nucleation

CASE-037: The parity triangle with uniform weights: a no-signalling, strongly contextual model beside a family of point sections that glues

3 min read

Properties9
StatusActive
Standingstipulated
Kindconstruction
Tagscontextuality, mathematics
Linepragmatic-transport
SupportsCLAIM-201 — Over a fixed scenario, a change of global compatibility at small local distortion needs contextual judgements, which the behavioural data read so far lack; the form of the headline those data leave open is a change of cover, in which a rendering makes some of the source's questions undecidable for its readers and decides others the source left open, CLAIM-319 — Over a scenario of n contexts in an odd cycle, changing the support of one context takes the contextual fraction from 1 to 0 while the other n − 1 contexts and every marginal are kept exactly, at a weighted observational distortion of 1/n, so within a scenario the fraction's Lipschitz constant against that distortion is at least n; and any change of global extension needs a contextual model on one side, CLAIM-320 — Within one fixed measurement scenario, two no-signalling empirical models can agree on all but one of their contexts and on every single-observable marginal while their contextual fractions are 1 and 0, so keeping most local judgement distributions does not fix their global compatibility
IllustratesCLAIM-131 — A compatible family of sections of the event sheaf always glues, so where a communicative object lacks a global realization the obstruction lies in its supports or distributions, not in the event sheaf, CLAIM-037 — Formal contextuality is a failure of global extension, distinct from ordinary context dependence, and needs Contextuality-by-Default when marginals shift with context, CLAIM-105 — In documented reconstruction chains, steps that keep most local judgements commonly change their global compatibility: translation preserves local meanings while changing the global structure of meaning
Variant ofCASE-035 — Solidarity, condemnation and humour: the parity obstruction
Cited as “grounds” byARG-032 — The parity triangle shows that translation changes the global structure of meaning

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The case

A129 §2, on X = {A, B, C} with the cover {AB, BC, AC} and binary outcomes:

  • The supports are S_AB = S_BC = {(0,0), (1,1)} and S_AC = {(0,1), (1,0)}. These are CASE-035’s constraints (S = M, M = H, H ≠ S) under new letters.
  • Each allowed outcome has probability 1/2: “if each context assigns probability 1/2 to both allowed outcomes, the marginal of each individual observable is uniform across its two contexts. Thus these local probability distributions are overlap-consistent.” So the model is no-signalling.
  • “Yet no global assignment satisfies all three support conditions”, because A = B = C contradicts A ≠ C. The model is strongly contextual, and its contextual fraction is 1.
  • Beside it, the point sections s_AB = (A=0, B=1), s_BC = (B=1, C=0) and s_AC = (A=0, C=0) agree on overlaps and “glue uniquely to s_X=(A=0,B=1,C=0)”.

The same model returns twice. A161: “In the three-context parity example, replacing each deterministic assignment with a probability distribution does not produce a nonnegative global joint distribution matching the specified pairwise laws.” A178 §7 uses it as a negative control: “Pairwise consistent binary empirical distributions may lack a nonnegative global joint extension.”

What it can show

That the obstruction is in the supports, not in the event sheaf (CLAIM-131): the same cover carries a family of point sections that glues and a family of supports that admits no global assignment. It makes CLAIM-037’s failure of global extension concrete for a model that does not signal.

It also gives the pair CLAIM-105 lost. This is the reader’s own addition (R1), not the exchange’s. Change S_AC to {(0,0), (1,1)}, still at 1/2 each. Two of the three contexts are unchanged, and so is every single-observable marginal. The new model has the global distribution that puts 1/2 on (0,0,0) and 1/2 on (1,1,1), so its contextual fraction is 0. The fraction falls from 1 to 0 within one scenario. The two AC tables have disjoint supports, a total-variation distance of 1 in one context, so the jump is consistent with THEORY-165’s Lipschitz bound.

What it cannot show

That pragmatic judgements behave this way (CLAIM-009). The weights are stipulated, as CASE-035’s constraints are: “This parity example demonstrates an obstruction, not a claim about actual judgments.” It also cannot show anything about the possibilistic grade between the two: a model with a locally possible section that extends to no global assignment, while other sections do, is a different case.

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