The $L\log L$ Fourier-convergence conjecture

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Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L(\mathbb T)$ converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the $L\log L$ scale. (OpenAI’s Lean note)

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Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1 (Almost-everywhere convergence in L\log L)

In Almost-everywhere Fourier convergence in L log L, 23 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.

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.

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.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

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Almost-everywhere Fourier convergence in L log L

23 September 2026, current edition.

Read it here PDF

We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L$ on the circle converge almost everywhere to the function along the full sequence. This establishes the classical $L\log L$ sufficiency conjecture.

One Lean statement in this family

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`FourierLLogL`

The paper states

Theorem 1.1 (Almost-everywhere convergence in L\log L)

In Almost-everywhere Fourier convergence in L log L, 23 September 2026

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 words they share: almost, along, complex, convergence, everywhere, full, measurable, sequence, valued.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Almost-everywhere convergence of full symmetric Fourier partial sums

FourierLLogL.lean, the release lists it as a main result.

theorem paperMain (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)) := by  sorry

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.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

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: 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; 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

“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.

Read the original assessment Read MathVet’s method

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.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The $L\log L$ Fourier-convergence conjecture asks whether ordinary symmetric Fourier partial sums converge almost everywhere at this integrability scale. 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 $N\to\infty$.

AI-drafted comparisons, with no recorded human review

Almost-everywhere convergence of full symmetric Fourier partial sums FourierLLogL
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: symmetric Fourier partial sums of every function in L log L of the circle converge almost everywhere.

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Almost-everywhere convergence of full symmetric Fourier partial sums `FourierLLogL`

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.	10 October 2026	Original report
MathVet	“comparator-sandboxed-pass”	Re-run through Comparator in a sandboxed virtual machine: accepted.	10 October 2026	Original report

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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23 references in Almost-everywhere Fourier convergence in L log L

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What people have said

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.
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