Negative Kähler curvature without bounded holomorphic coordinates
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Constructs a contractible domain in ℂ3 with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below. (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 only part of the result described above, or a narrower statement.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.
There exist a contractible domain M\subset\C^3, a smooth complete Kähler metric g on M, and finite constants 0<A\le B such that
-B\le K_g(\sigma)\le -A<0
for every real two-plane \sigma\subset T_pM and every p\in M. No bounded holomorphic map F:M\to\C^3 has a nowhere-vanishing Jacobian determinant. In particular, M is not biholomorphic to a bounded domain in \C^3.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
For some finite integer m\ge2 there is a domain \mathcal T\subset\mathbb C^m, diffeomorphic to \mathbb R^{2m}, with a complete Kähler metric g such that
\operatorname{Sec}_g\le-1,
\qquad H^\infty(\mathcal T)=\mathbb C.
The sectional curvatures of g are unbounded below.
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 construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ3 with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.
We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm diffeomorphic to \(\mathbb R^{2m}\) that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.
One Lean statement in this family
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There exist a contractible domain M\subset\C^3, a smooth complete Kähler metric g on M, and finite constants 0<A\le B such that
-B\le K_g(\sigma)\le -A<0
for every real two-plane \sigma\subset T_pM and every p\in M. No bounded holomorphic map F:M\to\C^3 has a nowhere-vanishing Jacobian determinant. In particular, M is not biholomorphic to a bounded domain in \C^3.
The words they share: biholomorphic, bounded, complete, constants, contractible, domain, holomorphic, metric, real.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
PinchedKahler.lean, the release lists it as a main result.
defMainTheorem : Prop := ∃ (M : Set Ambient) (g : Ambient → CMatrix) (A B : ℝ), IsOpen M ∧ M.Nonempty ∧ ContractibleSpace M ∧ IsKahlerMetric M g ∧ GeodesicallyComplete M g ∧ IsNegativelyPinched M g A B ∧ NoBoundedCoordinates M ∧ ¬ BiholomorphicToBoundedDomain M⋮theoremmain_theorem : MainTheorem := 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 part of the advertised result (their term: “partial”).
Their note:main_theorem gives a nonempty open contractible M ⊆ ℂ³ with a smooth Kähler metric that is geodesically complete (all geodesics extend for all time) and negatively pinched (-B ≤ sectional ≤ -A < 0), with no bounded holomorphic map with everywhere-nonvanishing Jacobian and no biholomorphism onto a bounded domain; controlled_potential and metric_transfer support it. The summary’s second claim (a higher-dimensional example with sectional curvature ≤ -1, only constant bounded holomorphic functions, curvature not bounded below) is not stated.
Definitions the group asks readers to compare with the paper:Kähler geometry is hand-coded on ℂ²×ℂ: dz/dbar from real derivatives, curvature g (-∂̄_b∂_a g_{c d̄} + g^{-1}∂g ∂̄g), sectional := (R(u,ū,v,v̄) - R(u,v̄,u...))/(2·realArea), IsKahlerMetric (symmetry, positivity, ∂_i g_{kj̄} = ∂_k g_{ij̄}), GeodesicallyComplete via smooth solutions of the geodesic ODE; NoBoundedCoordinates only excludes bounded holomorphic F with nonvanishing Jacobian (stronger than excluding injective coordinate charts)
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 formalized result constructs a nonempty contractible complex threefold with a complete Kähler metric whose real sectional curvatures lie between two fixed negative constants. Nevertheless, it has no system of bounded holomorphic coordinates and is not biholomorphic to a bounded domain. The metric is obtained from an explicitly controlled potential. This is the negatively pinched construction; the different companion with only one-sided curvature control is separate.
AI-drafted comparisons, with no recorded human review
Negatively pinched Kähler threefold without bounded coordinatesPinchedKahler
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a contractible complex threefold with a complete Kähler metric of negatively pinched sectional curvature that has no bounded holomorphic coordinates.
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.
Negatively pinched Kähler threefold without bounded coordinatesPinchedKahler
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
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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
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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 counterexample and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
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Identify the theorem the catalogue describes
Family 359The 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
Only a special case is stated, or a co-equal headline claim has no challenge statement.
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 less than the headline, and OpenAI’s Lean note says so.
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.