Positive-probability transience and the velocity hemisphere
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Theorem selected for verification:OAI.DirectionalTransience.positive_probability_transience_velocity_hemisphere
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theorempositive_probability_transience_velocity_hemisphere {d : ℕ} (hd : 3 ≤ d) (ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (ℓ : Vector d) (hℓ : ℓ ≠ 0) (hpos : 0 < annealedLaw ν (TransientPaths ℓ)) : ∃ v : Vector d, v ≠ 0 ∧ 0 < dot v ℓ ∧ annealedLaw ν (VelocityPaths v) = 1 ∧ (∀ w : Vector d, annealedLaw ν (VelocityPaths w) = 1 → w = v) ∧ {u : Vector d | dot u u = 1 ∧ annealedLaw ν (TransientPaths u) = 1} = {u : Vector d | dot u u = 1 ∧ 0 < annealedLaw ν (TransientPaths u)} ∧ {u : Vector d | dot u u = 1 ∧ 0 < annealedLaw ν (TransientPaths u)} = {u : Vector d | dot u u = 1 ∧ 0 < dot v u} ∧ (∀ u : Vector d, dot u u = 1 → dot v u ≤ 0 → annealedLaw ν (TransientPaths u) = 0) := bysorry
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.
defVelocityPaths {d : ℕ} (v : Vector d) : Set (Path d) := {X | 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: hemisphere, velocity, annealed, vector, uniform, transience, transition, rows and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonzero.
Corollary 1.2 of Directional transience implies ballisticity
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.
Other possible paper matches, closest wording first
Theorem 1.1 of Directional transience implies ballisticity. Words of the Lean statement, its names split into words, that also appear in the theorem: vector, uniform, transience, transition, rows, real and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: walk.
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. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: product.
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, positive and law.
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, positive and eq. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: walk.
Proposition 6.2 of A directional zero–one law for finite-range-dependent random environments. Words of the Lean statement, its names split into words, that also appear in the theorem: pi, path, positive, integers and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: origin, sample and zero.
Proposition 4.1 of A directional zero–one law for finite-range-dependent random environments. Words of the Lean statement, its names split into words, that also appear in the theorem: steps, path and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: retain, unique and product.
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 $d\ge3$, positive probability of transience in any nonzero direction already suffices. The limiting velocity is unique, and the unit directions with positive transience probability are exactly the open hemisphere having positive inner product with that velocity; transience has probability one on that hemisphere and zero on its complement. The environment is sampled once and retained along the walk, and the probabilities use the annealed law from the origin.”
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
102 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: 77 to its own declarations, 55 to Mathlib’s documentation. 22 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.
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 220 link it.
Solution module
OAI.Probability.Ballisticity.VelocityHemisphere
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
20 local vocabulary declarations
Lines of Lean, without comments or blank lines
74
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Positive-probability transience and the velocity hemisphere
Family 220: Directional zero–one laws beyond iid environments and iid ballisticity
Candidate pairing
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.
Directional transience implies ballisticity
TeX label cor:positive-transience, build/01-introduction.tex, line 205
The Lean statement says
theorempositive_probability_transience_velocity_hemisphere {d : ℕ} (hd : 3 ≤ d) (ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (ℓ : Vector d) (hℓ : ℓ ≠ 0) (hpos : 0 < annealedLaw ν (TransientPaths ℓ)) : ∃ v : Vector d, v ≠ 0 ∧ 0 < dot v ℓ ∧ annealedLaw ν (VelocityPaths v) = 1 ∧ (∀ w : Vector d, annealedLaw ν (VelocityPaths w) = 1 → w = v) ∧ {u : Vector d | dot u u = 1 ∧ annealedLaw ν (TransientPaths u) = 1} = {u : Vector d | dot u u = 1 ∧ 0 < annealedLaw ν (TransientPaths u)} ∧ {u : Vector d | dot u u = 1 ∧ 0 < annealedLaw ν (TransientPaths u)} = {u : Vector d | dot u u = 1 ∧ 0 < dot v u} ∧ (∀ u : Vector d, dot u u = 1 → dot v u ≤ 0 → annealedLaw ν (TransientPaths u) = 0) := bysorry
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.