Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational \(p/q\) satisfies \(|\pi-p/q|\ge q^{-2-\varepsilon}\). This also proves convergence of the Flint–Hills series \(\sum_{n\ge1}1/(n^3\sin^2 n)\), with angles in radians. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments
In The irrationality exponent of pi is 2, 24 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
The irrationality exponent of \pi is 2. More precisely, for every real \nu>2, there is an integer Q(\nu) such that
In The irrationality exponent of pi is 2, 24 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
The classical Flint–Hills series converges.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
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OpenAI released an abridged summary of the model’s reasoning for this family: The irrationality exponent of π
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
PiExponentThe paper states
In The irrationality exponent of pi is 2, 24 September 2026
The irrationality exponent of \pi is 2. More precisely, for every real \nu>2, there is an integer Q(\nu) such that
The words they share: pi, real, integer, irrationality, exponent, nu.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PiExponent.lean, the release lists it as a supporting statement.
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 sorryThis file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program
MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note: Headline (irrationality exponent of pi = 2, both forms) stated with Mathlib objects; the Flint-Hills series convergence corollary is not stated.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name: Convergence of the Flint–Hills series Σ 1/(n³ sin² n) is claimed in the headline but not stated by PiExponent.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean 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 $p$. It also states the exact supremum characterization using infinitely many rational approximations.
The paper’s convergence consequence for the Flint–Hills series is outside this selected statement.
The irrationality exponent of $\pi$ equals two PiExponent
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem that the irrationality exponent of π is 2. The paper’s corollary on convergence of the Flint–Hills series is outside the statement.
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
PiExponentTwo reports, all at the release’s recorded version.
| Reported by | Their word | Their account | Date | Original report |
|---|---|---|---|---|
| cert-machine | “CERTIFIED” | Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
Other people’s accounts of running proof-verification software, with links to their reports.
aidilamrym-ops, aidilamrym-ops/the-asymmetric-wager-msaf-family017: audit vault for the irrationality exponent of pi, 10 October 2026. Source
869 of 869 modules in the PiExponent subtree build with standard axioms only, per the repository’s own report.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
These reports examine claims in the papers using HOL Light, a different proof assistant, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
2.664·10⁻⁷ < 355/113 − π < 2.669·10⁻⁷ and 24.5 < 1/(355³ sin² 355) < 24.7, so the Flint–Hills partial sum through n = 355 exceeds 24
HOL Light card 017
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
mo271, Wikipedia/FlintCooksonHills: flint_hills_series_converges is true (issue 6909), 7 October 2026. Source
With the release’s mu(pi) = 2 (Lean proof of the exponent statement in PiExponent.lean) and Alekseyev’s implication from mu(pi) < 5/2, the Flint Hills series converges.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Kenta Kitamura, Add a conjecture: the irrationality exponent of log 2 is 2 (issue 6899; PR 6923), 7 October 2026. Source
Claimed proof that the irrationality exponent of log 2 equals 2, with Lean formalisation (PR 6923).
Ryan Matthew Casper, Mattie/math: irrationality exponent preprints with Lean, extending the release’s method for pi, 7 October 2026. Source
Irrationality exponent 2 for positive rational logarithms, via Formal Conjectures issue 6924 and PR 6925.
Ryan Matthew Casper, Mattie/math: irrationality exponent preprints with Lean, extending the release’s method for pi, 7 October 2026. Source
Irrationality exponent 2 for nonzero rational arctangents (issue 6927, PR 6928).
Ryan Matthew Casper, Mattie/math: irrationality exponent preprints with Lean, extending the release’s method for pi, 7 October 2026. Source
Exponent 2 for arctan(sqrt 2)/sqrt 2 (issue 6929) and log(3+2 sqrt 2)/sqrt 2 (issue 6941).
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Landmark of the century”. The page says that nothing on it is verified.
Author not stated. Source
eigendash, eigendash/openai-math-visualised: numerical probes of seven result families, 8 October 2026. Source
Over the first 40 continued-fraction convergents of pi, q^2 |pi - p/q| stays below 0.94 and the empirical exponent settles at 2.0, consistent with irrationality exponent 2.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In The irrationality exponent of pi is 2, 24 September 2026
Why this passage is shown
Corollary 1.2
In The irrationality exponent of pi is 2, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, PiExponent
Proof-verification report
MathVet, PiExponent
Report using another proof or calculation method
2.664·10⁻⁷ < 355/113 − π < 2.669·10⁻⁷ and 24.5 < 1/(355³ sin² 355) < 24.7, so the Flint–Hills partial sum through n = 355 exceeds 24
Proofs of selected steps in HOL Light, Ben Knill, openai-math-hol-cards
68a1228Lean statement
PiExponent