OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves that every type II1 factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion \(P\subset M\) of such factors, the unitaries u with \(M=W^*(P,u)\) form a dense Gδ subset in the trace 2-norm topology. (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.
Let P\subset M be an irreducible inclusion of type \mathrm{II}_1 factors, with M of separable predual. The set
\{u\in\mathcal U(M):W^*(P,u)=M\}
is a dense G_\delta subset of the unitary group \mathcal U(M) for the trace 2-norm topology.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Every type \mathrm{II}_1 factor M with separable predual is generated by two self-adjoint elements. Equivalently, there is an x\in M such that M=W^*(x).
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 every type II1 factor with separable predual is generated by one operator, equivalently by two self-adjoint operators. Combined with the established direct-integral reduction to this case, this gives an affirmative solution of the generator problem for von Neumann algebras with separable predual.
Two 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.
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.
Let P\subset M be an irreducible inclusion of type \mathrm{II}_1 factors, with M of separable predual. The set
\{u\in\mathcal U(M):W^*(P,u)=M\}
is a dense G_\delta subset of the unitary group \mathcal U(M) for the trace 2-norm topology.
The words they share: topology, dense, group, unitary, norm, trace, predual, inclusion, irreducible, separable, ii, factors, relative, generator, finite, delta.
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.
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:single_generation_of_II1_separable_predual (S) (hS : IsII1Factor S) (hsep : HasSeparablePredual S) : SinglyGenerated S (∃ x ∈ S, wstar {x} = S, wstar the smallest WOT-closed unital *-subalgebra containing x) and RelativeGeneration.main_theorem : MainTarget: for every irreducible inclusion small ≤ large of concrete II₁ factors with HasSeparablePredual large, there is a faithful normal normalized trace on large such that relativeGeneratorLocus small large = {u | wstar (small ∪ {u}) = large} is a dense Gδ in unitary large for the trace 2-norm topology (traceTopology). Both summary claims are stated; ‘II₁ factor’ is encoded as nonzero + WOT-closed + scalar center + finite (isometries unitary) + diffuse.
Definitions the group asks readers to compare with the paper:Hand-written von Neumann-algebra notions: IsII1Factor (not Mathlib’s VonNeumannAlgebra with trace), HasSeparablePredual := ∃ separable Banach X, Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] S), wstar := sInf {S | WOTClosed S ∧ s ⊆ S}, ultraweakTopology via summable vector functionals and traceTopology generated by traceDistance balls on the unitary group; standard formulations.
“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 formalization answers the generator problem affirmatively for type $\mathrm{II}_1$ factors with separable predual: one bounded operator generates the factor under adjoints and weak-operator closure. It imposes no separability assumption on the representing Hilbert space or on the operator-norm topology.
It also proves the paper’s relative-generation result. For an irreducible inclusion of type $\mathrm{II}_1$ factors with separable predual for the larger factor, the unitaries that generate the larger factor together with the smaller one form a dense $G_\delta$ set in the normalized trace two-norm topology. The normalized faithful normal trace is included in the assertion.
AI-drafted comparisons, with no recorded human review
Single generation of separable-predual $\mathrm{II}_1$ factorsFactorGeneration
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every type II₁ factor with separable predual is generated by one operator.
Dense relative generators for irreducible inclusions of finite factorsRelativeGeneration
Unreviewed AI draft: states a paper’s theorem as the paper does. States the relative-generation theorem: for an irreducible inclusion of II₁ factors, the unitaries generating the larger factor with the smaller one form a dense G_δ set.
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.
Single generation of separable-predual $\mathrm{II}_1$ factorsFactorGeneration
Two 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
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
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
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 296The 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 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.