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Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters

The standard map fk(x,y)=(x+y+ksin⁡(2πx), y+ksin⁡(2πx))f_k(x,y)=(x+y+k\sin(2\pi x),\,y+k\sin(2\pi x)) on T2\mathbb T^2 preserves area mm and was introduced by Chirikov as a model of conservative chaos. For large kk it has hyperbolic sets (Duarte), full-dimension transitive sets for residual parameters (Gorodetski) and positive topological entropy (Oliveira), but also elliptic islands, and none of this gives positive area. Sinai's conjecture, in the formulation of Berger and Turaev (2019, Conjecture 0.4), asks: is the metric entropy hm(fk)h_m(f_k), equivalently a positive Lyapunov exponent on positive area, positive for a set of parameters kk of positive Lebesgue measure?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Dynamical systems and ergodic theory
Posed by
Yakov Sinai; the manuscript cites the formulation of P. Berger and D. Turaev, On Herman's positive entropy conjecture, Adv. Math. 349 (2019), Conjecture 0.4
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is k0>0k_0>0 with hm(fk)>0h_m(f_k)>0 for every k≥k0k\ge k_0; equivalently, by Pesin's formula, the top Lyapunov exponent is positive on a set of positive area. Corollary 1.2: a positive-area ergodic component with nonzero exponents that splits into finitely many cyclic Bernoulli pieces. It does NOT claim ergodicity of area, positive exponents almost everywhere, a uniform lower bound on the entropy, or any explicit k0k_0, and says nothing about small or moderate kk.

What the AI did

The release README says the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result, out of roughly 4,000 problems posed; the output was aggregated into result families and manuscripts and kept if judged significant enough. This family has one manuscript, dated September 23, 2026. The manuscript is credited to 'OpenAI' alone and names no human author. The README's two exceptions to the fixed procedure (the Riemann zeta zero-free region work, whose Re(s) > 11/12 write-up was human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the TeX source read against Sinai's conjecture as cited; positivity for every k≥k0k\ge k_0 is stronger than the positive-measure parameter set the conjecture asks for. The proof was not refereed. Lean: three Comparator challenges, StandardMapEntropy (OAI.StandardMapEntropy.main_entropy), StandardMapLyapunov and StandardMapComponents, with solution modules present at the pinned commit; none is in the formalization catalogue. The StandardMapEntropy statement was read here: it defines the same map and normalized area on the torus and asserts some k0>0k_0>0 with positive metric entropy for all k≥k0k\ge k_0, which is the headline. Metric entropy is defined in the challenge itself (finite measurable partitions, block entropies), since Mathlib has none, so that definition is the prover's own. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.

Sources

Changelog1 change

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