Lean statement

Quasinilpotent operator with transitive commutant

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.BackwardIntertwiners.direct_algebra_corollary

Lean statement file: BackwardIntertwiners.leanShow on the mapFamily 293. OpenAI lists this as a main result in its Lean catalogue.

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 direct_algebra_corollary (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H]    [CompleteSpace H] [TopologicalSpace.SeparableSpace H] (hInf : ¬FiniteDimensional ℂ H) :    Hyperinvariant233.FullClaim (H:=H) := by  sorry

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.

definition OAI.Hyperinvariant233.commutant line 11

def commutant (T : H →L[ℂ] H) : Subalgebra ℂ (H →L[ℂ] H) :=  Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H))

definition OAI.Hyperinvariant233.TransitiveCommutant line 14

def TransitiveCommutant (T : H →L[ℂ] H) : Prop :=  ∀ K : Submodule ℂ H, IsClosed (K : Set H) →    (∀ A : H →L[ℂ] H, A * T = T * A → ∀ x ∈ K, A x ∈ K) →    K = ⊥ ∨ K = ⊤@[instance_reducible]

definition OAI.Hyperinvariant233.strongOperatorTopology line 20

def strongOperatorTopology : TopologicalSpace (H →L[ℂ] H) :=  TopologicalSpace.induced (fun A : H →L[ℂ] H => (fun x : H => A x)) inferInstance

definition OAI.Hyperinvariant233.FullClaim line 23

def FullClaim : Prop :=  ∃ T : H →L[ℂ] H, T ≠ 0 ∧    Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧    TransitiveCommutant T ∧ commutant T ≠ ⊤ ∧    @IsClosed (H →L[ℂ] H) strongOperatorTopology (commutant T : Set (H →L[ℂ] H))

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: dimensional, hyperinvariant, commutant, transitive, complex, space, separable, closed and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: infinite, hilbert, proper, subspace, quasinilpotent and nonzero.

Statements in Backward intertwiners and a transitive commutant come first because the release’s name for this Lean statement shares words with that paper’s title.

Corollary 1.2 of Backward intertwiners and a transitive commutant

Corollary

Every infinite-dimensional separable complex Hilbert space admits a nonzero bounded norm-quasinilpotent operator with no nonzero proper closed hyperinvariant subspace. Its commutant is a proper strongly closed unital transitive complex operator algebra.

Section 1, Introduction Read it in the paper TeX source, line 72

Show the TeX

Every infinite-dimensional separable complex Hilbert space admits a
nonzero bounded norm-quasinilpotent operator with no nonzero proper
closed hyperinvariant subspace. Its commutant is a proper strongly
closed unital transitive complex operator algebra.

Other possible paper matches, closest wording first

Corollary 1.4 of Invariant-projection counterexamples for every irrational rotation. Words of the Lean statement, its names split into words, that also appear in the theorem: dimensional, hyperinvariant, commutant, transitive, complex, space and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nontrivial, infinite, hilbert, proper, subspace, quasinilpotent and one more.
Theorem 1.1 of Backward intertwiners and a transitive commutant. Words of the Lean statement, its names split into words, that also appear in the theorem: transitive, complex, closed, algebra, integer and operator. Found in the paper’s title, which its main theorem states: commutant. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: invariant, subspaces and nonzero.
Proposition 3.3 of Backward intertwiners and a transitive commutant. Words of the Lean statement, its names split into words, that also appear in the theorem: integers, norm and operator. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: quasinilpotent and nonzero.
Proposition A.9 of Invariant-projection counterexamples for every irrational rotation. Words of the Lean statement, its names split into words, that also appear in the theorem: full, finite, algebra and operator.
Proposition 3.2 of Backward intertwiners and a transitive commutant. Words of the Lean statement, its names split into words, that also appear in the theorem: closed, norm and operator. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: invariant and subspaces.
Theorem A.1 of Invariant-projection counterexamples for every irrational rotation. Words of the Lean statement, its names split into words, that also appear in the theorem: product, separable, finite, algebra and norm. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: quasinilpotent and nonzero.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 165 s; peak memory 6.4 GiB).
Reported 10 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 293’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: Both summary claims are stated: Hyperinvariant.main_theorem (hInf : ¬FiniteDimensional ℂ H) (also BackwardIntertwiners.direct_algebra_corollary) for every separable infinite-dimensional complex Hilbert space: ∃ T ≠ 0, ‖T^n‖^(1/n) → 0 ∧ TransitiveCommutant T ∧ commutant T ≠ ⊤ ∧ SOT-closed commutant (no nonzero proper closed hyperinvariant subspace), and for the invariant-projection counterexamples FiniteFactor.main_theorem (a II₁ factor with separable predual and a nonzero quasinilpotent T ∈ M with (1 - p) T p = 0 → p = 0 ∨ p = 1), IrrationalRotation.prescribed_irrational_counterexample (every irrational θ: continuous weight f with Rotation.IsContinuousCounterexample θ f, T in the irrational-rotation algebra, nonzero, norm-quasinilpotent, trivial invariant projections) and ProductBrown (Brown measure δ₀). 'Hyperfinite' is only implicit (FiniteFactor asserts a II₁ factor with separable predual; the rotation algebra is the hyperfinite one by construction).
Definitions the group asks readers to compare with the paper: Hand-written operator-algebra notions: TransitiveCommutant T := ∀ K closed submodule, (∀ A commuting with T, A K ⊆ K) → K = ⊥ ∨ K = ⊤, HasOnlyTrivialInvariantProjections, NormQuasinilpotent, vectorFugledeKadisonDeterminant, IsVectorBrownMeasure, strongOperatorTopology, HasSeparablePredual M := ∃ separable Banach E, Nonempty (StrongDual ℂ E ≃ₗᵢ⋆[ℂ] M); quasinilpotence is via ‖T^n‖^(1/n) → 0 and ContinuousCircleWeight.MainStatement does not itself state quasinilpotence (only trivial invariant projections for a weight with log-integral -∞).
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

“The formalization for the companion paper [Backward intertwiners and a transitive commutant](../../preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf) gives, on every separable infinite-dimensional complex Hilbert space, a nonzero quasinilpotent operator whose commutant is transitive, proper, and closed in the strong operator topology. Thus no nontrivial closed subspace is invariant under every operator in the commutant.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

44 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: 5 to its own declarations, 21 to Mathlib’s documentation. 5 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves direct_algebra_corollary 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 4 other declarations of the file; with the theorem itself they span 17 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.Hyperinvariant233 are listed without it.

def commutantline 11library
def TransitiveCommutantline 14library
def strongOperatorTopologyline 20library
def FullClaimline 23in the statementlibrary
theorem OAI.BackwardIntertwiners.direct_algebra_corollaryline 36selected theoremlibrary

import Mathlibnamespace OAInamespace Hyperinvariant233open Filteropen scoped Topologyuniverse uvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]def commutant (T : H →L[ℂ] H) : Subalgebra ℂ (H →L[ℂ] H) :=  Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H))def TransitiveCommutant (T : H →L[ℂ] H) : Prop :=  ∀ K : Submodule ℂ H, IsClosed (K : Set H) →    (∀ A : H →L[ℂ] H, A * T = T * A → ∀ x ∈ K, A x ∈ K) →    K = ⊥ ∨ K = ⊤@[instance_reducible]def strongOperatorTopology : TopologicalSpace (H →L[ℂ] H) :=  TopologicalSpace.induced (fun A : H →L[ℂ] H => (fun x : H => A x)) inferInstancedef FullClaim : Prop :=  ∃ T : H →L[ℂ] H, T ≠ 0 ∧    Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧    TransitiveCommutant T ∧ commutant T ≠ ⊤ ∧    @IsClosed (H →L[ℂ] H) strongOperatorTopology (commutant T : Set (H →L[ℂ] H))end Hyperinvariant233namespace BackwardIntertwinersopen Filter Topology MeasureTheory Setnoncomputable sectionopen scoped ENNRealtheorem direct_algebra_corollary (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H]    [CompleteSpace H] [TopologicalSpace.SeparableSpace H] (hInf : ¬FiniteDimensional ℂ H) :    Hyperinvariant233.FullClaim (H:=H) := by  sorryendend BackwardIntertwinersend 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.

Invariant-projection counterexamples for every irrational rotation

Theorem

For every irrational \theta\in(0,1) there exists a continuous function f:\T\to[0,1] such that

f^{-1}(\{0\})=\{1\},\qquad \int_\T\log f\,dm=-\infty,

and every projection p\in R_\theta satisfying (1-p)Uf(V)p=0 is either 0 or 1.

Section 1, Introduction Read it in the paper TeX source, line 33

Show the TeX

For every irrational $\theta\in(0,1)$ there exists a continuous function
$f:\T\to[0,1]$ such that
\[
 f^{-1}(\{0\})=\{1\},\qquad \int_\T\log f\,dm=-\infty,
\]
and every projection $p\in R_\theta$ satisfying $(1-p)Uf(V)p=0$
is either $0$ or $1$.

Corollary

Every separable infinite-dimensional complex Hilbert space carries a nonzero norm-quasinilpotent operator with no nontrivial hyperinvariant subspace. Its commutant is a proper strongly closed transitive unital algebra.

Section 1, Introduction Read it in the paper TeX source, line 94

Show the TeX

Every separable infinite-dimensional complex Hilbert space carries a
nonzero norm-quasinilpotent operator with no nontrivial hyperinvariant
subspace.  Its commutant is a proper strongly closed transitive unital
algebra.

Backward intertwiners and a transitive commutant

Theorem

There are measurable positive weights \beta_j:X\to(0,\infty), j\in\Z, and a contraction V\in\B(H_0) with the following properties. The shifts

S_xe_j=\beta_j(x)e_{j+1},\qquad (Sf)(x)=S_xf(x),

define a bounded operator S on H_0, and:

1.
For every x\in X, the closed invariant subspaces of S_x are exactly \Ktail{h}, h\in\Z\cup\{-\infty,+\infty\}. • The operator S is nonzero, \|S\|=e^{-10}, and for every integer n\ge1,
\|S^n\|\le \exp\left(-10\left\lfloor\frac{(n+1)^2}{4}\right\rfloor\right).
• The operator V commutes with S and has the form
(Vf)(\theta x)=V_xf(x),\qquad V_xe_j=b_j(x)e_{j-1},
where 0\le b_j(x)\le1 are measurable and are positive on a common conull set. On that set,
\overline{V_x\Ktail{h}}=\Ktail{h-1}\quad(h\in\Z), \qquad \overline{V_xK}=K.
• The unital complex algebra generated by S, V, and all scalar multipliers M_g is transitive.

Section 1, Introduction Read it in the paper TeX source, line 35

Show the TeX

There are measurable positive weights $\beta_j:X\to(0,\infty)$,
$j\in\Z$, and a contraction $V\in\B(H_0)$ with the following properties.
The shifts
\[
 S_xe_j=\beta_j(x)e_{j+1},\qquad (Sf)(x)=S_xf(x),
\]
define a bounded operator $S$ on $H_0$, and:
\begin{enumerate}
\item For every $x\in X$, the closed invariant subspaces of $S_x$ are
exactly $\Ktail{h}$, $h\in\Z\cup\{-\infty,+\infty\}$.
\item The operator $S$ is nonzero, $\|S\|=e^{-10}$, and for every
integer $n\ge1$,
\begin{equation}
 \|S^n\|\le
 \exp\left(-10\left\lfloor\frac{(n+1)^2}{4}\right\rfloor\right).
\end{equation}
\item The operator $V$ commutes with $S$ and has the form
\[
 (Vf)(\theta x)=V_xf(x),\qquad V_xe_j=b_j(x)e_{j-1},
\]
where $0\le b_j(x)\le1$ are measurable and are positive on a common
conull set. On that set,
\[
 \overline{V_x\Ktail{h}}=\Ktail{h-1}\quad(h\in\Z),
 \qquad \overline{V_xK}=K.
\]
\item The unital complex algebra generated by $S$, $V$, and all
scalar multipliers $M_g$ is transitive.
\end{enumerate}

Corollary

Section 1, Introduction Read it in the paper TeX source, line 72

Show the TeX

Every infinite-dimensional separable complex Hilbert space admits a
nonzero bounded norm-quasinilpotent operator with no nonzero proper
closed hyperinvariant subspace. Its commutant is a proper strongly
closed unital transitive complex operator algebra.

Configuration and file facts

File: lean/ComparatorChallenges/BackwardIntertwiners.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/BackwardIntertwiners/AlgebraCorollary.lean
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: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 18
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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