Ehtibar N. Dzhafarov and Janne V. Kujala (2016), Journal of Mathematical Psychology 74:11–33 — ARXIV-1511.03516
Key takeaways
- Variables by content and context. A variable is identified by what it measures (content) and the conditions it is recorded under (context). Variables in one context are jointly distributed and form a bunch. Variables in different contexts are stochastically unrelated: their joint probabilities are undefined, not zero. The variables sharing a content form a connection. Which is content and which is context is decided outside the theory, and changing it changes the answer.
- Direct influences are not contextuality. If a content’s copies have different distributions in different contexts, the system is inconsistently connected. That is treated as ordinary dependence on the context. Contextuality is defined as follows. A maximal coupling of a connection makes its copies coincide with probability Σ_v min_i Pr[R^i = v] (Theorem 3.3). The system is noncontextual if one coupling of all bunches achieves these coincidences for every connection at once (Definition 3.4).
- A linear program decides it (Theorem 4.1): noncontextual if and only if MQ = P has a nonnegative solution over the hidden outcomes.
- Cyclic systems (Theorem 5.1, Kujala & Dzhafarov 2016, cited): a rank-n binary cyclic system is noncontextual if and only if s_odd(⟨R_i^i R_{i⊕1}^i⟩) ≤ n − 2 + Σ|⟨R_i^i⟩ − ⟨R_i^{i⊖1}⟩|. Under consistent connectedness this is the Leggett–Garg, CHSH or KCBS form. Question-order data in which the probability of the same answer does not depend on order cannot be contextual. The singlet state at the CHSH angles gives 2√2 > 2.
- A measure from negative probabilities (Section 6). Dropping nonnegativity, a solution always exists (Theorem 6.1), for signalling and non-signalling systems alike. The least total variation of such a signed quasi-coupling, minus 1, is the proposed degree of contextuality. It generalizes de Barros and Oas. In the paper’s rank-2 example it falls linearly, as 2(1 − p), to noncontextuality.
- Unlike the sheaf account (LIT-016), bunches never overlap, so the question is not whether overlapping marginals glue. It is whether the bunches are compatible with maximal couplings of the connections.
Standing in the record
Filed on 2026-10-09 at the owner’s request, as one of the works in the reference list of the owner’s working manuscript (October 2026) that the record did not yet hold. See the curation entry of that day.
Against the record’s contextuality theories. For consistently connected systems a maximally connected coupling is a global section, so CbD agrees with the sheaf-theoretic criterion of LIT-016 and THEORY-012 there (the reader’s inference, NOTE-600). It extends that criterion to data with context-dependent marginals, the case THEORY-012 leaves to CbD. Its Theorem 6.1 (a signed maximally connected quasi-coupling always exists) is a third instance of the shape in THEORY-014. Its measure is not the contextual fraction of LIT-265: both are linear programs, but the contextual fraction is defined only without signalling, and the two are not compared. It states the cyclic criterion that THEORY-013 and LIT-264 rest on, but does not prove it, so THEORY-013’s promote_when is not met by this reading.