Almost-everywhere convergence of full symmetric Fourier partial sums
Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.FourierLLogL.paperMain
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theorempaperMain (f : AddCircle (2 * Real.pi) → ℂ) (hfm : AEStronglyMeasurable f AddCircle.haarAddCircle) (hfi : Integrable (fun x ↦ ‖f x‖ * Real.log (2 + ‖f x‖)) AddCircle.haarAddCircle) : ∃ E : Set (AddCircle (2 * Real.pi)), MeasurableSet E ∧ AddCircle.haarAddCircle E = 0 ∧ ∀ x ∉ E, Tendsto (fun N : ℕ ↦ ∑ k ∈ Finset.Icc (-(N : ℤ)) (N : ℤ), fourierCoeff f k * fourier k x) atTop (𝓝 (f x)) := bysorry
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Possible paper match, with no recorded review
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: almost, along, complex, convergence, everywhere, full, measurable, sequence and one more.
Theorem 1.1 of Almost-everywhere Fourier convergence in L log L
Theorem(Almost-everywhere convergence in L\log L)
For every complex-valued f\in L\log L(\mathbb T), there is a measurable set N_f of measure zero such that
For every complex-valued $f\in L\log L(\mathbb T)$, there is a measurable
set $N_f$ of measure zero such that
\[
\lim_{N\to\infty}S_Nf(x)=f(x)
\qquad (x\in\mathbb T\setminus N_f).
\]
The limit is along the full sequence $N=0,1,2,\ldots$.
Other possible paper matches, closest wording first
Proposition 10.2 of Almost-everywhere Fourier convergence in L log L. Words of the Lean statement, its names split into words, that also appear in the theorem: fourier, symmetric, partial, convergence, complex, almost and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: function, sequence, full, along and valued.
Corollary 10.3 of Almost-everywhere Fourier convergence in L log L. Words of the Lean statement, its names split into words, that also appear in the theorem: real, pi, symmetric, partial, complex, integer and one more.
Proposition 6.1 of Almost-everywhere Fourier convergence in L log L. Words of the Lean statement, its names split into words, that also appear in the theorem: haar, measurable, top, pi, integers, almost and two more.
Theorem 10.1 of Almost-everywhere Fourier convergence in L log L. Words of the Lean statement, its names split into words, that also appear in the theorem: almost and everywhere. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: function.
Proposition 6.2 of Almost-everywhere Fourier convergence in L log L. Words of the Lean statement, its names split into words, that also appear in the theorem: top and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: full, one and along.
Proposition 7.2 of Almost-everywhere Fourier convergence in L log L. Words of the Lean statement, its names split into words, that also appear in the theorem: integers and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one and values.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
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 (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 075’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: The Lean theorem OAI.FourierLLogL.paperMain is the paper's Theorem 1.1 verbatim: for f on AddCircle (2*pi) with AEStronglyMeasurable and Integrable (||f|| * log(2+||f||)) w.r.t. the normalised Haar measure, there is a measurable null set E with S_N f x -> f x for x outside E along N : Nat. Only unpacking is needed: haarAddCircle is dx/(2pi), fourierCoeff/fourier are fhat(k) and e^{ikx}, and the Integrable hypothesis is the paper's definition of L log L (f is automatically in L^1, so fourierCoeff is the genuine integral).
Definitions the group asks readers to compare with the paper: AddCircle (2 * Real.pi): standard Mathlib quotient R/(2 pi Z); the paper's T ; AddCircle.haarAddCircle: standard Mathlib Haar measure normalised to total mass 1 (addHaarMeasure top, IsProbabilityMeasure), equal to dx/(2 pi) for period 2 pi; matches the paper's mu; standard ; fourierCoeff: standard Mathlib n-th coefficient = integral of fourier (-n) t * f t d(haarAddCircle) = integral of f(x) e^{-inx} dx/(2 pi); Bochner integral so junk value 0 for non-integrable f, but the hypotheses force f in L^1 (||f|| log 2 <= ||f|| log(2+||f||)); standard ; fourier k x: standard Mathlib exponential monomial toCircle(k . x) = exp(2 pi i k x / T) = e^{ikx} for T = 2 pi; standard ; Finset.Icc (-(N:Z)) (N:Z) with Filter.atTop on N : Nat: the symmetric partial sum over |k| <= N along the full sequence; standard ; Integrable (fun x => ||f x|| * Real.log (2 + ||f x||)) haarAddCircle: exactly the paper's L log L(T) class (introduction.tex:11-15); standard ; circlePeriod_pos: local instance Fact (0 < 2*Real.pi) (proved by mul_pos and Real.pi_pos); needed for haarAddCircle/fourier, adds no hypothesis
OpenAI’s own note on what this Lean statement covers
“The formalization proves this for every measurable complex-valued function on $\mathbb T=\mathbb R/(2\pi\mathbb Z)$ satisfying $\int_{\mathbb T}|f|\log(2+|f|)<\infty$. Outside one measurable null set,
$\sum_{k=-N}^{N}\widehat f(k)e^{ikx}\to f(x)$ along the full sequence $N\to\infty$.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
24 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 0 to its own declarations, 21 to Mathlib’s documentation. 2 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
For every complex-valued $f\in L\log L(\mathbb T)$, there is a measurable
set $N_f$ of measure zero such that
\[
\lim_{N\to\infty}S_Nf(x)=f(x)
\qquad (x\in\mathbb T\setminus N_f).
\]
The limit is along the full sequence $N=0,1,2,\ldots$.
The limit is along the full sequence N=0,1,2,\ldots.
Almost-everywhere Fourier convergence in L log L
TeX label thm:main, build/sections/01-introduction.tex, line 17
The Lean statement says
theorempaperMain (f : AddCircle (2 * Real.pi) → ℂ) (hfm : AEStronglyMeasurable f AddCircle.haarAddCircle) (hfi : Integrable (fun x ↦ ‖f x‖ * Real.log (2 + ‖f x‖)) AddCircle.haarAddCircle) : ∃ E : Set (AddCircle (2 * Real.pi)), MeasurableSet E ∧ AddCircle.haarAddCircle E = 0 ∧ ∀ x ∉ E, Tendsto (fun N : ℕ ↦ ∑ k ∈ Finset.Icc (-(N : ℤ)) (N : ℤ), fourierCoeff f k * fourier k x) atTop (𝓝 (f x)) := bysorry
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.