OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F∞: a classifying space with finitely many cells in each dimension. A single group of type F∞ can also contain every finitely presented group. (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.
Four other possible matches are listed below the card, among them Theorem 1.2 in the same paper and Theorem 1.1 in Simple F∞ overgroups of groups with decidable word problem.
For every finitely generated group G, the following are equivalent:
(i)
G has decidable word problem.
(ii)
G embeds in a finitely presented simple group.
Four more possible matches
Theorem 1.2in Finite algebraic envelopes and the Boone–Higman conjecture.
Theorem 1.1in Simple F∞ overgroups of groups with decidable word problem.
We prove the Boone–Higman conjecture. A finitely generated group has decidable word problem if and only if it embeds in a finitely presented simple group.
Every finitely generated group with decidable word problem embeds in a simple group of type F∞. This resolves the higher-finiteness strengthening of the Boone–Higman conjecture.
We construct a single group of type F∞ containing every finitely presented group. Its finitely generated subgroups, up to isomorphism, are exactly the finitely generated recursively presented groups. This answers the F∞ form of the higher-dimensional Higman embedding question.
Three Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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.
UniversalFInfinity.lean, the release lists it as a main result.
theoremuniversal_group_of_type_FInfinity : ∃ (H : Type u) (groupH : Group H), @HasTypeFInfinity H groupH ∧ ∀ (G : Type u) [Group G] [Group.IsFinitelyPresented G], ∃ φ : G →* H, Function.Injective φ := 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”.
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”.
SimpleOvergroups.lean, the release lists it as a supporting statement.
theoremmain (G : Type u) [Group G] [Group.FG G] (hG : HasDecidableWordProblem G) : ∃ (H : Type u) (_ : Group H), Nontrivial H ∧ IsSimpleGroup H ∧ HasTypeFInfty H ∧ ∃ f : G →* H, Function.Injective f := 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.
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?
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:BooneHigman.main (G) [Group.FG G] : HasDecidableWordProblem G ↔ EmbedsInFinitelyPresentedSimpleGroup G (ComputablePred of evalWord g w = 1 for some finite generating tuple; Mathlib Group.IsFinitelyPresented, IsSimpleGroup), SimpleOvergroups.main (G f.g. with decidable word problem ⇒ ∃ nontrivial simple H of type F∞ and injective G →* H) and UniversalFInfinity.universal_group_of_type_FInfinity (∃ H of type F∞ into which every Group.IsFinitelyPresented group embeds). All three claims are stated; type F∞ = aspherical Hausdorff connected CW complex with FiniteType (finitely many cells per dimension).
“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.
The Boone–Higman conjecture characterizes finitely generated groups with decidable word problem by embeddings into finitely presented simple groups. The formalization proves the equivalence in full: a finitely generated group has a computable word problem for a finite generating set exactly when it embeds by an injective homomorphism into a finitely presented simple group.
The formalization proves the higher-finiteness strengthening of the Boone–Higman conjecture: every finitely generated group with decidable word problem embeds by an injective homomorphism into a nontrivial simple group of type $F_\infty$. Here type $F_\infty$ means that the group has a classifying CW complex with finitely many cells in each dimension. No additional finiteness property of the original group is assumed.
The formalized result constructs one group $H$ of type $F_\infty$ containing every finitely presented group. Here type $F_\infty$ means that $H$ has a classifying space with finitely many cells in each dimension.
The group $H$ is fixed before the groups embedded into it are chosen. No word-problem assumption is imposed, and the classifying space need not be finite-dimensional or have finitely many cells in total. The paper’s converse about recursively presented subgroups is not included.
AI-drafted comparisons, with no recorded human review
Universal group of type $F_\infty$UniversalFInfinity
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: one group of type F_∞ containing every finitely presented group. The corollary characterising its finitely generated subgroups is not included.
Boone–Higman equivalenceBooneHigman
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group.
Simple type $F_\infty$ overgroups of groups with decidable word problemSimpleOvergroups
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the higher-finiteness paper: every finitely generated group with decidable word problem embeds in a nontrivial simple group of type F_∞.
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.
Universal group of type $F_\infty$UniversalFInfinity
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.
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 “Major math journal”. The page says that nothing on it is verified.
Matt Zaremsky says OpenAI’s Boone-Higman result shows their team’s sufficient condition always works, via a mechanism close to their 2025 breakthrough.
Questions awaiting review
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
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 250The 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
Compare the proposed paper and Lean matches
Family 250Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In Finite algebraic envelopes and the Boone–Higman conjecture
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
decidable, embeds, finitely, generated, group, presented, problem, simple, word
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
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 147 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 113 s; peak memory 6.3 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.
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.