The claim
Within one scenario, global compatibility can change maximally while most local judgements are kept exactly. Take n odd, n ≥ 3, and observables a_1, …, a_n with outcomes ±1. The contexts are C_i = {a_i, a_(i+1)} for i = 1, …, n, indices mod n, so that C_n = {a_n, a_1}. Weight every context 1/n.
- Model e. In every context the two observables disagree: the outcomes (+, −) and (−, +) each have probability 1/2. A global assignment consistent with every context’s support would need a_(i+1) = −a_i for every i, so going round the cycle a_1 = (−1)^n a_1 = −a_1, since n is odd. No ±1 value satisfies that, so no global assignment is consistent with the supports. The model is strongly contextual, and its contextual fraction is 1.
- Model e′. The same, except in C_n, where a_n = a_1: the outcomes (+, +) and (−, −) each have probability 1/2. Now the constraints are a_(i+1) = −a_i for i = 1, …, n − 1 and a_n = a_1. The first n − 1 give a_n = (−1)^(n−1) a_1 = a_1, since n − 1 is even, so they agree with the last. The two assignments a_i = s(−1)^(i−1), s = +1 or −1, satisfy every constraint. Their equal mixture reproduces every context: in C_i for i < n it gives (s(−1)^(i−1), −s(−1)^(i−1)), which is (+, −) or (−, +) with probability 1/2 each; in C_n it gives (s, s), which is (+, +) or (−, −) with probability 1/2 each. So e′ has a global distribution, and its contextual fraction is 0.
- What changed. In both models every observable is +1 or −1 with probability 1/2 in each context that measures it, so neither signals, and overlap consistency is the same. The contexts C_1, …, C_(n−1) are identical. C_n changed from support {(+, −), (−, +)} to the disjoint support {(+, +), (−, −)}, a total-variation distance of 1. With equal weights the weighted observational distortion is D_obs = (1/n) · 1 = 1/n.
For n = 3 this is CASE-037 with the outcomes of one observable relabelled, the pair CLAIM-320 states for one scenario; this claim is its general form. So within a fixed scenario the contextual fraction can fall from 1 to 0 at a weighted local distortion of 1/n, which is as small as one likes for large n. THEORY-165 says the fraction is Lipschitz within a scenario and that its constant “depends on the scenario and is not bounded here”. Measured against weighted D_obs, the example shows the constant is at least n.
The bound that moving every context by at most ε moves each overlap discrepancy by at most 2ε (CLAIM-219) is respected: no marginal moves at all. It concerns overlap consistency and every context moved a little, not global extension with most contexts kept exactly.
Every such case needs a contextual model on one side, here e: if both models had a global extension, global extension would not have changed.
What it does not say
- That any data are contextual, or that a rendering does this. Whether behavioural data supply the contextual side is CLAIM-325, and what follows for CLAIM-105’s headline is CLAIM-201’s, which rests on this claim.
- Anything across scenarios. The construction keeps the scenario fixed; a change of scenario, where the fraction can jump, is a different case.
- That the bound of n is tight. It is a lower bound on the constant, measured against weighted D_obs with equal weights.