Cross-axis obstruction staircase

Exact statement

In a commuting bicomplex, if $[\partial x]=0$ in $\delta$ -cohomology, choose $h$ with $\delta h=\partial x$ . The corrected total cochain has next defect $\partial h$ ; the choice-independent secondary obstruction is the class of $[\partial h]$ modulo the image induced by $\partial$ on $H_\delta$ , equivalently the corresponding $E_2$ -class.

StatusProved, sign-audited, and scope-corrected

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

Direct calculation with the pinned totalization sign verifies cancellation and closure; a finite choice-dependence control verifies that only the quotient class is canonical.

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 raw class $[\partial h]$ depends on the primitive.
Changing totalization convention changes displayed signs, not the invariant quotient class.

Open work

State the secondary receipt in its natural quotient after pinning both axes and signs.