OpenAI reports an integral counterexample to Gersten’s conjecture

There is a two-dimensional ramified regular local ring A of mixed characteristic (0,5) for which K₅(A)→K₅(Frac A) has a nonzero kernel.

Sources for this record

What was already known

Gersten’s conjecture had positive equicharacteristic and smooth or unramified mixed-characteristic cases. A preceding finite-coefficient ramified example did not yield an integral generic kernel.

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

Disproves the unrestricted integral Gersten conjecture with a ramified regular local ring.

Limitations

The counterexample uses a specific mixed-characteristic ramified ring in degree five; it does not contradict smooth or unramified positive cases. No completed proof check was documented in the available sources.

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, and prior-work review for non-notifying local inclusion; technical check remains undocumented.

Read this published revision