Lean statement

Dense relative generators for irreducible inclusions of finite factors

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Theorem selected for verification: OAI.RelativeGeneration.main_theorem

Lean statement file: RelativeGeneration.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 296 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main_theorem : MainTarget.{level} := by  sorry

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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.

definition OAI.RelativeGeneration.WOTClosed line 15

def WOTClosed (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop :=  IsClosed {operator : Hilbert →WOT[ℂ] Hilbert | operator.toCLM ∈ algebra}

definition OAI.RelativeGeneration.wstar line 19

def wstar (operators : Set (Hilbert →L[ℂ] Hilbert)) :    StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert) :=  sInf {algebra | WOTClosed algebra ∧ operators ⊆ algebra}

structure OAI.RelativeGeneration.IsII1Factor line 24

structure IsII1Factor (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop where  nonzero : (0 : Hilbert →L[ℂ] Hilbert) ≠ 1  weaklyClosed : WOTClosed algebra  factor : ∀ element : algebra, (∀ other : algebra, other * element = element * other) →    ∃ scalar : ℂ, element = algebraMap ℂ algebra scalar  finite : ∀ isometry : algebra, star isometry * isometry = 1 → isometry * star isometry = 1  diffuse : ∀ projection : algebra, IsStarProjection projection → projection ≠ 0 →    ∃ subprojection : algebra, IsStarProjection subprojection ∧ subprojection ≠ 0 ∧      subprojection ≠ projection ∧ projection * subprojection = subprojection ∧      subprojection * projection = subprojection

definition OAI.RelativeGeneration.HasSeparablePredual line 37

def HasSeparablePredual (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop :=  ∃ (Banach : Type level) (_ : NormedAddCommGroup Banach) (_ : NormedSpace ℂ Banach)    (_ : CompleteSpace Banach) (_ : TopologicalSpace.SeparableSpace Banach),    Nonempty (StrongDual ℂ Banach ≃ₗᵢ⋆[ℂ] algebra)

abbreviation OAI.RelativeGeneration.ultraweakTopology line 43

abbrev ultraweakTopology (Hilbert : Type level) [NormedAddCommGroup Hilbert]    [InnerProductSpace ℂ Hilbert] [CompleteSpace Hilbert] :    TopologicalSpace (Hilbert →L[ℂ] Hilbert) :=  ⨅ (left : ℕ → Hilbert) (right : ℕ → Hilbert)    (_ : Summable (fun index => ‖left index‖ * ‖right index‖)),    TopologicalSpace.induced      (fun operator : Hilbert →L[ℂ] Hilbert =>        ∑' index, inner ℂ (left index) (operator (right index))) inferInstance

abbreviation OAI.RelativeGeneration.algebraUltraweakTopology line 53

abbrev algebraUltraweakTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :    TopologicalSpace algebra :=  TopologicalSpace.induced (fun operator : algebra => (operator : Hilbert →L[ℂ] Hilbert))    (ultraweakTopology Hilbert)

structure OAI.RelativeGeneration.IsNormalizedTrace line 59

structure IsNormalizedTrace (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))    (trace : algebra →L[ℂ] ℂ) : Prop where  normalized : trace 1 = 1  positive : ∀ element : algebra, 0 ≤ (trace (star element * element)).re  star_preserving : ∀ element : algebra, trace (star element) = star (trace element)  tracial : ∀ left right : algebra, trace (left * right) = trace (right * left)  faithful : ∀ element : algebra, trace (star element * element) = 0 → element = 0  normal : @Continuous algebra ℂ (algebraUltraweakTopology algebra) inferInstance trace

definition OAI.RelativeGeneration.traceDistance line 69

def traceDistance (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))    (trace : algebra →L[ℂ] ℂ) (left right : unitary algebra) : ℝ :=  Real.sqrt (trace (star ((left : algebra) - (right : algebra)) *    ((left : algebra) - (right : algebra)))).re

abbreviation OAI.RelativeGeneration.traceTopology line 75

abbrev traceTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))    (trace : algebra →L[ℂ] ℂ) : TopologicalSpace (unitary algebra) :=  TopologicalSpace.generateFrom {ball | ∃ center : unitary algebra, ∃ radius : ℝ,    0 < radius ∧ ball = {point | traceDistance algebra trace center point < radius}}

definition OAI.RelativeGeneration.relativeGeneratorLocus line 81

def relativeGeneratorLocus (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :    Set (unitary large) :=  {element | wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪    {((element : large) : Hilbert →L[ℂ] Hilbert)}) = large}

definition OAI.RelativeGeneration.MainTarget line 90

def MainTarget : Prop :=

Show the remaining 11 lines

  ∀ (Hilbert : Type level) [NormedAddCommGroup Hilbert] [InnerProductSpace ℂ Hilbert]    [CompleteSpace Hilbert] (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)),    IsII1Factor small → IsII1Factor large → ∀ inclusion : small ≤ large,      (∀ element : large,        (∀ member : small, StarSubalgebra.inclusion inclusion member * element =          element * StarSubalgebra.inclusion inclusion member) →        ∃ scalar : ℂ, element = algebraMap ℂ large scalar) →      HasSeparablePredual large →      ∃ trace : large →L[ℂ] ℂ, IsNormalizedTrace large trace ∧        @IsGδ (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large) ∧        @Dense (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large)

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: topology, dense, group, unitary, norm, trace, predual, inclusion and four more. Found in the paper’s title, which its main theorem states: relative, generator and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: delta.

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.

Section 1, Introduction Read it in the paper TeX source, line 19

Show the TeX

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: two, elements, generated, predual, separable, ii and one more. Found in the paper’s title, which its main theorem states: relative, generator and finite.
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, linear, algebra, norm, trace, inclusion and four more.
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: tracial, algebra, two, integer, finite, generating and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: delta.
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: sqrt, products, indexed, unitaries, integer, norm and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
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, inclusion, irreducible, separable, ii and one more.
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: elements, generated, predual, separable, ii and factor.

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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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.

MathVet
Finds the whole advertised result stated or implied Their 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.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

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

110 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: 15 to its own declarations, 63 to Mathlib’s documentation. 12 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main_theorem exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 11 other declarations of the file; with the theorem itself they span 65 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.RelativeGeneration are listed without it.

def WOTClosedline 15library
def wstarline 19library
structure IsII1Factorline 24library
def HasSeparablePredualline 37library
abbrev ultraweakTopologyline 43library
abbrev algebraUltraweakTopologyline 53library
structure IsNormalizedTraceline 59library
def traceDistanceline 69library
abbrev traceTopologyline 75library
def relativeGeneratorLocusline 81library
def MainTargetline 90in the statementlibrary
theorem main_theoremline 103selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionuniverse levelnamespace RelativeGenerationvariable {Hilbert : Type level} [NormedAddCommGroup Hilbert]  [InnerProductSpace ℂ Hilbert] [CompleteSpace Hilbert]/-- Weak operator closedness of a concrete unital star subalgebra. -/def WOTClosed (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop :=  IsClosed {operator : Hilbert →WOT[ℂ] Hilbert | operator.toCLM ∈ algebra}/-- The weakly closed unital star algebra generated by a set of operators. -/def wstar (operators : Set (Hilbert →L[ℂ] Hilbert)) :    StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert) :=  sInf {algebra | WOTClosed algebra ∧ operators ⊆ algebra}/-- A nonzero, weakly closed, finite diffuse algebra with scalar center. -/structure IsII1Factor (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop where  nonzero : (0 : Hilbert →L[ℂ] Hilbert) ≠ 1  weaklyClosed : WOTClosed algebra  factor : ∀ element : algebra, (∀ other : algebra, other * element = element * other) →    ∃ scalar : ℂ, element = algebraMap ℂ algebra scalar  finite : ∀ isometry : algebra, star isometry * isometry = 1 → isometry * star isometry = 1  diffuse : ∀ projection : algebra, IsStarProjection projection → projection ≠ 0 →    ∃ subprojection : algebra, IsStarProjection subprojection ∧ subprojection ≠ 0 ∧      subprojection ≠ projection ∧ projection * subprojection = subprojection ∧      subprojection * projection = subprojection/-- A same-universe separable Banach predual, with the conjugate-linear dual convention.This imposes no separability on the Hilbert representation or the operator norm topology. -/def HasSeparablePredual (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) : Prop :=  ∃ (Banach : Type level) (_ : NormedAddCommGroup Banach) (_ : NormedSpace ℂ Banach)    (_ : CompleteSpace Banach) (_ : TopologicalSpace.SeparableSpace Banach),    Nonempty (StrongDual ℂ Banach ≃ₗᵢ⋆[ℂ] algebra)/-- The ultraweak topology is generated by absolutely summable vector functionals. -/abbrev ultraweakTopology (Hilbert : Type level) [NormedAddCommGroup Hilbert]    [InnerProductSpace ℂ Hilbert] [CompleteSpace Hilbert] :    TopologicalSpace (Hilbert →L[ℂ] Hilbert) :=  ⨅ (left : ℕ → Hilbert) (right : ℕ → Hilbert)    (_ : Summable (fun index => ‖left index‖ * ‖right index‖)),    TopologicalSpace.induced      (fun operator : Hilbert →L[ℂ] Hilbert =>        ∑' index, inner ℂ (left index) (operator (right index))) inferInstance/-- The concrete algebra carries the induced ultraweak topology. -/abbrev algebraUltraweakTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :    TopologicalSpace algebra :=  TopologicalSpace.induced (fun operator : algebra => (operator : Hilbert →L[ℂ] Hilbert))    (ultraweakTopology Hilbert)/-- A faithful normal normalized trace, not an additional hypothesis on the inclusion. -/structure IsNormalizedTrace (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))    (trace : algebra →L[ℂ] ℂ) : Prop where  normalized : trace 1 = 1  positive : ∀ element : algebra, 0 ≤ (trace (star element * element)).re  star_preserving : ∀ element : algebra, trace (star element) = star (trace element)  tracial : ∀ left right : algebra, trace (left * right) = trace (right * left)  faithful : ∀ element : algebra, trace (star element * element) = 0 → element = 0  normal : @Continuous algebra ℂ (algebraUltraweakTopology algebra) inferInstance trace/-- The normalized trace two-distance between genuine unitaries of the original algebra. -/def traceDistance (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))    (trace : algebra →L[ℂ] ℂ) (left right : unitary algebra) : ℝ :=  Real.sqrt (trace (star ((left : algebra) - (right : algebra)) *    ((left : algebra) - (right : algebra)))).re/-- The topology generated by trace two-distance balls, not the inherited operator norm topology. -/abbrev traceTopology (algebra : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert))    (trace : algebra →L[ℂ] ℂ) : TopologicalSpace (unitary algebra) :=  TopologicalSpace.generateFrom {ball | ∃ center : unitary algebra, ∃ radius : ℝ,    0 < radius ∧ ball = {point | traceDistance algebra trace center point < radius}}/-- The entire relative-generator locus in the original unitary group. -/def relativeGeneratorLocus (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)) :    Set (unitary large) :=  {element | wstar ((small : Set (Hilbert →L[ℂ] Hilbert)) ∪    {((element : large) : Hilbert →L[ℂ] Hilbert)}) = large}/-- Every irreducible inclusion of concrete type II₁ factors with separable predualof the larger factor has a dense Gδ full relative-generator locus in the normalizedtrace two-topology. The normalized faithful normal trace is asserted to exist;no representation, transport, cyclicity, or perturbation premise is imposed. -/def MainTarget : Prop :=  ∀ (Hilbert : Type level) [NormedAddCommGroup Hilbert] [InnerProductSpace ℂ Hilbert]    [CompleteSpace Hilbert] (small large : StarSubalgebra ℂ (Hilbert →L[ℂ] Hilbert)),    IsII1Factor small → IsII1Factor large → ∀ inclusion : small ≤ large,      (∀ element : large,        (∀ member : small, StarSubalgebra.inclusion inclusion member * element =          element * StarSubalgebra.inclusion inclusion member) →        ∃ scalar : ℂ, element = algebraMap ℂ large scalar) →      HasSeparablePredual large →      ∃ trace : large →L[ℂ] ℂ, IsNormalizedTrace large trace ∧        @IsGδ (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large) ∧        @Dense (unitary large) (traceTopology large trace) (relativeGeneratorLocus small large)theorem main_theorem : MainTarget.{level} := by  sorryend RelativeGenerationendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

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.

Section 1, Introduction Read it in the paper TeX source, line 19

Show the TeX

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).

Section 1, Introduction Read it in the paper TeX source, line 29

Show the TeX

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 $M=W^*(x)$.

Configuration and file facts

File: lean/ComparatorChallenges/RelativeGeneration.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 296 link it.
Solution module: OAI.Analysis.RelativeGeneration.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 11 local vocabulary declarations
Lines of Lean, without comments or blank lines: 66
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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