Lean statement

Removable-boundary rigidity and the circle-domain theorem

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.

Theorems selected for verification: OAI.Problem047.koebe_circle_domain OAI.Problem047.removability_implies_rigidity

Lean statement file: KoebeCircleDomains.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 071 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 koebe_circle_domain :    ∀ U : Set Sphere, IsOpen U → IsConnected U →      ∃ (V : Set Sphere) (f : Sphere → Sphere),        IsCircleDomain V ∧ IsConformalEquivalence f U V := by  sorry⋮theorem removability_implies_rigidity :    ∀ (U V : Set Sphere) (f : Sphere → Sphere),      IsCircleDomain U → IsCircleDomain V →      IsConformallyRemovable (frontier U) → IsConformalEquivalence f U V →      ∃ M : Mobius, ∀ p ∈ U, f p = M • p := 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.CircleDomainRigidity.Sphere line 13
```
abbrev Sphere := OnePoint ℂ
```

abbreviation OAI.CircleDomainRigidity.Mobius line 14

abbrev Mobius := Matrix.GeneralLinearGroup (Fin 2) ℂ

definition OAI.CircleDomainRigidity.sphereInv line 16

def sphereInv (p : Sphere) : Sphere :=  p.elim (0 : ℂ) (fun z ↦ if z = 0 then ∞ else (z⁻¹ : ℂ))

definition OAI.CircleDomainRigidity.finiteCoord line 19

def finiteCoord (p : Sphere) : ℂ := p.elim 0 id

definition OAI.CircleDomainRigidity.chart line 21

def chart (p q : Sphere) : ℂ :=  if p = ∞ then finiteCoord (sphereInv q) else finiteCoord q

definition OAI.CircleDomainRigidity.chartSymm line 24

def chartSymm (p : Sphere) (z : ℂ) : Sphere :=  if p = ∞ then sphereInv (z : Sphere) else (z : Sphere)

definition OAI.CircleDomainRigidity.IsConformalAt line 27

def IsConformalAt (f : Sphere → Sphere) (p : Sphere) : Prop :=  ContinuousAt f p ∧ ∃ d : ℂ, d ≠ 0 ∧    HasDerivAt (fun z ↦ chart (f p) (f (chartSymm p z))) d (chart p p)

definition OAI.CircleDomainRigidity.IsConformalOn line 31

def IsConformalOn (f : Sphere → Sphere) (U : Set Sphere) : Prop :=  ∀ p ∈ U, IsConformalAt f p

definition OAI.CircleDomainRigidity.IsConformalEquivalence line 34

def IsConformalEquivalence (f : Sphere → Sphere) (U V : Set Sphere) : Prop :=  IsConformalOn f U ∧ MapsTo f U V ∧ ∃ g : Sphere → Sphere,    IsConformalOn g V ∧ MapsTo g V U ∧    (∀ z ∈ U, g (f z) = z) ∧ (∀ w ∈ V, f (g w) = w)

definition OAI.CircleDomainRigidity.IsRoundClosedDisk line 39

def IsRoundClosedDisk (K : Set Sphere) : Prop :=  ∃ M : Mobius,    K = (fun p : Sphere ↦ M • p) ''      (((↑) : ℂ → Sphere) '' Metric.closedBall (0 : ℂ) 1)

definition OAI.CircleDomainRigidity.IsCircleDomain line 44

def IsCircleDomain (U : Set Sphere) : Prop :=  IsOpen U ∧ IsConnected U ∧    ∀ p ∈ Uᶜ, (∃ q : Sphere, connectedComponentIn Uᶜ p = {q}) ∨      IsRoundClosedDisk (connectedComponentIn Uᶜ p)

definition OAI.CircleDomainRigidity.OrientationPreserving line 49

def OrientationPreserving (h : Sphere ≃ₜ Sphere) : Prop :=  ContinuousMap.Homotopic (⟨h, h.continuous⟩ : C(Sphere, Sphere)) (ContinuousMap.id Sphere)

definition OAI.CircleDomainRigidity.IsConformallyRemovable line 52

def IsConformallyRemovable (E : Set Sphere) : Prop :=  IsCompact E ∧ ∀ h : Sphere ≃ₜ Sphere, OrientationPreserving h →    IsConformalOn h Eᶜ → ∃ M : Mobius, ∀ p : Sphere, h p = M • p

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: equivalence, removable, conformally, circle, domain and conformal. Found in the paper’s title, which its main theorem states: rigidity. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: boundary.

Statements in Removable Boundaries and Rigidity of Circle Domains come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Removable Boundaries and Rigidity of Circle Domains

Theorem

Every circle domain with conformally removable boundary is conformally rigid. More explicitly, if \Omega,\Omega'\subset\Chat are circle domains, \partial\Omega is conformally removable, and f\colon\Omega\to\Omega' is a conformal equivalence, then there is a Möbius transformation M such that f=M|_\Omega.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

Every circle domain with conformally removable boundary is conformally
rigid.  More explicitly, if $\Omega,\Omega'\subset\Chat$ are circle
domains, $\partial\Omega$ is conformally removable, and
$f\colon\Omega\to\Omega'$ is a conformal equivalence, then there is a
M\"obius transformation $M$ such that $f=M|_\Omega$.

Other possible paper matches, closest wording first

Proposition 6.2 of Removable Boundaries and Rigidity of Circle Domains. Words of the Lean statement, its names split into words, that also appear in the theorem: equivalence, removable, conformally, points, components, circle and two more.
Theorem 1.1 of Koebe’s Circle-Domain Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: connected, conformally, domain and circle. Found in the paper’s title, which its main theorem states: koebes.
Corollary 8.1 of Koebe’s Circle-Domain Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: closed, connected, orientable, round, metric, points and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: boundary.
Theorem 5.1 of Removable Boundaries and Rigidity of Circle Domains. Words of the Lean statement, its names split into words, that also appear in the theorem: continuous, removable, conformally and compact.
Corollary 7.6 of Koebe’s Circle-Domain Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: maps, round, sphere, metric, point, disks and six more.
Proposition 3.2 of Removable Boundaries and Rigidity of Circle Domains. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, circle and domain.

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; 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

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 783 s; peak memory 7.4 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 071’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: None; both the existence part of Koebe's conjecture and removability => rigidity are stated for arbitrary domains. Conformality is encoded chartwise on OnePoint C (custom).
Definitions the group asks readers to compare with the paper: Custom conformality on the sphere: IsConformalAt f p := ContinuousAt f p ∧ ∃ d ≠ 0, HasDerivAt (fun z => chart (f p) (f (chartSymm p z))) d (chart p p) (hand-rolled charts at ∞ via sphereInv; equals holomorphic-with-nonzero-derivative; fine)
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 same Comparator file also states Koebe's circle-domain theorem, which gives a circle-domain representative for every domain in the Riemann sphere.”
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

79 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: 62 to its own declarations, 27 to Mathlib’s documentation. 15 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 koebe_circle_domain 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 13 other declarations of the file; with the theorem itself they span 42 lines. 5 of them are named in the statement itself.

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

abbrev Sphereline 13in the statementlibrary
abbrev Mobiusline 14in the statementlibrary
def sphereInvline 16library
def finiteCoordline 19library
def chartline 21library
def chartSymmline 24library
def IsConformalAtline 27library
def IsConformalOnline 31library
def IsConformalEquivalenceline 34in the statementlibrary
def IsRoundClosedDiskline 39library
def IsCircleDomainline 44in the statementlibrary
def OrientationPreservingline 49library
def IsConformallyRemovableline 52in the statementlibrary
theorem OAI.Problem047.koebe_circle_domainline 62selected theoremlibrary
theorem OAI.Problem047.removability_implies_rigidityline 68selected theoremlibrary

import Mathlibnamespace OAI/-! Circle-domain uniformization and rigidity under conformal removability. -/noncomputable sectionopen Set Classicalopen scoped OnePointnamespace CircleDomainRigidityabbrev Sphere := OnePoint ℂabbrev Mobius := Matrix.GeneralLinearGroup (Fin 2) ℂdef sphereInv (p : Sphere) : Sphere :=  p.elim (0 : ℂ) (fun z ↦ if z = 0 then ∞ else (z⁻¹ : ℂ))def finiteCoord (p : Sphere) : ℂ := p.elim 0 iddef chart (p q : Sphere) : ℂ :=  if p = ∞ then finiteCoord (sphereInv q) else finiteCoord qdef chartSymm (p : Sphere) (z : ℂ) : Sphere :=  if p = ∞ then sphereInv (z : Sphere) else (z : Sphere)def IsConformalAt (f : Sphere → Sphere) (p : Sphere) : Prop :=  ContinuousAt f p ∧ ∃ d : ℂ, d ≠ 0 ∧    HasDerivAt (fun z ↦ chart (f p) (f (chartSymm p z))) d (chart p p)def IsConformalOn (f : Sphere → Sphere) (U : Set Sphere) : Prop :=  ∀ p ∈ U, IsConformalAt f pdef IsConformalEquivalence (f : Sphere → Sphere) (U V : Set Sphere) : Prop :=  IsConformalOn f U ∧ MapsTo f U V ∧ ∃ g : Sphere → Sphere,    IsConformalOn g V ∧ MapsTo g V U ∧    (∀ z ∈ U, g (f z) = z) ∧ (∀ w ∈ V, f (g w) = w)def IsRoundClosedDisk (K : Set Sphere) : Prop :=  ∃ M : Mobius,    K = (fun p : Sphere ↦ M • p) ''      (((↑) : ℂ → Sphere) '' Metric.closedBall (0 : ℂ) 1)def IsCircleDomain (U : Set Sphere) : Prop :=  IsOpen U ∧ IsConnected U ∧    ∀ p ∈ Uᶜ, (∃ q : Sphere, connectedComponentIn Uᶜ p = {q}) ∨      IsRoundClosedDisk (connectedComponentIn Uᶜ p)def OrientationPreserving (h : Sphere ≃ₜ Sphere) : Prop :=  ContinuousMap.Homotopic (⟨h, h.continuous⟩ : C(Sphere, Sphere)) (ContinuousMap.id Sphere)def IsConformallyRemovable (E : Set Sphere) : Prop :=  IsCompact E ∧ ∀ h : Sphere ≃ₜ Sphere, OrientationPreserving h →    IsConformalOn h Eᶜ → ∃ M : Mobius, ∀ p : Sphere, h p = M • pend CircleDomainRigidityopen CircleDomainRigiditynamespace Problem047theorem koebe_circle_domain :    ∀ U : Set Sphere, IsOpen U → IsConnected U →      ∃ (V : Set Sphere) (f : Sphere → Sphere),        IsCircleDomain V ∧ IsConformalEquivalence f U V := by  sorrytheorem removability_implies_rigidity :    ∀ (U V : Set Sphere) (f : Sphere → Sphere),      IsCircleDomain U → IsCircleDomain V →      IsConformallyRemovable (frontier U) → IsConformalEquivalence f U V →      ∃ M : Mobius, ∀ p ∈ U, f p = M • p := by  sorryend Problem047endend 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.

Removable Boundaries and Rigidity of Circle Domains

Theorem

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

Every circle domain with conformally removable boundary is conformally
rigid.  More explicitly, if $\Omega,\Omega'\subset\Chat$ are circle
domains, $\partial\Omega$ is conformally removable, and
$f\colon\Omega\to\Omega'$ is a conformal equivalence, then there is a
M\"obius transformation $M$ such that $f=M|_\Omega$.

Koebe’s Circle-Domain Conjecture

Theorem (Circle-domain uniformization)

Every nonempty connected open set G\subset\Chat is conformally equivalent to a circle domain.

Section 1, Introduction Read it in the paper TeX source, line 9

Show the TeX

Every nonempty connected open set $G\subset\Chat$ is conformally equivalent
to a circle domain.

Configuration and file facts

File: lean/ComparatorChallenges/KoebeCircleDomains.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 071 link it.
Solution module: OAI.Analysis.CircleDomains.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: 13 local vocabulary declarations
Lines of Lean, without comments or blank lines: 42
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.

Plectis Atlas