Lean statement

Positive metric entropy for all sufficiently large parameters

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.StandardMapEntropy.main_entropy

Lean statement file: StandardMapEntropy.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 146 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

def MainObligation : Prop :=  ∃ k₀ : ℝ, 0 < k₀ ∧ ∀ k : ℝ, k₀ ≤ k → 0 < metricEntropy area (standardMap k)⋮theorem main_entropy : MainObligation := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.StandardMapEntropy.Circle line 10
```
abbrev Circle := AddCircle (1 : ℝ)
```
abbreviation OAI.StandardMapEntropy.Torus line 11
```
abbrev Torus := Circle × Circle
```

definition OAI.StandardMapEntropy.sine line 19

noncomputable def sine (x : Circle) : ℝ := sine_periodic.lift x

definition OAI.StandardMapEntropy.standardMap line 21

noncomputable def standardMap (k : ℝ) (z : Torus) : Torus :=  (z.1 + z.2 + (k * sine z.1 : ℝ), z.2 + (k * sine z.1 : ℝ))

definition OAI.StandardMapEntropy.area line 24

noncomputable def area : Measure Torus :=  (volume : Measure Circle).prod (volume : Measure Circle)

definition OAI.StandardMapEntropy.FinitePartition line 27

def FinitePartition (r : ℕ) := { p : Torus → Fin (r + 1) // Measurable p }

definition OAI.StandardMapEntropy.blockAtom line 29

def blockAtom {r : ℕ} (f : Torus → Torus) (p : FinitePartition r)    (n : ℕ) (w : Fin n → Fin (r + 1)) : Set Torus :=  { z | ∀ i : Fin n, p.val (f^[i.val] z) = w i }

definition OAI.StandardMapEntropy.blockEntropy line 33

noncomputable def blockEntropy {r : ℕ} (μ : Measure Torus) (f : Torus → Torus)    (p : FinitePartition r) (n : ℕ) : ℝ :=  - ∑ w : Fin n → Fin (r + 1),    (μ (blockAtom f p n w)).toReal * Real.log (μ (blockAtom f p n w)).toReal

definition OAI.StandardMapEntropy.partitionEntropy line 38

noncomputable def partitionEntropy {r : ℕ} (μ : Measure Torus) (f : Torus → Torus)    (p : FinitePartition r) : ℝ≥0∞ :=  ⨅ n : ℕ, ENNReal.ofReal (blockEntropy μ f p (n + 1) / (n + 1 : ℝ))

definition OAI.StandardMapEntropy.metricEntropy line 42

noncomputable def metricEntropy (μ : Measure Torus) (f : Torus → Torus) : ℝ≥0∞ :=  ⨆ r : ℕ, ⨆ p : FinitePartition r, partitionEntropy μ f p

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: measure. Found in the paper’s title, which its main theorem states: metric, standard, map and entropy. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: largest, exponent, lyapunov and positive.

Theorem 1.1 of Positive Metric Entropy for the Standard Map at Large Parameters

Theorem

There exists k_0>0 such that

h_m(f_k)>0\qquad\text{for every }k\ge k_0.

Equivalently, for every such k the largest Lyapunov exponent of f_k is positive on a set of positive m-measure.

Section 1, Introduction Read it in the paper TeX source, line 15

Show the TeX

There exists $k_0>0$ such that
\[
 h_m(f_k)>0\qquad\text{for every }k\ge k_0.
\]
Equivalently, for every such $k$ the largest Lyapunov exponent of $f_k$
is positive on a set of positive $m$-measure.

Other possible paper matches, closest wording first

Corollary 1.2 of Positive Metric Entropy for the Standard Map at Large Parameters. Words of the Lean statement, its names split into words, that also appear in the theorem: partition, area, integer and measurable. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: exponent, lyapunov and positive.
Theorem 8.4 of Positive Metric Entropy for the Standard Map at Large Parameters. Words of the Lean statement, its names split into words, that also appear in the theorem: entropy. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: largest, exponent, lyapunov and positive.
Theorem 5.1 of Positive Metric Entropy for the Standard Map at Large Parameters. Words of the Lean statement, its names split into words, that also appear in the theorem: functions, two and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sufficiently and large.
Proposition 6.8 of Positive Metric Entropy for the Standard Map at Large Parameters. Words of the Lean statement, its names split into words, that also appear in the theorem: finite and measures. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
Proposition 6.7 of Positive Metric Entropy for the Standard Map at Large Parameters. Words of the Lean statement, its names split into words, that also appear in the theorem: real, finite and measures. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: limiting.
Proposition 7.1 of Positive Metric Entropy for the Standard Map at Large Parameters. Words of the Lean statement, its names split into words, that also appear in the theorem: real and measurable. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: full and positive.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 146’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: main_entropy : ∃ k₀ > 0, ∀ k ≥ k₀, 0 < metricEntropy area (standardMap k) is the headline verbatim (full parameter tail); the Lyapunov and hyperbolic Bernoulli-component claims are also stated (Lyapunov, Components). Parameter is k·sin(2πx) on (ℝ/ℤ)², i.e. Chirikov K = 2πk, immaterial for 'k large'.
Definitions the group asks readers to compare with the paper: metricEntropy μ f := ⨆ r, ⨆ p : FinitePartition r, partitionEntropy μ f p with partitionEntropy := ⨅ n, ofReal (blockEntropy μ f p (n+1)/(n+1)): custom (not in Mathlib); the inf form equals the usual limit only for measure-preserving f, which holds here (checked: the map is a composition of shears) but is not part of the statement ; Two statement files (StandardMapComponents, StandardMapLyapunov) contain the meta command run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {} (clears the aux-lemma cache; not a fidelity issue, but non-standard in a Comparator statement)
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

53 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 36 to its own declarations, 26 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_entropy exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 12 other declarations of the file; with the theorem itself they span 29 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.StandardMapEntropy are listed without it.

abbrev Circleline 10library
abbrev Torusline 11library
lemma sine_periodicline 13library
def sineline 19library
def standardMapline 21library
def arealine 24library
def FinitePartitionline 27library
def blockAtomline 29library
def blockEntropyline 33library
def partitionEntropyline 38library
def metricEntropyline 42library
def MainObligationline 45in the statementlibrary
theorem main_entropyline 48selected theoremlibrary

import Mathlibnamespace OAIopen MeasureTheory Setopen scoped ENNReal BigOperatorsnamespace StandardMapEntropyabbrev Circle := AddCircle (1 : ℝ)abbrev Torus := Circle × Circlelemma sine_periodic : Function.Periodic (fun x : ℝ => Real.sin (2 * Real.pi * x)) 1 := by  intro x  dsimp only  rw [show 2 * Real.pi * (x + 1) = 2 * Real.pi * x + 2 * Real.pi by ring]  exact Real.sin_add_two_pi _noncomputable def sine (x : Circle) : ℝ := sine_periodic.lift xnoncomputable def standardMap (k : ℝ) (z : Torus) : Torus :=  (z.1 + z.2 + (k * sine z.1 : ℝ), z.2 + (k * sine z.1 : ℝ))noncomputable def area : Measure Torus :=  (volume : Measure Circle).prod (volume : Measure Circle)def FinitePartition (r : ℕ) := { p : Torus → Fin (r + 1) // Measurable p }def blockAtom {r : ℕ} (f : Torus → Torus) (p : FinitePartition r)    (n : ℕ) (w : Fin n → Fin (r + 1)) : Set Torus :=  { z | ∀ i : Fin n, p.val (f^[i.val] z) = w i }noncomputable def blockEntropy {r : ℕ} (μ : Measure Torus) (f : Torus → Torus)    (p : FinitePartition r) (n : ℕ) : ℝ :=  - ∑ w : Fin n → Fin (r + 1),    (μ (blockAtom f p n w)).toReal * Real.log (μ (blockAtom f p n w)).toRealnoncomputable def partitionEntropy {r : ℕ} (μ : Measure Torus) (f : Torus → Torus)    (p : FinitePartition r) : ℝ≥0∞ :=  ⨅ n : ℕ, ENNReal.ofReal (blockEntropy μ f p (n + 1) / (n + 1 : ℝ))noncomputable def metricEntropy (μ : Measure Torus) (f : Torus → Torus) : ℝ≥0∞ :=  ⨆ r : ℕ, ⨆ p : FinitePartition r, partitionEntropy μ f pdef MainObligation : Prop :=  ∃ k₀ : ℝ, 0 < k₀ ∧ ∀ k : ℝ, k₀ ≤ k → 0 < metricEntropy area (standardMap k)theorem main_entropy : MainObligation := by  sorryend StandardMapEntropyend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Positive Metric Entropy for the Standard Map at Large Parameters

Theorem

There exists k_0>0 such that

h_m(f_k)>0\qquad\text{for every }k\ge k_0.

Equivalently, for every such k the largest Lyapunov exponent of f_k is positive on a set of positive m-measure.

Section 1, Introduction Read it in the paper TeX source, line 15

Show the TeX

There exists $k_0>0$ such that
\[
 h_m(f_k)>0\qquad\text{for every }k\ge k_0.
\]
Equivalently, for every such $k$ the largest Lyapunov exponent of $f_k$
is positive on a set of positive $m$-measure.

Corollary (Nonuniformly hyperbolic ergodic component)

For each k\ge k_0 there is a measurable f_k-invariant set E=E(k) with m(E)>0 such that (E,m(E)^{-1}m|_E,f_k) is ergodic and f_k has one positive and one negative Lyapunov exponent almost everywhere on E. 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. All invariance and partition statements are modulo m-null sets; each E_j has area m(E)/N>0.

Section 1, Introduction Read it in the paper TeX source, line 88

Show the TeX

For each $k\ge k_0$ there is a measurable $f_k$-invariant set $E=E(k)$
with $m(E)>0$ such that $(E,m(E)^{-1}m|_E,f_k)$ is ergodic and $f_k$ has
one positive and one negative Lyapunov exponent almost everywhere on $E$.
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 $j$.
All invariance and partition statements are modulo $m$-null sets;
each $E_j$ has area $m(E)/N>0$.

Configuration and file facts

File: lean/ComparatorChallenges/StandardMapEntropy.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 146 link it.
Solution module: OAI.Dynamics.StandardMap.EntropyEndpoint
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 11 local vocabulary declarations
Lines of Lean, without comments or blank lines: 29
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas