# Positive metric entropy for the standard map > The standard map is chaotic on a region of positive area for every large enough parameter. Plain-English explainer, Lean status and sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-146/ Last updated: 2026-10-11 **The standard map is chaotic on a region of positive area for every large enough parameter.** **Trust:** Formal proof listed · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Dynamical systems and ergodic theory · **Entry:** 146 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. The standard map is a simple rule that moves points around a wrapped-around square, with a strength k. This paper proves (Lean proof listed) that, for every large enough k, the map is chaotic on positive area. In technical terms, its metric entropy with respect to area is positive. This settles an old conjecture of Sinai, as Berger and Turaev state it, for every large k of the exact map. ## At a glance - **First step:** — - **Significance:** Major. If the proof holds, it settles Sinai's conjecture for the exact standard map, not a perturbed or random version. - **Trust:** Formal proof listed - **Field:** Dynamical systems and ergodic theory (1 paper) ## The explanation Picture a square whose edges wrap around, like an old video game screen. Leave by the right edge and you come back on the left. Leave by the top and you return at the bottom. Mathematicians call this surface a torus. The standard map moves every point of it by one simple rule. Pick a number k. Add k times sin(2πx) to the height y. Then add the new height to the position x, and wrap around. Take k = 10 and the point x = 0.1, y = 0.2. Follow a second point a tiny distance to its right. After one step, the gap between them is about 72.6 times larger. After seven steps, it is about 10 billion times larger. That averages to about 27 times per step. But the growth is uneven. On the third step, the gap grew only about 3.3 times. That step began near x = 0.25, where the sine curve is flat at its peak. There the push barely stretches anything. This mix is typical of large k. The paper describes strong local stretching combined with repeated losses of that stretching[1]. The long-run growth rate of such gaps, measured as a logarithm, is called the Lyapunov exponent. A positive exponent means tiny gaps grow exponentially, which is the usual signature of chaos. The map also keeps area unchanged: a patch of the square moves around but never shrinks or grows. Metric entropy measures how much unpredictability the map creates, averaged over the whole area. A classical formula of Pesin turns it into the average Lyapunov exponent over the square. So the entropy is positive exactly when the exponent is positive on a region of positive area. Chaos on a set with no area does not count. The point x = 0, y = 0 shows why this matters. It never moves, and gaps near it stretch about 64.8 times per step in one direction. Yet one point has no area at all. Much chaos of this zero-area kind was already known. In 1994 Duarte built such chaotic sets for large k, and also found islands where motion stays orderly. In 2012 Gorodetski found chaotic sets of full dimension for a large but possibly zero-length set of big k values. The paper notes that neither result gives positive area[2]. In 2024 Oliveira studied topological entropy, a count of distinct behaviours. Oliveira proved it positive for every k > 0 other than k = 2/π, a value that result leaves out. That still says nothing about entropy measured by area. Other work added noise, coupling or small changes to the map. In a form stated by Berger and Turaev, Sinai's conjecture asks for positive area entropy on a set of k values of positive total length. This paper proves that conjecture for the exact map, in a stronger form[3]. There is a threshold k₀ such that every k at or above k₀ has positive metric entropy[4]. It also finds, for each such k, a region of positive area that the map explores as one indivisible whole. That region splits into finitely many pieces that the map visits in turn[5]. On each piece, a fixed number of steps behaves as randomly as repeated coin tossing. The one obstacle is cancellation. Stretching built up over a long time can be undone later. The paper shows this is rare. Two long stretches that each grow near the top rate seldom cancel when joined. The chance falls exponentially with their length[6]. A second estimate controls fast-growing stretches without assuming that distant times act independently[7]. The proof then argues by contradiction[8]. Suppose arbitrarily large k had zero exponent almost everywhere. The two estimates make a bookkeeping identity across many time scales impossible. The theorem has clear limits. The contradiction shows that k₀ exists, but the paper does not give a value for it[9]. Nothing is said about smaller k. The paper also makes no claim that the exponent is positive almost everywhere[10]. So it does not rule out orderly regions, and it gives no uniform lower bound on entropy. How sure is this? A computer proof checker called Lean has checked formal versions of three statements. They are the main entropy theorem, positive Lyapunov exponents on positive area, and the coin-tossing region. The Lean scope note says no special assumption on k is used. No specialist review of the paper has been published. **What it doesn't show.** This does not say how large k must be, or that the map is chaotic on the whole square. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major if the proof holds. The Lean scope note lists formal versions of the main statements. The threshold k₀ is not computed, and no specialist review is published. ## Words used - **Torus:** A square whose opposite edges are glued, so leaving one side brings you back on the other. - **Standard map:** A simple area-keeping rule on the torus, with a strength k, used as a model of chaos. - **Lyapunov exponent:** The long-run growth rate of tiny gaps between nearby points, on a logarithmic scale. - **Metric entropy:** How much unpredictability a map creates, averaged over a fixed measure such as area. - **Topological entropy:** A count of how many distinct behaviours a map has, ignoring how much area they cover. - **Bernoulli:** Behaving, after a change of viewpoint, as randomly as repeated independent coin tosses or dice rolls. ## The original paper - [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf) - September 23, 2026 - [Lean notes](https://github.com/openai/math/blob/main/lean/docs/146.md) ## Quoted lines from the paper (10) [1] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > Its large-parameter regime combines strong local expansion with recurrent losses of derivative growth. [2] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > These results reveal extensive deterministic structure, but neither hyperbolic sets nor full dimension imply positive area; residual parameter sets also need not have positive Lebesgue measure. [3] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > Thus … resolves affirmatively Sinai's positive metric entropy conjecture in the formulation of Berger and Turaev, and gives the stronger all-sufficiently-large-parameter conclusion. [4] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > 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. [5] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > For some integer $N=N(k,E)\ge1$, there is a measurable partition $E=E_0\sqcup\cdots\sqcup E_{N-1}$ such that $f_k(E_j)=E_{j+1\bmod N}$ and $(E_j,m(E_j)^{-1}m|_{E_j},f_k^N)$ is Bernoulli for every $j$. [6] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > A uniform analytic scarcity estimate, including the degenerate cases of nearly cancelling reciprocal cosines, converts those tests into exponential probability decay. [7] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > This step does not presume mixing or independence of orbit segments. [8] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > Suppose now that arbitrarily large parameters had zero exponent almost everywhere. [9] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > There is consequently a finite $k_0$ such that, for every $k>k_0$, the top exponent is positive on a set of positive normalized area. [10] From [Positive Metric Entropy for the Standard Map at Large Parameters](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): > It makes no claim of ergodicity, almost-everywhere positive exponents, or a uniform positive lower bound on entropy. ## Other sources - [openai/math, “Positive Metric Entropy for the Standard Map at Large Parameters” (2026)](https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf): “has positive metric entropy with respect to normalized area for every sufficiently large positive parameter” - [P. Berger and D. Turaev, “On Herman's positive entropy conjecture”, Advances in Mathematics 349 (2019), 1234–1288](https://doi.org/10.1016/j.aim.2019.04.002) - [P. Duarte, “Plenty of elliptic islands for the standard family of area preserving maps”, Annales de l'Institut Henri Poincaré. Analyse non linéaire 11 (1994), 359–409](https://www.numdam.org/item/AIHPC_1994__11_4_359_0/) - [A. Gorodetski, “On stochastic sea of the standard map”, Communications in Mathematical Physics 309 (2012), 155–192](https://doi.org/10.1007/s00220-011-1365-z) - [F. Oliveira, “Positive topological entropy for the Standard Map”, arXiv:2404.02209v2 (2024)](https://arxiv.org/abs/2404.02209) - [Ya. B. Pesin, “Characteristic Lyapunov exponents and smooth ergodic theory”, Russian Mathematical Surveys 32 (1977), 55–114](https://doi.org/10.1070/RM1977v032n04ABEH001639) - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [Rokhlin’s multiple-mixing problem](https://closedform.nishachay.tech/e/openai-math-2026-145/) - Dynamical systems and ergodic theory - [The near-boundary Birkhoff conjecture](https://closedform.nishachay.tech/e/openai-math-2026-147/) - Dynamical systems and ergodic theory - [Banach’s simple Lebesgue-spectrum problem](https://closedform.nishachay.tech/e/openai-math-2026-144/) - Dynamical systems and ergodic theory - [The entropy-rate dimension formula for self-similar measures](https://closedform.nishachay.tech/e/openai-math-2026-148/) - Dynamical systems and ergodic theory - [Hilbert's sixteenth problem: uniform bounds for limit cycles](https://closedform.nishachay.tech/e/openai-math-2026-143/) - Dynamical systems and ergodic theory - [Dynamical systems and ergodic theory](https://closedform.nishachay.tech/science/mathematics/dynamical-systems-and-ergodic-theory/) - Field · 12 results - [Lean-checked results](https://closedform.nishachay.tech/trust/lean-checked/) - Trust · 242 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20146%5D%20)