Two claimed proofs of the LeBrun–Salamon conjecture
Hu, Liu & Wan (arXiv, ChatGPT as ‘auxiliary tool’) and OpenAI’s catalogue (family 062)
Awaiting review
Importance: major (4 of 5)The takeaway
The 1994 conjecture that every positive quaternion-Kähler manifold is a symmetric Wolf space now has two independent claimed proofs, one by human geometers who used ChatGPT and one in OpenAI’s AI-generated catalogue. Neither has been refereed.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 8 October 2026
Your AI and this story
- GPT-6 Astra160 days after its cutoff
- Claude Opus 5.599 days after its cutoff
- Gemini 3.8 Flash190 days after its cutoff
- Grok 4.7129 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 99 days before it.
Key facts
- Conjecture: LeBrun & Salamon, ‘Strong rigidity of positive quaternionic–Kähler manifolds’ (Invent. Math. 118, 1994); equivalent, via twistor spaces, to the homogeneity of contact Fano manifolds (every contact Fano manifold is the adjoint variety of a simple Lie group). Previously known in low dimensions and under large-symmetry assumptions (Buczyński–Wiśniewski–Weber; Occhetta–Romano–Solá Conde–Wiśniewski)
- Hu–Liu–Wan method: transverse rational curves (stable domains of contact degree two) on the twistor space, an intrinsic two-point tensor extended across the nodal boundary, an algebraic additive action from a nilpotent endomorphism, first-jet interpolation, then Hwang–Mok, Beauville’s theorem and LeBrun’s uniqueness theorem
- AI declaration (verbatim): ‘The authors used ChatGPT as an auxiliary tool in this work. The authors verified and completed all mathematical arguments and take full responsibility for the content of this paper.’
- Priority remark (verbatim): ‘This project was developed over several months, and its main result and core argument were obtained well before the appearance of the OpenAI manuscript [21]. Our proof was developed independently and follows a different approach.’ They describe OpenAI’s proof as smoothing a pair of contact lines into very free conics and using a diagonal-jet construction
- OpenAI family 062, ‘Projective contact classification and the LeBrun–Salamon conjecture’ (manuscript ‘Contact Fano manifolds and the LeBrun–Salamon conjecture’, 23 Sep 2026): also claims contact-Fano homogeneity and a classification of projective contact manifolds of dimension ≥ 3 (b₂ = 1: adjoint varieties; b₂ ≥ 2: ℙ(T*Z)). CONTENTS.md shows no Lean link for this family
What happened
Positive quaternion-Kähler manifolds are Einstein manifolds with holonomy in Sp(n)Sp(1). LeBrun and Salamon conjectured in 1994 that the only compact ones are the symmetric Wolf spaces, such as quaternionic projective space and Gr₂(ℂⁿ⁺²). Through the twistor construction the problem is equivalent to showing that every contact Fano manifold is homogeneous, an adjoint variety. Earlier work settled low dimensions and cases with large torus symmetry.
On 7 Oct 2026 Hu, Liu and Wan posted a proof in all dimensions. They study rational curves on the twistor space that are transverse to the contact distribution, build an additive group action from an intrinsic tensor, and then invoke Hwang–Mok, Beauville and LeBrun to identify the twistor space and the metric. The paper declares ChatGPT use as an “auxiliary tool” without details. It also includes a priority remark about OpenAI’s catalogue, released the day before, which claims the same conjecture in family 062 by a different argument.
Why it matters
If either proof holds, it closes a well-known classification problem in Riemannian and contact geometry. Like the exact overlaps conjecture the same day, it shows a new pattern: human groups posting their own proofs with priority notes as soon as OpenAI’s catalogue makes a problem look “taken”. The AI’s share in the human paper is undisclosed beyond one sentence, so this entry counts it as human-led.
What is disputed or not yet verified
| Verification | Unrefereed preprints; no formal proof; no expert assessment found as of 8 Oct |
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Sources
4 sources from 4 sites. Numbers match the chips in the text.
4 sources: 4 primary
Primary
- Hu, Liu & Wan: Transverse rational curves and the LeBrun–Salamon conjecture (arXiv 2610.10410)arxiv.org, paper
- OpenAI math catalogue (CONTENTS.md, family 062)github.com, paper
- LeBrun & Salamon (1994): Strong rigidity of positive quaternion-Kähler manifoldsdoi.org, paper
- Buczyński, Wiśniewski & Weber: Algebraic torus actions on contact manifolds (J. Differential Geom.)projecteuclid.org, paper
Changes
- Filed from the arXiv PDF (abstract, Remark 1.2, AI declaration, references) and OpenAI’s CONTENTS.md