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Curtis's conjecture: the mod-2 stable Hurewicz image of the sphere is spanned by the Hopf- and Kervaire-invariant-one classes

For d>0d>0 let hd:πdS→Hd(Q0S0;F2)h_d:\pi_d^S\to H_d(Q_0S^0;\mathbb F_2) be the mod-2 Hurewicz homomorphism into the homology of the basepoint component of QS0=colimrΩrSrQS^0=\mathrm{colim}_r\Omega^rS^r. The obvious elements with nonzero image are the Hopf-invariant-one classes η,ν,σ\eta,\nu,\sigma in degrees 1, 3, 7 and the Kervaire-invariant-one classes θj\theta_j in degrees 2j+1−22^{j+1}-2, when they exist. Curtis (1975), in his study of the Dyer-Lashof and lambda algebras, conjectured that nothing else is detected; May (1977) recorded a gap, found by Wellington, in the proposed proof. The conjecture implies Eccles's conjecture for spheres. Is the image of ⨁d>0hd\bigoplus_{d>0}h_d spanned over F2\mathbb F_2 by the Hurewicz images of η,ν,σ\eta,\nu,\sigma and of the Kervaire-invariant-one classes that exist?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Stable homotopy theory
Posed by
Edward B. Curtis
Year posed
1975
Years open
51y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: the image of ⨁d>0hd\bigoplus_{d>0}h_d is spanned over F2\mathbb F_2 by the Hurewicz images of η,ν,σ\eta,\nu,\sigma and of the Kervaire-invariant-one classes θj\theta_j (degree 2j+1−22^{j+1}-2) whenever they exist; no existence is posited. Corollary 1.2: Eccles's conjecture holds for every sphere SnS^n, n>0n>0: each nonzero spherical class in H∗(QSn;F2)H_*(QS^n;\mathbb F_2) is the bottom class or a suspended Hopf-invariant-one image. Corollary 1.3 (with Hill-Hopkins-Ravenel): hd=0h_d=0 for d>0d>0 outside {1,2,3,6,7,14,30,62,126}\{1,2,3,6,7,14,30,62,126\}, so the positive image vanishes above degree 126. Not shown: an exact dimension of the image or nonvanishing in each listed degree, odd primes, or Eccles's conjecture for QXQX with XX other than a sphere.

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, and that some outputs build on earlier model results. 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). The manuscripts are authored 'OpenAI' and name no human author. The single manuscript (September 25, 2026) is the whole family.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 with Corollaries 1.2 and 1.3 were read against Curtis's conjecture as the manuscript cites it; Theorem 1.1 states the conjecture in every positive degree. The proof (last nonzero homology suspension, weight decomposition through Dickson-algebra modules, a Steenrod-module obstruction, Kuhn's lifting theorem and Browder's identification of Kervaire classes) was not refereed. No Lean formalization accompanies this manuscript. The paper says its exclusion of fourth powers (Section 4) is already present in Eccles-Zare, and Corollary 1.3 uses the Hill-Hopkins-Ravenel nonexistence theorem as an input.

Source

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