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Lean statement

Finite-factor invariant-projection counterexample

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.FiniteFactor.main_theorem

Lean statement file: FiniteFactor.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 293 link it.

See the candidate theoremWho ran the proof-verification programReport 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.{0} := 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.FiniteFactor.HasScalarCenter line 14

    def HasScalarCenter (M : VonNeumannAlgebra H) : Prop :=  ∀ a ∈ M, (∀ b ∈ M, a * b = b * a) → ∃ z : ℂ, a = z • (1 : H →L[ℂ] H)

  • definition OAI.FiniteFactor.IsFiniteAlgebra line 18

    def IsFiniteAlgebra (M : VonNeumannAlgebra H) : Prop :=  ∀ v ∈ M, star v * v = 1 → v * star v = 1

  • definition OAI.FiniteFactor.HasNoMinimalProjection line 22

    def HasNoMinimalProjection (M : VonNeumannAlgebra H) : Prop :=  ∀ p ∈ M, IsStarProjection p → p ≠ 0 →    ∃ q ∈ M, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ q * p = q

  • definition OAI.FiniteFactor.HasSeparablePredual line 27

    def HasSeparablePredual (M : VonNeumannAlgebra H) : Prop :=  ∃ (E : Type u) (_ : NormedAddCommGroup E) (_ : NormedSpace ℂ E)    (_ : CompleteSpace E) (_ : TopologicalSpace.SeparableSpace E),    Nonempty (StrongDual ℂ E ≃ₗᵢ⋆[ℂ] M.toStarSubalgebra)

  • definition OAI.FiniteFactor.MainTarget line 32

    def MainTarget : Prop :=  ∃ (H : Type u) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H)    (_ : CompleteSpace H) (M : VonNeumannAlgebra H),    HasScalarCenter M ∧ IsFiniteAlgebra M ∧ HasNoMinimalProjection M ∧    HasSeparablePredual M ∧    ∃ T : H →L[ℂ] H, T ∈ M ∧ T ≠ 0 ∧      Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧      ∀ p ∈ M, IsStarProjection p → (1 - p) * T * p = 0 → p = 0 ∨ p = 1

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: conjugates, predual, factor, product, separable, finite, algebra, nonzero and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: delta, realization and measurable.

Statements in Invariant-projection counterexamples for every irrational rotation come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem A.1 of Invariant-projection counterexamples for every irrational rotation

Theorem (Product realization)

There are angles 0<\theta_i<1 and costs c_i>0, with \sum_i\theta_i<\infty and 1,\theta_1,\theta_2,\ldots rationally independent, such that the measurable weight

f(x)=\exp\left(-\sum_{i\ge1}c_i\lfloor x_i+\theta_i\rfloor\right)

is positive almost everywhere and has \int_X\log f\,d\mu=-\infty. The algebra M is a type \mathrm{II}_1 factor with separable predual, and T=U\pi(f) is nonzero and norm-quasinilpotent, with \Delta(T)=0. Every projection p\in M satisfying (1-p)Tp=0 is 0 or 1.

Moreover, |T| has pure-point spectral type. The abelian algebra generated by any finite family of polar conjugates U^n|T|U^{-n} is atomic, whereas W^*(U^n|T|U^{-n}:n\in\Z) is diffuse.

Appendix, A product-rotation realization Read it in the paper TeX source, line 27
Show the TeX
There are angles $0<\theta_i<1$ and costs $c_i>0$, with
$\sum_i\theta_i<\infty$ and $1,\theta_1,\theta_2,\ldots$ rationally
independent, such that the measurable weight
\[
 f(x)=\exp\left(-\sum_{i\ge1}c_i\lfloor x_i+\theta_i\rfloor\right)
\]
is positive almost everywhere and has $\int_X\log f\,d\mu=-\infty$.
The algebra $M$ is a type $\mathrm{II}_1$ factor with separable predual,
and $T=U\pi(f)$ is nonzero and norm-quasinilpotent, with
$\Delta(T)=0$.  Every projection $p\in M$ satisfying $(1-p)Tp=0$ is
$0$ or $1$.

Moreover, $|T|$ has pure-point spectral type.  The abelian algebra
generated by any finite family of polar conjugates
$U^n|T|U^{-n}$ is atomic, whereas
$W^*(U^n|T|U^{-n}:n\in\Z)$ is diffuse.

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: complex, space, separable, algebra, nonzero and norm.
  • 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: von, neumann, finite and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measure.
  • Corollary 1.2 of Backward intertwiners and a transitive commutant. Words of the Lean statement, its names split into words, that also appear in the theorem: complex, space, separable, algebra, nonzero and norm.
  • 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: scalar, complex, algebra, integer and nonzero. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: invariant and measurable.
  • Proposition A.3 of Invariant-projection counterexamples for every irrational rotation. Words of the Lean statement, its names split into words, that also appear in the theorem: space, finite, integers and norms. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: realization and measures.
  • Proposition A.6 of Invariant-projection counterexamples for every irrational rotation. Words of the Lean statement, its names split into words, that also appear in the theorem: finite and projection.

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 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 linked statements also include an earlier existence construction, a product realization, and the selected product model's Brown measure $\delta_0$.”

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

Supporting material

The complete Lean statement file

46 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, 24 to Mathlib’s documentation. 6 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.

import Mathlib.Analysis.VonNeumannAlgebra.Basicimport Mathlib.Analysis.SpecialFunctions.Pow.Realnamespace OAInamespace FiniteFactoropen Filteropen scoped Topologyuniverse uvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]/-- A concrete factor has scalar center. -/def HasScalarCenter (M : VonNeumannAlgebra H) : Prop :=  ∀ a ∈ M, (∀ b ∈ M, a * b = b * a) → ∃ z : ℂ, a = z • (1 : H →L[ℂ] H)/-- Finiteness: every isometry in the algebra is unitary. -/def IsFiniteAlgebra (M : VonNeumannAlgebra H) : Prop :=  ∀ v ∈ M, star v * v = 1 → v * star v = 1/-- No nonzero projection is minimal, excluding matrix factors. -/def HasNoMinimalProjection (M : VonNeumannAlgebra H) : Prop :=  ∀ p ∈ M, IsStarProjection p → p ≠ 0 →    ∃ q ∈ M, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ q * p = q/-- A separable Banach space whose dual is conjugate-linearly isometric to M. -/def HasSeparablePredual (M : VonNeumannAlgebra H) : Prop :=  ∃ (E : Type u) (_ : NormedAddCommGroup E) (_ : NormedSpace ℂ E)    (_ : CompleteSpace E) (_ : TopologicalSpace.SeparableSpace E),    Nonempty (StrongDual ℂ E ≃ₗᵢ⋆[ℂ] M.toStarSubalgebra)def MainTarget : Prop :=  ∃ (H : Type u) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H)    (_ : CompleteSpace H) (M : VonNeumannAlgebra H),    HasScalarCenter M ∧ IsFiniteAlgebra M ∧ HasNoMinimalProjection M ∧    HasSeparablePredual M ∧    ∃ T : H →L[ℂ] H, T ∈ M ∧ T ≠ 0 ∧      Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧      ∀ p ∈ M, IsStarProjection p → (1 - p) * T * p = 0 → p = 0 ∨ p = 1theorem main_theorem : MainTarget.{0} := by  sorryend FiniteFactorend 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. 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

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.
Configuration and file facts
OpenAI’s Lean catalogue
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 293 link it.
Solution module
OAI.Analysis.FiniteFactor.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.Analysis.VonNeumannAlgebra.Basic, Mathlib.Analysis.SpecialFunctions.Pow.Real
Imports from outside Mathlib
None
Local vocabulary
5 local vocabulary declarations
Lines of Lean, without comments or blank lines
21
Second proof-verification program (nanoda)
Off in OpenAI's configuration.

Statement comparison

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

Plectis Atlas

Finite-factor invariant-projection counterexample

Family 293: Invariant projections, hyperinvariant subspaces, and transitive algebras

Candidate pairing

Theorem (Product realization)

There are angles 0<\theta_i<1 and costs c_i>0, with \sum_i\theta_i<\infty and 1,\theta_1,\theta_2,\ldots rationally independent, such that the measurable weight

f(x)=\exp\left(-\sum_{i\ge1}c_i\lfloor x_i+\theta_i\rfloor\right)

is positive almost everywhere and has \int_X\log f\,d\mu=-\infty. The algebra M is a type \mathrm{II}_1 factor with separable predual, and T=U\pi(f) is nonzero and norm-quasinilpotent, with \Delta(T)=0. Every projection p\in M satisfying (1-p)Tp=0 is 0 or 1.

Moreover, |T| has pure-point spectral type. The abelian algebra generated by any finite family of polar conjugates U^n|T|U^{-n} is atomic, whereas W^*(U^n|T|U^{-n}:n\in\Z) is diffuse.

Invariant-projection counterexamples for every irrational rotation

TeX label p:thm:model, build/appendix-product.tex, line 27

The Lean statement says

theorem main_theorem : MainTarget.{0} := 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.

Atlas selected this possible match because the two statements share words. No review of whether they mean the same thing is recorded.two independent groups report that Comparator accepted runs for this file at the release version shown here.Statement review: none recorded

5 local vocabulary declarationsAllowed axioms: propext, Quot.sound, Classical.choiceopenai/math at fd4aeebwcook04.github.io/plectis-atlas/statements/FiniteFactor.html

Recorded comparisons with the paper

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