Lean statement

Exact two-cycle bound for quintic Liénard systems

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.QuinticLienard.main

Lean statement file: QuinticLienard.leanShow on the mapFamily 143. 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 main :    (∀ F : Polynomial ℝ, F.degree ≤ 5 → {C : Set Plane | IsLimitCycle F C}.encard ≤ 2) ∧    (∃ F : Polynomial ℝ, F.degree ≤ 5 ∧ {C : Set Plane | IsLimitCycle F C}.encard = 2) := 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.

abbreviation OAI.QuinticLienard.Plane line 7
```
abbrev Plane := ℝ × ℝ
```

definition OAI.QuinticLienard.vectorField line 9

def vectorField (F : Polynomial ℝ) (z : Plane) : Plane :=  (z.2 - F.eval z.1, -z.1)

definition OAI.QuinticLienard.IsSolution line 12

def IsSolution (F : Polynomial ℝ) (z : ℝ → Plane) : Prop :=  ∀ t : ℝ, HasDerivAt z (vectorField F (z t)) t

definition OAI.QuinticLienard.IsPeriodicOrbit line 15

def IsPeriodicOrbit (F : Polynomial ℝ) (C : Set Plane) : Prop :=  ∃ (z : ℝ → Plane) (T : ℝ),    IsSolution F z ∧ 0 < T ∧ Function.Periodic z T ∧    (∃ s t : ℝ, z s ≠ z t) ∧ C = Set.range z

definition OAI.QuinticLienard.IsLimitCycle line 20

def IsLimitCycle (F : Polynomial ℝ) (C : Set Plane) : Prop :=  IsPeriodicOrbit F C ∧    ∃ U : Set Plane, IsOpen U ∧ C ⊆ U ∧      ∀ C' : Set Plane, IsPeriodicOrbit F C' → C' ⊆ U → C' = C

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: orbits, periodic and solutions. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: exactly, images, nonconstant, isolated and system.

Statements in Two limit cycles for quintic Liénard systems come first because the release lists only that paper for this Lean statement.

Proposition 2.5 of Two limit cycles for quintic Liénard systems

Proposition (Periodic-orbit matching)

For the system

\dot x=y-F(x),\qquad \dot y=-x,

the images of nonconstant periodic solutions are in one-to-one correspondence with the zeros of

\Delta(r)=M_+(0,r)-M_-(0,r)

on the single open interval \mathcal I=\mathcal I_+\cap\mathcal I_-. If this interval is empty, there are no such periodic solutions. Each periodic orbit has exactly one positive and one negative intersection with the y-axis. Under the correspondence, isolated periodic orbits are exactly isolated zeros of \Delta.

Section 2, Arc coordinates and periodic-orbit matching Read it in the paper TeX source, line 498

Show the TeX

For the system
\begin{equation}
  \dot x=y-F(x),\qquad \dot y=-x,
\end{equation}
the images of nonconstant periodic solutions are in one-to-one
correspondence with the zeros of
\begin{equation}
  \Delta(r)=M_+(0,r)-M_-(0,r)
\end{equation}
on the single open interval
\(\mathcal I=\mathcal I_+\cap\mathcal I_-\).
If this interval is empty, there are no such periodic solutions.
Each periodic orbit has exactly one positive and one negative
intersection with the \(y\)-axis. Under the correspondence,
isolated periodic orbits are exactly isolated zeros of \(\Delta\).

Other possible paper matches, closest wording first

Theorem 1.1 of Uniform bounds for planar polynomial limit cycles. Words of the Lean statement, its names split into words, that also appear in the theorem: plane, vector, cycles, degree, field, polynomial and two more.
Theorem 1.1 of Two limit cycles for quintic Liénard systems. Words of the Lean statement, its names split into words, that also appear in the theorem: cycles and limit. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two and system.
Theorem 9.7 of Uniform bounds for planar polynomial limit cycles. Words of the Lean statement, its names split into words, that also appear in the theorem: orbits, periodic, degree, polynomial, field and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: resulting, nonconstant, sign and bound.
Theorem 11.4 of Uniform bounds for planar polynomial limit cycles. Words of the Lean statement, its names split into words, that also appear in the theorem: orbits, periodic, degree, polynomial and field. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: restrictions, systems and bound.
Theorem 5.7 of Uniform bounds for planar polynomial limit cycles. Words of the Lean statement, its names split into words, that also appear in the theorem: solutions, functions, polynomial, limit and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isolated, systems and bound.
Theorem 10.1 of Uniform bounds for planar polynomial limit cycles. Words of the Lean statement, its names split into words, that also appear in the theorem: solutions, limiting and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: imposed, isolated, amplitude, sign and bounded.

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 143’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Only the quintic Lienard system x' = y - F(x), y' = -x (at most 2 limit cycles, exactly 2 attained); the general uniform boundedness theorem for planar polynomial vector fields of degree n is not stated.
Definitions the group asks readers to compare with the paper: IsLimitCycle F C := IsPeriodicOrbit F C ∧ ∃ U open ⊇ C, ∀ periodic orbits C' ⊆ U, C' = C (isolated nonconstant periodic orbit, standard); counts via Set.encard
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The headline's main clause — the Hilbert's-sixteenth uniform bound on limit cycles of every real planar polynomial vector field by a constant depending only on the degree — is not stated; only the quintic Liénard count is.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For the quintic Liénard system $x'=y-F(x)$, $y'=-x$, the formalized result proves that every real polynomial $F$ of degree at most five yields at most two limit cycles, and that some such $F$ yields exactly two. Limit cycles are isolated images of nonconstant periodic solutions. No sign, parity, hyperbolicity, or amplitude restriction is imposed.”
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

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: 16 to its own declarations, 16 to Mathlib’s documentation. 6 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 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 5 other declarations of the file; with the theorem itself they span 17 lines. 2 of them are named in the statement itself.

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

abbrev Planeline 7in the statementlibrary
def vectorFieldline 9library
def IsSolutionline 12library
def IsPeriodicOrbitline 15library
def IsLimitCycleline 20in the statementlibrary
theorem mainline 39selected theoremlibrary

import Mathlibnamespace OAInamespace QuinticLienardabbrev Plane := ℝ × ℝdef vectorField (F : Polynomial ℝ) (z : Plane) : Plane :=  (z.2 - F.eval z.1, -z.1)def IsSolution (F : Polynomial ℝ) (z : ℝ → Plane) : Prop :=  ∀ t : ℝ, HasDerivAt z (vectorField F (z t)) tdef IsPeriodicOrbit (F : Polynomial ℝ) (C : Set Plane) : Prop :=  ∃ (z : ℝ → Plane) (T : ℝ),    IsSolution F z ∧ 0 < T ∧ Function.Periodic z T ∧    (∃ s t : ℝ, z s ≠ z t) ∧ C = Set.range zdef IsLimitCycle (F : Polynomial ℝ) (C : Set Plane) : Prop :=  IsPeriodicOrbit F C ∧    ∃ U : Set Plane, IsOpen U ∧ C ⊆ U ∧      ∀ C' : Set Plane, IsPeriodicOrbit F C' → C' ⊆ U → C' = Copen scoped Topology NNReal ContDiff Manifoldopen Filter Setopen Set Filter Metric MeasureTheoryopen scoped Topology NNReal ContDiffopen scoped Topology ENNRealopen Set Filter MeasureTheoryopen Set Filter Asymptoticsopen Set Filter Metricopen scoped Topology NNRealopen scoped Topologyopen scoped Topology ContDiffopen Set Filteropen scoped Topology ContDiff NNRealtheorem main :    (∀ F : Polynomial ℝ, F.degree ≤ 5 → {C : Set Plane | IsLimitCycle F C}.encard ≤ 2) ∧    (∃ F : Polynomial ℝ, F.degree ≤ 5 ∧ {C : Set Plane | IsLimitCycle F C}.encard = 2) := by  sorryend QuinticLienardend 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.

Two limit cycles for quintic Liénard systems

Theorem

If \deg F\le5, the system (eq:original-system) has at most two limit cycles in \R^2. Some members of this class have two limit cycles. Thus its exact maximum is two.

Section 1, Introduction Read it in the paper TeX source, line 13

Show the TeX

If $\deg F\le5$, the system \eqref{eq:original-system} has at most
two limit cycles in $\R^2$. Some members of this class have two
limit cycles. Thus its exact maximum is two.

Configuration and file facts

File: lean/ComparatorChallenges/QuinticLienard.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/LienardCycles/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: 5 local vocabulary declarations
Lines of Lean, without comments or blank lines: 17
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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