Nonnegative-curvature Einstein classification and an L2 topological gap
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Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S4, Fubini–Study \(\mathbb{CP}^2\), or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L2 norm of its trace-free Ricci curvature lies below a universal positive constant. (OpenAI’s Lean note)
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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.
Three other possible matches are listed below the card, among them Corollary 1.2 in Zero-Plane Rigidity for Einstein Four-Manifolds and Theorem 1.2 in An L² Einstein Gap for Nonnegatively Curved Four-Manifolds.
Let (M,g) be a connected smooth closed Einstein four-manifold with strictly positive sectional curvature, without an orientability assumption. There is a constant a>0 such that (M,ag) is isometric to the standard round S^4, to \mathbb{CP}^2 with its Fubini–Study metric, or to \mathbb{RP}^4 with its standard round metric.
Three more possible matches
Corollary 1.2in Zero-Plane Rigidity for Einstein Four-Manifolds, 4 October 2026.
Theorem 1.2in An L² Einstein Gap for Nonnegatively Curved Four-Manifolds, 5 October 2026.
Theorem 1.1(Zero-plane rigidity)in Zero-Plane Rigidity for Einstein Four-Manifolds, 4 October 2026.
Reported proof verification
One Lean statement in the release; one independent group reports that Comparator accepted a run for this file at the version shown here.
We prove that a closed Einstein four-manifold with positive Einstein constant, nonnegative sectional curvature, and a zero-curvature plane has universal Riemannian cover isometric to a product of two round two-spheres. Under the normalization \(\mathop{\mathrm{Ric}}\nolimits =3g\), both spheres have radius \(1/\sqrt3\). The proof extends coupled estimates for the two Weyl curvature blocks to the boundary of the sectional-curvature cone and determines their equality case.
We prove a universal, scale-invariant L2 gap for the trace-free Ricci tensor on simply connected closed four-manifolds with nonnegative sectional curvature. If the trace-free Ricci energy is sufficiently small, the manifold is diffeomorphic to S4, \(\mathbb{CP}^2\), or \(S^2\times S^2\). The proof uses the classification of positive-Einstein, nonnegatively curved four-manifolds supplied by the companion zero-plane rigidity theorem, stated explicitly as the classification premise of our result. No auxiliary curvature, volume, diameter, injectivity-radius, or Sobolev bound is required. The conclusion concerns the smooth manifold, not an isometry of the original metric.
We prove the classification conjecture for connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or real projective four-space with its round metric. No orientability assumption is needed.
One Lean statement in this family
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Let (M,g) be a connected smooth closed Einstein four-manifold with strictly positive sectional curvature, without an orientability assumption. There is a constant a>0 such that (M,ag) is isometric to the standard round S^4, to \mathbb{CP}^2 with its Fubini–Study metric, or to \mathbb{RP}^4 with its standard round metric.
The words they share: closed, connected, curvature, einstein, four, fubini, manifold, metric, orientability, positive, round, sectional, smooth, standard, strictly, study.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
EinsteinFour.lean, the release lists it as a supporting statement.
theoremorientedClassification (X : ClosedOrientedMetricFour.{u}) (hEinstein : IsEinstein X.metric) (hpositive : HasPositiveSectionalCurvature X.metric) : ∃ a : ℝ, 0 < a ∧ (ScaledIsometricTo X.metric a roundDistance ∨ ScaledIsometricTo X.metric a fubiniStudyDistance) := 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:The Lean classification assumes strictly positive sectional curvature (HasPositiveSectionalCurvature) and concludes scaled isometry to round S⁴, Fubini-Study ℂP² or round ℝP⁴ (oriented case: S⁴ or ℂP²). That is a special case of the summary’s nonnegative-curvature classification (which also has the S²×S² product); the L² trace-free-Ricci topological gap theorem is not stated.
Definitions the group asks readers to compare with the paper:Curvature is computed by hand in the centre chart of each point (christoffel, curvatureComponent, ricciInChart, sectionalNumerator, IsEinstein := ∃ lam, ricci = lam·g at chartAt c c), and model spaces are only given as distance functions (roundDistance := arccos ⟨p,q⟩, fubiniStudyDistance := arccos(‖⟨p,q⟩‖/(‖p‖‖q‖)), realProjectiveDistance) compared to Manifold.riemannianEDist scaled by √a (ScaledIsometricTo); two statement-level run_cmd Lean.modifyEnv... auxLemmasExt meta commands; external package RellichKondrachov in the solution cone
“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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:The nonnegative-curvature classification (zero-plane rigidity: universal cover S²×S² with equal radii) and the L² trace-free-Ricci gap theorem — the two October manuscripts, neither linked by the scope note.
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalization classifies connected smooth closed Einstein four-manifolds with strictly positive sectional curvature. Up to positive scaling and isometry, every such manifold is the round four-sphere, the complex projective plane with its Fubini–Study metric, or round real projective four-space. No orientability hypothesis is required. In the oriented case, only the sphere and complex projective plane occur.
AI-drafted comparisons, with no recorded human review
Classification of positively curved Einstein four-manifoldsEinsteinFour
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: closed Einstein four-manifolds of positive sectional curvature are, up to scaling and isometry, S⁴, CP² with the Fubini–Study metric or RP⁴.
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.
Classification of positively curved Einstein four-manifoldsEinsteinFour
One report, 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.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4’s sine minorant, upper-deficit integrals and moment-obstruction coefficient table; not the classification theorem”
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4’s sine minorant, upper-deficit integrals and moment-obstruction coefficient table and the conditional lower-volume rows; not the classification theorem of F-667”
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.
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Identify the theorem the catalogue describes
Family 348The 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; the missing part comes from a paper the Lean note does not link.
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…
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4’s sine minorant, upper-deficit integrals and moment-obstruction coefficient table; not the classification theorem”
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: every exact polynomial certificate the paper prints -- the appendix lemmas scalar (8 interval boxes), matrix-and-boundary (12 triangle rows) and potential F (20 interval boxes), with the polynomials re-derived from Section 6 and the coverings -- plus Section 4’s sine minorant, upper-deficit integrals and moment-obstruction coefficient table and the conditional lower-volume rows; not the classification theorem of F-667”