Lean statement

Threshold parallel repetition for entangled games

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Theorem selected for verification: OAI.ThresholdParallelRepetition.threshold_parallel_repetition

Lean statement file: EntangledGames.leanShow on the mapFamily 277. 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.

theorem threshold_parallel_repetition :    ∃ κ₀ : ℝ, 0 < κ₀ ∧      ∀ (x y a b : ℕ) (G : Game x y a b), entangledValue G < 1 →      ∀ (δ : ℝ), 0 < δ → δ < 1 - entangledValue G →      ∀ (k : ℕ), 1 ≤ k →        (∀ S : RepeatedStrategy x y a b k,          thresholdProbability G δ S ≤            Real.exp (-κ₀ * δ ^ 13 / (1 + Real.log (((a + 1) * (b + 1) : ℕ) : ℝ)) * k)) ∧        thresholdValue (k := k) G δ ≤          Real.exp (-κ₀ * δ ^ 13 / (1 + Real.log (((a + 1) * (b + 1) : ℕ) : ℝ)) * k) := 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.

structure OAI.ThresholdParallelRepetition.Game line 10

structure Game (x y a b : ℕ) where  questionProb : Fin (x + 1) × Fin (y + 1) → ℝ  questionProb_nonneg : ∀ z, 0 ≤ questionProb z  questionProb_sum : ∑ z, questionProb z = 1  accepts : Fin (x + 1) → Fin (y + 1) → Fin (a + 1) → Fin (b + 1) → Bool

structure OAI.ThresholdParallelRepetition.Strategy line 16

structure Strategy (X Y A B : Type) [Fintype A] [Fintype B] where

Show the remaining 10 lines

  dimA : ℕ  dimB : ℕ  state : Fin (dimA + 1) × Fin (dimB + 1) → ℂ  state_unit : ∑ i, Complex.normSq (state i) = 1  alice : X → A → Matrix (Fin (dimA + 1)) (Fin (dimA + 1)) ℂ  bob : Y → B → Matrix (Fin (dimB + 1)) (Fin (dimB + 1)) ℂ  alice_pos : ∀ x a, (alice x a).PosSemidef  bob_pos : ∀ y b, (bob y b).PosSemidef  alice_total : ∀ x, ∑ a, alice x a = 1  bob_total : ∀ y, ∑ b, bob y b = 1

definition OAI.ThresholdParallelRepetition.Strategy.born line 28

def Strategy.born {X Y A B : Type} [Fintype A] [Fintype B]    (S : Strategy X Y A B) (x : X) (y : Y) (a : A) (b : B) : ℝ :=  ∑ i, ∑ j, (star (S.state i) *    (S.alice x a i.1 j.1 * S.bob y b i.2 j.2) * S.state j).re

abbreviation OAI.ThresholdParallelRepetition.SingleStrategy line 33

abbrev SingleStrategy (x y a b : ℕ) :=  Strategy (Fin (x + 1)) (Fin (y + 1)) (Fin (a + 1)) (Fin (b + 1))

definition OAI.ThresholdParallelRepetition.successProbability line 36

def successProbability {x y a b : ℕ} (G : Game x y a b) (S : SingleStrategy x y a b) : ℝ :=  ∑ z : Fin (x + 1) × Fin (y + 1), G.questionProb z *    ∑ w : Fin (a + 1) × Fin (b + 1),      if G.accepts z.1 z.2 w.1 w.2 then S.born z.1 z.2 w.1 w.2 else 0

definition OAI.ThresholdParallelRepetition.entangledValue line 41

def entangledValue {x y a b : ℕ} (G : Game x y a b) : ℝ :=  sSup (Set.range (successProbability G))

abbreviation OAI.ThresholdParallelRepetition.RepeatedStrategy line 44

abbrev RepeatedStrategy (x y a b k : ℕ) :=  Strategy (Fin k → Fin (x + 1)) (Fin k → Fin (y + 1))    (Fin k → Fin (a + 1)) (Fin k → Fin (b + 1))

definition OAI.ThresholdParallelRepetition.wins line 48

def wins {x y a b k : ℕ} (G : Game x y a b)    (xs : Fin k → Fin (x + 1)) (ys : Fin k → Fin (y + 1))    (as : Fin k → Fin (a + 1)) (bs : Fin k → Fin (b + 1)) : ℕ :=  ∑ i, if G.accepts (xs i) (ys i) (as i) (bs i) then 1 else 0

definition OAI.ThresholdParallelRepetition.thresholdProbability line 53

def thresholdProbability {x y a b k : ℕ} (G : Game x y a b) (δ : ℝ)    (S : RepeatedStrategy x y a b k) : ℝ :=  ∑ xs : Fin k → Fin (x + 1), ∑ ys : Fin k → Fin (y + 1),    (∏ i, G.questionProb (xs i, ys i)) *      ∑ as : Fin k → Fin (a + 1), ∑ bs : Fin k → Fin (b + 1),        if ⌈(entangledValue G + δ) * (k : ℝ)⌉₊ ≤ wins G xs ys as bs        then S.born xs ys as bs else 0

definition OAI.ThresholdParallelRepetition.thresholdValue line 61

def thresholdValue {x y a b k : ℕ} (G : Game x y a b) (δ : ℝ) : ℝ :=  sSup (Set.range (thresholdProbability (k := k) G δ))

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: parallel, integer, threshold, value, entangled, exp, repetitions, strategy and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: joint, one, universal, supremum, delta, kappa and dimensional.

Theorem 1.1 of Threshold parallel repetition for finite-dimensional entangled games

Theorem (Threshold parallel repetition)

There is a universal constant \kappa_0>0 such that the following holds. Let G be any finite two-player one-round game with Boolean acceptance predicate and entangled value v=\omega^*(G)<1. For every 0<\delta<1-v, every integer k\ge1, and every finite-dimensional joint strategy for k repetitions,

\Pr\!\left[W_k\ge\lceil(v+\delta)k\rceil\right] \le \exp\!\left(-\frac{\kappa_0\delta^5} {1+\log d}\,k\right).

The same inequality holds for the supremum over those strategies.

Section 1, Introduction Read it in the paper TeX source, line 50

Show the TeX

There is a universal constant $\kappa_0>0$ such that the following holds.
Let $G$ be any finite two-player one-round game with Boolean acceptance
predicate and entangled value $v=\omega^*(G)<1$. For every
$0<\delta<1-v$, every integer $k\ge1$, and every finite-dimensional
joint strategy for $k$ repetitions,
\begin{equation}
 \Pr\!\left[W_k\ge\lceil(v+\delta)k\rceil\right]
 \le \exp\!\left(-\frac{\kappa_0\delta^5}
                         {1+\log d}\,k\right).
\end{equation}
The same inequality holds for the supremum over those strategies.

Other possible paper matches, closest wording first

Corollary 5.2 of Threshold parallel repetition for finite-dimensional entangled games. Words of the Lean statement, its names split into words, that also appear in the theorem: value, question, entangled, exp, repetitions, strategy and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: joint, bound, supremum, delta, kappa and dimensional.
Proposition 2.3 of Threshold parallel repetition for finite-dimensional entangled games. Words of the Lean statement, its names split into words, that also appear in the theorem: threshold, exp, repetitions, strategy, finite, game and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound, supremum, delta and dimensional.
Corollary 6.1 of Threshold parallel repetition for finite-dimensional entangled games. Words of the Lean statement, its names split into words, that also appear in the theorem: accept, integer, question, exp, finite, game and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one, universal, delta and kappa.
Proposition 2.4 of Threshold parallel repetition for finite-dimensional entangled games. Words of the Lean statement, its names split into words, that also appear in the theorem: value, entangled, repetitions, strategy, finite, game and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: universal and dimensional.
Proposition 5.1 of Threshold parallel repetition for finite-dimensional entangled games. Words of the Lean statement, its names split into words, that also appear in the theorem: question, strategy, finite, game and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: kappa and dimensional.

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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.

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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 169 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

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Do family 277’s Lean statements express the advertised result?

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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: threshold_parallel_repetition : ∃ κ₀ > 0, ∀ games G with entangledValue G < 1, ∀ 0 < δ < 1 - v, ∀ k ≥ 1, (∀ joint finite-dimensional k-fold strategies S, thresholdProbability G δ S ≤ exp(-κ₀ δ^13 k / (1 + log((a+1)(b+1))))) ∧ thresholdValue G δ ≤ the same bound, with Game an arbitrary correlated question distribution plus predicate, Strategy a finite-dimensional bipartite state with POVMs and entangledValue := sSup of single-copy success, and thresholdProbability = probability of at least ⌈(v+δ)k⌉ wins. Exactly the summary's exponential threshold repetition (explicit δ^13 exponent).
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

“If its finite-dimensional entangled value is $v<1$, its answer sets are $A,B$, and $0<\delta<1-v$, then the probability of winning at least the fraction $v+\delta$ of $k$ repetitions is at most $\exp(-\kappa\delta^{13}k/(1+\log(|A||B|)))$ for one universal $\kappa>0$. The bound applies to every joint finite-dimensional strategy and to their supremum, without assuming attainment.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

80 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: 20 to its own declarations, 61 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 threshold_parallel_repetition 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 55 lines. 5 of them are named in the statement itself.

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

structure Gameline 10in the statementlibrary
structure Strategyline 16library
def Strategy.bornline 28library
abbrev SingleStrategyline 33library
def successProbabilityline 36library
def entangledValueline 41in the statementlibrary
abbrev RepeatedStrategyline 44in the statementlibrary
def winsline 48library
def thresholdProbabilityline 53in the statementlibrary
def thresholdValueline 61in the statementlibrary
theorem threshold_parallel_repetitionline 64selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperators ComplexOrdernamespace ThresholdParallelRepetitionstructure Game (x y a b : ℕ) where  questionProb : Fin (x + 1) × Fin (y + 1) → ℝ  questionProb_nonneg : ∀ z, 0 ≤ questionProb z  questionProb_sum : ∑ z, questionProb z = 1  accepts : Fin (x + 1) → Fin (y + 1) → Fin (a + 1) → Fin (b + 1) → Boolstructure Strategy (X Y A B : Type) [Fintype A] [Fintype B] where  dimA : ℕ  dimB : ℕ  state : Fin (dimA + 1) × Fin (dimB + 1) → ℂ  state_unit : ∑ i, Complex.normSq (state i) = 1  alice : X → A → Matrix (Fin (dimA + 1)) (Fin (dimA + 1)) ℂ  bob : Y → B → Matrix (Fin (dimB + 1)) (Fin (dimB + 1)) ℂ  alice_pos : ∀ x a, (alice x a).PosSemidef  bob_pos : ∀ y b, (bob y b).PosSemidef  alice_total : ∀ x, ∑ a, alice x a = 1  bob_total : ∀ y, ∑ b, bob y b = 1def Strategy.born {X Y A B : Type} [Fintype A] [Fintype B]    (S : Strategy X Y A B) (x : X) (y : Y) (a : A) (b : B) : ℝ :=  ∑ i, ∑ j, (star (S.state i) *    (S.alice x a i.1 j.1 * S.bob y b i.2 j.2) * S.state j).reabbrev SingleStrategy (x y a b : ℕ) :=  Strategy (Fin (x + 1)) (Fin (y + 1)) (Fin (a + 1)) (Fin (b + 1))def successProbability {x y a b : ℕ} (G : Game x y a b) (S : SingleStrategy x y a b) : ℝ :=  ∑ z : Fin (x + 1) × Fin (y + 1), G.questionProb z *    ∑ w : Fin (a + 1) × Fin (b + 1),      if G.accepts z.1 z.2 w.1 w.2 then S.born z.1 z.2 w.1 w.2 else 0def entangledValue {x y a b : ℕ} (G : Game x y a b) : ℝ :=  sSup (Set.range (successProbability G))abbrev RepeatedStrategy (x y a b k : ℕ) :=  Strategy (Fin k → Fin (x + 1)) (Fin k → Fin (y + 1))    (Fin k → Fin (a + 1)) (Fin k → Fin (b + 1))def wins {x y a b k : ℕ} (G : Game x y a b)    (xs : Fin k → Fin (x + 1)) (ys : Fin k → Fin (y + 1))    (as : Fin k → Fin (a + 1)) (bs : Fin k → Fin (b + 1)) : ℕ :=  ∑ i, if G.accepts (xs i) (ys i) (as i) (bs i) then 1 else 0def thresholdProbability {x y a b k : ℕ} (G : Game x y a b) (δ : ℝ)    (S : RepeatedStrategy x y a b k) : ℝ :=  ∑ xs : Fin k → Fin (x + 1), ∑ ys : Fin k → Fin (y + 1),    (∏ i, G.questionProb (xs i, ys i)) *      ∑ as : Fin k → Fin (a + 1), ∑ bs : Fin k → Fin (b + 1),        if ⌈(entangledValue G + δ) * (k : ℝ)⌉₊ ≤ wins G xs ys as bs        then S.born xs ys as bs else 0def thresholdValue {x y a b k : ℕ} (G : Game x y a b) (δ : ℝ) : ℝ :=  sSup (Set.range (thresholdProbability (k := k) G δ))theorem threshold_parallel_repetition :    ∃ κ₀ : ℝ, 0 < κ₀ ∧      ∀ (x y a b : ℕ) (G : Game x y a b), entangledValue G < 1 →      ∀ (δ : ℝ), 0 < δ → δ < 1 - entangledValue G →      ∀ (k : ℕ), 1 ≤ k →        (∀ S : RepeatedStrategy x y a b k,          thresholdProbability G δ S ≤            Real.exp (-κ₀ * δ ^ 13 / (1 + Real.log (((a + 1) * (b + 1) : ℕ) : ℝ)) * k)) ∧        thresholdValue (k := k) G δ ≤          Real.exp (-κ₀ * δ ^ 13 / (1 + Real.log (((a + 1) * (b + 1) : ℕ) : ℝ)) * k) := by  sorryend ThresholdParallelRepetitionendend 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.

Threshold parallel repetition for finite-dimensional entangled games

Theorem (Threshold parallel repetition)

\Pr\!\left[W_k\ge\lceil(v+\delta)k\rceil\right] \le \exp\!\left(-\frac{\kappa_0\delta^5} {1+\log d}\,k\right).

The same inequality holds for the supremum over those strategies.

Section 1, Introduction Read it in the paper TeX source, line 50

Show the TeX

There is a universal constant $\kappa_0>0$ such that the following holds.
Let $G$ be any finite two-player one-round game with Boolean acceptance
predicate and entangled value $v=\omega^*(G)<1$. For every
$0<\delta<1-v$, every integer $k\ge1$, and every finite-dimensional
joint strategy for $k$ repetitions,
\begin{equation}
 \Pr\!\left[W_k\ge\lceil(v+\delta)k\rceil\right]
 \le \exp\!\left(-\frac{\kappa_0\delta^5}
                         {1+\log d}\,k\right).
\end{equation}
The same inequality holds for the supremum over those strategies.

Configuration and file facts

File: lean/ComparatorChallenges/EntangledGames.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Probability/EntangledGames/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: 10 local vocabulary declarations
Lines of Lean, without comments or blank lines: 55
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.

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