OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction \(v+\delta\) of k independent repetitions decays exponentially in k, for \(0\lt \delta\lt 1-v\). Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
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,
For every finite two-player game with entangled value v < 1, we prove exponential decay for the probability of winning at least a \(v+\delta\) fraction of k independent repetitions, uniformly over all finite-dimensional joint strategies. The result allows arbitrary correlated question distributions and holds for every \(0\lt \delta\lt 1-v\) and k ≥ 1. Its universal rate is proportional to \(\delta^5/(1+\log(|\mathcal A||\mathcal B|))\), where \(\mathcal A,\mathcal B\) are the answer alphabets. For each fixed question distribution, we also obtain an explicit cubic rate in δ.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
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,
EntangledGames.lean, the release lists it as a main result.
theoremthreshold_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) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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).
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
The formalization gives exponential threshold parallel repetition for every finite two-player game with arbitrary correlated questions. 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. The separate distribution-dependent cubic bound is outside this scope.
AI-drafted comparisons, with no recorded human review
Threshold parallel repetition for entangled gamesEntangledGames
Unreviewed AI draft: states a weaker form of a paper’s theorem. States threshold parallel repetition with decay rate κδ^13/(1 + log(|A||B|)), whereas the main theorem gives κ₀δ^5/(1 + log d) with d = |A||B|; the paper notes that the δ^13 rate follows from its theorem.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Threshold parallel repetition for entangled gamesEntangledGames
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The Lean statement gives a weaker decay rate than the paper’s main theorem, a rate the paper says already follows from it.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 277The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 277Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1 (Threshold parallel repetition)
In Threshold parallel repetition for finite-dimensional entangled games, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
dimensional, entangled, finite, game, joint, player, repetitions, round, strategies, value
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 169 s; peak memory 6.4 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.