--- id: "2026-10-07-lebrun-salamon-conjecture-proof" url: "https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/" as_of: "2026-10-08T23:45:00+02:00" date: "2026-10-07" date_precision: day category: science importance: 4 confidence: medium status: [Awaiting review] verification: Preprint only sources: 4 editor: Adam Bicz human_review: null version: "2026-10-08" --- As of: 2026-10-08 23:45 CEST. Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz. Canonical page: https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/ # Two claimed proofs of the LeBrun–Salamon conjecture Full title: Two claimed proofs of the LeBrun–Salamon conjecture: Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062) On 7 Oct 2026 Zhengyu Hu, Kefeng Liu and Xueyuan Wan (Mathematical Sciences Research Center, Chongqing University of Technology) posted arXiv 2610.10410, claiming a proof of the LeBrun–Salamon conjecture: every compact connected positive quaternion-Kähler manifold of real dimension 4n, n ≥ 2, is a Wolf space after rescaling. Their AI declaration says only that "the authors used ChatGPT as an auxiliary tool". They state that their argument was "obtained well before" OpenAI's 6 Oct catalogue, which also claims the conjecture (family 062, manuscript dated 23 Sep) by a different route. Both proofs are unrefereed, and neither has a Lean formalization. ## 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 ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 160 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 99 days after its cutoff - Gemini 3.8 Flash (training cutoff March 2026): 190 days after its cutoff - Grok 4.7 (training cutoff May 2026): 129 days after its cutoff ## Sources 1. [Hu, Liu & Wan: Transverse rational curves and the LeBrun–Salamon conjecture (arXiv 2610.10410)](https://arxiv.org/abs/2610.10410) (arxiv.org, paper) 2. [OpenAI math catalogue (CONTENTS.md, family 062)](https://github.com/openai/math/blob/main/CONTENTS.md) (github.com, paper) 3. [LeBrun & Salamon (1994): Strong rigidity of positive quaternion-Kähler manifolds](https://doi.org/10.1007/BF01231528) (doi.org, paper) 4. [Buczyński, Wiśniewski & Weber: Algebraic torus actions on contact manifolds (J. Differential Geom.)](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-121/issue-2/Algebraic-torus-actions-on-contact-manifolds/10.4310/jdg/1659987892.full) (projecteuclid.org, paper) ## Changes - 2026-10-08 (filed): Created from the arXiv PDF (abstract, Remark 1.2, AI declaration, references) and OpenAI's CONTENTS.md ## Related - 2026-10-07: [Exact overlaps conjecture for self-similar measures on the line proved](https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/index.md) - 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md) - 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)