A three-manifold without conjugate points or nonpositive curvature
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second. (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.
There exists a closed connected orientable smooth three-manifold M with a C^\infty Riemannian metric without conjugate points, such that M admits no C^\infty Riemannian metric whose sectional curvature is everywhere at most zero.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We construct a closed connected orientable smooth three-manifold that admits a smooth Riemannian metric without conjugate points but admits no smooth Riemannian metric of nonpositive sectional curvature.
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.
There exists a closed connected orientable smooth three-manifold M with a C^\infty Riemannian metric without conjugate points, such that M admits no C^\infty Riemannian metric whose sectional curvature is everywhere at most zero.
The words they share: conjugate, orientable, points, connected, riemannian, sectional, curvature, metric, smooth, manifold, without, everywhere, three, admits, closed.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
ConjugatePoints.lean, the release lists it as a main result.
defMainStatement : Prop := ∃ (M : Type) (top : TopologicalSpace M),letI := top ∃ (_ : T2Space M) (_ : SecondCountableTopology M) (_ : CompactSpace M) (_ : ConnectedSpace M) (charts : ChartedSpace E3 M),letI := charts ∃ (smooth : IsManifold I3 ∞ M),letI := smooth OrientedAtlas (M := M) ∧ (∃ g : SmoothMetric M, NoConjugatePoints g) ∧ ¬ ∃ g : SmoothMetric M, NonpositiveSectionalCurvature g⋮theoremmain_theorem : MainStatement := 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
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:MainStatement: there is a compact connected Hausdorff second-countable C^∞ 3-manifold with oriented atlas carrying a smooth metric with NoConjugatePoints (for every geodesic and Jacobi field on any open interval, J vanishing at two distinct parameters vanishes identically, constant geodesics included) while no smooth metric satisfies NonpositiveSectionalCurvature.
Definitions the group asks readers to compare with the paper:Hand-written chart-wise Christoffel symbols, curvature curvatureTerm, geodesics and Jacobi fields (IsGeodesicOn, IsJacobiFieldOn); sign conventions checked: ⟨R(u,v)v,u⟩ ≤ 0 is sectional curvature ≤ 0 and the Jacobi equation J'' + R(J,γ')γ' = 0 is the standard one
“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 formalized result separates absence of conjugate points from nonpositive sectional curvature. It constructs a closed connected orientable smooth three-manifold with a smooth Riemannian metric having no conjugate points, while the same manifold admits no smooth metric of everywhere nonpositive sectional curvature. The no-conjugate-points assertion ranges over all geodesics and tangent-bundle Jacobi fields, including constant geodesics and unbounded time intervals. The separate no-focal-points and CAT(0) consequences are not included.
AI-drafted comparisons, with no recorded human review
No conjugate points without a nonpositively curved metricConjugatePoints
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a closed orientable three-manifold with a metric without conjugate points that admits no metric of nonpositive curvature. The no-focal-points and CAT(0) consequences are not included.
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.
No conjugate points without a nonpositively curved metricConjugatePoints
Three 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 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.
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 358The 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 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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 288 s; peak memory 6.6 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.
No conjugate points without a nonpositively curved metric
Role
the release lists it as a main result
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.