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_components
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.
defLyapunovSpectrumAt (k : ℝ) (z : Torus) (l : ℝ) : Prop :=0 ≤ l ∧ ((l = 0 ∧ ∀ v : ℂ, v ≠ 0 → VectorGrowth k z v 0) ∨ (0 < l ∧ ∃ s : ℂ, s ≠ 0 ∧ ∀ v : ℂ, v ≠ 0 → VectorGrowth k z v (if wedge s v = 0then -l else l)))
∃ E : Set Torus, MeasurableSet E ∧ 0 < area E ∧ ((standardMap k) ⁻¹' E =ᵐ[area] E) ∧ Ergodic (standardMap k) (normalizedArea E) ∧ (∀ᵐ z ∂normalizedArea E, ∃ l : ℝ, 0 < l ∧ LyapunovSpectrumAt k z l) ∧ ∃ (N : ℕ) (hN : 0 < N) (P : Fin N → Set Torus), (∀ j, MeasurableSet (P j)) ∧ ((⋃ j, P j) =ᵐ[area] E) ∧ (∀ i j, i ≠ j → area (P i ∩ P j) = 0) ∧ (∀ j : Fin N, (standardMap k '' P j) =ᵐ[area] P ⟨(j.val+1)%N, Nat.mod_lt _ hN⟩) ∧ (∀ j, area (P j) = area E / (N : ℝ≥0∞) ∧ 0 < area (P j)) ∧ (∀ j, IsBernoulli ((standardMap k)^[N]) (normalizedArea (P j)))
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: hyperbolic, bernoulli, component, area, ergodic, integer, lyapunov and measurable. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: invariant and positive.
Corollary 1.2 of Positive Metric Entropy for the Standard Map at Large Parameters
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.
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 $m(E)/N>0$.
Other possible paper matches, closest wording first
Theorem 1.1 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: lyapunov and measure. Found in the paper’s title, which its main theorem states: standard and map. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 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, product, two and log.
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: lyapunov. 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: continuous, real and measures. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: invariant.
Theorem 3.1 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: log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: parameter.
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: measures. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: invariant 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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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.
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.
Finds the whole advertised result stated or impliedTheir 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)
OpenAI’s own note on what this Lean statement covers
“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.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
110 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: 59 to its own declarations, 62 to Mathlib’s documentation. 22 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
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.
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.
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 $m(E)/N>0$.
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.
Positive Metric Entropy for the Standard Map at Large Parameters
TeX label main:components, build/sections/introduction.tex, line 88
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.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.