Čech, transport, and connecting receipts agree

Exact statement

For a fibrewise short exact sequence of abelian local systems on a finite graph, the edge-overlap disagreement of local lifts is the cochain connecting cocycle. Its class vanishes exactly when the quotient section has a globally transport-compatible lift.

StatusProved and exhaustively 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

All 625 three-cycle transport instances over $\mathbb Z/5$ reproduce the equivalence; an independent $\mathbb Z/7$ implementation checks 2,401 lift/twist cases.

Self-contained cohomological sewing proof and evidence →

Verification notes

The exact archived proofs were reconstructed, the displayed hypotheses and degree conventions were checked, the original finite controls were replayed, and an independent implementation tested the connecting, graph-transport, staged, and missing-subcomplex boundaries.

Limitations

The theorem is pinned to the stated finite graph/local-system model.

Open work

Extend to higher-dimensional nerves through totalization.