Lean statement

Unbounded violations of the square-root Fourier-degree bound

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.SquareRootDegree.main

Lean statement file: SquareRootDegree.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 192 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by  sorry

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.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.SquareRootDegree.Cube line 16
```
abbrev Cube (ι : Type*) := ι → Bool
```

definition OAI.SquareRootDegree.sign line 18

def sign (b : Bool) : ℝ := if b then -1 else 1

definition OAI.SquareRootDegree.average line 20

def average {α : Type*} [Fintype α] (f : α → ℝ) : ℝ :=  (Fintype.card α : ℝ)⁻¹ * ∑ x, f x

definition OAI.SquareRootDegree.character line 23

def character {ι : Type*} (s : Finset ι) (x : Cube ι) : ℝ :=  ∏ i ∈ s, sign (x i)

definition OAI.SquareRootDegree.fourierCoeff line 26

def fourierCoeff {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) (s : Finset ι) : ℝ := by  classical  exact average (fun x => f x * character s x)

definition OAI.SquareRootDegree.fourierDegree line 30

def fourierDegree {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℕ := by  classical  exact (Finset.univ.filter (fun s : Finset ι => fourierCoeff f s ≠ 0)).sup Finset.card

definition OAI.SquareRootDegree.IsBoolean line 34

def IsBoolean {α : Type*} (f : α → ℝ) : Prop :=  ∀ x, f x = -1 ∨ f x = 1

definition OAI.SquareRootDegree.Nonconstant line 37

def Nonconstant {α : Type*} (f : α → ℝ) : Prop :=  ∃ x y, f x ≠ f y

definition OAI.SquareRootDegree.singletonSum line 40

def singletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=  ∑ i, fourierCoeff f {i}

definition OAI.SquareRootDegree.absoluteSingletonSum line 43

def absoluteSingletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=  ∑ i, |fourierCoeff f {i}|

definition OAI.SquareRootDegree.SignedViolations line 46

def SignedViolations : Prop :=  ∀ C : ℝ, 0 < C → ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,    IsBoolean f ∧ Nonconstant f ∧ C * Real.sqrt (fourierDegree f) < singletonSum f

definition OAI.SquareRootDegree.AbsoluteRatios line 50

def AbsoluteRatios : Set ℝ :=  {r | ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,    IsBoolean f ∧ 0 < fourierDegree f ∧      r = absoluteSingletonSum f / Real.sqrt (fourierDegree f)}

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, nonconstant, positive, real, boolean and integer. Found in the paper’s title, which its main theorem states: violations and degree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: deg and function.

Theorem 1.1 of Unbounded Violations of the Square-Root Degree Bound

Theorem

For every real C>0, there are a positive integer n and a nonconstant Boolean function f:\{-1,1\}^n\to\{-1,1\} such that

\sum_{i=1}^n\widehat f(\{i\})>C\sqrt{\deg(f)}.

Consequently,

\begin{gathered}\sup_{\substack{n\ge1,\ f:\{-1,1\}^n\to\{-1,1\}\\\deg(f)>0}} \frac{\sum_i|\widehat f(\{i\})|}{\sqrt{\deg(f)}}=\infty.\end{gathered}

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

For every real \(C>0\), there are a positive integer \(n\) and a
nonconstant Boolean function \(f:\{-1,1\}^n\to\{-1,1\}\) such that
\[
 \sum_{i=1}^n\widehat f(\{i\})>C\sqrt{\deg(f)}.
\]
Consequently,
\[
 \sup_{\substack{n\ge1,\ f:\{-1,1\}^n\to\{-1,1\}\\\deg(f)>0}}
 \frac{\sum_i|\widehat f(\{i\})|}{\sqrt{\deg(f)}}=\infty.
\]

Other possible paper matches, closest wording first

Theorem 5.2 of Unbounded Violations of the Square-Root Degree Bound. Words of the Lean statement, its names split into words, that also appear in the theorem: averaging, sign, boolean, finite, ratios and degree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: linear, bound and functions.
Proposition 2.3 of Unbounded Violations of the Square-Root Degree Bound. Words of the Lean statement, its names split into words, that also appear in the theorem: value, real, integers, finite, ratios and degree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: linear, bounded and functions.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 423 s; peak memory 6.4 GiB).
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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 192’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.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios with SignedViolations := ∀ C > 0, ∃ n > 0, ∃ f : Cube (Fin n) → ℝ, IsBoolean f ∧ Nonconstant f ∧ C * √(fourierDegree f) < singletonSum f; Fourier coefficients are the uniform-average E[f * χ_s] over the ±1 cube and fourierDegree is the max |s| with nonzero coefficient (real multilinear degree, as the file comment says). Standard definitions, matches the summary exactly.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization disproves the Gopalan–Servedio square-root degree conjecture by an unbounded factor. For every $C>0$, it gives a nonconstant Boolean function on a finite sign cube for which the sum of its linear Fourier coefficients exceeds $C\sqrt{\deg f}$, where $\deg f$ is its real multilinear degree.”
Quoted from OpenAI’s Lean note for this family

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

64 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: 23 to its own declarations, 19 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves main exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 12 other declarations of the file; with the theorem itself they span 29 lines. 2 of them are named in the statement itself.

Names in the namespace OAI.SquareRootDegree are listed without it.

abbrev Cubeline 16library
def signline 18library
def averageline 20library
def characterline 23library
def fourierCoeffline 26library
def fourierDegreeline 30library
def IsBooleanline 34library
def Nonconstantline 37library
def singletonSumline 40library
def absoluteSingletonSumline 43library
def SignedViolationsline 46in the statementlibrary
def AbsoluteRatiosline 50in the statementlibrary
theorem mainline 57selected theoremlibrary

import Mathlibnamespace OAI/-!Bool encodes a sign: false is +1 and true is -1. The average is uniform,and degree is ordinary real Fourier degree, not F₂ degree or threshold degree.-/open scoped BigOperatorsnoncomputable sectionnamespace SquareRootDegreeabbrev Cube (ι : Type*) := ι → Booldef sign (b : Bool) : ℝ := if b then -1 else 1def average {α : Type*} [Fintype α] (f : α → ℝ) : ℝ :=  (Fintype.card α : ℝ)⁻¹ * ∑ x, f xdef character {ι : Type*} (s : Finset ι) (x : Cube ι) : ℝ :=  ∏ i ∈ s, sign (x i)def fourierCoeff {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) (s : Finset ι) : ℝ := by  classical  exact average (fun x => f x * character s x)def fourierDegree {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℕ := by  classical  exact (Finset.univ.filter (fun s : Finset ι => fourierCoeff f s ≠ 0)).sup Finset.carddef IsBoolean {α : Type*} (f : α → ℝ) : Prop :=  ∀ x, f x = -1 ∨ f x = 1def Nonconstant {α : Type*} (f : α → ℝ) : Prop :=  ∃ x y, f x ≠ f ydef singletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=  ∑ i, fourierCoeff f {i}def absoluteSingletonSum {ι : Type*} [Fintype ι] (f : Cube ι → ℝ) : ℝ :=  ∑ i, |fourierCoeff f {i}|def SignedViolations : Prop :=  ∀ C : ℝ, 0 < C → ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,    IsBoolean f ∧ Nonconstant f ∧ C * Real.sqrt (fourierDegree f) < singletonSum fdef AbsoluteRatios : Set ℝ :=  {r | ∃ n : ℕ, 0 < n ∧ ∃ f : Cube (Fin n) → ℝ,    IsBoolean f ∧ 0 < fourierDegree f ∧      r = absoluteSingletonSum f / Real.sqrt (fourierDegree f)}/-- The two clauses of the literal main theorem. Unboundedness above is thereal-valued formulation of the displayed supremum being positive infinity. -/theorem main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios := by  sorryend SquareRootDegreeendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Unbounded Violations of the Square-Root Degree Bound

Theorem

For every real C>0, there are a positive integer n and a nonconstant Boolean function f:\{-1,1\}^n\to\{-1,1\} such that

\sum_{i=1}^n\widehat f(\{i\})>C\sqrt{\deg(f)}.

Consequently,

\begin{gathered}\sup_{\substack{n\ge1,\ f:\{-1,1\}^n\to\{-1,1\}\\\deg(f)>0}} \frac{\sum_i|\widehat f(\{i\})|}{\sqrt{\deg(f)}}=\infty.\end{gathered}

Section 1, Introduction Read it in the paper TeX source, line 23

Show the TeX

For every real \(C>0\), there are a positive integer \(n\) and a
nonconstant Boolean function \(f:\{-1,1\}^n\to\{-1,1\}\) such that
\[
 \sum_{i=1}^n\widehat f(\{i\})>C\sqrt{\deg(f)}.
\]
Consequently,
\[
 \sup_{\substack{n\ge1,\ f:\{-1,1\}^n\to\{-1,1\}\\\deg(f)>0}}
 \frac{\sum_i|\widehat f(\{i\})|}{\sqrt{\deg(f)}}=\infty.
\]

Configuration and file facts

File: lean/ComparatorChallenges/SquareRootDegree.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 192 link it.
Solution module: OAI.Combinatorics.BooleanFunctions.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 29
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas