What was already known
A separate theorem gives a linear bound for four-generated semigroups. Shifted-semigroup work provides a periodicity window, but neither supplies this exact five-generator family.
AI involvement
Ben Keene set the project and reviewed correspondence. GPT-6 Astra supplied the family presentation, minimality argument and formal minimality proofs; Claude supplied scaffolding.
Astra supplied the retained informal presentation and minimality argument within Keene’s directed family project; the source does not establish full autonomy.
- Supporting source (github.com) — Reported three-statement Comparator run, as linked from pinned metadata
Importance
Gives an unbounded five-generator family with an exact first Betti count and presentation.
Limitations
The proposed bound β₁≤2w+2 for all five-generated semigroups remains conjectural. Machine checks are reported for the family statements; no external human proof check is documented. The novelty search is incomplete, and the reported multi-system check predates the final documentation revision.
Technical verification
The cited record reports sorry-free Lean proof and successful Comparator checks with Lean, NanoDa and con-ron for all three F statements at the later packaging run. Theorem W lies outside the solution closure with an admitted proof and is not rated as proved. Source-reported; this review did not reproduce kernels.
Independent checking
Comparator statement check and multi-kernel proof replay
- Checker
- Lean, NanoDa and con-ron on the reported GitHub Actions run 39036226197
- Scope
- F: family spanRank 4k+2, minimal five-generation and explicit nonredundant presentation
- Basis
- Reported by the linked checker
Human understanding
Default for undocumented human explanation of the complete family proof: Keene read correspondence, while the named mathematical reviews and reconstruction were by Astra.
Correction and publication history
Gives an unbounded five-generator family with an exact first Betti count and presentation.
Read this published revision