Source
Dzhafarov & Kujala (2017), Contextuality-by-Default 2.0 (LIT-831), Section 4 and Examples 1–3, as read in NOTE-643 (claims C4, “strong (immediate from the definitions)”, and C6, “strong (counterexamples)”).
What was actually shown
Contextuality-by-Default gives each measurement a separate random variable in each context. A connection is the set of variables that measure one content in different contexts. CbD 2.0 calls a system noncontextual when some coupling of all its variables agrees with each context’s joint distribution and is multimaximal on each connection. Multimaximal means every subset of the connection is maximally coupled, which is equivalent to every pair being maximally coupled (LIT-831 Theorem 5).
Coarse-graining can create contextuality (Examples 1–3). Take one connection of three 3-valued variables, each alone in its context, with masses (0, ½, ½), (½, 0, ½) and (½, ½, 0) on {a, b, c}. A maximal coupling of a pair puts on agreement the sum of the smaller masses, here ½: at c for the first two variables, at b for the first and third, and at a for the second and third. The three events {X = Y = c}, {X = Z = b} and {Y = Z = a} are pairwise disjoint, since any two of them would require one variable to take two values. Three disjoint events cannot each have probability ½, so no coupling is pairwise maximal, and the system is contextual. In LIT-831 this connection is the lumping of a 6-valued one that has multimaximal couplings (Examples 2 and 3).
A witness of the record’s own shows the same thing. Split each value in two: X has mass ½ on b and ½ on c, Y has ½ on a′ and ½ on c′, and Z has ½ on a and ½ on b′. No two variables share a value, so every pairwise maximum is 0 and every coupling is multimaximal. That system is noncontextual. Lumping a with a′, b with b′ and c with c′ gives the contextual system above.
Where the two versions agree (Section 4). For a connection of two variables, the only subset of two or more variables is the whole connection, so a maximal coupling of the connection (CbD 1.0) and a multimaximal one (CbD 2.0) are the same thing. Cyclic systems have such connections throughout. In a consistently connected system every connection’s variables are identically distributed, so both versions require them to be equal with probability 1, and both reduce to the traditional definition.
What could have come out otherwise: multimaximality could have survived the lumping of values. It does not, because lumping can raise each pair’s maximal agreement, and the raised agreements of different pairs can fall at values that no single outcome reaches at once.
What this does not say
- That maximal and multimaximal couplings disagree on larger connections. LIT-831 says they do on Kochen–Specker systems, and that the 1.0 definition is not hereditary, but gives no worked example. That stays in THEORY-168.
- Anything about dichotomizations. The nominal-dominance theorem for all splits of a content-sharing pair (LIT-790, Theorem 4.6) is THEORY-168’s, and is waiting on a checked second proof.
- That a verdict is wrong. The papers present the representation as the analyst’s choice. This result says only that the choice matters.
- Anything about the sheaf framework. There, coarse-graining outcomes cannot raise the contextual fraction (LIT-265, Theorem 2). The contrast is between frameworks, and these papers do not draw it.