nucleation

CLAIM-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

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Properties9
StatusActive
Rolegranted
Warrantdemonstrated
Tagscontextuality, mathematics
Linepragmatic-transport, formal-apparatus
GroundsTHEORY-012 — An empirical model is Kochen–Specker-noncontextual exactly when its context distributions glue to one global distribution, and since signed global sections always exist for no-signalling models, negative probability does not mark contextuality, LIT-016 — The sheaf-theoretic structure of non-locality and contextuality
ComplementsCLAIM-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-038 — A communicative object is a context-indexed system of empirical constraints over a cover of contexts, and these need not extend to one globally realizable interpretation, CLAIM-310 — Sorting schemes over objects with determinate properties never conflict as descriptions, since the product of any sortings refines them all; they conflict as arrangements, since a fixed number of bins realizes one partition when each item is stored once, and which costs least depends on what is searched for and on what stands out within a bin
UsesTERM-042 — communicative object, as a context-indexed system of empirical constraints over a cover
Illustrated byCASE-037 — The parity triangle with uniform weights: a no-signalling, strongly contextual model beside a family of point sections that glues, CASE-095 — Two builders sorting one collection: the brick that one bins as red and two-wide and the other as a curved connector has a bin that respects both, and what the builders cannot share is an arrangement, not a description

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The claim

A125 §2: “In the ordinary event sheaf, a compatible family of local sections already glues to a global section. What may fail to possess a global section is the empirically constrained structure: local support sets or probability distributions may not admit a jointly satisfying global assignment or distribution.”

A129 §2 shows both halves on X = {A, B, C} with the cover {AB, BC, AC}. The point sections (A=0, B=1), (B=1, C=0) and (A=0, C=0) agree on overlaps, and “glue uniquely to s_X=(A=0,B=1,C=0)”. The parity supports admit no global assignment, and “The obstruction arises from the empirically constrained supports, not from any failure of the event sheaf’s gluing axiom.” The two contrasts are CASE-037.

In Abramsky and Brandenburger’s terms (THEORY-012, LIT-016), E(U) = ∏_(x∈U) O_x is a sheaf on the observables. An empirical model is a compatible family of sections of the distribution presheaf D_R∘ℰ, and that presheaf is not a sheaf. So the question whether a communicative object (TERM-042) has a global realization is a question about its supports or distributions.

Two extension questions follow, as A129 §2 sets them.

  • Over the supports. Is there s ∈ ℰ(X) with s|_C ∈ S_C for every context C?
  • Over the distributions. Is there p ∈ D(ℰ(X)) with p|_C = e_C for every C?

A negative answer to the first gives a negative answer to the second. A global p would be supported on sections whose restrictions all lie in the supports. A129: “Failure of the first is a stronger obstruction than failure of the second.”

The claim is used against TERM-002’s wording, “a compatible family of local sections … with no global assignment presupposed”, which is vacuous for ℰ. The manuscript §3 already put the obstruction in “empirical supports or distributions”. It complements CLAIM-037, which says contextuality is a failure of global extension, and CLAIM-038, whose version 2 is restated in these terms. Restated at A151 §2, A173, A178 §4.2 (“A compatible family of unrestricted event assignments glues.”), A203 §11 and A218 §6 (“The event sheaf glues compatible unrestricted local assignments.”).

What it does not say

It does not say that deterministic local data always glue. In causal measurement scenarios the strategy presheaf is not a sheaf, and deterministic strategies that are compatible in every context can fail to glue (LIT-843 Example 7.2; THEORY-171; CLAIM-038, “A further kind of non-gluing”). The claim is about the flat event sheaf. A129 §5 recommends that very paper and proposes “a separate causal layer, not a replacement of the sheaf layer”, without noticing that the causal layer can break this premise.

It also does not grade the obstruction. Abramsky and Brandenburger have three levels, probabilistic, possibilistic and strong (THEORY-012), and A129’s “possibilistic extension” question tests only the strongest. A model in which some locally admissible section belongs to no compatible family (possibilistic contextuality, Hardy’s kind) passes A129’s first question and fails its second. So A129’s two questions do not separate the possibilistic grade from the probabilistic one.

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