OpenAI reports Lech’s multiplicity conjecture

Under every flat local homomorphism of nonzero Noetherian local rings, Hilbert–Samuel multiplicity of the source does not exceed that of the target.

Sources for this record

What was already known

Lech established low-dimensional and special-fiber cases, while later work covered equal-characteristic dimension three and gave weaker general comparison constants.

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Major

Resolves Lech’s multiplicity conjecture in arbitrary dimension and characteristic.

Limitations

The linked Lean scope checks a supporting complete-domain Dutta-multiplicity comparison in characteristic p, not the full flat-local theorem in arbitrary characteristic.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The OpenAI-authored manuscript reports and argues the result, but the reviewed sources do not document a completed check of the full assessed claim.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, prior-work, and Lean-scope mismatch review for non-notifying local inclusion.

Read this published revision