Lean statement

Average sensitivity of polynomial threshold functions

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.LeanBlast.GotsmanLinial.gotsmanLinialStatement

Lean statement file: GotsmanLinial.leanShow on the mapFamily 127. OpenAI lists this as a main result in its Lean catalogue.

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.

def GotsmanLinialStatement : Prop :=  ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d → d ≤ n →    ∀ p : MvPolynomial (Fin n) ℝ,      IsMultilinear p → p.totalDegree ≤ d →        averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ)⋮theorem gotsmanLinialStatement : GotsmanLinialStatement := 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.LeanBlast.GotsmanLinial.Cube line 13
```
abbrev Cube (n : ℕ) := Fin n → Bool
```

definition OAI.LeanBlast.GotsmanLinial.cubeCoord line 15

def cubeCoord {n : ℕ} (x : Cube n) (i : Fin n) : ℝ :=  if x i then 1 else -1

definition OAI.LeanBlast.GotsmanLinial.flip line 18

def flip {n : ℕ} (i : Fin n) (x : Cube n) : Cube n :=  Function.update x i (!(x i))

definition OAI.LeanBlast.GotsmanLinial.thresholdSign line 21

noncomputable def thresholdSign (t : ℝ) : ℝ :=  if 0 ≤ t then 1 else -1

definition OAI.LeanBlast.GotsmanLinial.sensitiveVertices line 24

noncomputable def sensitiveVertices {n : ℕ} (f : Cube n → ℝ) (i : Fin n) :    Finset (Cube n) := by  classical  exact Finset.univ.filter fun x => f x ≠ f (flip i x)

definition OAI.LeanBlast.GotsmanLinial.averageSensitivity line 29

noncomputable def averageSensitivity {n : ℕ} (f : Cube n → ℝ) : ℝ :=  ∑ i : Fin n, ((sensitiveVertices f i).card : ℝ) / (2 : ℝ) ^ n

definition OAI.LeanBlast.GotsmanLinial.IsMultilinear line 32

def IsMultilinear {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : Prop :=  ∀ m ∈ p.support, ∀ i : Fin n, m i ≤ 1

definition OAI.LeanBlast.GotsmanLinial.polynomialValue line 35

noncomputable def polynomialValue {n : ℕ} (p : MvPolynomial (Fin n) ℝ)    (x : Cube n) : ℝ :=  MvPolynomial.eval (cubeCoord x) p

definition OAI.LeanBlast.GotsmanLinial.polynomialThreshold line 39

noncomputable def polynomialThreshold {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :    Cube n → ℝ :=  fun x => thresholdSign (polynomialValue p x)

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: multilinear, average, sensitivity, real, sqrt, degree, polynomial and integers. Found in the paper’s title, which its main theorem states: threshold and functions.

Theorem 1 of Average sensitivity of polynomial threshold functions

Theorem

Let n\ge1 and 1\le d\le n be integers. Let p be a real multilinear polynomial of degree at most d, and define f:\{-1,1\}^n\to\{-1,1\} by f(x)=\sgn(p(x)), with \sgn(0)=1. Then, for average sensitivity under the uniform law,

I(f)\le 8d\sqrt n.

Section 1, Introduction Read it in the paper TeX source, line 20

Show the TeX

Let $n\ge1$ and $1\le d\le n$ be integers. Let $p$ be a real multilinear
polynomial of degree at most $d$, and define
$f:\{-1,1\}^n\to\{-1,1\}$ by $f(x)=\sgn(p(x))$, with $\sgn(0)=1$.
Then, for average sensitivity under the uniform law,
\[
 I(f)\le 8d\sqrt n.
\]

Other possible paper matches, closest wording first

Corollary 4 of Average sensitivity of polynomial threshold functions. Words of the Lean statement, its names split into words, that also appear in the theorem: sensitivity, real, sqrt, degree, polynomial and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: boolean.
Corollary 5 of Average sensitivity of polynomial threshold functions. Words of the Lean statement, its names split into words, that also appear in the theorem: functions, polynomial and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: boolean.

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 143 s; peak memory 3.3 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 127’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: None; the exact-extremizer form of the Gotsman-Linial conjecture is not claimed.
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 formalized result proves the average-sensitivity bound $8d\sqrt n$ for every Boolean threshold function defined by a real multilinear polynomial of degree at most $d$ on the $n$-dimensional cube. The sign convention assigns value $1$ at zero. This establishes the stated asymptotic bound, rather than the separate conjecture identifying exact extremizers.”
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

54 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: 17 to its own declarations, 20 to Mathlib’s documentation. 11 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 gotsmanLinialStatement 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 10 other declarations of the file; with the theorem itself they span 28 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.LeanBlast.GotsmanLinial are listed without it.

abbrev Cubeline 13library
def cubeCoordline 15library
def flipline 18library
def thresholdSignline 21library
def sensitiveVerticesline 24library
def averageSensitivityline 29library
def IsMultilinearline 32library
def polynomialValueline 35library
def polynomialThresholdline 39library
def GotsmanLinialStatementline 43in the statementlibrary
theorem gotsmanLinialStatementline 49selected theoremlibrary

import Mathlib.Algebra.MvPolynomial.Degreesimport Mathlib.Algebra.BigOperators.Fieldimport Mathlib.Data.Fintype.BigOperatorsimport Mathlib.Analysis.Real.Sqrtimport Mathlib.Tactic.NormNumnamespace OAIopen scoped BigOperatorsnamespace LeanBlast.GotsmanLinialabbrev Cube (n : ℕ) := Fin n → Booldef cubeCoord {n : ℕ} (x : Cube n) (i : Fin n) : ℝ :=  if x i then 1 else -1def flip {n : ℕ} (i : Fin n) (x : Cube n) : Cube n :=  Function.update x i (!(x i))noncomputable def thresholdSign (t : ℝ) : ℝ :=  if 0 ≤ t then 1 else -1noncomputable def sensitiveVertices {n : ℕ} (f : Cube n → ℝ) (i : Fin n) :    Finset (Cube n) := by  classical  exact Finset.univ.filter fun x => f x ≠ f (flip i x)noncomputable def averageSensitivity {n : ℕ} (f : Cube n → ℝ) : ℝ :=  ∑ i : Fin n, ((sensitiveVertices f i).card : ℝ) / (2 : ℝ) ^ ndef IsMultilinear {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : Prop :=  ∀ m ∈ p.support, ∀ i : Fin n, m i ≤ 1noncomputable def polynomialValue {n : ℕ} (p : MvPolynomial (Fin n) ℝ)    (x : Cube n) : ℝ :=  MvPolynomial.eval (cubeCoord x) pnoncomputable def polynomialThreshold {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :    Cube n → ℝ :=  fun x => thresholdSign (polynomialValue p x)def GotsmanLinialStatement : Prop :=  ∀ (n d : ℕ), 1 ≤ n → 1 ≤ d → d ≤ n →    ∀ p : MvPolynomial (Fin n) ℝ,      IsMultilinear p → p.totalDegree ≤ d →        averageSensitivity (polynomialThreshold p) ≤ 8 * (d : ℝ) * Real.sqrt (n : ℝ)theorem gotsmanLinialStatement : GotsmanLinialStatement := by  sorryend LeanBlast.GotsmanLinialend 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.