Primary and secondary receipts in stages

Exact statement

For nested subcomplexes $N\subset M\subset D$ , a class in $H^k(D/M)$ lifts to $H^k(D)$ exactly when its primary class in $H^{k+1}(M/N)$ vanishes and the resulting secondary coset in $H^{k+1}(N)/\operatorname{im}\delta_{N,M}$ vanishes.

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

A $\mathbb Z/8$ model exhibits primary-zero/secondary-nonzero behavior and confirms choice independence only after quotienting by the correction image.

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

A raw secondary representative depends on the first-stage lift.

Open work

Identify higher-stage indeterminacy when more layers are present.