Source
Dzhafarov, Cervantes & Kujala (2017), LIT-790, Sections 1, 3–5 and the supplement (NOTE-640). Dzhafarov & Kujala (2017), LIT-831, Sections 1, 4 and 6 (NOTE-643). Dzhafarov & Kujala (2016), LIT-777, Definition 3.4 (NOTE-600).
What was actually shown
Contextuality-by-Default gives each measurement a separate random variable in each context. A system is noncontextual when some coupling of the contexts restricts, on each content, to a prescribed coupling of that content’s copies. The verdict therefore depends on two choices: which coupling is prescribed, and which variables are in the system. Each choice has been made in more than one way:
- Maximal vs multimaximal couplings. CbD 1.0 (LIT-777) prescribes a maximal coupling of each whole connection. CbD 2.0 (LIT-831) prescribes a multimaximal one, with every subset (equivalently every pair) maximally coupled. The two agree for connections of two variables, hence for cyclic systems, and for consistently connected systems. The authors say they disagree on Kochen–Specker systems with more variables per connection. They state, without an example, that the 1.0 definition is not hereditary; the note of 2026-10-29 gives one.
- Coarse-graining non-binary variables. Under multimaximality, coarse-graining can turn a noncontextual system into a contextual one. This is proved by example: a 6-valued connection with two multimaximal couplings becomes, after lumping, a 3-valued connection with none (LIT-831, Examples 1–3).
- Which dichotomizations to include. The canonical representation (LIT-790) replaces each variable by binary splits, and the analyst chooses which coarsenings to add. With all splits of two content-sharing k-valued variables, the pair is noncontextual if and only if Pr[R = x] < Pr[R′ = x] holds for at most one value x, in one direction or the other (Theorem 4.6, proved in the supplement). With only the k one-value splits, the same pair is always noncontextual. For continuous densities with all splits, any difference in distribution is contextual (Section 5, an argument rather than a theorem).
What could have come out otherwise: Theorem 4.6 could have found that the all-splits system of one pair is always noncontextual, as the unsplit pair is. The proof shows instead that the maximal couplings of the 1- and 2-splits over-determine the k × k joint table.
What this does not say
- Not that CbD is inconsistent. Each version is internally coherent. The authors adopt multimaximality and the canonical form deliberately, and present the expansion as the analyst’s statement of interest.
- Not that every verdict in the CbD literature changes. Binary cyclic systems, which carry most published analyses, are judged the same under 1.0, 2.0 and the canonical form (their variables are already binary, and each connection has two). THEORY-013’s findings are of this kind. They are untouched for binary data. They do not carry over to multi-valued responses analysed canonically, where Theorem 4.6 makes contextuality easy to find.
- Not a statement about the sheaf framework. There, a model is no-signalling by definition, and coarse-graining outcomes cannot raise the contextual fraction (LIT-265, Theorem 2). The contrast concerns how each framework handles coarse-graining, and the papers do not compare them.
- Not a rule for choosing. The papers suggest that intervals or cuts may be more natural for ordered values. They do not develop it.
Note of 2026-10-19: the proved part filed apart
Item 2 above, the coarse-graining failure, and the agreement of CbD 1.0 and 2.0 stated in item 1 are now THEORY-722, Active, sourced to LIT-831. What keeps this THEORY Proposed is the rest of the title’s first clause. LIT-831 asserts that maximal and multimaximal couplings disagree on larger connections, with no worked example, and Theorem 4.6 is waiting on a second proof. The promote_when says what would settle both.
Note of 2026-10-29: Theorem 4.6 re-proved, and the CbD 1.0 non-heredity case worked
Theorem 4.6, by a second route. With all splits of R and R′ (values 1, …, k, with laws p and p′), a coupling is a joint table J. The split pair for a set S is maximally coupled exactly when the probability that the splits differ, J(S, Sᶜ) + J(Sᶜ, S), equals |p(S) − p′(S)| = |J(S, Sᶜ) − J(Sᶜ, S)|, that is, when one of the two is zero. Call x → y, with x ≠ y and J(x, y) > 0, a transfer. S = {x} forbids a value that both sends and receives. Two transfers a → b and c → d with a ≠ c and b ≠ d are forbidden by S = {a, d}. So any two transfers share a source or a target, and then all of them share one source or all share one target. All from one source x₀ means p(x) ≤ p′(x) except at x₀; all into one target means p(x) ≥ p′(x) except at one value. Conversely, if p > p′ at most at x₀, the table with diagonal min(p, p′) and x₀’s excess sent to the deficits satisfies every split, and symmetrically. With only the one-value splits, sending each surplus value’s excess to the deficit values always works, so the pair is always noncontextual. The proof does not go through the 1-2 subsystem or Eq. (23), as the supplement’s does. For k = 3 one of the two conditions always holds, so the effect needs four values. Checked by brute force (numpy, run by the reviewing session: every choice of forbidden direction per split, feasibility by maximum flow) on 400, 300 and 12 random pairs for k = 3, 4 and 5. It agreed with nominal dominance on all of them, and found 89 and 5 contextual pairs for k = 4 and 5.
A CbD 1.0-noncontextual system with a contextual subsystem. A binary content q is measured in contexts c₁, …, c₄ with Pr[V = 1] = 0.1, 0.4, 0.6 and 0.9. A binary content s is measured in c₂ and c₃, with W₂ = V₂ in c₂, and W₃ distributed as Bernoulli(0.4) independently of V₃ in c₃. The s-connection is consistent, so its maximal coupling is W₂ = W₃, which forces V₂ = W₃ and so makes V₂ independent of V₃: Pr[V₂ ≠ V₃] = 0.52. Under CbD 1.0 the whole q-connection needs only Pr[all four equal] = 0.1 + 0.1 = 0.2, and a coupling attains it with V₂ and V₃ independent, V₁ = 1 on a 0.1 sub-event of V₂ = V₃ = 1 and V₄ = 0 on a 0.1 sub-event of V₂ = V₃ = 0. So the system is noncontextual under 1.0. Under CbD 2.0 the pair (V₂, V₃) must be maximally coupled, with Pr[V₂ ≠ V₃] = 0.2, so the system is contextual; and the subsystem without V₁ and V₄ is contextual under 1.0 as well. With binary variables a connection of three has a unique maximal coupling, the staircase of LIT-831’s Theorem 1, so 1.0 and 2.0 first disagree at four contexts.