Direct-product reduction stages agree

Exact statement

For commuting Hamiltonian actions of a direct product $N\times H$ with split moment map, the Koszul and curvature receipts split and direct versus two-stage reduction are compatible. At the simplicial level, diagonal and iterated totalizations are naturally chain-homotopy equivalent rather than literally identical.

StatusProved and computationally 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

Exterior-algebra dimensions, the split moment-map model, and zero extension curvature agree; the simplicial comparison uses the classical Alexander–Whitney/Eilenberg–Zilber equivalence.

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

The claim is scoped to direct-product Hamiltonian actions and their stated complexes.
Split compatibility does not extend automatically to semidirect or nonsplit extensions.

Open work

Carry action and curvature receipts before applying reduction in stages beyond direct products.