Lean statement

Left/right finitistic-dimension asymmetry

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.LittleFinitistic.extreme_left_right_asymmetry

Lean statement file: FinitisticAsymmetry.leanShow on the mapFamily 198. 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 extreme_left_right_asymmetry :    ∃ (Λ : Type) (_ : Ring Λ) (_ : Algebra ℂ Λ),      FiniteDimensional ℂ Λ ∧      littleFinitisticDimension Λ = ⊤ ∧ bigFinitisticDimension Λ = ⊤ ∧      littleFinitisticDimension Λᵐᵒᵖ = 0 ∧ bigFinitisticDimension Λᵐᵒᵖ = 0 ∧      ¬ InjectivesGenerate Λ ∧ InjectivesGenerate Λᵐᵒᵖ := 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.LittleFinitistic.littleFinitisticDimension line 13

def littleFinitisticDimension : WithBot ℕ∞ :=  ⨆ (M : ModuleCat.{u} A) (_ : Module.Finite A M) (_ : projectiveDimension M < ⊤),    projectiveDimension M

definition OAI.LittleFinitistic.bigFinitisticDimension line 17

def bigFinitisticDimension : WithBot ℕ∞ :=  ⨆ (M : ModuleCat.{u} A) (_ : projectiveDimension M < ⊤), projectiveDimension M

definition OAI.LittleFinitistic.InjectivesGenerate line 20

def InjectivesGenerate : Prop :=  letI := HasDerivedCategory.standard (ModuleCat.{u} A)  ∀ (P : ObjectProperty (DerivedCategory (ModuleCat.{u} A))),    P.IsClosedUnderIsomorphisms → P.IsTriangulated →    (∀ I : Type u, P.IsClosedUnderColimitsOfShape (Discrete I)) →    (∀ (M : ModuleCat.{u} A), Injective M →      P ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj M)) →    ∀ X, P X

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: injective, asymmetry, extreme, finitistic, generate, little, derived, category and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unbounded, zero and infinite.

Corollary 1.2 of An algebra of infinite little finitistic dimension

Corollary (Extreme left–right asymmetry)

There is a finite-dimensional unital complex algebra \Lambda such that

\findim \Lambda=\Findim \Lambda=\infty, \qquad \findim(\Lambda^{\mathrm{op}})=\Findim(\Lambda^{\mathrm{op}})=0.

All four dimensions here use left modules. Thus the little and big finitistic dimensions of \Lambda are infinite on the left and zero on the right. Injective left \Lambda-modules do not generate the unbounded derived category of all left \Lambda-modules, whereas injective right \Lambda-modules generate the unbounded derived category of all right \Lambda-modules.

Section 1, Introduction Read it in the paper TeX source, line 91

Show the TeX

There is a finite-dimensional unital complex algebra \(\Lambda\) such that
\[
\findim \Lambda=\Findim \Lambda=\infty,
\qquad
\findim(\Lambda^{\mathrm{op}})=\Findim(\Lambda^{\mathrm{op}})=0.
\]
All four dimensions here use left modules. Thus the little and big
finitistic dimensions of \(\Lambda\) are infinite on the left and zero
on the right. Injective left \(\Lambda\)-modules do not generate the
unbounded derived category of all left \(\Lambda\)-modules, whereas
injective right \(\Lambda\)-modules generate the unbounded derived
category of all right \(\Lambda\)-modules.

Other possible paper matches, closest wording first

Theorem 1.1 of An algebra of infinite little finitistic dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, complex, algebra, dimensional, module and finite. Found in the paper’s title, which its main theorem states: little, finitistic and dimension.
Proposition 3.4 of An algebra of infinite little finitistic dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: objects, generated, projective, complexes, algebra, modules and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose.
Proposition 5.1 of An algebra of infinite little finitistic dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: isomorphism, dimension, complexes, algebra, modules and finite.
Proposition 6.1 of An algebra of infinite little finitistic dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: functor, derived, category, isomorphism, algebra, dimensional and two more.
Corollary 6.2 of An algebra of infinite little finitistic dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: projective, dimension, dimensional and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: infinite.
Theorem 4.1 of An algebra of infinite little finitistic dimension. Words of the Lean statement, its names split into words, that also appear in the theorem: single, projective, complex, algebra, dimensional, modules and one more.

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 198’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: exists_counterexample : ∃ A (_ : Ring A) (_ : Algebra ℂ A), FiniteDimensional ℂ A ∧ littleFinitisticDimension A = ⊤ ∧ ∀ m > 0, ∃ N, Module.Finite A N ∧ (2m-2 : WithBot ℕ∞) ≤ projectiveDimension N ∧ projectiveDimension N < ⊤, where littleFinitisticDimension A := ⨆ (M) (_ : Module.Finite A M) (_ : projectiveDimension M < ⊤), projectiveDimension M (Mathlib projectiveDimension): unbounded finite projective dimensions over a finite-dimensional complex algebra, as in the summary. FinitisticAsymmetry is an extra secondary result.
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

“A separate formalized result gives a finite-dimensional complex algebra whose little and big finitistic dimensions are infinite on the left and zero on the right. Injectives fail to generate its unbounded derived category on the left and generate it on the right.”
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

41 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, 24 to Mathlib’s documentation. 4 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 extreme_left_right_asymmetry 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 3 other declarations of the file; with the theorem itself they span 20 lines. 3 of them are named in the statement itself.

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

def littleFinitisticDimensionline 13in the statementlibrary
def bigFinitisticDimensionline 17in the statementlibrary
def InjectivesGenerateline 20in the statementlibrary
theorem extreme_left_right_asymmetryline 29selected theoremlibrary

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

attribute [-instance] CompletePartialOrder.toSupSet in line 12

import Mathlibnamespace OAInoncomputable sectionopen CategoryTheorynamespace LittleFinitisticuniverse uvariable (A : Type u) [Ring A]attribute [-instance] CompletePartialOrder.toSupSet indef littleFinitisticDimension : WithBot ℕ∞ :=  ⨆ (M : ModuleCat.{u} A) (_ : Module.Finite A M) (_ : projectiveDimension M < ⊤),    projectiveDimension Mdef bigFinitisticDimension : WithBot ℕ∞ :=  ⨆ (M : ModuleCat.{u} A) (_ : projectiveDimension M < ⊤), projectiveDimension Mdef InjectivesGenerate : Prop :=  letI := HasDerivedCategory.standard (ModuleCat.{u} A)  ∀ (P : ObjectProperty (DerivedCategory (ModuleCat.{u} A))),    P.IsClosedUnderIsomorphisms → P.IsTriangulated →    (∀ I : Type u, P.IsClosedUnderColimitsOfShape (Discrete I)) →    (∀ (M : ModuleCat.{u} A), Injective M →      P ((DerivedCategory.singleFunctor (ModuleCat.{u} A) 0).obj M)) →    ∀ X, P Xtheorem extreme_left_right_asymmetry :    ∃ (Λ : Type) (_ : Ring Λ) (_ : Algebra ℂ Λ),      FiniteDimensional ℂ Λ ∧      littleFinitisticDimension Λ = ⊤ ∧ bigFinitisticDimension Λ = ⊤ ∧      littleFinitisticDimension Λᵐᵒᵖ = 0 ∧ bigFinitisticDimension Λᵐᵒᵖ = 0 ∧      ¬ InjectivesGenerate Λ ∧ InjectivesGenerate Λᵐᵒᵖ := by  sorryend LittleFinitisticendend 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.

An algebra of infinite little finitistic dimension

Theorem

There is a finite-dimensional unital complex algebra A such that, for every integer m\ge1, there is a finite-dimensional left A-module N_m with

2m-2\le\pd_A N_m<\infty.

In particular, \findim A=\infty.

Section 1, Introduction Read it in the paper TeX source, line 24

Show the TeX

There is a finite-dimensional unital complex algebra \(A\) such that,
for every integer \(m\ge1\), there is a finite-dimensional left
\(A\)-module \(N_m\) with
\[
2m-2\le\pd_A N_m<\infty.
\]
In particular, \(\findim A=\infty\).

Corollary (Extreme left–right asymmetry)

There is a finite-dimensional unital complex algebra \Lambda such that

\findim \Lambda=\Findim \Lambda=\infty, \qquad \findim(\Lambda^{\mathrm{op}})=\Findim(\Lambda^{\mathrm{op}})=0.

Section 1, Introduction Read it in the paper TeX source, line 91

Show the TeX

There is a finite-dimensional unital complex algebra \(\Lambda\) such that
\[
\findim \Lambda=\Findim \Lambda=\infty,
\qquad
\findim(\Lambda^{\mathrm{op}})=\Findim(\Lambda^{\mathrm{op}})=0.
\]
All four dimensions here use left modules. Thus the little and big
finitistic dimensions of \(\Lambda\) are infinite on the left and zero
on the right. Injective left \(\Lambda\)-modules do not generate the
unbounded derived category of all left \(\Lambda\)-modules, whereas
injective right \(\Lambda\)-modules generate the unbounded derived
category of all right \(\Lambda\)-modules.

Configuration and file facts

File: lean/ComparatorChallenges/FinitisticAsymmetry.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Algebra/FinitisticAsymmetry/Main.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: 3 local vocabulary declarations
Lines of Lean, without comments or blank lines: 21
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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