Secondary cross-axis data has indeterminacy

Exact statement

If two primary primitives differ by a $\delta$ -closed cochain $z$ , their raw secondary defects differ by $\partial z$ . Thus the unquotiented secondary class is not canonical; only its coset modulo the induced image of $\partial$ on $H_\delta$ is.

StatusExact and independently 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

A $\mathbb Z/4$ control with induced map $u\mapsto2u$ gives raw representatives 0 and 2 in the same quotient coset.

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

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

Open work

Record the secondary obstruction in its natural quotient or spectral-sequence term.