Amenability, unitarizability, and strong Ulam stability
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves Dixmier’s problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert 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.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.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.
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.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
A countable discrete group is strongly Ulam stable if and only if it is amenable.
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 closer match to the description’s wording first. The atlas has not reviewed either choice.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that a discrete group is amenable if and only if every uniformly bounded representation on a complex Hilbert space is similar to a unitary representation. This resolves Dixmier’s unitarizability problem affirmatively for discrete groups.
A countable discrete group is amenable if and only if it is strongly Ulam stable: every sufficiently accurate unitary almost representation, on any complex Hilbert space, is uniformly close in operator norm to a genuine representation on the same space. We prove the converse to Kazhdan’s amenable stability theorem, answering the question of Burger, Ozawa, and Thom. The inclusion of infinite-dimensional Hilbert spaces is essential.
Two Lean statements in this family
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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.
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.
The words they share: unitarizable, pi, amenable, similar, complex, space, separable, discrete, unitary, group, countable.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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”.
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.
The words they share: pi, amenable, hilbert, similar, complex, space, separable, discrete, unitary, representation, nonamenable, group, countable, chosen.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
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”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 part of the advertised result (their term: “partial”).
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).
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 — UNLINKED”).
The gap they name:the strong-Ulam-stability characterization of amenability for countable groups
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
Dixmier’s unitarizability problem asks whether a discrete group is amenable exactly when every uniformly bounded Hilbert-space representation is similar to a unitary one. The formalization establishes this equivalence for every discrete group.
For every nonamenable group and every $\varepsilon>0$, it also gives a nonunitarizable representation with uniform operator bound at most $1+\varepsilon$. The Hilbert space can be chosen separable for countable groups; a separate linked statement records a separable witness with uniform bound $101$ in that case.
AI-drafted comparisons, with no recorded human review
Unitarizability for all discrete groupsDixmierAllDiscrete
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a discrete group is amenable exactly when it is unitarisable, with nonunitarisable representations of bound at most 1 + ε for every nonamenable group.
Separable nonunitarizable witnesses for countable nonamenable groupsDixmier
Unreviewed AI draft: states a weaker form of a paper’s theorem. States that every countable nonamenable group has a nonunitarisable representation on a separable Hilbert space with uniform bound 101; the main theorem gives the bound 1 + ε, the converse direction and all discrete groups.
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.
Unitarizability for all discrete groupsDixmierAllDiscrete
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
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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
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 251The 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
Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 less than the headline; the missing part comes from a paper the Lean note does not link.
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 122 s; peak memory 6.4 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.
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 74 s; peak memory 3.1 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.
Separable nonunitarizable witnesses for countable nonamenable groups
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.