Lean statement

Binary edit-distance distortion lower bound

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_lower_bound

Lean statement file: BinaryEditLower.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 099 link it.

Read the cited theorem in the paper Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem binary_lower_bound :    ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →      ∃ n : ℕ, 1 ≤ n ∧ n ≤ d ∧        ∃ W : Finset (Word Bool n), 2 ≤ W.card ∧          growth c d ≤ binarySetDistortion W := by  sorry

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.

inductive type OAI.TreeEdit.EditStep line 10

inductive EditStep {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)

inductive type OAI.TreeEdit.EditScript line 16

inductive EditScript {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)

definition OAI.TreeEdit.ed line 21

noncomputable def ed {α : Type u} (x y : List α) : ℕ :=  sInf {n | EditScript x y n}

abbreviation OAI.TreeEdit.RealL1 line 24

abbrev RealL1 := lp (fun _ : ℕ => ℝ) 1

abbreviation OAI.TreeEdit.Word line 25

abbrev Word (α : Type u) (n : ℕ) := {x : List α // x.length = n}

definition OAI.TreeEdit.BinaryLower.distortion line 30

noncomputable def distortion {X : Type u} (ρ : X → X → ℝ) (f : X → RealL1) : ℝ :=  sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ‖f x - f y‖ / ρ x y} *  sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ρ x y / ‖f x - f y‖}

definition OAI.TreeEdit.BinaryLower.c1 line 35

noncomputable def c1 (X : Type u) (ρ : X → X → ℝ) : ℝ :=  sInf {D : ℝ | ∃ f : X → RealL1, Function.Injective f ∧ D = distortion ρ f}

definition OAI.TreeEdit.BinaryLower.binarySetDistortion line 38

noncomputable def binarySetDistortion {n : ℕ} (W : Finset (Word Bool n)) : ℝ :=  c1 {x : Word Bool n // x ∈ W}    (fun x y => (ed x.val.val y.val.val : ℝ))

definition OAI.TreeEdit.BinaryLower.growth line 42

noncomputable def growth (c : ℝ) (d : ℕ) : ℝ :=  Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ))))

Paper theorem cited in the Lean source

A source comment for OAI.TreeEdit.binary_lower_bound names this theorem’s label in the paper’s TeX source. Read the source comment from line 10

/-- Introduction, `thm:binary`: actual equal-length binary witnesses for every
sufficiently large cap, from the finite-tree family and quadratic block code. -/

Theorem 1.1 of Tree Constructions for the l1 Distortion of Binary Edit Distance

Theorem (Binary lower bound)

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).

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

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
\[
 c_1(\mathcal W,\ED)
 \geq \exp\!\left(c\sqrt{\log d\,\log\log d}\right).
\]

Other possible paper matches, closest wording first

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, cost, deletion, substitution, word and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ordinary, one, ell and absolute.
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: two, length, lower, constructions, distortion, binary 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, least, one, ell and one more.
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: large, sufficiently, edit, distortion, word, distance and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ordinary, least and absolute.
Theorem 1.1 of Edit Distance in l1: Matching Bounds up to Constants in the Exponent. Words of the Lean statement, its names split into words, that also appear in the theorem: witnessed, length, lower, binary, bound, sqrt and four 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: common, one 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: length, binary, words, map, ed and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose and absolute.
Theorem 5.1 of Edit Distance in l1: Matching Bounds up to Constants in the Exponent. Words of the Lean statement, its names split into words, that also appear in the theorem: insertions, deletions, unit, cost, substitutions, map and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ell and absolute.

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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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.

MathVet
Finds the whole advertised result stated or implied Their 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)
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

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 $c>0$.”
Quoted from OpenAI’s Lean note for this family

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

60 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: 18 to its own declarations, 27 to Mathlib’s documentation. 10 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves binary_lower_bound exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 9 other declarations of the file; with the theorem itself they span 28 lines. 3 of them are named in the statement itself.

Names in the namespace OAI.TreeEdit are listed without it.

inductive EditStepline 10library
inductive EditScriptline 16library
def edline 21library
abbrev RealL1line 24library
abbrev Wordline 25in the statementlibrary
def BinaryLower.distortionline 30library
def BinaryLower.c1line 35library
def BinaryLower.binarySetDistortionline 38in the statementlibrary
def BinaryLower.growthline 42in the statementlibrary
theorem binary_lower_boundline 51selected theoremlibrary

import Mathlibnamespace OAInamespace TreeEdituniverse u/-- A single unit-cost insertion, deletion, or substitution at any position. -/inductive EditStep {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)/-- An edit script with no restriction on intermediate word lengths. -/inductive EditScript {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)noncomputable def ed {α : Type u} (x y : List α) : ℕ :=  sInf {n | EditScript x y n}abbrev RealL1 := lp (fun _ : ℕ => ℝ) 1abbrev Word (α : Type u) (n : ℕ) := {x : List α // x.length = n}namespace BinaryLower/-- The product of the two maximal pairwise distance ratios. -/noncomputable def distortion {X : Type u} (ρ : X → X → ℝ) (f : X → RealL1) : ℝ :=  sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ‖f x - f y‖ / ρ x y} *  sSup {r : ℝ | ∃ x y : X, x ≠ y ∧ r = ρ x y / ‖f x - f y‖}/-- The infimum over every injective map into the full real sequence space ℓ₁. -/noncomputable def c1 (X : Type u) (ρ : X → X → ℝ) : ℝ :=  sInf {D : ℝ | ∃ f : X → RealL1, Function.Injective f ∧ D = distortion ρ f}noncomputable def binarySetDistortion {n : ℕ} (W : Finset (Word Bool n)) : ℝ :=  c1 {x : Word Bool n // x ∈ W}    (fun x y => (ed x.val.val y.val.val : ℝ))noncomputable def growth (c : ℝ) (d : ℕ) : ℝ :=  Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ))))end BinaryLoweropen BinaryLower/-- The binary lower-bound theorem of OpenAI's September 27, 2026 tree-constructionsmanuscript: finite equal-length witnesses for every sufficiently large length cap. -/theorem binary_lower_bound :    ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d →      ∃ n : ℕ, 1 ≤ n ∧ n ≤ d ∧        ∃ W : Finset (Word Bool n), 2 ≤ W.card ∧          growth c d ≤ binarySetDistortion W := by  sorryend 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.

Edit Distance in l1: Matching Bounds up to Constants in the Exponent

Theorem

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,

\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).

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).

Section 1, Introduction Read it in the paper TeX source, line 28

Show the TeX

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
$\sup_{\Sigma:\,2\le|\Sigma|<\infty}E_\Sigma(d)$.

Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance

Theorem

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,

\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).

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.

Section 1, Introduction Read it in the paper TeX source, line 28

Show the TeX

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.

Tree Constructions for the l1 Distortion of Binary Edit Distance

Theorem (Binary lower bound)

c_1(\mathcal W,\ED) \geq \exp\!\left(c\sqrt{\log d\,\log\log d}\right).

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

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
\[
 c_1(\mathcal W,\ED)
 \geq \exp\!\left(c\sqrt{\log d\,\log\log d}\right).
\]

Configuration and file facts

File: lean/ComparatorChallenges/BinaryEditLower.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 099 link it.
Solution module: OAI.Combinatorics.BinaryEdit.BinaryLowerBound
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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 28
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas