Single generation of separable-predual $\mathrm{II}_1$ factors
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Theorem selected for verification:OAI.Generator.single_generation_of_II1_separable_predual
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.
structureIsII1Factor (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Propwhere nonzero : (0 : H →L[ℂ] H) ≠ 1 weaklyClosed : WOTClosed S factor : ∀ x : S, (∀ y : S, y * x = x * y) → ∃ c : ℂ, x = algebraMap ℂ S c finite : ∀ v : S, star v * v = 1 → v * star v = 1 diffuse : ∀ p : S, IsStarProjection p → p ≠ 0 → ∃ q : S, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ p * q = q ∧ q * p = q
defSinglyGenerated (S : StarSubalgebra ℂ (H →L[ℂ] H)) : Prop := ∃ x ∈ S, wstar {x} = S
Possible paper match, with no recorded review
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: dense, factors, inclusion, irreducible, norm, predual, separable, topology and two more.
Theorem 1.1 of Relative generation and the generator problem for finite factors
Theorem
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.
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.
Other possible paper matches, closest wording first
Theorem 1.2 of Relative generation and the generator problem for finite factors. Words of the Lean statement, its names split into words, that also appear in the theorem: generated, predual, separable, ii and factor. Found in the paper’s title, which its main theorem states: generation and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: adjoint.
Theorem 5.1 of Relative generation and the generator problem for finite factors. Words of the Lean statement, its names split into words, that also appear in the theorem: complex, preduals, separable, ii and factors.
Proposition 5.3 of Relative generation and the generator problem for finite factors. Words of the Lean statement, its names split into words, that also appear in the theorem: generated, predual, separable, ii and factor. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: adjoint.
Corollary 6.1 of Relative generation and the generator problem for finite factors. Words of the Lean statement, its names split into words, that also appear in the theorem: algebra, finite, generating and factor. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: adjoint.
Theorem 2.2 of Relative generation and the generator problem for finite factors. Words of the Lean statement, its names split into words, that also appear in the theorem: diffuse, algebra, norm, separable, ii and factors.
Proposition 3.2 of Relative generation and the generator problem for finite factors. Words of the Lean statement, its names split into words, that also appear in the theorem: norm, ii and factors. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
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
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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.
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 296’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.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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.
OpenAI’s own note on what this Lean statement covers
“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.”
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
50 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: 6 to its own declarations, 23 to Mathlib’s documentation. 6 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
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.
Theorem
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).
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 $M=W^*(x)$.
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.