Semialgebraic universal covers and bounded domains
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products \(D\times\mathbb C^m\times F\), with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂn has a finite étale cover by an abelian variety. (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.
The independent assessments find that the Lean statements express a supporting step or a different result.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.
Let X be a connected normal projective variety over \C, and let \widetilde X be its universal cover. The following conditions are equivalent:
(i)
\widetilde X is biholomorphic to a semialgebraic open subset of a projective variety.
(ii)
There are an integer m\geq0, a bounded symmetric domain D, and a simply connected normal projective variety F such that
\widetilde X\simeq D\times\C^m\times F.
A point is allowed as either D or F. No algebraicity of the deck transformations is assumed.
No Lean statement in the release is paired with this theorem.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let U be a nonempty connected semialgebraic bounded open subset of a complex affine variety. Suppose that a group \Gamma of biholomorphisms acts properly discontinuously on U and that U/\Gamma is compact. Then U is smooth and is biholomorphic to a bounded symmetric domain.
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 the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.
Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.
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. The outside readers find that the release’s Lean states a supporting or different result, so the theorem it is paired with is shown under its own heading.
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.
Let U be a nonempty connected semialgebraic bounded open subset of a complex affine variety. Suppose that a group \Gamma of biholomorphisms acts properly discontinuously on U and that U/\Gamma is compact. Then U is smooth and is biholomorphic to a bounded symmetric domain.
The words they share: group, affine, nonempty, symmetric, semialgebraic, smooth, domain, compact, bounded, complex, connected, quotient, properly, acts, biholomorphic.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SymmetricDomains.lean, the release lists it as a main result.
theoremmain {n : ℕ} (V U : Set (Affine n)) (hV : IsAffineAlgebraic V) (hUV : U ⊆ V) (hopen : IsOpen ((Subtype.val : V → Affine n) ⁻¹' U)) (hconn : IsConnected U) (hsa : IsSemialgebraic U) (hbounded : Bornology.IsBounded U) (Γ : Type*) [Group Γ] [TopologicalSpace Γ] [DiscreteTopology Γ] [MulAction Γ U] [ProperSMul Γ U] [CompactSpace (Quotient (MulAction.orbitRel Γ U))] (hhol : ∀ γ : Γ, HolomorphicOnSubset U (fun p => (γ • p : U).val)) : IsSmooth U ∧ ∃ (m : ℕ) (D : Set (Affine m)), IsBoundedSymmetricDomain D ∧ Nonempty (Biholomorph U D) := 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.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 a supporting result (their term: “supporting-only”).
Their note:Only the bounded-domain symmetry lemma (companion paper); the Kollar-Pardon classification of semialgebraic universal covers D x C^m x F, the quasi-projective criterion and the ‘covered by C^n => finite etale abelian cover’ corollary are not stated.
Definitions the group asks readers to compare with the paper:Custom IsSemialgebraic (inductive closure of p = 0, 0 < p, complement, union), HolomorphicOnSubset, Biholomorph, IsSmooth, IsBoundedSymmetricDomain (open, connected, bounded, every point an isolated fixed point of a holomorphic involution); plausible, standard-looking
“supporting-only”, in their words: The challenge statements state a lemma or auxiliary result, not the headline.
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.
cert-machine finds that the Lean statements express a different result (their term: “DIFFERENT”).
The gap they name:All three headline clauses (the classification, the quasi-projectivity criterion, the C^n corollary). The single challenge states the companion paper’s theorem instead: a connected bounded semialgebraic domain, relatively open in an affine algebraic set and carrying a proper cocompact holomorphic discrete action, is smooth and biholomorphic to a bounded symmetric domain. It neither implies the classification nor follows from it as stated. U need not be simply connected, the action need not be free, and the ambient set may be singular, so U is not presented as the universal cover of a normal projective variety.
“DIFFERENT”, in their words: The challenge statements state a different theorem from the headline.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalized result answers the paper’s symmetry question affirmatively. A nonempty connected bounded semialgebraic subset, relatively open in a complex affine algebraic set, is smooth and biholomorphic to a bounded symmetric domain whenever a discrete group acts properly and holomorphically with compact quotient.
The action need not be free, and the ambient algebraic set may be singular.
AI-drafted comparisons, with no recorded human review
Symmetry of semialgebraic bounded domainsSymmetricDomains
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a connected bounded semialgebraic domain with a properly discontinuous cocompact group of biholomorphisms is biholomorphic to a bounded symmetric domain.
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.
Symmetry of semialgebraic bounded domainsSymmetricDomains
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.
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 “Solid publication, could lead to tenure”. 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 058The 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 state a lemma or auxiliary result, not the headline.
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 a different theorem from the headline.
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 in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
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.