Log-concavity of codimension-three pure O-sequences

The question asks whether every pure $O$-sequence of codimension $3$ and type $2$ is log-concave. The question arose from earlier work of Iarrobino, and the 2022 literature identifies this as the principal unresolved case.

Known results

• Log-concavity holds for pure $O$-sequences of type $1$.
• Non-log-concave examples exist for $(r,t)=(3,4)$ and for all $r\geq 4$ with $2\leq t\leq r+1$.
• All level Hilbert functions of codimension $2$, and all Gorenstein Hilbert functions of codimension $3$, are settled cases.

May 2026 AI-generated proof claim

A newly reported AI-driven formal-proof-search agent claims a proof for the codimension-$3$, type-$2$ case, using a reformulation of the Hilbert function and case analysis of second-difference inequalities. The claim is not independently verified by a published proof or other corroborating source.

Current status (as of May 2026): the case remains unresolved in the literature, despite an unverified claim of an AI-generated proof.