OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT\((0)\) spaces. (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
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
Let n\ge2, let \kappa\le0, and let (M^n,g) be a complete simply connected smooth Riemannian manifold with \Sec_g\le\kappa. Every measurable set E\subset M with \vol_g(E)<\infty and P_g(E)<\infty satisfies
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let (M^n,g) be complete, simply connected and smooth, with n\ge2 and \Sec_g\le0. Let E\subset M be bounded and measurable, with 0<\vol_g(E)<\infty and P_g(E)<\infty. Then
P_g(E)=n\omega_n^{1/n}\vol_g(E)^{(n-1)/n}
if and only if E agrees up to ambient volume zero with an open region whose induced Riemannian metric is isometric to a round Euclidean ball.
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 paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
Three more possible matches
Corollary 1.3(Model-ball Faber–Krahn and Saint–Venant comparisons)in Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity, 23 September 2026.
Corollary 1.2in Sharp integral fillings in CAT(0) spaces, 23 September 2026.
Theorem 1.1(Sharp integral filling)in Sharp integral fillings in CAT(0) spaces, 23 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.
Every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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 X be a proper CAT(0) space and n\ge2 an integer. Every compactly supported T\in\I_n(X) with \partial T=0 admits a compactly supported S\in\I_{n+1}(X) such that
FillingCoefficient.lean, the release lists it as a supporting statement.
theoremcoefficient_optimal (n : ℕ) (hn : 2 ≤ n) : ∃ T : IntegralCurrent (Euc (n + 1)) n, CompactlySupported T.val ∧ IsCycle T.val ∧ 0 < mass T.val ∧ ∀ S : IntegralCurrent (Euc (n + 1)) (n + 1), CompactlySupported S.val → boundarySucc S.val = T.val → fillingCoefficient n * (mass T.val) ^ fillingPower n ≤ mass S.val := 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.
Let X be a proper CAT(0) space and n\ge2 an integer. Every compactly supported T\in\I_n(X) with \partial T=0 admits a compactly supported S\in\I_{n+1}(X) such that
SharpCAT0Filling.lean, the release lists it as a supporting statement.
theoremsharp_integral_filling (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] (hX : IsCAT0 X) (n : ℕ) (hn : 2 ≤ n) (T : IntegralCurrent X n) (hTc : CompactlySupported T.val) (hTz : IsCycle T.val) : ∃ S : IntegralCurrent X (n + 1), CompactlySupported S.val ∧ boundarySucc S.val = T.val ∧ mass S.val ≤ fillingCoefficient n * (mass T.val) ^ fillingPower n := 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.
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 part of the advertised result (their term: “partial”).
Their note:sharp_integral_filling (every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space has a compactly supported integral filling of mass ≤ C_n·mass^((n+1)/n)) and coefficient_optimal (Euclidean sharpness) cover the summary’s second claim only. The headline, the sharp Cartan-Hadamard isoperimetric inequality for finite-perimeter sets in complete simply connected manifolds with curvature ≤ κ ≤ 0 (and the Euclidean-ball equality case for κ = 0), has no statement; the docs concede only ‘the selected integral-current form’ is proved, and the Lean form does not imply the set-based inequality.
Definitions the group asks readers to compare with the paper:Hand-built metric currents: Functional X k := (X → ℝ) → (Fin k → X → ℝ) → ℝ, IsMetricCurrent (off-domain zero, multilinearity, sequential continuity, locality, finite mass), mass T := sInf {μ.real univ | Controls T μ}, IntegerRectifiable via countably many bi-Lipschitz IntegerCharts with integer multiplicity, boundarySucc T b π := if Admissible b π then T 1 (b, π) else 0; a Lipschitz-functional model of Ambrosio-Kirchheim integral currents, not in Mathlib; IsCAT0 X := ∃ segment, geodesic-parametrization ∧ CN-comparison inequality (hand-built)
“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:No Riemannian statement: neither the κ ≤ 0 isoperimetric comparison for finite-perimeter sets nor the κ = 0 equality rigidity is stated. (The κ = 0 inequality in dimension ≥ 3 would follow from the filling theorem via Hadamard manifolds being CAT(0) and the currents of finite-perimeter sets, but that bridge is not stated; κ < 0 and rigidity do not follow.)
“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 proves the sharp Euclidean filling inequality for every compactly supported integral $n$-cycle in a proper $\mathrm{CAT}(0)$ space, for $n\ge2$. It constructs a compactly supported integral filling whose mass is at most $C_n$ times the boundary mass to the power $(n+1)/n$, where $C_n$ is the Euclidean isoperimetric coefficient. Integer multiplicities and the ambient dimension are unrestricted.
A second statement proves that this coefficient is optimal by giving, in Euclidean space, a cycle for which every compactly supported integral filling has at least that mass. This yields the selected integral-current form of the Cartan–Hadamard isoperimetric conjecture.
AI-drafted comparisons, with no recorded human review
Optimality of the Euclidean filling coefficientFillingCoefficient
Unreviewed AI draft: states part of a paper’s theorem. States the optimality clause of the main theorem: a Euclidean cycle every integral filling of which has at least the stated mass. The filling inequality itself is SharpCAT0Filling.
Sharp integral fillings in proper $\mathrm{CAT}(0)$ spacesSharpCAT0Filling
Unreviewed AI draft: states part of a paper’s theorem. States the sharp filling inequality of the main theorem for integral cycles in every proper CAT(0) space. The theorem’s final clause, optimality of the coefficient, is the separate statement FillingCoefficient.
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.
Optimality of the Euclidean filling coefficientFillingCoefficient
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.
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 proof and rates its significance, if it holds up, as “Career-defining publication”. 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
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 337The 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…
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 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.