Lean statement

The irrationality exponent of $\pi$ equals two

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.PiExponent.main

Lean statement file: PiExponent.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 017 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.

theorem main :  (∀ ν : ℝ, 2 < ν → ∃ Q : ℤ, 2 ≤ Q ∧    ∀ p q : ℤ, Q ≤ q →      (q : ℝ) ^ (-ν) ≤ |Real.pi - (p : ℝ) / (q : ℝ)|) ∧  sSup {ν : ℝ | 0 < ν ∧    Set.Infinite {r : ℚ | 2 ≤ r.den ∧      0 < |Real.pi - (r : ℝ)| ∧      |Real.pi - (r : ℝ)| < (r.den : ℝ) ^ (-ν)}} = 2 := 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.

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: pi, real and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: irrationality, exponent and nu.

Theorem 1.1 of The irrationality exponent of pi is 2

Theorem

The irrationality exponent of \pi is 2. More precisely, for every real \nu>2, there is an integer Q(\nu) such that

\left|\pi-\frac pq\right|\ge q^{-\nu} \qquad(p\in\ZZ,\ q\in\ZZ,\ q\ge Q(\nu)).

Section 1, Introduction Read it in the paper TeX source, line 64

Show the TeX

The irrationality exponent of \(\pi\) is \(2\). More precisely, for every
real \(\nu>2\), there is an integer \(Q(\nu)\) such that
\[
\left|\pi-\frac pq\right|\ge q^{-\nu}
\qquad(p\in\ZZ,\ q\in\ZZ,\ q\ge Q(\nu)).
\]

Other possible paper matches, closest wording first

Theorem 2.1 of The irrationality exponent of pi is 2. Words of the Lean statement, its names split into words, that also appear in the theorem: rational and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive, sufficiently and large.
Corollary 5.2 of The irrationality exponent of pi is 2. Words of the Lean statement, its names split into words, that also appear in the theorem: real.
Proposition 3.4 of The irrationality exponent of pi is 2. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nu.

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 017’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 disagree 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: Headline (irrationality exponent of pi = 2, both forms) stated with Mathlib objects; the Flint-Hills series convergence corollary is not stated.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: Convergence of the Flint–Hills series Σ 1/(n³ sin² n) is claimed in the headline but not stated by PiExponent.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that the irrationality exponent of $\pi$ is exactly two. For every $\nu>2$, all sufficiently large positive denominators $q$ satisfy $|\pi-p/q|\ge q^{-\nu}$ for every integer numerator $p$.”
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

19 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: 0 to its own declarations, 5 to Mathlib’s documentation. 1 declared name also links 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.

import Mathlibnamespace OAInamespace PiExponenttheorem main :  (∀ ν : ℝ, 2 < ν → ∃ Q : ℤ, 2 ≤ Q ∧    ∀ p q : ℤ, Q ≤ q →      (q : ℝ) ^ (-ν) ≤ |Real.pi - (p : ℝ) / (q : ℝ)|) ∧  sSup {ν : ℝ | 0 < ν ∧    Set.Infinite {r : ℚ | 2 ≤ r.den ∧      0 < |Real.pi - (r : ℝ)| ∧      |Real.pi - (r : ℝ)| < (r.den : ℝ) ^ (-ν)}} = 2 := by  sorryend PiExponentend 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.

The irrationality exponent of pi is 2

Theorem

The irrationality exponent of \pi is 2. More precisely, for every real \nu>2, there is an integer Q(\nu) such that

\left|\pi-\frac pq\right|\ge q^{-\nu} \qquad(p\in\ZZ,\ q\in\ZZ,\ q\ge Q(\nu)).

Section 1, Introduction Read it in the paper TeX source, line 64

Show the TeX

The irrationality exponent of \(\pi\) is \(2\). More precisely, for every
real \(\nu>2\), there is an integer \(Q(\nu)\) such that
\[
\left|\pi-\frac pq\right|\ge q^{-\nu}
\qquad(p\in\ZZ,\ q\in\ZZ,\ q\ge Q(\nu)).
\]

Corollary

The classical Flint–Hills series converges.

Section 1, Introduction Read it in the paper TeX source, line 90

Show the TeX

The classical Flint--Hills series converges.

Configuration and file facts

File: lean/ComparatorChallenges/PiExponent.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 017 link it.
Solution module: OAI.NumberTheory.PiExponent.Main
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: Imports only Mathlib; no local vocabulary added
Lines of Lean, without comments or blank lines: 9
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