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Counterexample · necessary hypothesis

FPRD-QA-FAIL06

Null probes must form a subcomplex

Exact statement

If the null family is not closed under the differential, the proposed quotient differential is not well-defined.

StatusProved by finite example
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

Two representatives of one quotient class acquire distinct differentials in the Z/4\mathbb Z/4 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

Audit differential closure before defining any probe quotient.