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.Dixmier.current_main_theorem
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.
defCurrentMainTheorem : Prop := ∀ (G : Type u) [Group G] [TopologicalSpace G] [DiscreteTopology G], (Amenable G ↔ Unitarizable.{u, max u v} G) ∧ ∀ (ε : ℝ), 0 < ε → ¬ Amenable G → ∃ (H : Type (max u v)) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H) (_ : CompleteSpace H) (π : G →* (H →L[ℂ] H)), (Countable G → TopologicalSpace.SeparableSpace H) ∧ (∀ g, ‖π g‖ ≤ 1 + ε) ∧ ¬ SimilarToUnitary π
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: unitarizable, pi, amenable, similar, complex, space, separable, discrete and three more.
Statements in Unitarizability implies amenability for discrete groups come first because the release lists only that paper for this Lean statement.
Theorem 1.1 of Unitarizability implies amenability for discrete groups
Theorem
A discrete group is amenable if and only if it is unitarizable. For every discrete nonamenable group G and every \varepsilon>0, there are a complex Hilbert space \mathcal K and a representation \pi:G\to\mathrm{GL}(\mathcal K) with |\pi|\le1+\varepsilon that is not similar to a unitary representation. If G is countable, \mathcal K may be chosen separable.
A discrete group is amenable if and only if it is unitarizable.
For every discrete nonamenable group \(G\) and every \(\varepsilon>0\),
there are a complex Hilbert space \(\mathcal K\) and a representation
\(\pi:G\to\mathrm{GL}(\mathcal K)\) with \(|\pi|\le1+\varepsilon\)
that is not similar to a unitary representation. If \(G\) is countable,
\(\mathcal K\) may be chosen separable.
Other possible paper matches, closest wording first
Proposition 2.2 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: space, separable, pi, similar, complex, unitary and four more.
Theorem 1.1 of Strong Ulam Stability Characterizes Amenability. Words of the Lean statement, its names split into words, that also appear in the theorem: amenable, discrete, group and countable.
Proposition 4.4 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: unitary, bounds and norm.
Proposition 3.2 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: products, function and countable.
Corollary 4.3 of Unitarizability implies amenability for discrete groups. Words of the Lean statement, its names split into words, that also appear in the theorem: bounds and norm.
Proposition 2.2 of Strong Ulam Stability Characterizes Amenability. Words of the Lean statement, its names split into words, that also appear in the theorem: discrete and group.
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
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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.
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; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 122 s; peak memory 6.4 GiB).
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 251’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.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Stated: Dixmier's equivalence for all discrete groups (CurrentMainTheorem : ∀ G discrete, (Amenable G ↔ Unitarizable G) ∧ ∀ ε > 0, ¬ Amenable G → ∃ nonunitarizable π with ‖π g‖ ≤ 1 + ε on a complete Hilbert space, separable if G is countable) and the countable-group separable witness with bound 101 (Dixmier.main_theorem_all_universes). Not stated: the summary's second claim, equivalence of amenability with strong Ulam stability for countable discrete groups (it is not mentioned in the docs scope either, although the family title names Ulam stability).
Definitions the group asks readers to compare with the paper: Custom 'amenable': def Amenable (G) [DiscreteTopology G] : Prop := ∃ m : (G →ᵇ ℂ) →ₗ[ℂ] ℂ, m 1 = 1 ∧ (∀ f, (∀ x, 0 ≤ (f x).re ∧ (f x).im = 0) → 0 ≤ (m f).re ∧ (m f).im = 0) ∧ ∀ g f, m (leftTranslate g f) = m f, the standard positive left-invariant mean on ℓ^∞(G, ℂ); SimilarToUnitary π := ∃ S : H ≃L[ℂ] H, ∀ g x, ‖S (π g (S.symm x))‖ = ‖x‖ and Unitarizable quantify over Hilbert spaces in one universe (the theorem is universe polymorphic).
OpenAI’s own note on what this Lean statement covers
“The formalization establishes this equivalence for every discrete group.”
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
49 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: 7 to its own declarations, 23 to Mathlib’s documentation. 6 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.
A discrete group is amenable if and only if it is unitarizable. For every discrete nonamenable group G and every \varepsilon>0, there are a complex Hilbert space \mathcal K and a representation \pi:G\to\mathrm{GL}(\mathcal K) with |\pi|\le1+\varepsilon that is not similar to a unitary representation. If G is countable, \mathcal K may be chosen separable.
A discrete group is amenable if and only if it is unitarizable.
For every discrete nonamenable group \(G\) and every \(\varepsilon>0\),
there are a complex Hilbert space \(\mathcal K\) and a representation
\(\pi:G\to\mathrm{GL}(\mathcal K)\) with \(|\pi|\le1+\varepsilon\)
that is not similar to a unitary representation. If \(G\) is countable,
\(\mathcal K\) may be chosen separable.
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Unitarizability for all discrete groups
Family 251: Amenability, unitarizability, and strong Ulam stability
Candidate pairing
Theorem
A discrete group is amenable if and only if it is unitarizable. For every discrete nonamenable group G and every \varepsilon>0, there are a complex Hilbert space \mathcal K and a representation \pi:G\to\mathrm{GL}(\mathcal K) with |\pi|\le1+\varepsilon that is not similar to a unitary representation. If G is countable, \mathcal K may be chosen separable.
Unitarizability implies amenability for discrete groups
TeX label thm:main, build/sections/introduction.tex, line 38
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.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.