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Gigli’s characterization of Alexandrov curvature

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves Gigli’s conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support \(\mathop{\mathrm{RCD}}\nolimits ((n-1)\kappa,n)\) condition with reference measure \(\mathcal H^n\) and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced. (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

What the papers state

Theorem 1.1 (Gigli’s characterization)

In Gigli’s distributional curvature characterization of Alexandrov spaces, 24 September 2026

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 n\ge2 be an integer, let \kappa\in\R, and let (M,d) be a complete separable metric space. The following conditions are equivalent.

  1. (A)

    (M,d) is an n-dimensional Alexandrov space with curvature bounded below by \kappa.

  2. (B)

    With m=\mathcal H^n_d, the space (M,d,m) has full support, is \RCD((n-1)\kappa,n), and satisfies

    R(X,Y,Y,X)(f)\ge \kappa\int_M f\bigl(|X|^2|Y|^2-\langle X,Y\rangle^2\bigr)\dd m

    for every X,Y\in\TestV(M) and every nonnegative f\in\Test(M).

The \RCD condition is unreduced, and the test classes and curvature sign are exactly those defined above.

No Lean statement in the release is paired with this theorem.

Theorem 1.1 (Weak Hessian bounds along every geodesic)

In Weak Hessian bounds along every geodesic in RCD spaces, 24 September 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

Let (M,d,m) be a full-support \RCD(K,N) space, where K\in\mathbb R and 1<N<\infty. Let F\colon M\to\mathbb R be bounded and globally Lipschitz, and let G\colon M\to\mathbb R be bounded and continuous. Suppose that for every compactly supported g\in\Test(M) and every nonnegative h\in\Lip_c(M),

H_F(\nabla g,\nabla g)(h) \le\int_M hG\Gamma(g)\dd m.

Then every constant-speed minimizing geodesic \sigma\colon[0,1]\to M, of length \ell, satisfies

(F\circ\sigma)''\le\ell^2G\circ\sigma \qquad\text{in distributions on }(0,1).

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.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Gigli’s distributional curvature characterization of Alexandrov spaces

24 September 2026, current edition.

Read it herePDF

For every integer n ≥ 2 and κ ∈ ℝ, we prove that a complete separable metric space is an n-dimensional Alexandrov space of curvature at least κ if and only if, with reference measure \(\mathcal H^n\), it is a full-support \(\mathrm{RCD}((n-1)\kappa,n)\) space whose distributional sectional curvature is at least κ in Gigli’s original global test classes. This resolves Gigli’s characterization conjecture in dimensions at least two.

Weak Hessian bounds along every geodesic in RCD spaces

24 September 2026, current edition.

Read it herePDF

On a full-support \(\mathrm{RCD}(K,N)\) space with \(1\lt N\lt \infty\), we prove that a bounded globally Lipschitz function whose distributional Hessian is bounded above by a bounded continuous function satisfies the corresponding second-derivative inequality along every minimizing geodesic.

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.

WeakHessian

The paper states

Theorem 1.1 (Weak Hessian bounds along every geodesic)

In Weak Hessian bounds along every geodesic in RCD spaces, 24 September 2026

Let (M,d,m) be a full-support \RCD(K,N) space, where K\in\mathbb R and 1<N<\infty. Let F\colon M\to\mathbb R be bounded and globally Lipschitz, and let G\colon M\to\mathbb R be bounded and continuous. Suppose that for every compactly supported g\in\Test(M) and every nonnegative h\in\Lip_c(M),

H_F(\nabla g,\nabla g)(h) \le\int_M hG\Gamma(g)\dd m.

Then every constant-speed minimizing geodesic \sigma\colon[0,1]\to M, of length \ell, satisfies

(F\circ\sigma)''\le\ell^2G\circ\sigma \qquad\text{in distributions on }(0,1).

The words they share: minimizing, sigma, weak, geodesic, gamma, real, speed, constant, continuous, globally, along, hessian, rcd, lipschitz, full, space, bounds, supported, test, circ.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Weak Hessian bounds along every geodesic

WeakHessian.lean, the release lists it as a main result.

theorem every_geodesic [CompleteSpace X] [TopologicalSpace.SeparableSpace X]    (m : Measure X) (K N : ℝ) (hN : 1 < N)    (hfull : FullSupport m) (hfinite : FiniteOnBoundedSets m) (hrcd : RCD m K N)    (F G : X → ℝ) (hFbound : BoundedFunction F) (hFlip : GloballyLipschitz F)    (hGbound : BoundedFunction G) (hGcont : Continuous G)    (hHess : WeakHessianUpperBound m F G)    (σ : Curve X) (hσ : IsGeodesic σ) : CurveSecondDerivativeBound F G σ := by  sorry

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; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

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 every_geodesic: in a complete separable RCD(K,N) space (N > 1, full-support reference measure finite on bounded sets), a bounded Lipschitz F whose weak Hessian is ≤ G·metric (G bounded continuous) satisfies the distributional bound (F∘σ)'' ≤ G(σ)·d(σ(0),σ(1))² along every constant-speed geodesic. Gigli’s conjecture (Alexandrov curvature ≥ κ ⇔ RCD((n-1)κ,n) with H^n and distributional sectional curvature ≥ κ) is not stated.

Definitions the group asks readers to compare with the paper: Entire RCD/Sobolev calculus is hand-built: CurvatureDimension (unreduced CD(K,N) via optimal dynamical plans, tauCoeff/sigmaCoeff), QuadraticCheegerEnergy, IsTestPlan, IsLocalWeakUpperGradient, weakGradient/laplacian defined by Classical.choose with fallback 0, weakHessian; the cone is 65,542 OAI lines here (recomputed; tmp/import_cones.json says 64,010)

“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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a different result (their term: “DIFFERENT”).

The gap they name: The challenge states a supporting result (on a full-support RCD(K,N) space with 1 < N < ∞, a bounded Lipschitz F with weak Hessian ≤ G·g has (F∘γ)'' ≤ G(γ)·d(γ₀,γ₁)² distributionally along every constant-speed geodesic). It is an ingredient of the characterization, but neither implies it nor is implied by it. Gigli’s characterization (the headline) is not stated.

“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.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization transfers a weak Hessian upper bound to every prescribed minimizing geodesic in an $\mathrm{RCD}(K,N)$ space with finite $N>1$. If a bounded globally Lipschitz function $F$ has weak Hessian bounded above by $G$ times the metric, where $G$ is bounded and continuous, then every constant-speed geodesic $\gamma:[0,1]\to X$ satisfies $(F\circ\gamma)''\le G(\gamma)\,d(\gamma(0),\gamma(1))^2$ in the distributional sense. Constant geodesics are included. The space is complete and separable with full support and measure finite on bounded sets; compactness and metric nonbranching are not assumed.

AI-drafted comparisons, with no recorded human review

  • Weak Hessian bounds along every geodesic WeakHessian

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the paper’s main theorem: in RCD(K, N) spaces a weak Hessian upper bound controls the second distributional derivative along every constant-speed geodesic. Gigli’s characterisation of the family’s other paper is not part of it.

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.

Weak Hessian bounds along every geodesic WeakHessian

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal 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.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

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.

Within this family

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

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What people have said

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.

    Author not stated. Source

Questions awaiting review

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Read the Lean statement against the paper

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Identify the theorem the catalogue describes

  • Family 356 The 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

Compare the proposed paper and Lean matches

  • Family 356 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form