Connecting obstruction and correction receipt

Exact statement

For a short exact sequence $0\to N\xrightarrow{i}D\to Q\to0$ , lift a quotient cocycle $q\in Z^k(Q)$ to $x\in D^k$ and write $d_Dx=i(c)$ . A closed lift exists exactly when $[c]\in H^{k+1}(N)$ vanishes; if $c=d_Na$ , then $x-i(a)$ is a closed lift.

StatusProved and internally checked

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 standard connecting-map proof was replayed over explicit $\mathbb Z/4$ and $\mathbb Z/2$ complexes, including a nonzero $\mathbb{RP}^2$ control.

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

No novelty or specialist-review claim is made; the statement is limited to the displayed presentation and hypotheses.

Open work

Track minimal or resource-weighted primitives in applications.