OpenAI reports the prime-three Kervaire family

In the standard mod-three Adams Kervaire family, exactly the classes indexed 0, 2 and 3 survive to stable homotopy, in stems 10, 106 and 322, and each surviving detection coset contains an element of additive order three.

Sources for this record

What was already known

Toda excluded the class indexed one at every odd prime, while the remaining prime-three permanence and exclusion pattern was unsettled. The prime-three problem is distinct from the classical prime-two framed Kervaire-invariant-one problem.

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 Notable

Determines survival and order in the standard odd-primary Kervaire family at prime three.

Limitations

The claim concerns this specified mod-three Adams family, not all order-three elements of stable stems or the prime-two Kervaire problem.

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 verification remains Reported.

Read this published revision