Finite Čech totalization adapter

Exact statement

For a finite cover and a short exact sequence of abelian cochain presheaves, local lifts $a$ of a global closed quotient cochain determine the closed total cochain $O(a)=da+(-1)^k\check\delta a$ . A strict global closed lift exists exactly when $O(a)=-D_{\mathrm{tot}}n$ for a null receipt $n$ of Čech degree zero.

StatusProved and independently audited

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 total-differential calculation separates the internal and overlap components; the independent finite audit checks the sign cancellation and a nonextendable compatible presheaf cocycle.

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

A primitive with positive Čech degree gives homotopy-coherent data, not automatically one strict global section.
The public audit is finite corroboration; the displayed proof supplies the general argument.

Open work

Keep strict and homotopy-coherent lift targets distinct.