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.TreeEdit.binary_torus
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
defBinaryLowerBound : Prop := ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∃ n : ℕ, n ≤ d ∧ ∃ W : Finset (List Bool),2 ≤ W.card ∧ (∀ w ∈ W, w.length = n) ∧ Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ)))) ≤ c1 W⋮theorembinary_torus : BinaryLowerBound := 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.
inductiveEditScript {α : Type u} : List α → List α → ℕ → Prop | nil (x : List α) : EditScript x x 0 | cons {x y z : List α} {n : ℕ} : EditStep x y → EditScript y z n → EditScript x z (n + 1)
noncomputabledefc1 (W : Finset (List Bool)) : ℝ := sInf {D : ℝ | ∃ f : W → RealL1, Function.Injective f ∧ distortion W f = D}
No confident match with a paper statement
No paper theorem is paired with this Lean statement.
Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements in Tree Constructions for the l1 Distortion of Binary Edit Distance come first because the release’s name for this Lean statement shares words with that paper’s title.
Other possible paper matches, closest wording first
Theorem 1.1 of Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance. Words of the Lean statement, its names split into words, that also appear in the theorem: dimensional, two, length, least, lower, distortion and eight more. Found in the paper’s title, which its main theorem states: edit and distance. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: supplies, whose, common, one, constructions, words and two more.
Theorem 1.1 of Tree Constructions for the l1 Distortion of Binary Edit Distance. Words of the Lean statement, its names split into words, that also appear in the theorem: lower, binary, bound, ed, sqrt, exp and three more. Found in the paper’s title, which its main theorem states: edit, distortion and distance. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: absolute.
Proposition 3.5 of Tree Constructions for the l1 Distortion of Binary Edit Distance. Words of the Lean statement, its names split into words, that also appear in the theorem: ordinary, least, large, sufficiently, edit, distortion and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: word and absolute.
Theorem 6.1 of Tree Constructions for the l1 Distortion of Binary Edit Distance. Words of the Lean statement, its names split into words, that also appear in the theorem: insertion, unit, ordinary, cost, deletion, substitution and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one, word, ell and absolute.
Theorem A.1 of Tree Constructions for the l1 Distortion of Binary Edit Distance. Words of the Lean statement, its names split into words, that also appear in the theorem: precisely, length, binary, map, ed and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose, words and absolute.
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
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.
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.
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).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 108 s; peak memory 6.3 GiB).
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 099’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
Definitions the group asks readers to compare with the paper: Hand-rolled distortion ρ f := sSup {‖f x - f y‖/ρ x y} * sSup {ρ x y/‖f x - f y‖} and leastDistortion := sInf (range distortion) over injective maps into lp 1 (standard distortion; junk values only for degenerate sets with fewer than two points)
OpenAI’s own note on what this Lean statement covers
“The formalization gives two lower-bound constructions for the $\ell_1$ distortion of ordinary edit distance on binary words. For every sufficiently large length cap $d$, each construction supplies a finite set of at least two binary words of one common length at most $d$ whose least $\ell_1$ distortion is at least $\exp(c\sqrt{\log d\,\log\log d})$ for an absolute $c>0$.”
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
51 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: 15 to its own declarations, 32 to Mathlib’s documentation. 8 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.
import Mathlibnamespace OAInamespace TreeEdituniverse u/-- A single unit-cost insertion, deletion or substitution, at any position. -/inductiveEditStep {α : Type u} : List α → List α → Prop | insert (p q : List α) (a : α) : EditStep (p ++ q) (p ++ a :: q) | delete (p q : List α) (a : α) : EditStep (p ++ a :: q) (p ++ q) | substitute (p q : List α) (a b : α) : EditStep (p ++ a :: q) (p ++ b :: q)/-- A script of precisely `n` edits. There is no restriction on its intermediate lengths. -/inductiveEditScript {α : Type u} : List α → List α → ℕ → Prop | nil (x : List α) : EditScript x x 0 | cons {x y z : List α} {n : ℕ} :EditStep x y → EditScript y z n → EditScript x z (n + 1)/-- Ordinary unit-cost edit distance. The set is nonempty (delete then insert). -/noncomputabledefed {α : Type u} (x y : List α) : ℕ :=sInf {n | EditScript x y n}/-- The full infinite-dimensional real ℓ₁ space. -/abbrevRealL1 := lp (fun _ : ℕ => ℝ) 1/-- Product of the two maximal pairwise distance ratios. -/noncomputabledefdistortion (W : Finset (ListBool)) (f : W → RealL1) : ℝ :=sSup {r : ℝ | ∃ x y : W, x ≠ y ∧ r = dist (f x) (f y) / (ed x.val y.val : ℝ)} *sSup {r : ℝ | ∃ x y : W, x ≠ y ∧ r = (ed x.val y.val : ℝ) / dist (f x) (f y)}/-- Infimum over all injective ℓ₁ maps, used on finite sets with at least two points. -/noncomputabledefc1 (W : Finset (ListBool)) : ℝ :=sInf {D : ℝ | ∃ f : W → RealL1, Function.Injective f ∧ distortion W f = D}/-- The binary main endpoint, for every sufficiently large cap (natural logarithms). -/defBinaryLowerBound : Prop := ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∃ n : ℕ, n ≤ d ∧ ∃ W : Finset (ListBool),2 ≤ W.card ∧ (∀ w ∈ W, w.length = n) ∧Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ)))) ≤ c1 Wtheorembinary_torus : BinaryLowerBound := bysorryend TreeEditend 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.
The lower bound is witnessed by a subset of binary strings of one common length at most d. The same bounds hold for \sup_{\Sigma:\,2\le|\Sigma|<\infty}E_\Sigma(d).
There are absolute constants $c,C>0$ and an integer $d_0$ such that, for
every integer $d\ge d_0$ and every finite alphabet $\Sigma$ with
$|\Sigma|\ge2$,
\begin{equation}
\exp\!\bigl(c\sqrt{\log d\,\log\log d}\bigr)
\le E_\Sigma(d)\le
\exp\!\bigl(C\sqrt{\log d\,\log\log d}\bigr).
\end{equation}
The lower bound is witnessed by a subset of binary strings of one common
length at most $d$. The same bounds hold for
$\sup_{\Sigma:\,2\le|\Sigma|<\infty}E_\Sigma(d)$.
Each of our two lower constructions supplies a binary subset whose words have one common length at most d and whose least \ell_1 distortion satisfies the lower bound. The upper bound is realized by a deterministic finite-dimensional map.
There are absolute constants $c,C>0$ and an integer $d_0$ such that, for
every integer $d\geq d_0$ and every finite alphabet $\Sigma$ with
$|\Sigma|\geq2$,
\begin{equation}
\exp\!\bigl(c\sqrt{\log d\,\log\log d}\bigr)
\leq E_\Sigma(d)\leq
\exp\!\bigl(C\sqrt{\log d\,\log\log d}\bigr).
\end{equation}
Each of our two lower constructions supplies a binary subset whose words
have one common length at most $d$ and whose least $\ell_1$ distortion
satisfies the lower bound. The upper bound is realized by a deterministic
finite-dimensional map.
There are an absolute constant c>0 and an integer d_0 such that, for every integer d\geq d_0, there exist an integer 1\leq n\leq d and a finite set \mathcal W\subseteq\{0,1\}^n with |\mathcal W|\geq2 satisfying
There are an absolute constant $c>0$ and an integer $d_0$ such that,
for every integer $d\geq d_0$, there exist an integer $1\leq n\leq d$ and a
finite set $\mathcal W\subseteq\{0,1\}^n$ with $|\mathcal W|\geq2$
satisfying
\[
c_1(\mathcal W,\ED)
\geq \exp\!\left(c\sqrt{\log d\,\log\log d}\right).
\]
Family 099: The sharp exponential scale of edit-distance distortion
No paper statement selected
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
The Lean statement says
defBinaryLowerBound : Prop := ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∃ n : ℕ, n ≤ d ∧ ∃ W : Finset (List Bool),2 ≤ W.card ∧ (∀ w ∈ W, w.length = n) ∧ Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ)))) ≤ c1 W⋮theorembinary_torus : BinaryLowerBound := 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.