OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy \(\|T_\mu-T_\nu\|_{L^2(\rho)}\le C W_2(\mu,\nu)^{1/3}\) uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit’s conjectured square-root bound. (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.
For every K,Y as above there is a finite constant C(K,Y) such that
The targets may be atomic, singular, or absolutely continuous. The exponent 1/3 cannot be replaced by any larger exponent in general, even for K=Y=[-1,1]^d and targets with exactly three atoms.
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.
For the uniform probability measure ρ on a compact convex body in ℝd, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in \(L^2(\rho)\) with respect to the target’s 2-Wasserstein distance. The constant is uniform over all targets supported in a fixed compact set, and the exponent is optimal. A three-atom family on a fixed cube disproves Letrouit’s conjectured uniform square-root estimate.
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.
The targets may be atomic, singular, or absolutely continuous. The exponent 1/3 cannot be replaced by any larger exponent in general, even for K=Y=[-1,1]^d and targets with exactly three atoms.
The words they share: exactly, constant, atoms, three, exponent, 1/3, targets, prob, one, third, sharp, stability, maps.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
Brenier.lean, the release lists it as a supporting statement.
theoremone_third_stability : ∀ (dimension : ℕ) (source container : Set (E dimension)),2 ≤ dimension → IsCompact source → Convex ℝ source → (interior source).Nonempty → IsCompact container → container.Nonempty → ∃ constant : ℝ, 0 ≤ constant ∧ ∀ (first second : Measure (E dimension)), IsProbabilityMeasure first → IsProbabilityMeasure second → IsSupported first container → IsSupported second container → ∃ firstMap secondMap : E dimension → E dimension, IsUniqueQuadraticOptimalMap (uniformMeasure source) first firstMap ∧ IsUniqueQuadraticOptimalMap (uniformMeasure source) second secondMap ∧ mapL2Dist (uniformMeasure source) firstMap secondMap ≤ constant * Real.rpow (wasserstein2 first second) (1 / 3 : ℝ) := 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?
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:All 12 definition_names were read: uniformMeasure ((vol S)⁻¹·vol|_S), IsCoupling, quadraticPlanCost (∫⁻ ofReal ‖x-y‖²), wasserstein2 (√ toReal sInf of coupling costs), graphPlan, IsQuadraticOptimalMap, IsUniqueQuadraticOptimalMap (optimal and every coupling of cost ≤ it equals the graph plan), mapL2Dist, cube, HasExactlyThreeAtoms, IsSupported, E; all standard. one_third_stability gives, for d ≥ 2, any compact convex source with interior and any compact nonempty container, one constant with ‖T_μ - T_ν‖_{L²(uniform)} ≤ C·W₂(μ,ν)^(1/3) for all probability targets in the container (unique optimal maps exist), and one_third_exponent_is_sharp shows for every exponent > 1/3 and constant, three-atom targets in the cube violate the bound (so the conjectured square root fails).
Definitions the group asks readers to compare with the paper:wasserstein2 and mapL2Dist use ENNReal.toReal/Bochner integrals (value 0 if infinite or non-integrable), harmless here because targets lie in a compact container so all quantities are finite
“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 formalization proves one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior in $\mathbb R^d$, for every $d\ge2$. For targets supported in one fixed nonempty compact set, the $L^2$ distance between the unique quadratic optimal maps is at most a constant times the one-third power of the targets' $2$-Wasserstein distance. The constant is uniform over those targets.
It also proves optimality of the exponent: on a fixed cube, pairs of three-atom target measures violate every analogous bound with exponent greater than $1/3$. In particular, the conjectured uniform square-root estimate fails.
AI-drafted comparisons, with no recorded human review
One-third stability of Brenier maps and sharpnessBrenier
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (one-third Hölder stability of Brenier maps from a convex body) and sharpness of the exponent. Twelve definitions, including the Wasserstein distance and optimal maps, are Comparator holes filled from OpenAI’s library.
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.
One-third stability of Brenier maps and sharpnessBrenier
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.
Other people’s accounts of running proof-verification software, with links to their reports.
Jack McCarthy, Re-running Comparator with the definition holes pinned, 9 October 2026. Source
With the definition holes pinned so the solver’s definitions must match the challenge, Comparator reports the solution is okay for this challenge, so the hole was not being abused by OpenAI’s own solution.
Jack McCarthy, OpenAI math: Comparator fixes (mirror of openai/math at fd4aeeb), 9 October 2026. Source
With all 12 definition holes removed from the configuration, so that Comparator compares the definition bodies too, the release’s solution still passes; run with the Lean kernel only, on macOS without Comparator’s sandbox.
From github.com/BenKnill/openai-math-hol-cards, 8 October 2026. Source
Strict Comparator re-check with the definition holes removed, so hole bodies are compared too, accepted the solution with the three standard axioms. All 12 of its hole bodies matched the challenge. Run at adc7f124; the challenge file is unchanged at fd4aeeb2.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
Problems reported by others
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 8 October 2026. Source
Reported as broken in the same way: its definitions are listed as holes, so a trivial replacement makes the checked statement vacuous.
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 “A strong professor’s paper of the year”. The page says that nothing on it is verified.
Songbo Wang, Xiaozhen Wang, Quantitative stability of optimal transport via regularization, 7 October 2026. Source
Wang and Wang describe OpenAI’s Brenier-map result as proving the sharp one-third stability estimate for uniform sources on compact convex bodies, and prove complementary optimal exponents for other sources.
Bang-Xian Han, Zhuo-Nan Zhu, Stability of optimal transport on sub-Riemannian manifolds, 7 October 2026. Source
Han and Zhu note that OpenAI’s example with three target atoms and a fixed cube as source disproves Conjecture 3 of Letrouit’s 2026 paper and rules out W1 map exponents above 1/4.
Questions awaiting review
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
Read the definitions the solution supplies
OpenAI’s proof file supplies some definitions. Atlas records no comparison confirming that they mean the same objects as the paper’s definitions.E, uniformMeasure, IsSupported, IsCoupling, quadraticPlanCost, wasserstein2, graphPlan, IsQuadraticOptimalMap and 4 moreOpen the form
Identify the theorem the catalogue describes
Family 374The 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.