View source for Curtis's conjecture

'''Curtis's conjecture''' describes the image of the mod 2 stable Hurewicz homomorphism from the stable homotopy groups of spheres to the homology of the infinite loop space ''Q''<sub>0</sub>''S''<sup>0</sup>: in positive degrees it should be spanned by the images of the Hopf invariant one classes η, ν, σ and of the Kervaire invariant one classes. A manuscript dated 25 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims a proof in all positive degrees, with Eccles's conjecture for spheres as a consequence. 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:
** Curtis proposed the description in his study of the Dyer–Lashof and lambda algebras (1975). May's later account records a gap, found by Wellington, in the proposed proof.
** Adams's Hopf invariant one theorem identifies the Hopf cases, and Browder related the framed Kervaire invariant to the classes ''h''<sub>''j''</sub><sup>2</sup> in the mod 2 Adams spectral sequence. Hill, Hopkins and Ravenel showed that Kervaire invariant one classes can exist only up to dimension 126; the manuscript cites a preprint of Lin, Wang and Xu for existence in that dimension.
** Eccles and Zare gave restrictions on spherical squares and their roots, including the exclusion of higher iterated squares.
** Eccles's conjecture concerns the analogous realization problem for ''QX'': a sphere map with nonzero mod 2 Hurewicz image should have stable adjoint detected by homology or by a primary cohomology operation in its mapping cone.

== The original results ==
* '''Curtis's conjecture (25 September 2026).''' Theorem 1.1 of the manuscript states that the image of the mod 2 stable Hurewicz homomorphism in positive degrees is spanned over the field with two elements by the images of η, ν, σ, in degrees 1, 3 and 7, and of the Kervaire invariant one classes θ<sub>''j''</sub>, in degrees 2<sup>''j''+1</sup> − 2, whenever they exist. The manuscript notes that it does not assert existence of θ<sub>''j''</sub> in any additional degree.
* '''Corollaries.''' Eccles's conjecture holds for every sphere ''S''<sup>''n''</sup> with ''n'' > 0. Combined with the theorem of Hill, Hopkins and Ravenel, the positive Hurewicz image vanishes outside the degrees 1, 2, 3, 6, 7, 14, 30, 62 and 126; the manuscript states that this is an upper bound on the support, not a claim of nonvanishing in each listed degree.
* ''Method, as described:'' the argument follows the last nonzero homology suspension of a spherical class, shows that the terminal class is a square whose root keeps nonzero suspension, and turns the mapping cone of the next adjoint into a nonzero extension of Steenrod modules. An algebraic obstruction for the modules dual to primitives, built from Dickson algebras, allows such an extension only in one low case. At the lowest level, fourth powers are excluded by spectrum pairings and integral characteristic classes, and Kuhn's lifting theorem bounds the Adams filtration of the remaining classes by two, which gives Kervaire detection through Browder's identification.
* '''Sources.''' Commit <code>1581d4c</code> of the openai/math repository, folder [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Stable-Hurewicz-Image-of-the-Sphere-at-Two-September-25-2026 The stable Hurewicz image of the sphere at two].
* '''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)
|-
| 2026-10-16 || J. Jardine, Lattice24, [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_293.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 after reading the whole proof. Its one minor note says that the bound on the Adams filtration rests on the cited statement of Kuhn's 2018 lifting theorem, whose exact hypotheses the review could not check offline, and on a vanishing claim for the derived eighth power in a truncated algebra. || 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]]