Compatible local cocycles need not extend globally

Exact statement

Overlap compatibility alone does not imply global extendability: on a two-set cover, the presheaf with global group $0$ , both local and overlap groups $\mathbb Z/2$ , identity local restrictions, and zero global restrictions has the compatible local pair $(1,1)$ , but no global section maps to it.

StatusExact and independently enumerated

External reviewNo documented external or specialist review of this Quotient Arsenal result is recorded.

Context

The general calculus treats a failed lift as a cocycle, its cohomology class as the obstruction, and a primitive—when one exists—as the receipt that repairs the presentation.

Proof or evidence

The public checker enumerates all four local pairs: exactly $(0,0)$ and $(1,1)$ agree on the overlap, while only $(0,0)$ lies in the image of the global restriction map.

Local-to-global and reduction-in-stages proof and evidence →

Verification notes

The archived totalization and stage proofs were reconstructed, their domains and choice indeterminacies were checked, and an independent finite implementation tested strict presheaf descent, secondary-coset invariance, split exterior dimensions, and Heisenberg curvature.

Limitations

This is a presheaf counterexample; additional sheaf or descent hypotheses can restore local-to-global existence.
No novelty or specialist-review claim is made; the statement is limited to the displayed presentation and hypotheses.

Open work

State the relevant descent or global-surjectivity hypothesis when using compatible local data.