What was already known
Boij, Migliore, Miró-Roig, Nagel and Zanello had proved unimodality for this class; its log-concavity remained open.
AI involvement
DeepMind’s team formalized the target; AlphaProof Nexus generated the substantive proof. The authors supplied a natural-language explanation of the proof.
The agent generated the substantive proof from a given theorem target; no human mathematical repair of the result is documented.
Importance
The proof strengthens unimodality to log-concavity for this precise class.
Limitations
The report identifies a Lean proof, but does not document a separate comparison of its definitions with the natural-language pure O-sequence statement.
Technical verification
The authors report a Lean proof of the exact codimension-three, type-two statement and publish the natural-language theorem. This is source-reported machine checking; the Lean project was not rebuilt here.
Formal checking
Formal proof in the linked AlgebraicGeometry Lean output
- Checker
- AlphaProof Nexus Lean validator
- Scope
- Every pure O-sequence of codimension three and type two is log-concave
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The paper’s supplementary proof explains the order-ideal union of two maximal monomials and the case analysis proving the log-concavity inequalities.
Correction and publication history
The proof strengthens unimodality to log-concavity for this precise class.
Read this published revision