Lean statement

Courtade–Kumar inequality and attainment

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.CourtadeKumar.courtadeKumarAndAttainment

Lean statement file: CourtadeKumar.leanShow on the mapFamily 119. 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 FullStatement : Prop := CourtadeKumarStatement ∧ DictatorAttainmentStatement⋮theorem courtadeKumarAndAttainment : FullStatement := 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 9
```
abbrev Cube (n : ℕ) := Fin n → Bool
```

definition OAI.LeanBlast.CourtadeKumar.cubeAverage line 21

def cubeAverage {n : ℕ} (g : Cube n → ℝ) : ℝ :=  (∑ x, g x) / (2 : ℝ) ^ n

definition OAI.LeanBlast.CourtadeKumar.ell line 24
```
def ell : ℝ := Real.log 2
```

definition OAI.LeanBlast.CourtadeKumar.bitFlipKernel line 26

def bitFlipKernel {n : ℕ} (ε : ℝ) (x y : Cube n) : ℝ :=  ∏ i, if x i = y i then 1 - ε else ε

definition OAI.LeanBlast.CourtadeKumar.jointProbability line 29

def jointProbability {n : ℕ} (ε : ℝ) (f : Cube n → Bool) (b : Bool) (y : Cube n) : ℝ :=  cubeAverage (fun x => if f x = b then bitFlipKernel ε x y else 0)

definition OAI.LeanBlast.CourtadeKumar.functionMarginal line 32

def functionMarginal {n : ℕ} (ε : ℝ) (f : Cube n → Bool) (b : Bool) : ℝ :=  ∑ y, jointProbability ε f b y

definition OAI.LeanBlast.CourtadeKumar.observationMarginal line 35

def observationMarginal {n : ℕ} (ε : ℝ) (f : Cube n → Bool) (y : Cube n) : ℝ :=  ∑ b : Bool, jointProbability ε f b y

definition OAI.LeanBlast.CourtadeKumar.xlogx line 38
```
def xlogx (t : ℝ) : ℝ := t * Real.log t
```

definition OAI.LeanBlast.CourtadeKumar.binaryEntropy line 40

def binaryEntropy (ε : ℝ) : ℝ :=  -(xlogx ε + xlogx (1 - ε)) / ell

definition OAI.LeanBlast.CourtadeKumar.mutualInformation line 43

def mutualInformation {n : ℕ} (ε : ℝ) (f : Cube n → Bool) : ℝ :=  (∑ b : Bool, ∑ y : Cube n,    if 0 < jointProbability ε f b y then      jointProbability ε f b y *        Real.log (jointProbability ε f b y /          (functionMarginal ε f b * observationMarginal ε f y))    else 0) / ell

definition OAI.LeanBlast.CourtadeKumar.dictator line 51

def dictator {n : ℕ} (i : Fin n) (x : Cube n) : Bool := x i

definition OAI.LeanBlast.CourtadeKumar.complementDictator line 53

def complementDictator {n : ℕ} (i : Fin n) (x : Cube n) : Bool := !(x i)

definition OAI.LeanBlast.CourtadeKumar.CourtadeKumarStatement line 55

def CourtadeKumarStatement : Prop :=  ∀ (n : ℕ), 1 ≤ n →    ∀ ε : ℝ, 0 ≤ ε → ε ≤ (1 : ℝ) / 2 →      ∀ f : Cube n → Bool, mutualInformation ε f ≤ 1 - binaryEntropy ε

definition OAI.LeanBlast.CourtadeKumar.DictatorAttainmentStatement line 60

def DictatorAttainmentStatement : Prop :=  ∀ (n : ℕ), 1 ≤ n →    ∀ ε : ℝ, 0 ≤ ε → ε ≤ (1 : ℝ) / 2 →      ∀ i : Fin n,        mutualInformation ε (dictator i) = 1 - binaryEntropy ε ∧        mutualInformation ε (complementDictator i) = 1 - binaryEntropy ε

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: attain, bias, boolean, bound, coordinate, functions, output and refined.

Statements in Sharp binary-information contraction on the discrete cube come first because the release lists only that paper for this Lean statement.

Corollary 1.2 of Sharp binary-information contraction on the discrete cube

Corollary (The Boolean bound with output bias)

For every finite n\ge0, \rho\in[-1,1], and f:\Omega_n\to\{-1,1\}, put m=\E f and r_m=\psi^{-1}(H(m)). Then

I(f(X);Y)\le\psi(|\rho|r_m)\le\psi(|\rho|).

For n\ge1, signed coordinate functions attain the bound at m=0. If 0<|m|<1 and 0<|\rho|\le1, the refined upper curve is strictly below the universal curve \psi(|\rho|).

Section 1, The contraction theorem and its mechanism Read it in the paper TeX source, line 92

Show the TeX

For every finite $n\ge0$, $\rho\in[-1,1]$, and
$f:\Omega_n\to\{-1,1\}$, put
$m=\E f$ and $r_m=\psi^{-1}(H(m))$.  Then
\begin{equation}
 I(f(X);Y)\le\psi(|\rho|r_m)\le\psi(|\rho|).
 
\end{equation}
For $n\ge1$, signed coordinate functions attain the bound at $m=0$.
If $0<|m|<1$ and $0<|\rho|\le1$, the refined upper curve is
strictly below the universal curve $\psi(|\rho|)$.

Other possible paper matches, closest wording first

Theorem 1.1 of Sharp binary-information contraction on the discrete cube. Found in the paper’s title, which its main theorem states: cube, binary and information. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equality, attains and coordinate.
Corollary 8.1 of Hellinger contraction with arbitrary Boolean output bias. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/2, kumar and courtade. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: varepsilon, equality, coordinates, attain and boolean.
Proposition 3.2 of Sharp binary-information contraction on the discrete cube. Words of the Lean statement, its names split into words, that also appear in the theorem: average and entropy. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: attains.
Theorem 1.1 of Hellinger contraction with arbitrary Boolean output bias. Words of the Lean statement, its names split into words, that also appear in the theorem: probability and function. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: boolean.
Theorem 6.4 of Sharp binary-information contraction on the discrete cube. Words of the Lean statement, its names split into words, that also appear in the theorem: entropy.
Theorem 5.3 of Sharp binary-information contraction on the discrete cube. Words of the Lean statement, its names split into words, that also appear in the theorem: joint.

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 341 s; peak memory 6.4 GiB).
Reported 11 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 119’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Courtade-Kumar (with attainment) and the soft/biased-Boolean mutual-information contraction are stated; the Hellinger conjecture is not.
Definitions the group asks readers to compare with the paper: Information-theoretic quantities are re-implemented by hand (mutualInformation with an if 0 < joint guard and natural logs divided by ell := Real.log 2, psi, H, psiInv := Function.invFunOn psi (Set.Icc 0 1), production, hybrid), not Mathlib objects; the SoftChannel statement also bundles seven properties (FullMain)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The Hellinger conjecture (the October/September Hellinger paper's main theorem) is not stated anywhere in the family's challenges.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For Boolean functions this includes the Courtade–Kumar inequality $I(f(X);Y)\le1-h_2(\varepsilon)$ in bits at crossover probability $0\le\varepsilon\le1/2$, where $h_2$ is binary entropy. A coordinate and its complement attain equality.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

76 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: 37 to its own declarations, 19 to Mathlib’s documentation. 17 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 courtadeKumarAndAttainment 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 16 other declarations of the file; with the theorem itself they span 38 lines. 1 of them is named in the statement itself.

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

abbrev OAI.LeanBlast.GotsmanLinial.Cubeline 9library
abbrev Cubeline 19library
def cubeAverageline 21library
def ellline 24library
def bitFlipKernelline 26library
def jointProbabilityline 29library
def functionMarginalline 32library
def observationMarginalline 35library
def xlogxline 38library
def binaryEntropyline 40library
def mutualInformationline 43library
def dictatorline 51library
def complementDictatorline 53library
def CourtadeKumarStatementline 55library
def DictatorAttainmentStatementline 60library
def FullStatementline 67in the statementlibrary
theorem courtadeKumarAndAttainmentline 69selected theoremlibrary

import Mathlibnamespace OAIopen scoped BigOperatorsnamespace LeanBlast.GotsmanLinialabbrev Cube (n : ℕ) := Fin n → Boolend LeanBlast.GotsmanLinialnoncomputable sectionopen scoped BigOperatorsnamespace LeanBlast.CourtadeKumarabbrev Cube (n : ℕ) := LeanBlast.GotsmanLinial.Cube ndef cubeAverage {n : ℕ} (g : Cube n → ℝ) : ℝ :=  (∑ x, g x) / (2 : ℝ) ^ ndef ell : ℝ := Real.log 2def bitFlipKernel {n : ℕ} (ε : ℝ) (x y : Cube n) : ℝ :=  ∏ i, if x i = y i then 1 - ε else εdef jointProbability {n : ℕ} (ε : ℝ) (f : Cube n → Bool) (b : Bool) (y : Cube n) : ℝ :=  cubeAverage (fun x => if f x = b then bitFlipKernel ε x y else 0)def functionMarginal {n : ℕ} (ε : ℝ) (f : Cube n → Bool) (b : Bool) : ℝ :=  ∑ y, jointProbability ε f b ydef observationMarginal {n : ℕ} (ε : ℝ) (f : Cube n → Bool) (y : Cube n) : ℝ :=  ∑ b : Bool, jointProbability ε f b ydef xlogx (t : ℝ) : ℝ := t * Real.log tdef binaryEntropy (ε : ℝ) : ℝ :=  -(xlogx ε + xlogx (1 - ε)) / elldef mutualInformation {n : ℕ} (ε : ℝ) (f : Cube n → Bool) : ℝ :=  (∑ b : Bool, ∑ y : Cube n,    if 0 < jointProbability ε f b y then      jointProbability ε f b y *        Real.log (jointProbability ε f b y /          (functionMarginal ε f b * observationMarginal ε f y))    else 0) / elldef dictator {n : ℕ} (i : Fin n) (x : Cube n) : Bool := x idef complementDictator {n : ℕ} (i : Fin n) (x : Cube n) : Bool := !(x i)def CourtadeKumarStatement : Prop :=  ∀ (n : ℕ), 1 ≤ n →    ∀ ε : ℝ, 0 ≤ ε → ε ≤ (1 : ℝ) / 2 →      ∀ f : Cube n → Bool, mutualInformation ε f ≤ 1 - binaryEntropy εdef DictatorAttainmentStatement : Prop :=  ∀ (n : ℕ), 1 ≤ n →    ∀ ε : ℝ, 0 ≤ ε → ε ≤ (1 : ℝ) / 2 →      ∀ i : Fin n,        mutualInformation ε (dictator i) = 1 - binaryEntropy ε ∧        mutualInformation ε (complementDictator i) = 1 - binaryEntropy εdef FullStatement : Prop := CourtadeKumarStatement ∧ DictatorAttainmentStatementtheorem courtadeKumarAndAttainment : FullStatement := by  sorryend LeanBlast.CourtadeKumarendend 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.

Sharp binary-information contraction on the discrete cube

Theorem (Sharp soft-channel contraction)

For every finite n\ge0, every u:\Omega_n\to[-1,1], and -1\le\rho\le1,

I(T_\rho u) \le\psi\!\left(|\rho|\,\psi^{-1}(I(u))\right).

The inverse is the nonnegative increasing branch on [0,L]. For n\ge1, every soft coordinate u(x)=a x_i, |a|\le1, attains equality.

Section 1, The contraction theorem and its mechanism Read it in the paper TeX source, line 72

Show the TeX

For every finite $n\ge0$, every $u:\Omega_n\to[-1,1]$, and
$-1\le\rho\le1$,
\begin{equation}
 I(T_\rho u)
 \le\psi\!\left(|\rho|\,\psi^{-1}(I(u))\right).
 
\end{equation}
The inverse is the nonnegative increasing branch on $[0,L]$.
For $n\ge1$, every soft coordinate $u(x)=a x_i$, $|a|\le1$,
attains equality.

Corollary (The Boolean bound with output bias)

For every finite n\ge0, \rho\in[-1,1], and f:\Omega_n\to\{-1,1\}, put m=\E f and r_m=\psi^{-1}(H(m)). Then

I(f(X);Y)\le\psi(|\rho|r_m)\le\psi(|\rho|).

For n\ge1, signed coordinate functions attain the bound at m=0. If 0<|m|<1 and 0<|\rho|\le1, the refined upper curve is strictly below the universal curve \psi(|\rho|).

Section 1, The contraction theorem and its mechanism Read it in the paper TeX source, line 92

Show the TeX

For every finite $n\ge0$, $\rho\in[-1,1]$, and
$f:\Omega_n\to\{-1,1\}$, put
$m=\E f$ and $r_m=\psi^{-1}(H(m))$.  Then
\begin{equation}
 I(f(X);Y)\le\psi(|\rho|r_m)\le\psi(|\rho|).
 
\end{equation}
For $n\ge1$, signed coordinate functions attain the bound at $m=0$.
If $0<|m|<1$ and $0<|\rho|\le1$, the refined upper curve is
strictly below the universal curve $\psi(|\rho|)$.

Theorem (Sharp soft production)

For every finite n\ge0 and every g:\{-1,1\}^n\to(-1,1), with m=\E g,

D(g)\ge\widehat{\mathcal B}_m(I(g)) =\max\{\mathcal B_m(I(g)),F(I(g))\} \ge F(I(g)).

Section 1, The contraction theorem and its mechanism Read it in the paper TeX source, line 307

Show the TeX

For every finite $n\ge0$ and every $g:\{-1,1\}^n\to(-1,1)$,
with $m=\E g$,
\begin{equation}
 D(g)\ge\widehat{\mathcal B}_m(I(g))
       =\max\{\mathcal B_m(I(g)),F(I(g))\}
       \ge F(I(g)).
 
\end{equation}

Configuration and file facts

File: lean/ComparatorChallenges/CourtadeKumar.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/InformationTheory/BooleanNoise/Main.lean
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: 16 local vocabulary declarations
Lines of Lean, without comments or blank lines: 38
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