Source
Abramsky, Barbosa & Mansfield (2017), LIT-265, Eqs. (3)–(4), Theorem 1 and its supplemental proof, and Theorem 2 (mixing) with its supplemental proof, as read in NOTE-236. The continuity statement is the reader’s derivation, set out below.
What was actually shown
In a scenario ⟨X, M, O⟩ with incidence matrix M, an empirical model e is a vector v_e of context-wise probabilities whose marginals agree on overlaps. The paper defines NCF(e) = max {1·b : Mb ≤ v_e, b ≥ 0} and CF = 1 − NCF. It proves that strong duality holds, with dual min {y·v_e : Mᵀy ≥ 1, y ≥ 0}. Theorem 2 proves NCF(λe + (1 − λ)e′) ≥ λNCF(e) + (1 − λ)NCF(e′), so CF is convex along mixtures.
The derivation (the reader’s): the dual feasible set P = {y ≥ 0, Mᵀy ≥ 1} is a nonempty polyhedron inside the nonnegative orthant, so it has finitely many vertices y₁, …, y_K and its recession directions are nonnegative. Since v_e ≥ 0, the objective y·v_e cannot decrease along a recession direction, so the minimum is attained at a vertex: NCF(e) = min_k y_k·v_e. A minimum of finitely many linear functions is concave and piecewise linear, and |CF(e) − CF(e′)| ≤ (max_k ‖y_k‖₁) · ‖v_e − v_e′‖_∞. The constant depends only on the scenario.
A numerical check on the CHSH (2,2,2) scenario: along the line from uniform noise to the PR box, CF is 0 up to weight ½ and 2t − 1 after. It is continuous with a kink, as the derivation predicts. The Bell–CHSH table of the paper’s Table I gives CF = 1/4, its normalised CHSH violation (Theorem 1). The largest ‖y*‖₁ seen over 300 random mixtures was 8. That is a sample, not the constant.
What this does not say
- Not that small changes in the underlying states or chains give small changes in CF. The bound is in the empirical table. A drift result must first bound the change in every context’s distribution, and then pays the scenario’s constant, which can be large.
- Not anything across scenarios. The bound holds for a fixed scenario. CF is comparable across scenarios as a number in [0, 1] (the paper’s point), but a map between scenarios is a separate operation. Translations of measurements and coarse-grainings cannot raise it (Theorem 2), while other maps may.
- Not for signalling data. CF requires compatible marginals. An empirical table with context-dependent marginals has no CF, and Contextuality-by-Default’s measure (LIT-777) is a different quantity.
- Not that the strongly contextual / noncontextual decomposition varies continuously. The decomposition need not be unique (the paper’s Table II), even though its weight is continuous.
Note of 2026-10-19: split
The derivation in “What was actually shown”, that NCF is the minimum over the dual polyhedron’s vertices, and so is piecewise linear and Lipschitz in the table, is now CLAIM-372. It is Active, with a written proof. This THEORY keeps the linear programme and Theorem 2, which the paper proves. The CHSH check and the sample of dual norms above stay here as what the reader computed.