Hilbert’s 16th problem: uniform limit-cycle bound claimed (OpenAI)

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Key facts

• Main manuscript: ‘Uniform bounds for planar polynomial limit cycles’ (OpenAI, 24 Sept 2026, 160 pages). Theorem 1.1 (verbatim): ‘For each integer d ≥ 1, there is a finite nonnegative integer B(d) such that every real planar polynomial vector field of degree at most d has at most B(d) limit cycles in the whole plane.’ The paper adds: ‘It does not furnish an effective formula for B(d).’ Method (abstract): ‘separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument’
• Background as the paper recounts it: Dulac claimed individual finiteness (each fixed field has finitely many limit cycles) in 1923; after a gap was found, Écalle and Ilyashenko gave proofs; uniform boundedness (a bound depending only on the degree) stayed open, with partial results such as Bautin’s bound of three for quadratic weak foci, finite cyclicity of elementary polycycles (Ilyashenko–Yakovenko) and explicit bounds for zeros of Abelian integrals (Binyamini–Novikov–Yakovenko). Wikipedia: no upper bound is known for any degree n > 1
• Companion manuscript: ‘Two limit cycles for quintic Liénard systems’ (24 Sept, 40 pages): every system ẋ = y − F(x), ẏ = −x with real polynomial F of degree at most five ‘has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.’ Lean scope note (lean/docs/143.md): challenge QuinticLienard, no sign, parity, hyperbolicity or amplitude restriction; ‘accepted’ in the oai-math-revised comparator run of 25–26 Oct
• Carlos Toledo’s audit of the release (cert-machine, finding G1, 10 Oct): ‘Of the release’s 416 statements, only QuinticLienard defines periodic orbits or limit cycles at all’; the uniform-bound manuscript ‘is not linked by the scope note’, whose title nevertheless names the uniform bounds. He lists Hilbert’s sixteenth problem first among five headlines ‘carried by no machine-checked statement’
• Related human work: Chaoyang Qin and Xiaoming Sun’s lower bound H(d) = Ω(d² log² d) (arXiv 2610.12218, 8 Oct, obtained with GPT-5.6 Sol and formalized in Lean) is complementary; they write that when their results were obtained in August ‘the finiteness of the Hilbert number H(d) remained an open problem’ (see Hilbert’s 16th problem)
• Reactions: Alvaro Lozano-Robledo listed ‘the second part of Hilbert’s 16th’ among proposed solutions that ‘would represent huge contributions to their respective fields’ (Proofs and Prompts, 8 Oct). No assessment by a specialist in planar dynamics or limit cycles found (web, Hacker News, oai-math-revised notes) as of 29 Oct