{"schema":"postcutoff/event@1","as_of":"2026-10-12T07:22:00+02:00","url":"https://postcutoff.com/e/2026-10-09-nirenberg-conjecture-closed-asymptotic-curves-counterexample/","md":"https://postcutoff.com/e/2026-10-09-nirenberg-conjecture-closed-asymptotic-curves-counterexample/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":"2026-10-09-nirenberg-conjecture-closed-asymptotic-curves-counterexample","date":"2026-10-09","date_precision":"day","short_title":"Nirenberg's 1962 conjecture on closed asymptotic curves disproved","deck":"Claude and ChatGPT found the first, polygonal counterexample (Banks, Ghomi, Qiu, Raffaelli)","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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.13848: A counterexample to Nirenberg's conjecture on closed asymptotic curves","url":"https://arxiv.org/abs/2610.13848","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Georgia Institute of Technology","Anthropic","OpenAI"],"title":"Nirenberg's 1962 conjecture on closed asymptotic curves disproved; Claude and ChatGPT found the first, polygonal counterexample (Banks, Ghomi, Qiu, Raffaelli)","summary":"On 9 Oct 2026 Harrison Banks, Mohammad Ghomi, Hongda Qiu and Matteo Raffaelli posted arXiv 2610.13848. It constructs a smooth embedded annulus in R³ with negative curvature, bounded by two convex curves in tangent planes, that contains two closed asymptotic curves. This contradicts a conjecture Louis Nirenberg made in 1962. The acknowledgement says the work was \"partly carried out with AI tools Claude (Anthropic) and ChatGPT (OpenAI)\": on a discretized version with six-sided polygonal curves, \"the AI models found a counterexample\", which the authors smoothed and extended. Unrefereed preprint.","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"],"key_numbers":[],"tags":["math","differential-geometry","counterexample","nirenberg","surfaces","claude","chatgpt"],"science":{"field":"mathematics","subfield":"differential geometry of surfaces","problem":"Nirenberg's conjecture (1962): a negatively curved annulus bounded by convex curves in tangent planes contains no closed asymptotic curve","result":"Counterexample: a smooth embedded annulus with negative curvature in its interior, convex boundary curves in tangent parallel planes, and two closed asymptotic curves.","open_since":"1962","ai_system":["Claude","ChatGPT"],"human_role":"Human-led with AI tools: AI models found a first counterexample for a discretized (hexagonal) version; the authors smoothed it and separately built the explicit construction in the paper","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint; explicit construction with computations verified in a Mathematica notebook"},"status":"pending","shock":"The obstacle in Nirenberg's rigidity program was expected to be removable; instead it is a genuine exception.","short":"Nirenberg's closed asymptotic curves conjecture"},"body_md":"## What happened\n\nAsymptotic curves are the directions along which a saddle-shaped surface does not bend. In 1962 Louis Nirenberg, trying to prove\nthat smooth \"tight\" surfaces are rigid, needed to exclude closed asymptotic curves in the negatively curved parts of a surface,\nand conjectured that they cannot occur. Banks, Ghomi, Qiu and Raffaelli show that they can: their annulus contains two.\n\nThe acknowledgement describes two tracks. First the authors discretized a weaker form of the conjecture into polygons with six\nsides, and Claude and ChatGPT found a polygonal counterexample, which the authors smoothed and extended numerically. Later they\nfound the explicit construction via Lelieuvre's formula that the paper presents.\n\n## Why it matters\n\nIt settles a named conjecture of a major analyst after 64 years, in the negative. The AI role is specific and limited: the models\nsupplied the first evidence that the conjecture was false, and the published proof is a human, classical construction. The paper\ndoes not say which model versions were used.","disputed":[{"label":"Verification","text":"Unrefereed preprint; explicit construction with computations verified in a Mathematica notebook","sources":[]}],"related":[{"id":"2026-10-09-arxiv-ai-disclosed-results-early-october-2026","url":"https://postcutoff.com/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/","date":"2026-10-09","date_precision":"day","short_title":"arXiv, 5–9 October 2026","deck":"More conjectures settled with disclosed AI help (Quillen's local parameter conjecture in dimension 4, Laughlin gap, poset reconstruction), next to human-only results","takeaway":"The catalogue shows what the everyday use of frontier models in research mathematics looked like in October 2026: routine, disclosed in a sentence or two, and uneven in reliability.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-12","updated":"2026-10-12","orgs":["OpenAI","Anthropic","various mathematicians"]},{"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":"2026-10-12","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":162,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":101,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":192,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":131,"in_training_data":false}],"short_url":null}