Nirenberg’s 1962 conjecture on closed asymptotic curves disproved
Claude and ChatGPT found the first, polygonal counterexample (Banks, Ghomi, Qiu, Raffaelli)
Event confirmedAwaiting review
The takeaway
A 64-year-old conjecture from Louis Nirenberg’s program on rigidity of surfaces is false. The authors say AI models found the first discrete counterexample, which they then smoothed; the final construction in the paper is classical and checked in Mathematica.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 12 October 2026
Your AI and this story
- GPT-6 Astra162 days after its cutoff
- Claude Opus 5.5101 days after its cutoff
- Gemini 3.8 Flash192 days after its cutoff
- Grok 4.7131 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 101 days before it.
Key facts
- Paper: ‘A counterexample to Nirenberg’s conjecture on closed asymptotic curves’, arXiv 2610.13848 [math.DG], 10 pages, 3 figures, dated 6 Oct and posted 9 Oct 2026
- Theorem 1.1: ‘There exists a C∞ embedded annulus M ⊂ R³ whose interior has negative Gauss curvature and contains two distinct closed asymptotic curves. The boundary components of M are strictly convex curves in parallel planes, each tangent to M along the corresponding boundary component.’
- The conjecture: in his program to extend Alexandrov’s 1938 rigidity theorem for analytic tight surfaces to smooth surfaces, Nirenberg ‘conjectured that a negatively curved annulus bounded by convex curves in tangent planes cannot contain a closed asymptotic curve’. He had proved rigidity assuming at most one closed asymptotic curve per component
- Method: Lelieuvre’s classical formula yields a closed curve, realized as an asymptotic curve on a ruled annulus; the second closed asymptotic curve comes from the periodic solution of a Riccati equation. ‘All computations in this paper have been carried out and verified in a Mathematica notebook’
- AI disclosure (Acknowledgement): ‘This paper is the culmination of six months of work, partly carried out with AI tools Claude (Anthropic) and ChatGPT (OpenAI). Initially we investigated the weak form of the conjecture … We discretized the problem and considered polygonal curves with 6 sides. The AI models found a counterexample of this form, which we then smoothed and extended to a belt, and after a perturbation to an annulus by numerical means. Later, on a separate track, we investigated the conjecture via Lelieuvre’s formula … The authors have checked all AI assisted content and take full responsibility for this manuscript.’
- Model versions are not given. Not found as of 12 Oct 2026: expert reactions or press coverage
What happened
Asymptotic curves are the directions along which a saddle-shaped surface does not bend. In 1962 Louis Nirenberg, trying to prove that smooth “tight” surfaces are rigid, needed to exclude closed asymptotic curves in the negatively curved parts of a surface, and conjectured that they cannot occur. Banks, Ghomi, Qiu and Raffaelli show that they can: their annulus contains two.
The acknowledgement describes two tracks. First the authors discretized a weaker form of the conjecture into polygons with six sides, and Claude and ChatGPT found a polygonal counterexample, which the authors smoothed and extended numerically. Later they found the explicit construction via Lelieuvre’s formula that the paper presents.
Why it matters
It settles a named conjecture of a major analyst after 64 years, in the negative. The AI role is specific and limited: the models supplied the first evidence that the conjecture was false, and the published proof is a human, classical construction. The paper does not say which model versions were used.
What is disputed or not yet verified
| Verification | Unrefereed preprint; explicit construction with computations verified in a Mathematica notebook |
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Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed from the arXiv PDF