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.Problem367.free_ising_factor_iff_threshold
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.
defConnectedFinset {d : ℕ} (D : Finset (TreeVertex d)) : Prop := D.Nonempty ∧ ∀ u ∈ D, ∀ v ∈ D, Relation.ReflTransGen (fun x y : TreeVertex d => x ∈ D ∧ y ∈ D ∧ treeAdj x y) u v
defHasFreeIsingLaw {d : ℕ} (theta : ℝ) (mu : Measure (TreeVertex d → IsingSpin)) : Prop := ∀ (D : Finset (TreeVertex d)), ConnectedFinset D → ∀ eta : TreeVertex d → IsingSpin, mu {x | ∀ v ∈ D, x v = eta v} = freeCylinderWeight theta D eta
defIsFactorOfIID (d : ℕ) (theta : ℝ) : Prop := ∃ Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin), Measurable Phi ∧ HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)) ∧ ∀ g : TreeAut d, ∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U)
Possible paper match, with no recorded review
No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.
Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: automorphism, factor, field, free, ising and zero.
Theorem 1.1 of The sharp factor-of-IID threshold for the free Ising model on regular trees
Theorem
For every integer d\ge3 and every \beta\ge0, the free zero-field Ising law \mu_{d,\beta} is a factor of IID in the sense of (eq:equivariance) if and only if
\tanh\beta\le(d-1)^{-1/2}.
In this range the factor can be chosen to commute with every automorphism on every label input.
For every integer $d\ge3$ and every $\beta\ge0$, the free zero-field
Ising law $\mu_{d,\beta}$ is a factor of IID in the sense of
\eqref{eq:equivariance} if and only if
\[
\tanh\beta\le(d-1)^{-1/2}.
\]
In this range the factor can be chosen to commute with every
automorphism on every label input.
Other possible paper matches, closest wording first
Proposition 8.6 of The sharp factor-of-IID threshold for the free Ising model on regular trees. Words of the Lean statement, its names split into words, that also appear in the theorem: map, spins, iid, tanh, ising, aut and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: surely, almost and automorphism.
Proposition 6.1 of The sharp factor-of-IID threshold for the free Ising model on regular trees. Words of the Lean statement, its names split into words, that also appear in the theorem: real, pi, measurable, aut, finite, integer and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: independent and almost.
Proposition 7.3 of The sharp factor-of-IID threshold for the free Ising model on regular trees. Words of the Lean statement, its names split into words, that also appear in the theorem: length, weighted, finite, integer and eq. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounds.
Proposition 7.2 of The sharp factor-of-IID threshold for the free Ising model on regular trees. Words of the Lean statement, its names split into words, that also appear in the theorem: eta, first, uniform, vertex and integer.
Proposition 4.1 of The sharp factor-of-IID threshold for the free Ising model on regular trees. Words of the Lean statement, its names split into words, that also appear in the theorem: product, measurable, infinite, vertex, finite, 1/2 and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: zero, surely, bound, almost and field.
Proposition 5.3 of The sharp factor-of-IID threshold for the free Ising model on regular trees. Words of the Lean statement, its names split into words, that also appear in the theorem: threshold and theta.
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 236’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.
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: free_ising_factor_iff_threshold : ∀ d β, 3 ≤ d → 0 ≤ β → (IsFactorOfIID d (tanh β) ↔ tanh β ≤ 1 / √(d-1)), with TreeVertex d the reduced words in Fin d (the d-regular tree), HasFreeIsingLaw θ the free zero-field Ising cylinder weights 1/2 ∏_edges (1 ± θ)/2, and IsFactorOfIID = ∃ measurable Phi pushing iid Uniform[0,1] labels to that law and a.s. equivariant for each fixed tree automorphism g. Exactly the summary's iff, including equality.
OpenAI’s own note on what this Lean statement covers
“The formalized result determines the factor-of-IID threshold for the free zero-field Ising law on the infinite $d$-regular tree. For every $d\ge3$ and $\beta\ge0$, the law is a factor of IID exactly when $\tanh\beta\le1/\sqrt{d-1}$, including equality. The factor uses independent uniform vertex labels and is almost surely equivariant for each fixed tree automorphism. Finitary coding and coding-radius bounds are not included.”
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
95 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: 49 to its own declarations, 32 to Mathlib’s documentation. 18 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.
For every integer d\ge3 and every \beta\ge0, the free zero-field Ising law \mu_{d,\beta} is a factor of IID in the sense of (eq:equivariance) if and only if
\tanh\beta\le(d-1)^{-1/2}.
In this range the factor can be chosen to commute with every automorphism on every label input.
For every integer $d\ge3$ and every $\beta\ge0$, the free zero-field
Ising law $\mu_{d,\beta}$ is a factor of IID in the sense of
\eqref{eq:equivariance} if and only if
\[
\tanh\beta\le(d-1)^{-1/2}.
\]
In this range the factor can be chosen to commute with every
automorphism on every label input.
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
16 local vocabulary declarations
Lines of Lean, without comments or blank lines
61
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
Factor-of-IID threshold for the free Ising model
Family 236: The exact factor-of-IID threshold for free Ising spins on trees
Candidate pairing
Theorem
For every integer d\ge3 and every \beta\ge0, the free zero-field Ising law \mu_{d,\beta} is a factor of IID in the sense of (eq:equivariance) if and only if
\tanh\beta\le(d-1)^{-1/2}.
In this range the factor can be chosen to commute with every automorphism on every label input.
The sharp factor-of-IID threshold for the free Ising model on regular trees
TeX label thm:main, build/sections/01-introduction.tex, line 38
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.