Endpoint Sobolev regularity of centered disk averages
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real \(f\in W^{1,1}(\mathbb R^2)\), the centered disk maximal function satisfies \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) with an absolute constant. It belongs locally to \(W^{1,1}\) and has a globally integrable weak gradient. (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.
There is an absolute constant C<\infty such that, for every real-valued f\in W^{1,1}(\R^2), the function Mf is locally integrable, belongs to W^{1,1}_{\mathrm{loc}}(\R^2), and satisfies
\int_{\R^2}|\nabla Mf|\,\dd x
\le C\int_{\R^2}|\nabla f|\,\dd x.
In particular, its distributional derivatives are represented by globally integrable functions.
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
There is an absolute constant C such that, for every real g\in C_c^\infty(\R^2) and 0<a\le b with b/a a power of two, including one,
\int_{\R^2}|\nabla S_{a,b}g|\,\dd x
\le C\int_{\R^2}|\nabla g|\,\dd x.
The constant is independent of a,b, and the number of dyadic radius scales between them.
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.
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 prove the endpoint gradient bound \(\|\nabla Mf\|_1\le C\|\nabla f\|_1\) for every real-valued \(f\in W^{1,1}(\mathbb R^2)\), where \(Mf(x)\) is the supremum of the averages of \(|f|\) over disks centered at x, and C is an absolute constant. The maximal function belongs to \(W^{1,1}_{\mathrm{loc}}(\mathbb R^2)\) and has a globally integrable weak gradient. This gives a positive resolution of the planar centered-disk case of the endpoint question of Hajłasz and Onninen.
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.
There is an absolute constant C<\infty such that, for every real-valued f\in W^{1,1}(\R^2), the function Mf is locally integrable, belongs to W^{1,1}_{\mathrm{loc}}(\R^2), and satisfies
\int_{\R^2}|\nabla Mf|\,\dd x
\le C\int_{\R^2}|\nabla f|\,\dd x.
In particular, its distributional derivatives are represented by globally integrable functions.
The words they share: represented, valued, locally, integrable, functions, distributional, derivatives, real, gradient, disk, endpoint.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SignedFiniteBand.lean, the release lists it as a main result.
defSignedFiniteBandStatement : Prop := ∃ C : ℝ, 0 ≤ C ∧ ∀ (f : Plane → ℝ), ContDiff ℝ ∞ f → HasCompactSupport f → ∀ (a : ℝ), 0 < a → ∀ m : ℕ, LocallyIntegrable (signedBand f a ((2 : ℝ)^m * a)) ∧ ∃ G : Plane → Plane, Integrable G ∧ HasWeakGradient (signedBand f a ((2 : ℝ)^m * a)) G ∧ gradientNorm G ≤ C * gradientNorm (gradient f)⋮theoremmain_signed_finite_band : SignedFiniteBandStatement := 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
There is an absolute constant C<\infty such that, for every real-valued f\in W^{1,1}(\R^2), the function Mf is locally integrable, belongs to W^{1,1}_{\mathrm{loc}}(\R^2), and satisfies
\int_{\R^2}|\nabla Mf|\,\dd x
\le C\int_{\R^2}|\nabla f|\,\dd x.
In particular, its distributional derivatives are represented by globally integrable functions.
The words they share: represented, valued, locally, integrable, functions, distributional, loc, derivatives, real, absolute, gradient, disk, maximal, endpoint, centered, bound, belongs, mf, globally, nabla, constant.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
DiskMaximal.lean, the release lists it as a supporting statement.
defEndpointStatement : Prop := ∃ C : ℝ, 0 ≤ C ∧ ∀ (f : Plane → ℝ) (g : Plane → Plane), Integrable f → Integrable g → HasWeakGradient f g → (∀ᵐ x, maximal f x ≠ ⊤) ∧ LocallyIntegrable (maximalReal f) ∧ InW11Loc (maximalReal f) ∧ ∃ G : Plane → Plane, Integrable G ∧ HasWeakGradient (maximalReal f) G ∧ gradientNorm G ≤ C * gradientNorm g⋮theoremmain_endpoint : EndpointStatement := 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
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?
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:None; the BV extension is not claimed (docs). Weak gradients are encoded by hand (HasWeakGradient against C_c^∞ test functions), which is the standard definition.
“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 centered disk maximal operator takes the supremum of averages of $|f|$ over disks centered at each point. The formalization proves that every real $f\in W^{1,1}(\mathbb R^2)$ has a maximal function that is finite almost everywhere, belongs to $W^{1,1}_{\mathrm{loc}}$, and has a globally integrable weak gradient satisfying $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ for one absolute constant $C$.
The earlier signed finite-band estimate for smooth compactly supported functions is also retained. The general endpoint statement covers the Sobolev setting of the paper; a BV extension is outside these statements.
AI-drafted comparisons, with no recorded human review
Signed finite-band disk-maximal gradient boundSignedFiniteBand
Unreviewed AI draft: states a supporting result. States the paper’s signed finite-band proposition for smooth compactly supported functions, an earlier estimate the note keeps beside the endpoint statement; it is a step towards the main theorem rather than the theorem itself.
Endpoint gradient bound for the centered disk maximal operatorDiskMaximal
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for real f in W^{1,1}(R²) the centred disk maximal function is locally W^{1,1} with ‖∇Mf‖₁ ≤ C‖∇f‖₁. A BV extension is outside the statement.
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.
Signed finite-band disk-maximal gradient boundSignedFiniteBand
Three 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
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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.
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.
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 085The 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
Compare the proposed paper and Lean matches
Family 085Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In An Endpoint Gradient Bound for the Centered Disk Maximal Operator, 26 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 612 s; peak memory 6.5 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
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 with Lean’s kernel: accepted (the check took 277 s; peak memory 6.5 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Endpoint gradient bound for the centered disk maximal operator
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.