The claim
Take a system of finitely many observables q, each with finitely many outcomes, measured in contexts c. In context c, the observables it contains have a joint law P_c (the bunch), and R_q^c is q’s random variable there. The connection of q is {R_q^c : c contains q}. The system is consistently connected when, for every q, all R_q^c have the same distribution.
- Sheaf criterion (LIT-016, THEORY-012). The system is noncontextual when there is a law G on joint assignments of one outcome to every observable whose marginal on each context c is P_c.
- Contextuality-by-Default (LIT-777, Def. 3.4). The system is noncontextual when there is a coupling of all the bunches, one jointly distributed copy T_q^c for each R_q^c, with each bunch’s law preserved, whose subcoupling on every connection is a maximal coupling.
On a consistently connected system, the two agree.
Proof
By Theorem 3.3 of LIT-777 (Thorisson), the maximal coupling of a connection has Pr[T_q^(c₁) = … = T_q^(c_k) = v] = min_i Pr[R_q^(c_i) = v] for every value v. Under consistent connectedness each minimum is Pr[R_q = v], and these sum to 1. So a coupling of a connection is maximal exactly when its copies are equal with probability 1.
(CbD ⇒ sheaf.) Let T be a maximally connected coupling. On an event of probability 1, T_q^c = T_q^(c′) for every q and every pair of contexts containing it, so T_q is well defined there. The law G of (T_q)q is a law on global assignments. Its marginal on context c is the law of (T_q^c)(q∈c), which is P_c because T preserves each bunch.
(Sheaf ⇒ CbD.) Let G be a global law, and draw (X_q)q from it. Set T_q^c = X_q for each context c containing q. Each bunch (T_q^c)(q∈c) has law G’s marginal on c, which is P_c. Each connection’s copies are equal, so its coincidence probability is 1 and its subcoupling is maximal.
The argument uses Theorem 3.3 for connections of any size k, so it covers observables in more than two contexts. CLAIM-069 raises that case.
What it does not say
- Anything about inconsistently connected systems. There the criteria can diverge widely. LIT-844 finds 71.1% of instances contextual by Contextuality-by-Default and 0.148% by the sheaf criterion that allows for signalling. Which to use there is CLAIM-037’s question.
- That the sheaf criterion handles signalling. It does not apply when marginals shift. LIT-843’s causal-past route is a different answer.
- That a source states it. NOTE-600 calls it “my inference rather than the paper’s”. This claim is that inference, proved.