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The Kervaire invariant problem at the prime three: which standard classes bjb_j survive the mod-3 Adams spectral sequence

In the mod-pp Adams spectral sequence ExtAps,t(Fp,Fp)⇒π∗S\mathrm{Ext}_{\mathcal A_p}^{s,t}(\mathbb F_p,\mathbb F_p)\Rightarrow \pi_*^S, the odd-primary analogues of the Kervaire classes are bj∈E22,2(p−1)pj+1b_j\in E_2^{2,2(p-1)p^{j+1}}. Toda showed that b1b_1 supports a differential at every odd prime, and Ravenel (1978) proved that no bjb_j with j≥1j\ge1 survives for p≥5p\ge5. At p=3p=3 the argument fails: b0b_0 (stem 10) and b2b_2 (stem 106) survive, and the fate of the higher classes was open; Hill, Hopkins and Ravenel proposed attacking it through the action of C9C_9 on height-six Morava EE-theory. The strong form also asks for representatives of order pp. At p=3p=3, for which jj does bjb_j survive to E∞E_\infty, and when it does, is it detected by an element of order three?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Stable homotopy theory; odd-primary Kervaire invariant
Posed by
Douglas Ravenel (1978), whose odd-primary nonexistence theorem left p = 3 open; program of Hill, Hopkins and Ravenel (2011)
Year posed
1978
Years open
48y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: at p=3p=3, K=K3={0,2,3}K=K_3=\{0,2,3\}, i.e. bjb_j survives to E∞E_\infty exactly for j=0,2,3j=0,2,3 (stems 10, 106 and 322), and each surviving class is detected by an element of additive order three; b1b_1 and all bjb_j with j≥4j\ge4 die. The new content is the order-three element in stem 322 and the exclusion of every j≥4j\ge4. Corollaries via Amelotte and Selick: ΩS2n+1{3}\Omega S^{2n+1}\{3\} splits nontrivially exactly for n∈{3,27,81}n\in\{3,27,81\}, and the Anick space T2n+1(3)T^{2n+1}(3) is homotopy associative exactly for those nn. Side result: an infinite-order class in π2n(E6hC9)\pi_{2n}(E_6^{hC_9}) for every nn, refuting the full-gap clause of Belmont-Ray's Conjecture 0.1. Not covered: other odd primes (already settled for p≥5p\ge5) or the prime-two problem.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. Single manuscript dated September 24, 2026. The paper says it uses classical inputs (Toda's differentials, the Miller-Ravenel-Wilson two-line, Amelotte's and Selick's criteria) and proves Belmont-Ray's coefficient model itself.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1 and the reduction of the proof to its two branches were read against the problem as Ravenel and Hill-Hopkins-Ravenel framed it; the proof was not refereed. No Lean formalization exists for this family. The paper states its scope: the standard family bjb_j in the mod-3 Adams spectral sequence, not the classical prime-two Kervaire problem. The exclusion of j≥4j\ge4 runs through a height-six C9C_9 detector and finite-page outgoing differentials; the paper says it does not use the 972-periodicity or the degree −2-2 vanishing clause of Belmont-Ray's Conjecture 0.1, and it claims to refute the latter clause.

Source

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