Contribution
The paper turns the dichotomization proposal at the end of CbD 2.0 (LIT-831) into a definition. Contextuality is a property of the canonical (split) representation of a chosen expanded system. It then solves the smallest nontrivial case, a single pair of content-sharing categorical variables with every split kept, exactly. Before this paper, two variables measuring one content in two contexts could not be contextual in CbD by themselves. Here, with all splits, they are unless one nominally dominates the other.
Key insight
Splitting a k-valued variable into all its yes/no questions turns one connection into 2^{k−1} − 1 connections. Each must be coupled as tightly as its own two marginals allow, and all of them must come from one coupling of the two original variables. Maximal coupling of every one-value split fixes the diagonal of that coupling at min(p_i, q_i). Maximal coupling of every two-value split then forbids almost all off-diagonal mass. What is left is possible only when one distribution exceeds the other at no more than one value. Contextuality has here become a property of how two distributions differ, with no cross-content correlations involved.
Assumptions
- Categorical variables (finitely many values). The continuous case is discussed only informally in Section 5.
- Expansion is a choice. Which joinings and coarsenings to add is not dictated by the theory. Section 4 assumes the largest choice, all coarsenings, hence all splits.
- Multimaximality as pairwise maximality (Definition 3.8) is used for connections with more than two variables. The paper cites properties Multimax1–3 to LIT-831 and LIT-836. Note that Multimax1 (existence and uniqueness) holds because every canonical variable is binary.
- Section 4 treats a fragment, one connection of two variables. A system containing a contextual fragment is contextual (by heredity), but the converse needs the rest of the system.
Key results
- Definition 2.1 / 2.2. For couplings T = {T_q} of the connections, R is noncontextual with respect to T if some coupling S of the bunches has S_q ∼ T_q for every q. With a property C that picks a unique T_q, this is “noncontextual with respect to C”.
- Theorems 2.4–2.5 (from LIT-777). A quasi-coupling with connection marginals T always exists, and its total variation reaches a minimum. min‖X‖ − 1 is the degree of contextuality.
- Definitions 3.4–3.5. A split D_{qW}^c = 1 iff R_q^c ∈ W, with W normalised to the smaller side. The split representation of R_q^c is its k one-element splits. The canonical representation replaces every variable by its split representation.
- Remark 3.3. However many splits are added, a bunch’s support has at most as many states as the original variable has values.
- Equations (19)–(22). A coupling r_ij of the pair is a maximally connected coupling of the 1-2 system if and only if Σ_j r_ij = p_i, Σ_i r_ij = q_j, r_ii = min(p_i, q_i), and r_ii + r_ij + r_ji + r_jj = min(p_i + p_j, q_i + q_j) for i < j.
- Theorem 4.1. Noncontextuality of the all-splits system D implies that these 3k + C(k, 2) equations have a nonnegative solution. Theorem S.1 gives the equation system’s rank as 2k − 1 + C(k, 2).
- Theorem 4.3. In a maximally connected coupling (k ≥ 6), the probability that an m-split equals 1 is determined by the 1- and 2-splits: the sum of the m diagonal minima plus, for each pair, the pair’s excess min(p_i + p_j, q_i + q_j) − min(p_i, q_i) − min(p_j, q_j) (Eq. 23). Eq. (23) can fail for some distributions (Example S.2).
- Theorem 4.4. A maximally connected coupling of the 1-2 system is unique if it exists. Its off-diagonal mass lies in a single row or a single column.
- Corollary 4.5. One exists if and only if p_i > q_i for at most one i, or p_j < q_j for at most one j.
- Theorem 4.6. The all-splits system D is noncontextual if and only if its 1-2 subsystem is, i.e. if and only if one of R_1^1, R_1^2 nominally dominates the other. Two variables dominate each other if and only if they are identically distributed or k = 2.
- Section 5, continuous case (an argument, not a theorem). Take two densities f ≠ g with all splits. Then f exceeds g on some interval and g exceeds f on another. Any partition containing two subintervals of each violates Corollary 4.5. So the system is contextual unless f = g.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | With all splits of two content-sharing k-valued variables, the canonical system is noncontextual iff one nominally dominates the other | strong (proof in S) | Theorems 4.1, 4.3, 4.4, 4.6, Corollary 4.5 |
| C2 | Only the 1- and 2-splits matter for that verdict | strong (proof) | Theorems 4.3, 4.6 |
| C3 | The maximally connected coupling of the 1-2 system is unique when it exists | strong (proof) | Theorem 4.4 |
| C4 | With all splits, two continuous variables are contextual unless identically distributed | moderate (short argument, not a theorem) | Section 5 |
| C5 | Nominal dominance is likely to fail in many empirical systems | weak (no data analysed) | Section 5 |
| C6 | All dichotomous behavioural systems examined so far except one were noncontextual | weak here (cited) | Section 5, refs 7, 15, 17 |
Method
Both proofs reduce the problem to the k × k table r_ij of a coupling of the two original variables, since each bunch has at most k states. Theorem 4.4 then uses four rules about which off-diagonal cells can be positive (R1–R4, by “p-minimized” and “q-minimized” pairs). It concludes that positive off-diagonal mass is confined to one column (if p-minimized) or one row (if q-minimized). Corollary 4.5 exhibits the coupling. Theorem 4.6 checks Eq. (23) on that coupling for every higher-order split.
Concepts
- expansion; joining; coarsening. Adding new contents computed from existing ones in each context: tuples of variables (joining) or functions of one variable (coarsening). Any transformation is a joining followed by a coarsening.
- split; split representation; canonical representation. Definitions 3.4–3.5 above.
- m-split. A split whose set W has m elements (m ≤ k/2).
- 1-2 system. The subsystem of 1-splits and 2-splits.
- nominal dominance. R nominally dominates R′ if Pr[R = x] < Pr[R′ = x] for at most one value x. The authors call it a form of stochastic dominance for categorical variables that had not been identified before.
Connections
It completes LIT-831’s proposal and uses its multimaximality results and those of LIT-836. Its measure is LIT-777’s. Joining, which lets within-context dependence enter a connection, is presented as what the selective-influence theory and Abramsky and colleagues’ approach (LIT-016) always require. Most known behavioural CbD analyses are cited to the programme around LIT-264.
Bearing on the record
- THEORY-013. That theory reports CbD finding no contextuality in published behavioural data. Its data are binary and cyclic. This paper says the same of dichotomous systems “with the exception of one, very recent experiment”, which is consistent with THEORY-013. It also says that under canonical all-splits representations of multi-valued responses, noncontextuality requires nominal dominance, which “is likely to be violated in many empirical systems”. So THEORY-013’s finding is scoped to binary data under CbD 1.0/2.0 and does not carry over to multi-valued data analysed canonically. The theory’s “does not say” list should name this. This NOTE does not edit it.
- THEORY-012 and LIT-016. The sheaf framework’s contextual fraction is non-increasing under coarse-graining of outcomes (LIT-265 Theorem 2). Here, adding the coarsenings of a variable to the system can create contextuality where the original pair, a single connection, had none. The two frameworks treat coarse-graining differently: the sheaf framework coarse-grains a model, CbD adds coarsenings as further contents. The paper does not compare them.
- THEORY-168 is filed from this reading. It states that CbD’s verdict on a fixed set of measurements depends on the representation chosen, with this paper’s Theorem 4.6 as the sharpest instance.
- The manuscript. CLAIM-069’s “What it does not say” assumes that S, M and H are binary and each appear in two contexts. If any observable there is multi-valued and its coarsenings are of interest, this paper’s representation applies, and Theorem 4.6 can make the system contextual on a single connection. Reported for the coordinator; no claim file is edited here.
- No instruction for machine-learning practice.
Limitations
- Only one connection of two variables is solved. Systems with several contents, or with more than two variables per connection, are not analysed.
- The choice of expansion is left to the analyst. The paper argues that intervals or cuts are more natural for ordered values and leaves that theory to future work.
- C4 is an informal argument in the concluding remarks.
- Small slips in the supplement: in the proof of Theorem 4.4 the case “rij > 0 that is p-minimized” is announced and then argued as “q-minimized”; the zero cell “r15 = 0” is called p-minimized where the argument needs it positive. Neither affects the conclusion, which is rechecked by Corollary 4.5’s explicit construction.
- No data are analysed.
Open questions
- The verdicts under interval-only or cut-only splits for ordered responses, which the authors flag. Those would decide whether the continuous-case conclusion (C4) is an artefact of allowing every split.
- How CbD’s canonical verdicts relate to sheaf-theoretic monotonicity under coarse-graining for consistently connected multi-valued systems.