Source
Dzhafarov & Kujala (2017), “Probabilistic Foundations of Contextuality”, Fortschritte der Physik, arXiv 1604.08412v4, LIT-836, read in full in NOTE-623. The Kochen–Specker incidence matrices of Fig. 1 were not checked.
What was actually shown
Transitivity (§I.3). A random variable is a triple of domain probability space, codomain and measurable function, and two random variables are jointly distributed iff their domain spaces coincide (Eq. 7). If X, X′ are jointly distributed and so are X′, X″, then X, X″ are, since all three share X′‘s domain. So if the graph of contexts linked by shared variables has a path through every context, the traditional system has an overall joint distribution. The paper exhibits such paths for KCBS, KS-4D and KS-3D.
The reductio (§I.4). A contextuality contradiction shows that no joint distribution of the measurements has the observed context marginals. In the traditional representation such a joint distribution exists by transitivity. So what fails is the premise that made the contexts overlap: that a measurement of q is one random variable whatever else is measured with it (“Noncontextual Identification”).
What follows (§II). Index each variable by content and context. Contextuality is then whether some coupling of the now-disjoint contexts makes each content’s copies satisfy a property C. For binary systems C is multimaximality, and Theorem II.3 shows that a coupling of one content’s copies is multimaximal iff each adjacent pair, ordered by Pr[R = 1], is maximally coupled.
What could have come out otherwise: joint distributedness could have failed to propagate along chains of contexts. It is forced by the definition of a random variable.
What this does not say
- That the sheaf account is mistaken. Abramsky & Brandenburger (THEORY-012) keep one variable per measurement and ask whether context marginals glue to a global section; that is a statement about a presheaf over a cover, not about random variables on one space. The argument targets the random-variable reading and does not engage the sheaf formulation (NOTE-623, Limitations). Under consistent connectedness the two accounts agree (NOTE-600).
- That contextuality is an artefact. Under Contextuality-by-Default consistently connected Kochen–Specker and Bell systems are still contextual; the account moves the refuted assumption, not the phenomenon.
- That inconsistent connectedness is universal outside physics. Cited, not shown here (C6, weak).
- That the §II.4 properties are proved. Heredity, reduction to the traditional notion and agreement on cyclic systems are asserted as trivial.
Connections
- QUESTION-067 asks which noncontextuality assumption a test of pragmatic judgements would refute. This account answers part of it for the random-variable reading: what fails is the identification of a judgement with one variable regardless of what it is made with. It says nothing about Spekkens’s preparation notion, which the question also names.
- THEORY-168 (in Contextuality-by-Default the verdict depends on the couplings imposed) uses the multimaximality that Theorem II.3 characterises; CLAIM-069 cites this paper for it.
- THEORY-013 (behavioural data show no contextuality once marginals are separated) rests on the same programme.