What was already known
The cited width conjectures proposed a quadratic upper bound for this setting; a prior four-generator result covered sufficiently large width.
AI involvement
Ben Keene chose and supervised the target. GPT-6 Astra found the retained combinatorial argument, and Claude agents built its Lean proof.
Astra originated the retained proof, and Claude built its formal version, under Keene's target selection and supervision.
Importance
Replaces a quadratic bound with a linear one for all minimally four-generated numerical semigroups.
Limitations
The claims that the bound is attained at every width, a second proof, and reverse bridge equality are outside the formalized result. Comparator replay and external human proof review are not documented for the cited revision.
Technical verification
The two upper bounds are reported built and kernel-checked in Lean with permitted axioms. Comparator had not yet run at the cited source; sharpness and the reverse bridge are not included.
Formal checking
The two upper bounds are reported built and kernel-checked in Lean with permitted axioms. Comparator had not yet run at the pinned source; sharpness and the reverse bridge are not included.
- Checker
- Lean 4 kernel
- Scope
- SemigroupBetti.beta1_le_two_mul_normalizedWidth and beta1_le_two_mul_width
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The repository has a readable informal proof, but does not identify a human who explains its central fiber-graph mechanism; Few is the evidence-gap default, not a finding that Keene lacks understanding.
Correction and publication history
Replaces a quadratic bound with a linear one for all minimally four-generated numerical semigroups.
Read this published revision