View source for The Kervaire invariant problem at the prime three

''Not to be confused with [[The Kervaire conjecture]] on group presentations.''

The '''odd-primary Kervaire invariant problem''' asks which of the standard Kervaire classes ''b''<sub>''j''</sub> on the two-line of the mod ''p'' Adams spectral sequence represent stable homotopy classes of spheres; its strong form asks for representatives of additive order ''p''. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims a complete answer at the prime 3 for both forms: the classes survive exactly for ''j'' = 0, 2 and 3, each with a representative of order three, so that there is an order-three Kervaire element in stem 322. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the manuscript:
** The class ''b''<sub>''j''</sub> lies in Ext<sup>2, 4·3<sup>''j''+1</sup></sup> over the mod-3 Steenrod algebra, in stem 4·3<sup>''j''+1</sup> − 2. The classical prime-two Kervaire invariant one problem, which Browder identified with the permanence of the squares ''h''<sub>''j''</sub><sup>2</sup>, has a different family and dimension convention.
** Toda proved that ''b''<sub>1</sub> supports a nonzero differential at every odd prime. Ravenel excluded every ''b''<sub>''j''</sub> with ''j'' at least 1 for every prime ''p'' at least 5, using restriction to a cyclic subgroup of a Morava stabilizer group. At the prime 3 this conclusion fails: the classical surviving class in stem 106 is detected by a corrected combination of β<sub>9/9</sub> and β<sub>7</sub>, although β<sub>9/9</sub> itself supports a differential.
** Selick's criterion identifies an order-''p'' Kervaire representative with a stable map from a Moore spectrum whose cone has a specified nonzero reduced power. Amelotte related this criterion to product decompositions of looped degree-''p'' sphere fibres and recorded the order-three representative in stem 106. His class θ<sub>''r''</sub> is detected by ''b''<sub>''r''−1</sub>, so the stem-322 class corresponds to θ<sub>4</sub>.
** Hill, Hopkins and Ravenel proposed using the action of the cyclic group of order 9 on height-six Morava ''E''-theory at the prime 3. Belmont and Ray proved detection for this action and, assuming an explicit coefficient-action model, calculated the relevant group cohomology; their Conjectures 0.1 and 0.3 concern this approach.

== The original results ==
* '''Classification at the prime 3 (24 September 2026).''' Theorem 1.1 of the manuscript states that, at the prime 3, the classes ''b''<sub>''j''</sub> that survive to the ''E''<sub>∞</sub> page are exactly those with ''j'' = 0, 2 and 3, in stems 10, 106 and 322, and that each of them detects a class of additive order three. The class ''b''<sub>1</sub> and every ''b''<sub>''j''</sub> with ''j'' at least 4 do not survive. The cases ''j'' = 0, 1, 2 are classical; the exclusion of ''j'' at least 4 and the order-three class in stem 322 are the new assertions.
* '''Unstable consequences.''' Using Selick's and Amelotte's results, the manuscript deduces that, after localization at 3, for every integer ''n'' greater than 1 the loop space of the fibre of the degree-three map on ''S''<sup>2''n''+1</sup> has a nontrivial product decomposition, and Anick's space ''T''<sup>2''n''+1</sup>(3) admits a homotopy-associative H-space multiplication, exactly when ''n'' is 3, 27 or 81. At ''n'' = 81 it gives an H-space splitting into factors built from ''T''<sup>163</sup>(3) and ''T''<sup>487</sup>(3).
* '''On the Belmont–Ray conjectures.''' The manuscript states that it proves the coefficient-action model proposed in Conjecture 0.3 of Belmont and Ray, and that the full-gap clause of their Conjecture 0.1 is false: it exhibits an infinite-order element in every even degree of the homotopy fixed points of height-six Morava ''E''-theory under the cyclic group of order 9. It states that the periodicity and torsion questions of that conjecture are left untouched.
* ''Method, as described:'' the proof has two independent branches. The exclusion of ''j'' at least 4 maps a putative sphere class into the height-six detector, proves finite Euler-class bounds and uses the coefficient geometry to force an outgoing differential on every such detector. The stem-322 class is constructed from the order-three class in stem 106 by a truncated extended-cube construction after Toda, with a nonzero reduced power of degree 81 in the cone; the required corrections use finite Adams–Novikov computations over the field with three elements.
* '''Computations.''' The finite computations behind the stem-322 branch are specified in an appendix of the manuscript (integral structure formulas, a terminating resolution, a cobar comparison and elimination procedures) and their results are printed as tables. The largest printed system has 23,384 unknowns and 137,471 equations. The release folder contains the manuscript source and PDF but no program or data files for these computations.
* '''Sources.''' Commit <code>1581d4c</code> of the openai/math repository, folder [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026 The Kervaire invariant problem at the prime three].
* '''Formalization.''' The collection has no Lean scope note in <code>lean/docs/</code> for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 17 October 2026)
|-
| 2026-10-10 || [https://github.com/carlostoledo1891/cert-machine/blob/1e952c4a60883bbc974397e5ccae6cefe2c5e7e5/instruments/openaimath/finite/f605_kervaire_three.py cert-machine], openai/math lane F, check F-605 || Partial verification (finite computations) || Re-implements in exact arithmetic the structure constants obtained from the manuscript's recursion, an ordinary cobar complex reproducing the low-weight checkpoints printed in the appendix, the cochain dimensions of the manuscript's smaller resolution at three slots, the grid sizes and the rational inequalities of the high-filtration argument; the project's [https://github.com/carlostoledo1891/cert-machine/blob/1e952c4a60883bbc974397e5ccae6cefe2c5e7e5/certs/openai-math-finite.json ledger] records these checks as passing. The tables that carry the stem-322 argument, including the two large affine systems, the enclosure, cancellation and injection tests, the product ranks and the vanishing of a cube, are not re-derived, so the recorded verdict for the item is refused. The record estimates that an independent implementation would take days and notes that the topological deductions are theory. || Repository (checked 17 October 2026).
|-
| 2026-10-16 || J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_169.json referee review] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995]) || Review (AI-assisted audit) || The referee review reports no gap. It notes that the positive branch, the order-three class in stem 322, depends on large exact rank computations over the field with three elements that the review could not reproduce, and states that this describes what it could check, not a demonstrated error. It also notes that the new proof of the coefficient-action model was only spot-checked. || Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]