THEORY-177: In a binary cyclic system, the Contextuality-by-Default measure of signalling is twice the sum, over contents, of the least direct influence of context that a canonical causal model of the data must place on each content, and each content's least influence is the total-variation distance between its two marginals

Source

Wang, Sadrzadeh, Abramsky & Cervantes (2021), LIT-851, §3.1: Proposition 1, Lemma 1, Proposition 2 and Corollary 1, as read in NOTE-654.

What was actually shown

Take a cyclic system with variables valued in {±1}. Each content q is measured in exactly two contexts, c and c′. CbD’s degree of signalling is Δ = Σ_q |⟨R_q^c⟩ − ⟨R_q^{c′}⟩|. A canonical model in Jones’s sense has a context variable C and a latent background Λ that jointly determine each content F_q. The direct influence of context on q is the probability over Λ that F_q(λ, c) ≠ F_q(λ, c′). Let Δ*(F_q) be its least value over all canonical models that reproduce the observed distributions.

The proposition is Δ = 2 Σ_q Δ*(F_q), a sum of per-content minima, with Δ*(F_q) = 1 − Σ_v min(P[R_q^c = v], P[R_q^{c′} = v]). The proof has three steps:

1. The coupling that keeps a content’s copies equal as often as possible attains the sum of the pointwise minima (Lemma 1, from LIT-777’s Theorem 3.3).
2. Every coupling corresponds to a canonical model whose direct influences are its disagreement probabilities, and conversely (quoted from Jones 2019, Proposition 8.4).
3. For ±1 variables, |⟨R⟩ − ⟨R′⟩| = 2(1 − m₊ − m₋).

The paper’s worked example checks out: Δ = 1, and Δ* = 7/20 and 3/20. The identity is a proof. It could fail only through an error in the quoted step, which is why promote_when asks for that step to be read.

What this does not say

• Not a contextuality measure. Δ and Δ* measure how far the marginals move, the part CbD sets aside before asking about contextuality. A system can have large Δ and be noncontextual. In rank 2, Δ ≥ 2 forces noncontextuality.
• Not the true causal influence. Δ* is the minimum over all canonical models compatible with the data. An actual mechanism may have more direct influence. The data identify only the lower bound.
• Not for non-binary or non-cyclic systems as stated. The per-content formula for Δ* extends to any content in two contexts, but the sum Δ and the factor 2 need ±1 variables.
• Not that signalling in language is “really” causal. The canonical model is a representation that always exists. It is not an empirical causal claim about the speaker or the reader.
• Not about transport. It measures signalling within one scenario. Whether maps between scenarios (CLAIM-125) preserve or bound it is not addressed.

Note of 2026-10-19: the quoted step, derived

Derived for the promotion audit (batch 045, ADR-047 v2) from the causal-model side, as the promote_when allows. A canonical model has context C and background Λ, with Λ independent of C. The definition of direct influence as P(Λ ∈ {λ : F_q(λ, c) ≠ F_q(λ, c′)}), unconditional on the context, requires that. F_q(Λ, c) is distributed as the observed R_q

1. Every model gives a coupling. Take content q, measured in contexts c and c′. Since Λ is independent of C, the pair (F_q(Λ, c), F_q(Λ, c′)) has the two observed marginals, so it is a coupling of them. Its disagreement probability is q’s direct influence. So every model’s direct influence on q is at least the least disagreement over couplings. That is 1 − Σ_v min(P[R_q^c = v], P[R_q^{c′} = v]), the total-variation distance, reached by the maximal coupling (LIT-777, Theorem 3.3).
2. The minimal coupling gives a model, for one content at a time. Fix the maximal coupling of q’s two copies, and leave the other contents’ copies uncoupled. In a cyclic system the remaining constraints, the context bundles and q’s pair, form a path with no cycle. So they glue into one joint distribution by conditional independence along it. Taking Λ to be that joint reproduces every context’s observed distribution and gives q the least direct influence. So Δ*(F_q) equals the total-variation distance.
3. The factor 2. For ±1 variables with P(R = +1) = p, |⟨R⟩ − ⟨R′⟩| = 2|p − p′| = 2·TV. Summing over contents, Δ = 2 Σ_q Δ*(F_q).

Step 2 reaches each content’s minimum in its own model. A single model that reached every content’s minimum at once would be a coupling with every connection maximal. In Contextuality-by-Default that exists exactly when the system is noncontextual. So the least total direct influence of one model equals Δ/2 for a noncontextual system and exceeds it for a contextual one, which is why the title now says “the sum, over contents, of the least”.