OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves the existence part of Koebe’s circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
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.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Every nonempty connected open set G\subset\Chat is conformally equivalent to a circle domain.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.
We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe’s circle-domain conjecture positively.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s 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.
The words they share: equivalence, removable, conformally, circle, domain, conformal, rigidity, boundary.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
KoebeCircleDomains.lean, the release lists it as a supporting statement.
theoremkoebe_circle_domain : ∀ U : Set Sphere, IsOpen U → IsConnected U → ∃ (V : Set Sphere) (f : Sphere → Sphere), IsCircleDomain V ∧ IsConformalEquivalence f U V := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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)
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
The formalization proves the removability-to-rigidity direction of the He–Schramm conjecture. If a circle domain has conformally removable boundary, then every conformal equivalence from it to another circle domain agrees on the source with a Möbius transformation. There is no bound on the number of complementary components.
The same Comparator file also states Koebe’s circle-domain theorem, which gives a circle-domain representative for every domain in the Riemann sphere.
Koebe’s circle-domain conjecture asks whether every domain in the Riemann sphere is conformally equivalent to a circle domain, whose complementary components are round disks or points. The formalization proves this for every domain. It also proves rigidity under a separate hypothesis: a conformal equivalence between circle domains is a Möbius transformation when the source boundary is conformally removable.
AI-drafted comparisons, with no recorded human review
Removable-boundary rigidity and the circle-domain theoremKoebeCircleDomains
Unreviewed AI draft: states a paper’s theorem as the paper does. The file states the main theorems of both accompanying papers: Koebe’s circle-domain theorem for every domain in the Riemann sphere, and conformal rigidity of circle domains with conformally removable boundary.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Removable-boundary rigidity and the circle-domain theoremKoebeCircleDomains
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 071The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator with Lean’s kernel: accepted (the check took 783 s; peak memory 7.4 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
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.