle of length ≥ 4. The square is CHSH, which needs dimension 4 with sharp measurements. The pentagon is KCBS, the minimum in dimension 3 (§IV.A.2, §V.B.2, App. A.6).
- For nondegenerate observables, commutativity is transitive. The compatibility graph is then a disjoint union of cliques, and no contextuality is possible (Correggi & Morchio 2002; §V.F.2).
What could have failed: a tree-like scenario could have carried some higher-order inconsistency that pairwise agreement misses. The theorem says it cannot.
What this does not say
- Not proved in the source. The review states the theorem, cites it and illustrates it on a path of three variables. Its strength here is that of a well-cited classical result, read secondhand.
- Not a characterization. Acyclicity is sufficient for every consistent family to glue, not necessary for a particular family to glue. Many behaviours on cyclic scenarios are noncontextual.
- Not about data whose marginals disagree across contexts. Local consistency is a hypothesis. With context-dependent marginals the question changes (THEORY-013; the review’s §IV.C).
- Not about Spekkens’s generalized contextuality. A qubit is preparation contextual (THEORY-011), with no compatibility structure of this kind involved.
Connections
- THEORY-012 states noncontextuality as the existence of a global section. This account says on which covers a global section exists for every consistent family: the acyclic ones. Put in sheaf terms, which is the reader’s gloss and not the review’s, acyclic covers carry no obstruction.
- CLAIM-141 sets causal-loop consistency beside sheaf contextuality as local-to-global problems. Vorob’ev’s theorem is the measurement-side statement that obstructions need cycles.
- THEORY-309, from the same reading, is the operational precondition: the hypergraph means something only once contexts are fixed by more than joint measurability.