OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves the probability-preserving form of Banach’s simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L2 space: the bilateral iterates of one real observable form an orthonormal basis of that space. (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.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
There exist a C^\infty volume-preserving diffeomorphism T\colon\T^3\to\T^3 and a real function f\in L^2_0(\T^3,\mu) such that
\{f\circ T^n:n\in\Z\}
is an orthonormal basis of L^2_0(\T^3,\mu). Consequently T is ergodic, and U_T on this space is unitarily equivalent to multiplication by the coordinate w on L^2(S^1,m).
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We solve the probability-preserving form of Banach’s simple Lebesgue-spectrum problem in smooth dynamics. We construct a C∞ diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L2 space.
One Lean statement in this family
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.
There exist a C^\infty volume-preserving diffeomorphism T\colon\T^3\to\T^3 and a real function f\in L^2_0(\T^3,\mu) such that
\{f\circ T^n:n\in\Z\}
is an orthonormal basis of L^2_0(\T^3,\mu). Consequently T is ergodic, and U_T on this space is unitarily equivalent to multiplication by the coordinate w on L^2(S^1,m).
The words they share: orthonormal, ergodic, preserving, real, simple, torus, lebesgue, spectrum, smooth, multiplication, basis, diffeomorphism, space, coordinate, volume, function.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
ThreeTorus.lean, the release lists it as a supporting statement.
defMainConclusion : Prop := ∃ T : Equiv.Perm Torus, Smooth T ∧ Smooth T.symm ∧ MeasurePreserving T μ μ ∧ ∃ f : H₀, IsReal f ∧ ∃ v : ℤ → H₀, (∀ n : ℤ, ∀ᵐ x ∂μ, (v n).1 x = f.1 ((T ^ n) x)) ∧ Orthonormal ℂ v ∧ Dense (Submodule.span ℂ (Set.range v) : Set H₀) ∧ Ergodic T μ ∧ ∃ W : H₀ ≃ₗᵢ[ℂ] CircleH, ∀ g : H₀, ∃ h : H₀, (∀ᵐ x ∂μ, h.1 x = g.1 (T x)) ∧ (∀ᵐ z ∂(AddCircle.haarAddCircle (T := (1 : ℝ))), W h z = fourier 1 z * W g z)⋮theoremsimple_lebesgue_spectrum : MainConclusion := bysorry
This 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
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:None.
Definitions the group asks readers to compare with the paper:Smooth is hand-defined by local smooth lifts through project : (Fin 3 → ℝ) → Torus with ContDiffOn ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (C^∞, not analytic); meanZero is a hand-built Submodule of Lp ℂ 2 μ
“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.
Banach’s simple Lebesgue-spectrum problem asks for a probability-preserving transformation with simple Lebesgue spectrum on its mean-zero $L^2$ space. The formalization constructs a smooth volume-preserving diffeomorphism of the three-torus with this property. It gives a real-valued cyclic vector whose integer iterates form an orthonormal basis of the entire mean-zero space, and an isometric identification under which the Koopman operator becomes multiplication by the coordinate function on the circle. The transformation is also ergodic.
AI-drafted comparisons, with no recorded human review
Smooth simple Lebesgue spectrum on the three-torusThreeTorus
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a smooth volume-preserving diffeomorphism of the three-torus with simple Lebesgue spectrum, given by a cyclic vector whose iterates form an orthonormal basis of the mean-zero space.
Who ran the proof-verification program?
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.
Smooth simple Lebesgue spectrum on the three-torusThreeTorus
Three 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.
What these papers cite in the release, and what cites them
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.
Papers elsewhere in the release that this family’s papers cite
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
Explain a result, compare statements or propose a correction
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.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
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 “Career-defining publication”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 144The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 605 s; peak memory 6.6 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Smooth simple Lebesgue spectrum on the three-torus
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.