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.DirectionalTransience.directional_transience_implies_ballisticity
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.
defHasVelocity {d : ℕ} (ν : Measure (Row d)) (v : Vector d) : Prop := ∀ᵐ X ∂annealedLaw ν, Tendsto (fun n : ℕ => (fun i => (X n i : ℝ) / (n : ℝ))) atTop (𝓝 v)
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: ballisticity, implies, vector, uniform, transience, transition, rows, real and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nearest, neighbor and deterministic.
Statements in Directional transience implies ballisticity come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of Directional transience implies ballisticity
Let d\ge2 and let P=\nu^{\otimes\mathbb Z^d} be an iid law of nearest-neighbor transition rows satisfying uniform ellipticity. If \ell\in S^{d-1} and P_0(A_\ell)=1, then there exists a deterministic vector v=v(\nu)\in\mathbb R^d such that
Let $d\ge2$ and let $P=\nu^{\otimes\mathbb Z^d}$ be an iid law of
nearest-neighbor transition rows satisfying uniform ellipticity. If
$\ell\in S^{d-1}$ and $P_0(A_\ell)=1$, then there exists a deterministic
vector $v=v(\nu)\in\mathbb R^d$ such that
\[
v\cdot\ell>0,
\qquad
P_0\!\left(\lim_{n\to\infty}\frac{X_n}{n}=v\right)=1.
\]
In particular, the walk is ballistic in the same direction $\ell$.
Other possible paper matches, closest wording first
Corollary 1.2 of Directional transience implies ballisticity. Words of the Lean statement, its names split into words, that also appear in the theorem: velocity, annealed, vector, uniform, transience, transition and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nearest, neighbor, deterministic and positive.
Proposition 2.5 of Directional transience implies ballisticity. Words of the Lean statement, its names split into words, that also appear in the theorem: velocity, direction, probabilities and eq. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sure, projection, deterministic, almost and positive.
Proposition 3.7 of Directional transience implies ballisticity. Words of the Lean statement, its names split into words, that also appear in the theorem: measured, pi, environment, path, probability, integer and one more.
Theorem 1.1 of A directional zero–one law under strict ellipticity. Words of the Lean statement, its names split into words, that also appear in the theorem: transition, rows, eq and law. Found in the paper’s title, which its main theorem states: directional. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nearest and neighbor.
Proposition 6.5 of Directional transience implies ballisticity. Words of the Lean statement, its names split into words, that also appear in the theorem: row, environment, probability and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniformly and random.
Proposition 3.1 of A directional zero–one law under strict ellipticity. Words of the Lean statement, its names split into words, that also appear in the theorem: initial, transition, rows, environment and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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.
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 220’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.
Finds part of it stated, or a narrower or weaker statementTheir 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 directional zero–one law for stationary ergodic finite-range-dependent uniformly elliptic environments is claimed but not stated by any family-220 challenge.
OpenAI’s own note on what this Lean statement covers
“For an i.i.d. uniformly elliptic nearest-neighbor random environment on $\mathbb Z^d$, the formalization proves that almost-sure directional transience implies convergence of $X_n/n$ to a deterministic velocity with positive projection in that direction for every $d\ge2$.”
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
100 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: 56 to its own declarations, 55 to Mathlib’s documentation. 23 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
Let d\ge3, let \PP be an independent and identically distributed law of nearest-neighbor transition rows on \ZZ^d satisfying (eq:ellipticity), and let \ell\in\RR^d\setminus\{0\}. Then
Let $d\ge3$, let $\PP$ be an independent and identically distributed
law of nearest-neighbor transition rows on $\ZZ^d$ satisfying
\eqref{eq:ellipticity}, and let $\ell\in\RR^d\setminus\{0\}$.
Then
\[
P_0(A_\ell)\in\{0,1\}.
\]
Let d\ge2 and let P=\nu^{\otimes\mathbb Z^d} be an iid law of nearest-neighbor transition rows satisfying uniform ellipticity. If \ell\in S^{d-1} and P_0(A_\ell)=1, then there exists a deterministic vector v=v(\nu)\in\mathbb R^d such that
Let $d\ge2$ and let $P=\nu^{\otimes\mathbb Z^d}$ be an iid law of
nearest-neighbor transition rows satisfying uniform ellipticity. If
$\ell\in S^{d-1}$ and $P_0(A_\ell)=1$, then there exists a deterministic
vector $v=v(\nu)\in\mathbb R^d$ such that
\[
v\cdot\ell>0,
\qquad
P_0\!\left(\lim_{n\to\infty}\frac{X_n}{n}=v\right)=1.
\]
In particular, the walk is ballistic in the same direction $\ell$.
Corollary(Positive-probability transience and the velocity hemisphere)
Let d\ge3 and let P=\nu^{\otimes\mathbb Z^d} be an iid law of nearest-neighbor transition rows satisfying uniform ellipticity. For a fixed \ell\in\mathbb R^d\setminus\{0\}, set A_\ell=\{X_n\cdot\ell\longrightarrow+\infty\}. If P_0(A_\ell)>0, then there is a deterministic nonzero vector v=v(\nu) such that
In particular, P_0(A_u)=0 for each fixed u\in S^{d-1} with v\cdot u\le0, including every equator direction. The equality concerns annealed probabilities for each fixed deterministic direction, not a simultaneous pathwise assertion over all directions.
Let $d\ge3$ and let $P=\nu^{\otimes\mathbb Z^d}$ be an iid law of
nearest-neighbor transition rows satisfying uniform ellipticity. For a
fixed $\ell\in\mathbb R^d\setminus\{0\}$, set
$A_\ell=\{X_n\cdot\ell\longrightarrow+\infty\}$. If $P_0(A_\ell)>0$,
then there is a deterministic nonzero vector $v=v(\nu)$ such that
\[
v\cdot\ell>0,
\qquad
P_0\!\left(\lim_{n\to\infty}\frac{X_n}{n}=v\right)=1.
\]
If such a direction exists, then the deterministic sets of directions
satisfy
\[
\{u\in S^{d-1}:P_0(A_u)=1\}
=\{u\in S^{d-1}:P_0(A_u)>0\}
=\{u\in S^{d-1}:v\cdot u>0\}.
\]
In particular, $P_0(A_u)=0$ for each fixed $u\in S^{d-1}$ with
$v\cdot u\le0$, including every equator direction. The equality concerns
annealed probabilities for each fixed deterministic direction, not a
simultaneous pathwise assertion over all directions.
Let d\ge2 and let P=\nu^{\otimes\mathbb Z^d} be an iid law of nearest-neighbor transition rows satisfying uniform ellipticity. If \ell\in S^{d-1} and P_0(A_\ell)=1, then there exists a deterministic vector v=v(\nu)\in\mathbb R^d such that
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.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.