Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and \(2\le n\le d-2\), an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L2-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds. (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
In Riesz transforms and uniform rectifiability in higher codimension, 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.
In Riesz transforms and uniform rectifiability in higher codimension, 24 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
If D_E>0, there are a common \mu-null set N\subset E and families \mathcal D_k of Borel cells in E\setminus N such that:
each \mathcal D_k partitions E\setminus N, and cells of different generations are disjoint or nested;
writing \ell(Q)=2^{-k} for Q\in\mathcal D_k, there are centers z_Q\in Q and constants 0<c_0\le1/2, C_0\ge2 for which
every non-top cell has one parent, and every cell has finitely many children. When D_E=\infty, all integer generations are available. When D_E<\infty, there is a largest scale \ell_0 with D_E\le\ell_0<2D_E, and finitely many cells at that scale.
All constants depend only on d,n,C_{\rm AD}.
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RieszQuantitativeThe paper states
In Riesz transforms and uniform rectifiability in higher codimension, 24 September 2026
The words they share: rectifiable, bound, ad, regular, riesz, measure, integers, constants, higher, codimension, uniform, ahlfors, david.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
RieszQuantitative.lean, the release lists it as a main result.
def QuantitativeFullStatement : Prop := ∀ d n : ℕ, 4 ≤ d → 2 ≤ n → n + 2 ≤ d → ∀ C_AD : ℝ, 1 ≤ C_AD → ∀ C_R : ℝ≥0, ∃ θ : ℝ, 0 < θ ∧ ∃ M : ℝ≥0, ∀ μ : Measure (Ambient d), μ.Regular → ADRegularWithConstant n C_AD μ → RieszL2BoundedWithConstant n C_R μ → BallImageConclusion n μ θ Mtheorem quantitative_higher_codimension_riesz_rectifiability : QuantitativeFullStatement := by sorryThis 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.
One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program
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.
AdmissibleRadius Model.lean, line 12ADRegularWithConstant Model.lean, line 22RieszL2BoundedWithConstant Model.lean, line 28BallImageConclusion Model.lean, line 34QuantitativeFullStatement Model.lean, line 42MathVet 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: None for the headline; uniform n-rectifiability is encoded as big pieces of Lipschitz images (standard equivalent form). The variation / principal-value corollary is not stated.
Definitions the group asks readers to compare with the paper: ADRegularWithConstant, RieszL2BoundedWithConstant (real-valued f, vector-valued hard truncation truncated), BallImageConclusion are custom but standard in form; AdmissibleRadius uses Metric.ediam μ.support
“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.
cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean covers
The formalized result proves that bounded Riesz transforms force quantitative uniform rectifiability in higher codimension. For $d\ge4$ and $2\le n\le d-2$, an $n$-Ahlfors–David regular measure whose positive hard truncations have one uniform $L^2$ operator bound has big pieces of Lipschitz images. The mass fraction and Lipschitz constant depend only on the dimensions and the stated regularity and operator bounds, and work at every support point and admissible radius, including unbounded support. The variation and principal-value corollary is not included.
Quantitative higher-codimension Riesz rectifiability RieszQuantitative
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem in quantitative form: an n-Ahlfors–David regular measure in codimension at least two with bounded Riesz transforms has big pieces of Lipschitz images, with constants depending only on the stated data. The variation and principal-value corollary is not included.
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.
RieszQuantitativeOne 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. | 10 October 2026 | Original report |
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
RieszQuantitative: Tracked source lean/OAI/Analysis/RieszRectifiability/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeebThese 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.
This family’s papers cite no paper elsewhere in the release.
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An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Author not stated. Source
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Why this passage is shown
Theorem 1.3
In Riesz transforms and uniform rectifiability in higher codimension, 24 September 2026
Why this passage is shown
Proposition 1.8 (Regular dyadic cells)
In Riesz transforms and uniform rectifiability in higher codimension, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, RieszQuantitative
Lean statement
RieszQuantitative