Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously. (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
The paper states
In An infinite finitely presented simple amenable group, 23 September 2026
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 exists an infinite finitely presented simple amenable group.
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SimpleAmenableThe paper states
In An infinite finitely presented simple amenable group, 23 September 2026
There exists an infinite finitely presented simple amenable group.
The words they share: presented, infinite, simple, amenable, finitely, group.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SimpleAmenable.lean, the release lists it as a main result.
theorem main : ∃ (G : Type) (_ : Group G), Infinite G ∧ Group.IsFinitelyPresented G ∧ IsSimpleGroup G ∧ FolnerAmenable G := 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
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.
FolnerAmenable Model.lean, line 9MainStatement Model.lean, line 15MathVet 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: main : ∃ (G : Type) (_ : Group G), Infinite G ∧ Group.IsFinitelyPresented G ∧ IsSimpleGroup G ∧ FolnerAmenable G, with FolnerAmenable G := ∀ K : Finset G, ∀ ε > 0, ∃ D, D.Nonempty ∧ ∀ g ∈ K, |(g • D) ∆ D| < ε |D| (the left Folner criterion). Exactly the summary’s existence statement.
Definitions the group asks readers to compare with the paper: Custom ‘amenable’: FolnerAmenable G := ∀ K : Finset G, ∀ ε : ℝ, 0 < ε → ∃ D : Finset G, D.Nonempty ∧ ∀ g ∈ K, (((D.image (g * ·)) ∆ D).card : ℝ) < ε * D.card; this is Folner’s criterion (equivalent to amenability for discrete groups) but not Mathlib’s notion, and the group is existentially chosen.
“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 or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean covers
The formalized result answers the existence question affirmatively: there is one group that is infinite, finitely presented, simple, and amenable. Amenability is expressed by the Følner condition.
The later general claims about central kernels and enlargements across families are not included.
Infinite finitely presented simple amenable group SimpleAmenable
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: an infinite finitely presented simple amenable group, with amenability in Følner form.
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.
SimpleAmenableTwo 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 |
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
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Why this passage is shown
Theorem 1.1
In An infinite finitely presented simple amenable group, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, SimpleAmenable
Proof-verification report
MathVet, SimpleAmenable
Lean statement
SimpleAmenable