Smooth Carathéodory and Loewner conjectures disproved with Claude
Awaiting review
Importance: major (4 of 5)The takeaway
On 19 Aug 2026 Levent Alpöge (Anthropic) posted an explicit formula on X: a smooth strictly convex closed surface in ℝ³ with exactly one umbilic point, against Carathéodory’s conjecture that every such surface has at least two, and umbilics of index 1 + k/2, against Loewner’s index ≤ 1. He credited John-Paul Smith ‘and of course claude’. On 25 Oct the umbilic specialists Naoya Ando and Masaaki Umehara posted a simpler, self-contained construction on arXiv, found with GPT-6.1’s help. The analytic case, the classical form of both conjectures, is not affected.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 27 October 2026
Your AI and this story
- GPT-6 Astra111 days after its cutoff
- Claude Opus 5.550 days after its cutoff
- Gemini 4 Argon141 days after its cutoff
- Grok 4.780 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 50 days before it.
Key facts
- Alpöge’s post (19 Aug 2026, verbatim excerpt): ‘C^\infty caratheodory and loewner are false, let f(x + iy) := -\cos(2x) / 4 + 3\cos(2y) / 10 - \cos(4y) / 32 + \sin(x)\sin(y) … then g_k has an index 1 + k/2 umbilic point at 0, and g_2 … is the support function of a convex body with exactly one umbilic point.’ ~107k views, 460 likes (fxtwitter, 27 Oct)
- Provenance as told by Alpöge: the problem came up in a chat with the mathematician Micah Warren (@AchimWar); seeing ‘the complicated literature situation’, he asked whether he could ‘look at with a model’; ‘me, John-Paul Smith, and of course claude checked’, and a PDF write-up circulated for a week before the public post, which he wrote ‘in a form certifiable by models’
- Conjectures: Carathéodory’s conjecture says every closed convex surface in ℝ³ has at least two umbilics (points where the principal curvatures coincide). Loewner’s conjecture says an isolated umbilic has index at most 1 (for the principal line fields), which would imply Carathéodory’s, since on a closed convex surface the indices add up to 2
- Follow-up (arXiv 2610.34036, 25 Oct 2026; Naoya Ando, Kumamoto University; Masaaki Umehara, Institute of Science Tokyo): ‘Alpöge recently announced counterexamples to both conjectures in the C∞ category, developed with the aid of AI … The purpose of the present paper is to give a simpler construction through a variant of Alpöge’s construction.’ For every k they build a smooth function flat at the origin whose Hessian eigenline fields have an isolated singularity of index 1 + k/2, and a strictly convex closed smooth surface with exactly one umbilic, of index 2
- Ando–Umehara AI statement (verbatim): ‘We used ChatGPT (OpenAI, GPT-6.1) to assist in the search for candidate functions for our constructions and in revising the exposition. The authors have verified all arguments presented in this paper and take full responsibility for its mathematical content.’
- Scope: the counterexamples are C^∞ but not real-analytic (the support functions use factors like e^(−|z|^(−1/4)) that are flat at the origin). The analytic case of Carathéodory’s conjecture goes back to Hamburger’s proof in the 1940s and is untouched. Guilfoyle (2018) had already built single-umbilic spheres for Riemannian metrics arbitrarily close to the flat one, and Guilfoyle–Klingenberg (2025) describe the conjectures as hinging on symmetry
- Status: no Lean formalization and no refereed paper of Alpöge’s construction; Ando–Umehara’s arXiv paper is the first independent written verification by specialists (they work on umbilics and have published on them before). Not previously in this dataset: found on 27 Oct through the Ando–Umehara paper’s reference to the X post
What happened
An umbilic is a point of a surface where it curves equally in every direction, like every point of a sphere. Carathéodory conjectured that every closed convex surface has at least two of them. Loewner’s stronger conjecture limits how the curvature lines can swirl around an isolated umbilic (index at most 1). With indices summing to 2 on such a surface, it would rule out a surface with only one umbilic.
On 19 Aug 2026 Levent Alpöge, a number theorist at Anthropic, posted on X that both conjectures fail for smooth (C^∞) surfaces. The post gives the construction as explicit formulas, built from a short trigonometric seed function. He wrote that he had discussed the problem with Micah Warren a week earlier and looked at it “with a model”, and that “me, John-Paul Smith, and of course claude checked”. A PDF had circulated privately before the post.
Two months later, on 25 Oct, Naoya Ando and Masaaki Umehara, who work on umbilics, posted a self-contained arXiv paper with a simpler variant of Alpöge’s construction. It gives umbilics of index 1 + k/2 for every k and a strictly convex closed smooth surface with exactly one umbilic. They used GPT-6.1 to search for candidate functions.
We did not cover the August post when it appeared. It was found on 27 Oct through the arXiv paper’s reference to it.
Why it matters
These are two of the best-known open problems about surfaces in ℝ³. The constructions do not touch the analytic case, which is how the conjectures were classically posed and where Hamburger’s proof applies. But they settle the smooth case, whose literature Alpöge called “complicated”, with formulas short enough to check by hand or by a model. The paper by specialists turns a social-media announcement into a written proof that others can check.
What is disputed or not yet verified
| Verification | X-post announcement with explicit formulas; independent simplified construction in an unrefereed arXiv preprint by specialists (2610.34036); no formal proof |
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Sources
4 sources from 2 sites. Numbers match the chips in the text.
4 sources: 3 primary, 1 reaction
Primary
- arXiv 2610.34036: A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures (Ando, Umehara)arxiv.org, paper
- Guilfoyle (2018): On isolated umbilic points (counterexamples for near-flat Riemannian metrics)arxiv.org, paper
- Guilfoyle, Klingenberg (2025): The three obdurate conjectures of differential geometryarxiv.org, paper
Reactions
Changes
- Filed (science agent; found via arXiv 2610.34036’s citation of the X post; post read through fxtwitter, paper PDF read)