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The Kervaire invariant problem at the prime three
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Abstract
We solve the Kervaire invariant problem at the prime three for the standard Kervaire classes in the mod-three Adams spectral sequence. These classes survive precisely at indices 0, 2, and 3, and each surviving detection coset contains an element of additive order three. In particular, there is an order-three Kervaire element in stem 322.
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Comments and discussion
- Rin Ray, About Rin, research entry on the prime-three Kervaire problem (Good Fibrations, undated). Ray identifies the OpenAI manuscript as a claimed resolution using the results and method of joint work with Eva Belmont, without reporting a verification of its proof.
Related prior literature mentioned by others
- Eva Belmont and Rin Ray, Towards the p=3 Kervaire Invariant Problem: The E₂-page for the homotopy fixed points spectral sequence computing π∗(E₆ʰᶜ⁹) (arXiv:2507.10157, July 14, 2025). Identified by Ray as an input to the claimed resolution, this paper proves a detection theorem and computes positive-degree coefficient cohomology conditional on a model for the C₉-action.
Possibly relevant literature (found by the agent, not checked)
- Stephen D. Theriault, A Case When the Fiber of the Double Suspension is the Double Loops on Anick's Space (Canadian Mathematical Bulletin 53 (2010), 730–736). Proves the case n = p of the conjecture BW_n ≃ ΩT^{2np+1}(p); the paper's Corollary 1.2(i) asserts BW_81 ≃ ΩT^{487}(3), an instance of this conjecture, citing Amelotte for it.
- Brayton Gray and Stephen Theriault, An elementary construction of Anick's fibration (arXiv:0710.1024; Geometry & Topology 14 (2010), 243–275). Gives a construction of the Anick fibration that the abstract says is valid for p = 3; the paper's unstable corollary concerns the 3-local Anick spaces T^{2n+1}(3), which it introduces in Amelotte's notation.
- Donald M. Larson, The Adams–Novikov E₂-term for Behrens' spectrum Q(2) at the prime 3 (arXiv:1407.3423, revised March 23, 2015). Its computation and conjectural detection of the prime-three beta family offer a comparison with the manuscript's Adams–Novikov detection arguments for Kervaire classes.
Digests and expositions
- Latent Seminar, The prime-three Kervaire family: detection and order three (October 7, 2026). Explains the outgoing-differential argument and Moore-spectrum correction producing the claimed order-three class, while explicitly excluding independent verification of the complete proofs and finite matrix calculations.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 309
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026,
author = {{OpenAI}},
title = {{The Kervaire invariant problem at the prime three}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026/paper.pdf}{OAI:The-Kervaire-Invariant-Problem-at-the-Prime-Three-September-24-2026}},
year = {2026}
}Family 309: The Kervaire invariant problem at the prime three
Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.
This is the only paper in this family.