Contribution
The paper gives the probabilistic argument for indexing measurements by context. It is a diagnosis of the traditional reading, not a new theorem. It then states the CbD definition of contextuality for arbitrary binary systems, using multimaximal couplings, and supplies the adjacent-pairs characterization of those couplings (Theorem II.3) that the CbD 2.0 paper (LIT-831) cites as its Theorem 2.
Key insight
Within classical probability you cannot conclude “there is no joint distribution” from a contextuality contradiction, because sharing even one variable already puts two contexts on the same probability space. Joint distributedness then propagates along any chain of overlapping contexts. So what the contradiction refutes is the assumption that makes the contexts overlap: that a measurement of q is one random variable whatever else is measured with it. Drop that, and contextuality becomes a question about which couplings of the now-disjoint contexts are possible.
Assumptions
- Kolmogorovian probability with multiple domain probability spaces. A random variable is a triple of domain probability space, codomain and measurable function. Two random variables are jointly distributed if and only if their domain spaces coincide (Eq. 7).
- Finitely many contents and contexts. All examples and the Section II theory use binary measurements.
- Which conditions count as a context is part of the system’s specification. The KS-3D example’s contexts include an outcome condition, not just the co-measured observables.
Key results
- Transitivity (Section I.3). If X, X′ are jointly distributed and so are X′, X″, then X, X″ are, since all three share X′‘s domain space. An equivalent form: jointly distributed variables are functions of one random variable, and that representation also propagates. Hence if the graph of contexts linked by shared variables has a path through all contexts, the traditional system has an overall joint distribution. The paper exhibits Hamiltonian paths for KCBS, KS-4D and KS-3D.
- Reductio (Section I.4). The contradiction refutes Noncontextual Identification, not the existence of a joint distribution.
- Definition I.1. A system is traditionally noncontextual if it has a coupling in which same-content variables are equal with probability 1 (equivalently, adjacent pairs in any ordering, Eq. 22).
- Definition II.1. A binary system is noncontextual if it has a coupling in which every set of same-content variables is equal with maximal possible probability. The remark credits this to the CbD 2.0 paper and says CbD 1.0 applied the constraint only to the whole connection.
- Theorem II.2 (cited to LIT-831). A binary connection’s multimaximal coupling exists, is unique and has the staircase form of Eq. (30).
- Theorem II.3. A coupling of a binary connection sorted by p_i = Pr[R_q^i = 1] is multimaximal if and only if each adjacent pair is maximally coupled. Derived in Eqs. (31)–(34): adjacent maximality forces Pr[S^l = 1, S^{l+1} = −1] = 0, which propagates to the staircase.
- Section II.4. Three consequences, said to follow “trivially”: (1) deleting measurements preserves noncontextuality; (2) consistently connected systems get Definition I.1; (3) for cyclic systems the theory is the earlier one.
- Theorem II.4 (cited to Kujala & Dzhafarov 2016). A rank-n binary cyclic system is noncontextual if and only if s_odd of the n within-context product expectations ≤ n − 2 + Σ_i |⟨R_i^i⟩ − ⟨R_i
- Magic boxes (Section III). Specker’s three boxes, with one gem in each opened pair, form a rank-3 cyclic system. With consistent connectedness the system is contextual. Without it, the system is noncontextual if and only if the three marginal differences sum in absolute value to at least 2 (Eq. 42). In particular, any deterministic version is noncontextual.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | If a measurement is one random variable in all its contexts, overlapping contexts force an overall joint distribution | strong (elementary proof) | Section I.3, Eqs. 7–12 |
| C2 | Contextuality contradictions refute context-independent identification of measurements, not joint distributions | moderate (a philosophical argument resting on C1) | Section I.4 |
| C3 | A coupling of a binary connection is multimaximal if and only if its adjacent pairs, in the order of means, are maximally coupled | strong (derivation) | Theorem II.3, Eqs. 31–34 |
| C4 | Multimaximal noncontextuality is hereditary, reduces to the traditional notion under consistent connectedness, and leaves cyclic systems unchanged | strong (immediate) | Section II.4 |
| C5 | Specker’s magic boxes are noncontextual exactly when the marginal discrepancies sum to at least 2 | strong (Theorem II.4 at n = 3) | Eq. 42 |
| C6 | Inconsistent connectedness is essentially universal outside physics, and non-negligible in a KCBS experiment | weak here (cited) | Section II.1, refs 20, 21, 32, 35, 36 |
Concepts
- Noncontextual Identification. The assumption that the random variable representing a measurement is identified by the property it measures alone.
- Contextual Identification; Stochastic (Un)Relatedness. CbD’s replacements. A variable is identified by content and context, and it is jointly distributed only with variables in its own context.
- S1 / S2. The traditional determination (existence of a joint distribution of all measurements) and CbD’s (existence of a coupling whose same-content parts satisfy C).
- consistently / inconsistently connected. Whether same-content variables are identically distributed across contexts.
- multimaximal coupling. As in LIT-831.
Connections
The paper is the conceptual companion of LIT-831. Each cites the other for half of the binary multimaximality theory. It builds on LIT-777’s system notation and on the earlier “Contextuality is about identity of random variables” (2014). The transitivity point is aimed at the Khrennikov line, which locates contextuality in variables defined on different probability spaces. Fine and Suppes–Zanotti are cited for the joint-distribution criterion.
Bearing on the record
- LIT-016 and THEORY-012. The sheaf account keeps one variable per measurement and asks whether overlapping marginals glue. On that account, the traditional “no joint distribution” is a statement about a global section of a presheaf over a cover, not about random variables on one space. This paper’s transitivity argument targets the random-variable reading and does not engage the sheaf formulation. The two accounts agree under consistent connectedness (NOTE-600). This paper does not compare them.
- THEORY-013. C6 restates the premise THEORY-013’s programme rests on. Nothing here changes that theory.
- CLAIM-069 (the manuscript’s). This paper is the second of the two references the claim gives for the multimaximal version. It supports the claim as LIT-831 does.
- No new THEORY. No instruction for machine-learning practice.
Limitations
- C2 is an argument about how to read a reductio. A reader who treats quantum contextuality as a property of the measurement structure (the sheaf view) is not bound by it.
- Section II.4’s three properties are asserted as trivial, without proofs. The first needs the subset clause of Definition II.1, which holds.
- The Kochen–Specker matrices are not analysed under CbD; the paper says only that they are not cyclic.
- Theorem II.2’s citation goes to the companion paper, which in turn cites this one for the adjacent-pairs theorem.
Open questions
- A CbD analysis of the KS-4D and KS-3D systems with realistic inconsistent connectedness, which the paper sets up and does not do.