Lean statement

Sharp binary-channel contraction and entropy production

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.SoftChannel204.fullMain

Lean statement file: SoftChannel204.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 119 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.

def FullMain : Prop := SoftContraction ∧ SoftAttainment ∧ SoftInformationIdentity ∧  RefinedBoolean ∧ BooleanAttainment ∧ BitsConversion ∧ SharpProduction⋮theorem fullMain : FullMain := 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.SoftChannel204.Cube line 11
```
abbrev Cube (n : ℕ) := Fin n → Bool
```

definition OAI.SoftChannel204.sign line 13

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

definition OAI.SoftChannel204.avg line 15

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

definition OAI.SoftChannel204.L line 17
```
def L : ℝ := Real.log 2
```

definition OAI.SoftChannel204.psi line 19

def psi (r : ℝ) : ℝ :=  (1 + r) / 2 * Real.log (1 + r) + (1 - r) / 2 * Real.log (1 - r)

definition OAI.SoftChannel204.H line 22
```
def H (r : ℝ) : ℝ := L - psi r
```

definition OAI.SoftChannel204.info line 24

def info {n : ℕ} (u : Cube n → ℝ) : ℝ := H (avg u) - avg (H ∘ u)

definition OAI.SoftChannel204.psiInv line 26

def psiInv (s : ℝ) : ℝ := Function.invFunOn psi (Set.Icc 0 1) s

definition OAI.SoftChannel204.kernel line 28

def kernel {n : ℕ} (ρ : ℝ) (x y : Cube n) : ℝ :=  ∏ i, if x i = y i then (1 + ρ) / 2 else (1 - ρ) / 2

definition OAI.SoftChannel204.noise line 31

def noise {n : ℕ} (ρ : ℝ) (u : Cube n → ℝ) (y : Cube n) : ℝ :=  ∑ x, kernel ρ y x * u x

definition OAI.SoftChannel204.Closed line 34

def Closed {n : ℕ} (u : Cube n → ℝ) : Prop := ∀ x, u x ∈ Set.Icc (-1) 1

definition OAI.SoftChannel204.Interior line 36

def Interior {n : ℕ} (g : Cube n → ℝ) : Prop := ∀ x, g x ∈ Set.Ioo (-1) 1

definition OAI.SoftChannel204.softCoordinate line 38

def softCoordinate {n : ℕ} (i : Fin n) (a : ℝ) (x : Cube n) : ℝ := a * sign (x i)

definition OAI.SoftChannel204.flip line 40

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

definition OAI.SoftChannel204.numberOperator line 42

def numberOperator {n : ℕ} (g : Cube n → ℝ) (x : Cube n) : ℝ :=  (1 / 2 : ℝ) * ∑ i, (g x - g (flip i x))

definition OAI.SoftChannel204.production line 45

def production {n : ℕ} (g : Cube n → ℝ) : ℝ :=  avg (fun x => Real.artanh (g x) * numberOperator g x)

definition OAI.SoftChannel204.F line 49

def F (s : ℝ) : ℝ := psiInv s * Real.artanh (psiInv s)

definition OAI.SoftChannel204.S line 52

def S (s : ℝ) : ℝ := if s < 0 then 0 else F s - 2 * s

definition OAI.SoftChannel204.reserve line 55

def reserve (m I : ℝ) : ℝ :=  2 * I + (1 - m ^ 2) * S (L - (H m - I) / (1 - m ^ 2))

definition OAI.SoftChannel204.hybrid line 58

def hybrid (m I : ℝ) : ℝ := max (reserve m I) (F I)

definition OAI.SoftChannel204.softJoint line 60

def softJoint {n : ℕ} (ρ : ℝ) (u : Cube n → ℝ) (b : Bool) (y : Cube n) : ℝ :=  avg (fun x => (1 + sign b * u x) / 2 * kernel ρ x y)

definition OAI.SoftChannel204.boolJoint line 63

def boolJoint {n : ℕ} (ρ : ℝ) (f : Cube n → Bool) (b : Bool) (y : Cube n) : ℝ :=  avg (fun x => if f x = b then kernel ρ x y else 0)

definition OAI.SoftChannel204.leftMarginal line 66

def leftMarginal {n : ℕ} (p : Bool → Cube n → ℝ) (b : Bool) : ℝ := ∑ y, p b y

definition OAI.SoftChannel204.rightMarginal line 68

def rightMarginal {n : ℕ} (p : Bool → Cube n → ℝ) (y : Cube n) : ℝ := ∑ b, p b y

9 more definitions are reached the same way; the full file lists them.

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: soft, sharp, psi, contraction, finite and coordinate. 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: channel and attains.

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

Theorem 1.1 of 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.

Other possible paper matches, closest wording first

Theorem 1.3 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: production, soft, max, sharp and finite.
Corollary 1.2 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: refined, bound, psi, finite, signed, coordinate and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: output, bias and attain.
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: value and 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: reserve and joint.
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, bound, signed, coordinates and boolean. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: independent, uniform and attain.
Proposition 5.1 of Hellinger contraction with arbitrary Boolean output bias. Words of the Lean statement, its names split into words, that also appear in the theorem: signed, coordinate, function and boolean. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonconstant.

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.

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

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

“The formalization proves sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At each fixed initial information level, a noisy coordinate channel attains the maximum retained information. It also proves the refined Boolean bound that accounts for output bias, with strict improvement for nonconstant biased outputs at nonzero noise correlation, and the selected mean-dependent entropy-production inequality.”
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

121 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: 117 to its own declarations, 36 to Mathlib’s documentation. 35 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 fullMain 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 34 other declarations of the file; with the theorem itself they span 68 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.SoftChannel204 are listed without it.

abbrev Cubeline 11library
def signline 13library
def avgline 15library
def Lline 17library
def psiline 19library
def Hline 22library
def infoline 24library
def psiInvline 26library
def kernelline 28library
def noiseline 31library
def Closedline 34library
def Interiorline 36library
def softCoordinateline 38library
def flipline 40library
def numberOperatorline 42library
def productionline 45library
def Fline 49library
def Sline 52library
def reserveline 55library
def hybridline 58library
def softJointline 60library
def boolJointline 63library
def leftMarginalline 66library
def rightMarginalline 68library
def mutualInformationline 70library
def binaryEntropyBitsline 74library
def SoftContractionline 77library
def SoftAttainmentline 81library
def SoftInformationIdentityline 86library
def RefinedBooleanline 90library
def BooleanAttainmentline 98library
def BitsConversionline 103library
def SharpProductionline 108library
def FullMainline 111in the statementlibrary
theorem fullMainline 114selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsnamespace SoftChannel204abbrev Cube (n : ℕ) := Fin n → Booldef sign (b : Bool) : ℝ := if b then 1 else -1def avg {n : ℕ} (g : Cube n → ℝ) : ℝ := (∑ x, g x) / (2 : ℝ) ^ ndef L : ℝ := Real.log 2def psi (r : ℝ) : ℝ :=  (1 + r) / 2 * Real.log (1 + r) + (1 - r) / 2 * Real.log (1 - r)def H (r : ℝ) : ℝ := L - psi rdef info {n : ℕ} (u : Cube n → ℝ) : ℝ := H (avg u) - avg (H ∘ u)def psiInv (s : ℝ) : ℝ := Function.invFunOn psi (Set.Icc 0 1) sdef kernel {n : ℕ} (ρ : ℝ) (x y : Cube n) : ℝ :=  ∏ i, if x i = y i then (1 + ρ) / 2 else (1 - ρ) / 2def noise {n : ℕ} (ρ : ℝ) (u : Cube n → ℝ) (y : Cube n) : ℝ :=  ∑ x, kernel ρ y x * u xdef Closed {n : ℕ} (u : Cube n → ℝ) : Prop := ∀ x, u x ∈ Set.Icc (-1) 1def Interior {n : ℕ} (g : Cube n → ℝ) : Prop := ∀ x, g x ∈ Set.Ioo (-1) 1def softCoordinate {n : ℕ} (i : Fin n) (a : ℝ) (x : Cube n) : ℝ := a * sign (x i)def flip {n : ℕ} (i : Fin n) (x : Cube n) : Cube n := Function.update x i (!(x i))def numberOperator {n : ℕ} (g : Cube n → ℝ) (x : Cube n) : ℝ :=  (1 / 2 : ℝ) * ∑ i, (g x - g (flip i x))def production {n : ℕ} (g : Cube n → ℝ) : ℝ :=  avg (fun x => Real.artanh (g x) * numberOperator g x)/-- Defined by this formula on `[0, L)`; no finite endpoint value at `L` is asserted. -/def F (s : ℝ) : ℝ := psiInv s * Real.artanh (psiInv s)/-- The excess `F s - 2 * s`, used for `s < L`, is extended by zero to negative arguments. -/def S (s : ℝ) : ℝ := if s < 0 then 0 else F s - 2 * s/-- The reserve is used for `|m| < 1` and `0 ≤ I < H m`. -/def reserve (m I : ℝ) : ℝ :=  2 * I + (1 - m ^ 2) * S (L - (H m - I) / (1 - m ^ 2))def hybrid (m I : ℝ) : ℝ := max (reserve m I) (F I)def softJoint {n : ℕ} (ρ : ℝ) (u : Cube n → ℝ) (b : Bool) (y : Cube n) : ℝ :=  avg (fun x => (1 + sign b * u x) / 2 * kernel ρ x y)def boolJoint {n : ℕ} (ρ : ℝ) (f : Cube n → Bool) (b : Bool) (y : Cube n) : ℝ :=  avg (fun x => if f x = b then kernel ρ x y else 0)def leftMarginal {n : ℕ} (p : Bool → Cube n → ℝ) (b : Bool) : ℝ := ∑ y, p b ydef rightMarginal {n : ℕ} (p : Bool → Cube n → ℝ) (y : Cube n) : ℝ := ∑ b, p b ydef mutualInformation {n : ℕ} (p : Bool → Cube n → ℝ) : ℝ :=  ∑ b, ∑ y, if 0 < p b y then    p b y * Real.log (p b y / (leftMarginal p b * rightMarginal p y)) else 0def binaryEntropyBits (ε : ℝ) : ℝ :=  -(ε * Real.log ε + (1 - ε) * Real.log (1 - ε)) / Ldef SoftContraction : Prop := ∀ (n : ℕ) (u : Cube n → ℝ), Closed u →  ∀ ρ ∈ Set.Icc (-1 : ℝ) 1,    info (noise ρ u) ≤ psi (|ρ| * psiInv (info u))def SoftAttainment : Prop := ∀ (n : ℕ) (i : Fin n) (a : ℝ), |a| ≤ 1 →  ∀ ρ ∈ Set.Icc (-1 : ℝ) 1,    info (noise ρ (softCoordinate i a)) =      psi (|ρ| * psiInv (info (softCoordinate i a)))def SoftInformationIdentity : Prop := ∀ (n : ℕ) (u : Cube n → ℝ), Closed u →  ∀ ρ ∈ Set.Icc (-1 : ℝ) 1,    mutualInformation (softJoint ρ u) = info (noise ρ u)def RefinedBoolean : Prop := ∀ (n : ℕ) (f : Cube n → Bool)  (ρ : ℝ), ρ ∈ Set.Icc (-1 : ℝ) 1 →  let m := avg (sign ∘ f)  let bound := psi (|ρ| * psiInv (H m))  mutualInformation (boolJoint ρ f) = info (noise ρ (sign ∘ f)) ∧  mutualInformation (boolJoint ρ f) ≤ bound ∧ bound ≤ psi |ρ| ∧  (0 < |m| → |m| < 1 → 0 < |ρ| → bound < psi |ρ|)def BooleanAttainment : Prop := ∀ (n : ℕ) (i : Fin n)  (ρ : ℝ), ρ ∈ Set.Icc (-1 : ℝ) 1 →  mutualInformation (boolJoint ρ (fun x => x i)) = psi |ρ| ∧  mutualInformation (boolJoint ρ (fun x => !(x i))) = psi |ρ|def BitsConversion : Prop := ∀ (n : ℕ) (f : Cube n → Bool)  (ε : ℝ), ε ∈ Set.Icc (0 : ℝ) (1/2) →  psi (1 - 2 * ε) / L = 1 - binaryEntropyBits ε ∧  mutualInformation (boolJoint (1 - 2 * ε) f) / L ≤ 1 - binaryEntropyBits εdef SharpProduction : Prop := ∀ (n : ℕ) (g : Cube n → ℝ), Interior g →  hybrid (avg g) (info g) ≤ production g ∧ F (info g) ≤ hybrid (avg g) (info g)def FullMain : Prop := SoftContraction ∧ SoftAttainment ∧ SoftInformationIdentity ∧  RefinedBoolean ∧ BooleanAttainment ∧ BitsConversion ∧ SharpProductiontheorem fullMain : FullMain := by  sorryend SoftChannel204endend 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/SoftChannel204.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 119 link it.
Solution module: OAI.InformationTheory.SoftChannel.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: 34 local vocabulary declarations
Lines of Lean, without comments or blank lines: 68
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