Matching exponential-scale bounds for edit-distance distortion
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.edit_distance_main
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.
inductiveScript (substitutions : Bool) {Alpha : Type u} : ℕ → List Alpha → List Alpha → Prop | nil (x : List Alpha) : Script substitutions 0 x x | cons {n : ℕ} {x y z : List Alpha} (first : Step substitutions x y) (rest : Script substitutions n y z) : Script substitutions (n + 1) x z
noncomputabledefdistortion {X : Type u} (ρ : X → X → ℝ) (f : X ↪ L1) : 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‖}
defBinaryWitness (c : ℝ) (d : ℕ) : Prop := ∃ n : ℕ, n ≤ d ∧ ∃ S : Set (List Bool), S.Finite ∧ S.Nonempty ∧ (∀ w ∈ S, w.length = n) ∧ exponentScale c d ≤ leastDistortion (fun x y : S => (edit x.val y.val : ℝ))
defMainClaim : Prop := ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → (∀ (Alpha : Type u) [Fintype Alpha], 2 ≤ Fintype.card Alpha → exponentScale c d ≤ E Alpha d ∧ E Alpha d ≤ exponentScale C d) ∧ BinaryWitness c d ∧ exponentScale c d ≤ alphabetSup d ∧ alphabetSup d ≤ exponentScale C d
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: witnessed, common, length, one, binary, bound, sqrt, alphabet and five more. Found in the paper’s title, which its main theorem states: exponent, edit and distance. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: strings and absolute.
Statements in Edit Distance in l1: Matching Bounds up to Constants in the Exponent come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of 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,
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
$\sup_{\Sigma:\,2\le|\Sigma|<\infty}E_\Sigma(d)$.
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: two, common, length, least, one, distortion and eleven 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: ell 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.
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: threshold, insertion, unit, cost, deletion, one and twelve more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ell and absolute.
Proposition 4.2 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: least, distortion, exp, constant and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: large, sufficiently, ell and 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: least, edit, distortion, word, distance, constant and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: large, sufficiently and absolute.
Proposition 5.2 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: uniform, bound, sqrt, alphabet, exp, constant and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 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 69 s; peak memory 6.2 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 determines the exponential scale of the least $\ell_1$ distortion of unit-cost edit distance on strings of length at most $d$. For every sufficiently large $d$, uniformly over finite alphabets with at least two symbols, the distortion lies between $\exp(c\sqrt{\log d\,\log\log d})$ and $\exp(C\sqrt{\log d\,\log\log d})$ for absolute constants $c,C>0$.”
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
70 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: 27 to its own declarations, 33 to Mathlib’s documentation. 13 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 EditDistortionuniverse u/-- A single unit-cost insertion, deletion, or (when enabled) substitution. -/inductiveStep (substitutions : Bool) {Alpha : Type u} : List Alpha → List Alpha → Prop | insert (p q : List Alpha) (a : Alpha) : Step substitutions (p ++ q) (p ++ a :: q) | delete (p q : List Alpha) (a : Alpha) : Step substitutions (p ++ a :: q) (p ++ q) | substitute (h : substitutions = true) (p q : List Alpha) (a b : Alpha) :Step substitutions (p ++ a :: q) (p ++ b :: q)/-- An unrestricted script of exactly n unit edits. -/inductiveScript (substitutions : Bool) {Alpha : Type u} : ℕ → List Alpha → List Alpha → Prop | nil (x : List Alpha) : Script substitutions 0 x x | cons {n : ℕ} {x y z : List Alpha} (first : Step substitutions x y) (rest : Script substitutions n y z) : Script substitutions (n + 1) x z/-- Minimum script length. Scripts always exist (delete x and insert y). -/noncomputabledefedit {Alpha : Type u} (x y : List Alpha) : ℕ :=sInf {n | Scripttrue n x y}abbrevL1 := lp (fun _ : ℕ => ℝ) 1/-- The product of the two maximal distance ratios. -/noncomputabledefdistortion {X : Type u} (ρ : X → X → ℝ) (f : X ↪ L1) : 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 injective maps to real absolutely summable sequences. -/noncomputabledefleastDistortion {X : Type u} (ρ : X → X → ℝ) : ℝ :=sInf (Set.range (distortion ρ))abbrevWords (Alpha : Type u) (d : ℕ) := {x : List Alpha // x.length ≤ d}noncomputabledefE (Alpha : Type u) (d : ℕ) : ℝ :=leastDistortion (fun x y : Words Alpha d => (edit x.val y.val : ℝ))noncomputabledefexponentScale (c : ℝ) (d : ℕ) : ℝ :=Real.exp (c * Real.sqrt (Real.log (d : ℝ) * Real.log (Real.log (d : ℝ))))/-- Supremum over finite alphabet cardinalities. -/noncomputabledefalphabetSup (d : ℕ) : ℝ :=sSup {r : ℝ | ∃ n : ℕ, 2 ≤ n ∧ r = E (Fin n) d}/-- A finite nonempty subset of binary words of one common length. -/defBinaryWitness (c : ℝ) (d : ℕ) : Prop := ∃ n : ℕ, n ≤ d ∧ ∃ S : Set (ListBool), S.Finite ∧ S.Nonempty ∧ (∀ w ∈ S, w.length = n) ∧exponentScale c d ≤ leastDistortion (fun x y : S => (edit x.val y.val : ℝ))/-- Uniform two-sided bounds with common positive constants and threshold. -/defMainClaim : Prop := ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → (∀ (Alpha : Type u) [Fintype Alpha], 2 ≤ Fintype.card Alpha →exponentScale c d ≤ E Alpha d ∧ E Alpha d ≤ exponentScale C d) ∧BinaryWitness c d ∧exponentScale c d ≤ alphabetSup d ∧ alphabetSup d ≤ exponentScale C dend EditDistortionuniverse utheoremedit_distance_main : EditDistortion.MainClaim.{u} := bysorryend 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
\[
c_1(\mathcal W,\ED)
\geq \exp\!\left(c\sqrt{\log d\,\log\log d}\right).
\]
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).
Edit Distance in l1: Matching Bounds up to Constants in the Exponent
TeX label thm:main, build/sections/introduction.tex, line 28
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.