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Failed proof method

CRD-RW-F5

Vanishing first homology does not prove simple connectedness

Exact statement

For the five-leaf square–pentagon complex, the calculation H1=0H_1=0 cannot by itself establish π1=0\pi_1=0, because first homology records only the abelianization of the fundamental group.

StatusHomology-only proof withdrawn; conclusion recovered by an explicit collapse certificate
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

A correct invariant can still be insufficient for the desired conclusion. This entry keeps the valid homology computation separate from the invalid promotion to simple connectedness.

Definitions

  • For a connected CW complex, H1H_1 is the abelianization of π1\pi_1.

Hypotheses and scope

  • The fixed five-leaf square–pentagon complex after quotienting disjoint scheduling.

Proof or evidence

The boundary ranks rank⁡∂1=13\operatorname{rank}\partial_1=13 and rank⁡∂2=8\operatorname{rank}\partial_2=8 correctly yield H1=0H_1=0 and H2≅ZH_2\cong\mathbb Z, but a nontrivial perfect fundamental group would have the same first homology. The repaired proof removes one pentagon, collapses the other eight faces along recorded free edges, and leaves a connected graph with 14 vertices and 13 edges, hence a tree.

Proof-method boundary and collapse repair →

Verification notes

The ledger's invalid inference and successful replacement were checked separately. The matrix records L4 because the public page gives the complete logical obstruction and identifies the constructive proof, not because the abandoned homology argument was repaired rhetorically.

Limitations

  • The homology ranks remain valid and useful corroboration.
  • The collapse certificate proves the fixed complex simply connected; it is not an all-arity coherence theorem.
  • No external or specialist review is documented.

Open work

Use the explicit elementary-collapse certificate as the proof; retain the homology calculation only as corroboration.