Source
Abramsky & Brandenburger (2011), LIT-016, Thms 5.4, 5.9, 8.1. Schmid et al. (2020), LIT-003, Thm 2.8, Cor 3.5, Prop 4.4. NOTE-017 records that LIT-003 attempts no formal comparison with LIT-016.
What was actually shown
In the sheaf formalism every no-signalling model has a real signed global section, and noncontextuality is the existence of a nonnegative one (LIT-016 Thms 5.4, 5.9, 8.1). In the GPT formalism every tomographically local GPT has a real-linear diagram-preserving representation, and generalized noncontextuality is the existence of a positive quasiprobabilistic one (LIT-003 Thm 2.8, Cor 3.5, Prop 4.4). Both halves are proved in works this record has read. The shared shape — a linear representation that always exists, with classicality as its positivity — is the record’s inference. Contextuality-by-Default gives a third instance, and one that does not need no-signalling: for every finite system of categorical random variables, signalling or not, a signed maximally connected quasi-coupling always exists (LIT-777, Theorem 6.1), and noncontextuality is the existence of a nonnegative one (Theorem 4.1; NOTE-600).
What this does not say
- That the two representations are the same object. The sheaf section lives on global outcome assignments for one empirical model; the GPT representation lives on the state and effect spaces of a whole theory.
- That a sheaf-noncontextual empirical model yields a diagram-preserving ontological model of the corresponding GPT fragment, or the reverse.
- Anything beyond finite, tomographically local settings.
- Whether a map between the formalisms exists. A map from signed global sections to a GPT fragment’s quasiprobability representation, or a proof that none preserves positivity, is QUESTION-146.