{"schema":"postcutoff/event@1","as_of":"2026-10-09T19:24:00+02:00","url":"https://postcutoff.com/e/2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean/","md":"https://postcutoff.com/e/2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean/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":"2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean","date":"2026-10-08","date_precision":"day","short_title":"Hilbert's 16th problem","deck":"Qin & Sun prove H(d) = Ω(d² ln² d), the first log-factor gain since 1995, with GPT-5.6 Sol developing the proof and a Lean 4 certificate","takeaway":"The lower bound for the number of limit cycles of degree-d planar polynomial vector fields improves from d² ln d to d² ln² d. The authors say GPT-5.6 Sol developed the proof strategy and arguments, and the construction is formalized in Lean 4.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Qin & Sun: Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds (arXiv 2610.12218)","url":"https://arxiv.org/abs/2610.12218","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Lean formalization archive (Zenodo, doi:10.5281/zenodo.23219558)","url":"https://doi.org/10.5281/zenodo.23219558","type":"code","group":"primary","domain":"doi.org"},{"n":3,"title":"OpenAI math catalogue (CONTENTS.md, family 143)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"}],"official":3,"filed":"2026-10-09","updated":"2026-10-09","orgs":["Chinese Academy of Sciences","OpenAI"],"title":"Hilbert's 16th problem: Qin & Sun prove H(d) = Ω(d² ln² d), the first log-factor gain since 1995, with GPT-5.6 Sol developing the proof and a Lean 4 certificate","summary":"On 8 Oct 2026 Chaoyang Qin and Xiaoming Sun (Institute of Computing Technology, CAS) posted arXiv 2610.12218, proving that the Hilbert number satisfies H(d) = Ω(d² ln² d), a factor ln d above the d² ln d bound that follows from Christopher–Lloyd (1995). The construction (\"Hierarchical Melnikov Realization\") perturbs an anisotropic Chebyshev Hamiltonian so that the first-order Melnikov function has simple zeros in many disjoint period annuli. \"GPT-5.6 Sol was used to develop the proof strategy and arguments throughout this paper, assist with Lean formalization, and edit the manuscript.\" The authors say the Lean 4 development has no unproved placeholders or project-specific axioms. They note that OpenAI's 6 Oct catalogue (family 143) claims the finiteness of H(d), the opposite, upper-bound side of the problem.","key_facts":["Main theorem: H(d) = Ω(d² ln² d) as d → ∞ (lower bound on the maximum number of limit cycles of real planar polynomial vector fields of degree d)","Prior bounds: Otrokov 1954 (quadratic); Christopher & Lloyd 1995, order d² ln d along d = 2^k − 1, giving Ω(d² ln d) for all d; Li, Chan & Chung 2002 refined constants","Method: anisotropic Chebyshev Hamiltonian, hierarchically chosen perturbation coefficients, two 3-adic filtrations for the log factors, and a Borel–Gauss analysis for the local rank condition","Lean: the construction 'is fully formalized in Lean 4'; the formal statements assert a degree-bounded polynomial vector field with an injective finite family of hyperbolic limit cycles; 'no unproved placeholders or project-specific axioms'; code archived at Zenodo (doi:10.5281/zenodo.23219558)","AI disclosure (verbatim): 'GPT-5.6 Sol was used to develop the proof strategy and arguments throughout this paper, assist with Lean formalization, and edit the manuscript. The authors take full responsibility for the mathematical content and the final text.'","Priority remark: 'The main results were obtained by the end of August, at which time the finiteness of the Hilbert number H(d) remained an open problem'; OpenAI's family 143 (dated 24 Sep, released 6 Oct) claims uniform finiteness, with only its quintic Liénard case in Lean"],"key_numbers":[],"tags":["math","dynamical-systems","hilbert-16th-problem","limit-cycles","lean","gpt-5-6-sol","ai-assisted"],"science":{"field":"mathematics","subfield":"dynamical systems (Hilbert's 16th problem, second part)","problem":"Growth of the Hilbert number H(d), the maximal number of limit cycles of degree-d planar polynomial vector fields","result":"Lower bound improved from Ω(d² ln d) to Ω(d² ln² d), with a Lean 4 formalization of the construction.","open_since":"1900","ai_system":["GPT-5.6 Sol"],"human_role":"AI-assisted: GPT-5.6 Sol developed the strategy and arguments and helped with Lean; authors responsible for content","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 formalization (per the authors); unrefereed preprint"},"status":"pending","shock":"","short":"Hilbert number lower bound"},"body_md":"## What happened\n\nThe second part of Hilbert's 16th problem asks how many limit cycles a planar polynomial vector field of degree d can have.\nUpper bounds are out of reach, so most work builds systems with many cycles. The best growth rate, d² ln d, dates to\nChristopher and Lloyd in 1995. Qin and Sun gain another factor of ln d by placing simple zeros of the Melnikov function across\nmany period annuli at once. They formalize the full construction in Lean 4.\n\n## Why it matters\n\nIt is a quantitative improvement on a 30-year-old bound for one of Hilbert's problems, with a machine-checked construction and\nan AI model credited with developing the argument. It does not address finiteness, which OpenAI's catalogue separately claims\nwithout a full formalization.","disputed":[{"label":"Verification","text":"Lean 4 formalization (per the authors); unrefereed preprint","sources":[]}],"related":[{"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":58,"official":12,"filed":"2026-10-07","updated":"2026-10-09","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":"2026-10-09","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-09","type":"filed","text":"Created from the arXiv PDF"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-09","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-09","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":161,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":100,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":191,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":130,"in_training_data":false}],"short_url":null}