Result 146, Dynamical systems and ergodic theory

Positive metric entropy for the standard map

Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.

Lean formalization Proof

The bigger picture

Why it matters

Does chaos register when starting points are sampled uniformly, rather than appearing only on exceptional trajectories? The manuscript reports a positive answer for the standard sine map at every sufficiently large positive parameter, making the claim about area-weighted dynamics.

What changes?

The map acts on a two-dimensional torus, a space with two coordinates that wrap around. Metric entropy measures the average rate at which repeated evolution generates information, here using normalized area to weight starting points. The manuscript claims this entropy is positive for every positive parameter beyond some threshold. That is stronger than positivity on a set of parameters of positive measure, the assertion attributed to Sinai's conjecture. It does not cover all positive parameters.

What does that help mathematicians do?

Positive entropy with respect to area means that the information-generating behavior cannot be confined to a set of starting points of zero area. The claimed result therefore distinguishes area-relevant chaos from exceptional chaotic trajectories. Its full parameter tail would also rule out arbitrarily large positive parameter values with zero area-based metric entropy. It does not imply that every trajectory is chaotic or exclude regular regions.

Are there practical applications?

The immediate value is foundational: the claim identifies a whole parameter range in the original standard sine map where deterministic dynamics produce information at a positive average rate. This gives researchers a concrete constraint on descriptions of its large-parameter behavior. The supplied abstract reports no numerical entropy bound, computational method, or practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Positive Metric Entropy for the Standard Map at Large Parameters

September 23, 2026 45 pages

We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026,
  author = {{OpenAI}},
  title = {{Positive Metric Entropy for the Standard Map at Large Parameters}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf}{OAI:Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/146.md.

Positive metric entropy for the standard map

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves Sinai's positive-entropy conjecture for the standard sine map on the two-dimensional torus in the stronger form of a full positive parameter tail. For every sufficiently large parameter, normalized area has positive metric entropy, and a positive-area set has positive largest Lyapunov exponent with the stated derivative-growth limit.

It also constructs a positive-area invariant ergodic hyperbolic component. That component splits into finitely many cyclic pieces on each of which the corresponding iterate is Bernoulli. No genericity assumption on the parameter is used.

Comparator links

Result Comparator statement
Positive-area hyperbolic Bernoulli component StandardMapComponents.lean
Positive metric entropy for all sufficiently large parameters StandardMapEntropy.lean
Positive Lyapunov exponents and entropy StandardMapLyapunov.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.