{"schema":"postcutoff/event@1","as_of":"2026-10-08T23:45:00+02:00","url":"https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/","md":"https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-07-lebrun-salamon-conjecture-proof","date":"2026-10-07","date_precision":"day","short_title":"Two claimed proofs of the LeBrun–Salamon conjecture","deck":"Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062)","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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hu, Liu & Wan: Transverse rational curves and the LeBrun–Salamon conjecture (arXiv 2610.10410)","url":"https://arxiv.org/abs/2610.10410","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"OpenAI math catalogue (CONTENTS.md, family 062)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"LeBrun & Salamon (1994): Strong rigidity of positive quaternion-Kähler manifolds","url":"https://doi.org/10.1007/BF01231528","type":"paper","group":"primary","domain":"doi.org"},{"n":4,"title":"Buczyński, Wiśniewski & Weber: Algebraic torus actions on contact manifolds (J. Differential Geom.)","url":"https://projecteuclid.org/journals/journal-of-differential-geometry/volume-121/issue-2/Algebraic-torus-actions-on-contact-manifolds/10.4310/jdg/1659987892.full","type":"paper","group":"primary","domain":"projecteuclid.org"}],"official":4,"filed":"2026-10-08","updated":"2026-10-08","orgs":["Chongqing University of Technology","OpenAI"],"title":"Two claimed proofs of the LeBrun–Salamon conjecture: Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062)","summary":"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"],"key_numbers":[],"tags":["math","differential-geometry","algebraic-geometry","quaternion-kahler","contact-fano","chatgpt","openai","priority","unverified"],"science":{"field":"mathematics","subfield":"differential geometry / algebraic geometry (quaternion-Kähler manifolds, contact Fano manifolds)","problem":"LeBrun–Salamon conjecture: every compact positive quaternion-Kähler manifold is a symmetric Wolf space","result":"Claimed proof for all real dimensions 4n with n ≥ 2 (the 4-dimensional case was already known), via transverse rational curves on the twistor space; independently claimed in OpenAI's family 062.","open_since":"1994","ai_system":["ChatGPT","OpenAI internal model (unreleased)"],"human_role":"Human-led with AI tools for Hu–Liu–Wan (ChatGPT as an 'auxiliary tool', role not specified); autonomous for OpenAI's family 062","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprints; no formal proof; no expert assessment found as of 8 Oct"},"status":"pending","shock":"A classification problem open for three decades received two independent claimed proofs within two weeks, one of them AI-generated.","short":"LeBrun–Salamon conjecture"},"body_md":"## What happened\n\nPositive quaternion-Kähler manifolds are Einstein manifolds with holonomy in Sp(n)Sp(1). LeBrun and Salamon conjectured in\n1994 that the only compact ones are the symmetric Wolf spaces, such as quaternionic projective space and Gr₂(ℂⁿ⁺²). Through\nthe twistor construction the problem is equivalent to showing that every contact Fano manifold is homogeneous, an adjoint\nvariety. Earlier work settled low dimensions and cases with large torus symmetry.\n\nOn 7 Oct 2026 Hu, Liu and Wan posted a proof in all dimensions. They study rational curves on the twistor space that are\ntransverse to the contact distribution, build an additive group action from an intrinsic tensor, and then invoke Hwang–Mok,\nBeauville and LeBrun to identify the twistor space and the metric. The paper declares ChatGPT use as an \"auxiliary tool\"\nwithout details. It also includes a priority remark about OpenAI's catalogue, released the day before, which claims the same\nconjecture in family 062 by a different argument.\n\n## Why it matters\n\nIf either proof holds, it closes a well-known classification problem in Riemannian and contact geometry. Like the exact\noverlaps conjecture the same day, it shows a new pattern: human groups posting their own proofs with priority notes as soon as\nOpenAI's catalogue makes a problem look \"taken\". The AI's share in the human paper is undisclosed beyond one sentence, so this\nentry counts it as human-led.","disputed":[{"label":"Verification","text":"Unrefereed preprints; no formal proof; no expert assessment found as of 8 Oct","sources":[]}],"related":[{"id":"2026-10-07-exact-overlaps-conjecture-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/","date":"2026-10-07","date_precision":"day","short_title":"Exact overlaps conjecture for self-similar measures on the line proved","deck":"GPT-6 Astra found the first proof (Kittle & Kogler), 47,000-line Lean formalization; OpenAI's catalogue claims it independently","takeaway":"A central conjecture of fractal geometry, open in general since Hochman's 2014 breakthrough, now has a proof whose key part came from a GPT-6 Astra session. The authors rewrote and formalized it in Lean, and OpenAI's own release claims the same theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-08","updated":"2026-10-08","orgs":["University College London","Institute for Advanced Study","OpenAI"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":42,"official":12,"filed":"2026-10-07","updated":"2026-10-08","orgs":["OpenAI"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-09-30","updated":"2026-10-08","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-08","type":"filed","text":"Created from the arXiv PDF (abstract, Remark 1.2, AI declaration, references) and OpenAI's CONTENTS.md"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-08","run":null,"sources_read":"The arXiv PDF (abstract, Remark 1.2, AI declaration, references) and OpenAI's CONTENTS.md","updated":"2026-10-08","human_review":null,"version":{"date":"2026-10-08"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":160,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":99,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":190,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":129,"in_training_data":false}],"short_url":"https://postcutoff.com/s/two-proofs-lebrun-salamon"}