Lean statement

Finite-circle obstructions and histogram embeddings for edit distance

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.finiteCircle_fullMain

Lean statement file: FiniteCircle.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.

See the candidate theorem 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.

def FullMain : Prop :=  ∃ c C a Ccode δ ε : ℝ, ∃ d₀ : ℕ,    0 < c ∧ 0 < C ∧ 0 < a ∧ 0 < Ccode ∧ 0 < δ ∧ 0 < ε ∧    TwoCodes a Ccode δ ε ∧ UniformUpper C d₀ ∧    (∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α → ∀ d : ℕ, d₀ ≤ d →      Real.exp (c * exponentScale d) ≤ E α d ∧      E α d ≤ Real.exp (C * exponentScale d)) ∧    (∀ d : ℕ, d₀ ≤ d →      MaskWitness a Ccode δ c d ∧ ExcludedWitness a Ccode ε c d ∧      Real.exp (c * exponentScale d) ≤ alphabetSupremum d ∧      alphabetSupremum d ≤ Real.exp (C * exponentScale d))⋮theorem finiteCircle_fullMain : FiniteCircle.FullMain := 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.FiniteCircle.EditStep line 11

inductive EditStep {α : Type u} : List α → List α → Prop  | insert (u v : List α) (a : α) : EditStep (u ++ v) (u ++ a :: v)  | delete (u v : List α) (a : α) : EditStep (u ++ a :: v) (u ++ v)  | substitute (u v : List α) (a b : α) : EditStep (u ++ a :: v) (u ++ b :: v)

inductive type OAI.FiniteCircle.EditScript line 16

inductive EditScript {α : Type u} : ℕ → List α → List α → Prop  | nil (x : List α) : EditScript 0 x x  | cons {n : ℕ} {x y z : List α} :      EditStep x y → EditScript n y z → EditScript (n + 1) x z

definition OAI.FiniteCircle.editDistance line 21

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

definition OAI.FiniteCircle.lcs line 24

noncomputable def lcs {α : Type u} (x y : List α) : ℕ :=  sSup {n : ℕ | ∃ z : List α, z <+ x ∧ z <+ y ∧ z.length = n}

definition OAI.FiniteCircle.deficit line 27

noncomputable def deficit {α : Type u} (x y : List α) : ℕ := x.length - lcs x y

abbreviation OAI.FiniteCircle.Words line 29

abbrev Words (α : Type u) (d : ℕ) := {x : List α // x.length ≤ d}

abbreviation OAI.FiniteCircle.RealL1 line 30

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

definition OAI.FiniteCircle.distortion line 32

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

definition OAI.FiniteCircle.optimalDistortion line 39

noncomputable def optimalDistortion {α : Type u} {X : Type v}    (word : X → List α) : ℝ :=  sInf {D | ∃ f : X → RealL1, Function.Injective f ∧ D = distortion word f}

definition OAI.FiniteCircle.E line 43

noncomputable def E (α : Type u) (d : ℕ) : ℝ :=  optimalDistortion (fun x : Words α d => x.val)

definition OAI.FiniteCircle.setDistortion line 46

noncomputable def setDistortion (S : Set (List Bool)) : ℝ :=  optimalDistortion (fun x : S => x.val)

definition OAI.FiniteCircle.alphabetSupremum line 49

noncomputable def alphabetSupremum (d : ℕ) : ℝ :=  sSup {r | ∃ A : ℕ, 2 ≤ A ∧ r = E (Fin A) d}

definition OAI.FiniteCircle.exponentScale line 52

noncomputable def exponentScale (d : ℕ) : ℝ :=  Real.sqrt (Real.log d * Real.log (Real.log d))

definition OAI.FiniteCircle.encode line 55

def encode {α : Type u} {w : ℕ} (c : α → Fin w → Bool) (x : List α) : List Bool :=  x.flatMap (fun a => List.ofFn (c a))

definition OAI.FiniteCircle.interval line 58

def interval {α : Type u} {w : ℕ} (c : α → Fin w → Bool)    (a : α) (s len : ℕ) : List Bool :=  ((List.ofFn (c a)).drop s).take len

definition OAI.FiniteCircle.OffsetLocal line 62

def OffsetLocal {α : Type u} {w : ℕ} (δ : ℝ) (c : α → Fin w → Bool) : Prop :=  ∀ a b : α, a ≠ b → ∀ s t : ℕ,    s + w / 8 ≤ w → t + w / 8 ≤ w →    (lcs (interval c a s (w / 8)) (interval c b t (w / 8)) : ℝ) ≤      (1 - δ) * (w / 8 : ℕ)

definition OAI.FiniteCircle.indexSpan line 68

def indexSpan {w : ℕ} (is : List (Fin w)) : ℕ :=  match is with  | [] => 0  | i :: js => (js.getLastD i).val + 1 - i.val

definition OAI.FiniteCircle.AnchorLocal line 73

def AnchorLocal {α : Type u} {w : ℕ} (ε : ℝ) (c : α → Fin w → Bool) : Prop :=  ∀ a b : α, ∀ is js : List (Fin w),    is.Pairwise (fun i j => i.val < j.val) →    js.Pairwise (fun i j => i.val < j.val) →    is.length = js.length → (w : ℝ) / 4 ≤ is.length →    (indexSpan is - is.length : ℕ) ≤ ε * w →    (indexSpan js - js.length : ℕ) ≤ ε * w →    (∀ ij ∈ is.zip js, c a ij.1 = c b ij.2) →    a = b ∧ ∃ i : Fin w, (i, i) ∈ is.zip js

definition OAI.FiniteCircle.SubstitutionBounds line 83

def SubstitutionBounds {α : Type u} [Fintype α] {w : ℕ}    (a C : ℝ) (c : α → Fin w → Bool) : Prop :=  0 < w ∧ (w : ℝ) ≤ C * Real.log (2 * Fintype.card α) ∧  (∀ L : ℕ, ∀ x y : List α, x.length = L → y.length = L →    a * w * deficit x y ≤ (deficit (encode c x) (encode c y) : ℝ) ∧    deficit (encode c x) (encode c y) ≤ w * deficit x y) ∧  (∀ x y : List α, editDistance (encode c x) (encode c y) ≤ w * editDistance x y)

definition OAI.FiniteCircle.TwoCodes line 91

def TwoCodes (a C δ ε : ℝ) : Prop :=  ∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α →    (∃ (w : ℕ) (c : α → Fin w → Bool),      SubstitutionBounds a C c ∧ OffsetLocal δ c) ∧    (∃ (w : ℕ) (c : α → Fin w → Bool),      SubstitutionBounds a C c ∧ AnchorLocal ε c)

definition OAI.FiniteCircle.period line 98

def period (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) : ℕ := ∏ k : Fin h, p l k

definition OAI.FiniteCircle.circleSize line 100

def circleSize (h : ℕ) (p : ℕ → ℕ → ℕ) : ℕ :=  2 * ∏ l : Fin h, period h p (l.val + 1)

definition OAI.FiniteCircle.MaskPrimes line 103

def MaskPrimes (h : ℕ) (p : ℕ → ℕ → ℕ) : Prop :=  (∀ l k : Fin h, Nat.Prime (p (l.val + 1) k)) ∧  Function.Injective (fun lk : Fin h × Fin h => p (lk.1.val + 1) lk.2) ∧  ∀ l k : Fin h, h ^ 3 ≤ p (l.val + 1) k ∧ p (l.val + 1) k ≤ 2 * h ^ 3

definition OAI.FiniteCircle.ExcludedPrimes line 108

def ExcludedPrimes (m : ℕ) (p : ℕ → ℕ → ℕ) : Prop :=  (∀ l k : Fin m, Nat.Prime (p (l.val + 1) k)) ∧  Function.Injective (fun lk : Fin m × Fin m => p (lk.1.val + 1) lk.2) ∧  ∀ l k : Fin m, m ^ 3 < p (l.val + 1) k ∧ p (l.val + 1) k ≤ m ^ 5

15 more definitions are reached the same way; the full file lists them.

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: two, length, upper, lower, distortion, words, map, bound and seven more. Found in the paper’s title, which its main theorem states: codes, circle, edit and distance. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constants and absolute.

Statements in Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of 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.

Other possible paper matches, closest wording first

Theorem 8.1 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: real, upper, uniform, map, bound, sigma and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constants 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, bound, sigma, sqrt and five more. Found in the paper’s title, which its main theorem states: exponent, matching, edit and distance. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: strings, constants and absolute.
Theorem 4.1 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, substitutions, words, map, alphabet, log and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: gives, constant 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: upper, substitution, uniform, word, distance, map and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: empty, includes, embedding, constants and absolute.
Proposition 3.1 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: state, step, listed and shift.
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: encoded, gamma, length, words, map, alphabet and one 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.

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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 114 s; peak memory 6.3 GiB).
Reported 11 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

“It also gives a finite histogram embedding with distortion at most $\exp(C\sqrt{\log d\,\log\log d})$ for all strings of length at most $d$, uniformly over finite alphabets, including the empty string. Thus the lower and upper exponential scales agree up to absolute constants.”
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

228 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: 120 to its own declarations, 150 to Mathlib’s documentation. 43 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves finiteCircle_fullMain 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 40 of the file’s 42 other declarations; with the theorem itself they span 164 lines. 1 of them is named in the statement itself.

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

inductive EditStepline 11library
inductive EditScriptline 16library
def editDistanceline 21library
def lcsline 24library
def deficitline 27library
abbrev Wordsline 29library
abbrev RealL1line 30library
def distortionline 32library
def optimalDistortionline 39library
def Eline 43library
def setDistortionline 46library
def alphabetSupremumline 49library
def exponentScaleline 52library
def encodeline 55library
def intervalline 58library
def OffsetLocalline 62library
def indexSpanline 68library
def AnchorLocalline 73library
def SubstitutionBoundsline 83library
def TwoCodesline 91library
def periodline 98library
def circleSizeline 100library
def MaskPrimesline 103library
def ExcludedPrimesline 108library
abbrev MaskTagline 113library
def maskTagsline 115library
def MaskAlphabetline 118library
def MaskStateline 128library
def shiftMaskStateline 132library
def maskGammaline 137library
def maskWordline 142library
def ExcludedAlphabetline 151library
def ExcludedStateline 160library
def shiftExcludedStateline 164library
def excludedTranslationline 169library
def excludedWordline 172library
def MaskWitnessline 182library
def ExcludedWitnessline 193library
def UniformUpperline 204library
def FullMainline 211in the statementlibrary
theorem OAI.finiteCircle_fullMainline 225selected theoremlibrary

Other Lean rules that may affect the statement 2

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance maskAlphabetFintype line 122
instance excludedAlphabetFintype line 155

import Mathlibnamespace OAInamespace FiniteCircleopen scoped BigOperators Listuniverse u vinductive EditStep {α : Type u} : List α → List α → Prop  | insert (u v : List α) (a : α) : EditStep (u ++ v) (u ++ a :: v)  | delete (u v : List α) (a : α) : EditStep (u ++ a :: v) (u ++ v)  | substitute (u v : List α) (a b : α) : EditStep (u ++ a :: v) (u ++ b :: v)inductive EditScript {α : Type u} : ℕ → List α → List α → Prop  | nil (x : List α) : EditScript 0 x x  | cons {n : ℕ} {x y z : List α} :      EditStep x y → EditScript n y z → EditScript (n + 1) x znoncomputable def editDistance {α : Type u} (x y : List α) : ℕ :=  sInf {n : ℕ | EditScript n x y}noncomputable def lcs {α : Type u} (x y : List α) : ℕ :=  sSup {n : ℕ | ∃ z : List α, z <+ x ∧ z <+ y ∧ z.length = n}noncomputable def deficit {α : Type u} (x y : List α) : ℕ := x.length - lcs x yabbrev Words (α : Type u) (d : ℕ) := {x : List α // x.length ≤ d}abbrev RealL1 := lp (fun _ : ℕ => ℝ) 1noncomputable def distortion {α : Type u} {X : Type v} (word : X → List α)    (f : X → RealL1) : ℝ :=  (sSup {r : ℝ | ∃ x y : X, x ≠ y ∧    r = ‖f x - f y‖ / (editDistance (word x) (word y) : ℝ)}) *  (sSup {r : ℝ | ∃ x y : X, x ≠ y ∧    r = (editDistance (word x) (word y) : ℝ) / ‖f x - f y‖})noncomputable def optimalDistortion {α : Type u} {X : Type v}    (word : X → List α) : ℝ :=  sInf {D | ∃ f : X → RealL1, Function.Injective f ∧ D = distortion word f}noncomputable def E (α : Type u) (d : ℕ) : ℝ :=  optimalDistortion (fun x : Words α d => x.val)noncomputable def setDistortion (S : Set (List Bool)) : ℝ :=  optimalDistortion (fun x : S => x.val)noncomputable def alphabetSupremum (d : ℕ) : ℝ :=  sSup {r | ∃ A : ℕ, 2 ≤ A ∧ r = E (Fin A) d}noncomputable def exponentScale (d : ℕ) : ℝ :=  Real.sqrt (Real.log d * Real.log (Real.log d))def encode {α : Type u} {w : ℕ} (c : α → Fin w → Bool) (x : List α) : List Bool :=  x.flatMap (fun a => List.ofFn (c a))def interval {α : Type u} {w : ℕ} (c : α → Fin w → Bool)    (a : α) (s len : ℕ) : List Bool :=  ((List.ofFn (c a)).drop s).take lendef OffsetLocal {α : Type u} {w : ℕ} (δ : ℝ) (c : α → Fin w → Bool) : Prop :=  ∀ a b : α, a ≠ b → ∀ s t : ℕ,    s + w / 8 ≤ w → t + w / 8 ≤ w →    (lcs (interval c a s (w / 8)) (interval c b t (w / 8)) : ℝ) ≤      (1 - δ) * (w / 8 : ℕ)def indexSpan {w : ℕ} (is : List (Fin w)) : ℕ :=  match is with  | [] => 0  | i :: js => (js.getLastD i).val + 1 - i.valdef AnchorLocal {α : Type u} {w : ℕ} (ε : ℝ) (c : α → Fin w → Bool) : Prop :=  ∀ a b : α, ∀ is js : List (Fin w),    is.Pairwise (fun i j => i.val < j.val) →    js.Pairwise (fun i j => i.val < j.val) →    is.length = js.length → (w : ℝ) / 4 ≤ is.length →    (indexSpan is - is.length : ℕ) ≤ ε * w →    (indexSpan js - js.length : ℕ) ≤ ε * w →    (∀ ij ∈ is.zip js, c a ij.1 = c b ij.2) →    a = b ∧ ∃ i : Fin w, (i, i) ∈ is.zip jsdef SubstitutionBounds {α : Type u} [Fintype α] {w : ℕ}    (a C : ℝ) (c : α → Fin w → Bool) : Prop :=  0 < w ∧ (w : ℝ) ≤ C * Real.log (2 * Fintype.card α) ∧  (∀ L : ℕ, ∀ x y : List α, x.length = L → y.length = L →    a * w * deficit x y ≤ (deficit (encode c x) (encode c y) : ℝ) ∧    deficit (encode c x) (encode c y) ≤ w * deficit x y) ∧  (∀ x y : List α, editDistance (encode c x) (encode c y) ≤ w * editDistance x y)def TwoCodes (a C δ ε : ℝ) : Prop :=  ∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α →    (∃ (w : ℕ) (c : α → Fin w → Bool),      SubstitutionBounds a C c ∧ OffsetLocal δ c) ∧    (∃ (w : ℕ) (c : α → Fin w → Bool),      SubstitutionBounds a C c ∧ AnchorLocal ε c)def period (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) : ℕ := ∏ k : Fin h, p l kdef circleSize (h : ℕ) (p : ℕ → ℕ → ℕ) : ℕ :=  2 * ∏ l : Fin h, period h p (l.val + 1)def MaskPrimes (h : ℕ) (p : ℕ → ℕ → ℕ) : Prop :=  (∀ l k : Fin h, Nat.Prime (p (l.val + 1) k)) ∧  Function.Injective (fun lk : Fin h × Fin h => p (lk.1.val + 1) lk.2) ∧  ∀ l k : Fin h, h ^ 3 ≤ p (l.val + 1) k ∧ p (l.val + 1) k ≤ 2 * h ^ 3def ExcludedPrimes (m : ℕ) (p : ℕ → ℕ → ℕ) : Prop :=  (∀ l k : Fin m, Nat.Prime (p (l.val + 1) k)) ∧  Function.Injective (fun lk : Fin m × Fin m => p (lk.1.val + 1) lk.2) ∧  ∀ l k : Fin m, m ^ 3 < p (l.val + 1) k ∧ p (l.val + 1) k ≤ m ^ 5abbrev MaskTag (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) := Σ k : Fin h, Fin (p l k)def maskTags (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) : List (MaskTag h p l) :=  (List.ofFn fun k : Fin h => List.ofFn fun b : Fin (p l k) => Sigma.mk k b).flattendef MaskAlphabet (h : ℕ) (p : ℕ → ℕ → ℕ) : ℕ → Type  | 0 => Bool  | l + 1 => MaskTag h p (l + 1) × MaskAlphabet h p linstance maskAlphabetFintype (h : ℕ) (p : ℕ → ℕ → ℕ) (l : ℕ) :    Fintype (MaskAlphabet h p l) := by  induction l with  | zero => exact inferInstanceAs (Fintype Bool)  | succ l ih => exact @instFintypeProd _ _ inferInstance ihdef MaskState (h : ℕ) (p : ℕ → ℕ → ℕ) (M : ℕ) : ℕ → Type  | 0 => ZMod M  | l + 1 => MaskTag h p (l + 1) → MaskState h p M ldef shiftMaskState (h : ℕ) (p : ℕ → ℕ → ℕ) (M : ℕ) :    (l : ℕ) → MaskState h p M l → ZMod M → MaskState h p M l  | 0, v, θ => (show ZMod M from v) + θ  | l + 1, v, θ => fun j => shiftMaskState h p M l (v j) θdef maskGamma (h : ℕ) (p : ℕ → ℕ → ℕ) (M l a : ℕ)    (j : MaskTag h p l) : ZMod M :=  (∑ k : Fin h, ((M / p l k * (a % p l k) : ℕ) : ZMod M)) +    ((j.2.val * (M / p l j.1) * (a % p l j.1) : ℕ) : ZMod M)def maskWord (h : ℕ) (p : ℕ → ℕ → ℕ) (R M : ℕ) :    (l : ℕ) → MaskState h p M l → List (MaskAlphabet h p l)  | 0, v => (List.range M).map (fun b => decide (((b : ZMod M) - (show ZMod M from v)).val < M / 2))  | l + 1, v => (List.range (R * period h p (l + 1))).flatMap fun a =>      (maskTags h p (l + 1)).flatMap fun j =>        (maskWord h p R M l          (shiftMaskState h p M l (v j) (maskGamma h p M (l + 1) a j))).map          (fun x => (j, x))def ExcludedAlphabet (m L : ℕ) : ℕ → Type  | 0 => Fin L × Bool  | l + 1 => Fin m × ExcludedAlphabet m L linstance excludedAlphabetFintype (m L l : ℕ) : Fintype (ExcludedAlphabet m L l) := by  induction l with  | zero => exact inferInstanceAs (Fintype (Fin L × Bool))  | succ l ih => exact @instFintypeProd _ _ inferInstance ihdef ExcludedState (m L : ℕ) : ℕ → Type  | 0 => ZMod L  | l + 1 => Fin m → ExcludedState m L ldef shiftExcludedState (m L : ℕ) :    (l : ℕ) → ExcludedState m L l → ZMod L → ExcludedState m L l  | 0, v, θ => (show ZMod L from v) + θ  | l + 1, v, θ => fun i => shiftExcludedState m L l (v i) θdef excludedTranslation (m : ℕ) (p : ℕ → ℕ → ℕ) (L l s : ℕ) (i : Fin m) :    ZMod L := ∑ j ∈ Finset.univ.erase i, ((L / p l j * (s % p l j) : ℕ) : ZMod L)def excludedWord (m : ℕ) (p : ℕ → ℕ → ℕ) (R L : ℕ) :    (l : ℕ) → ExcludedState m L l → List (ExcludedAlphabet m L l)  | 0, v => List.ofFn fun b : Fin L =>      (b, decide (((b.val : ZMod L) - (show ZMod L from v)).val < L / 2))  | l + 1, v => (List.range (R * period m p (l + 1))).flatMap fun s =>      (List.ofFn fun i : Fin m =>        (excludedWord m p R L l          (shiftExcludedState m L l (v i) (excludedTranslation m p L (l + 1) s i))).map          (fun x => (i, x))).flattendef MaskWitness (a C δ lower : ℝ) (d : ℕ) : Prop :=  ∃ h : ℕ, 2 ≤ h ∧ ∃ p : ℕ → ℕ → ℕ, MaskPrimes h p ∧    ∃ (w : ℕ) (c : MaskAlphabet h p h → Fin w → Bool),      SubstitutionBounds a C c ∧ OffsetLocal δ c ∧      ∃ n : ℕ, 0 < n ∧ n ≤ d ∧        (∀ v : MaskState h p (circleSize h p) h,          (encode c (maskWord h p (h ^ (2 * h)) (circleSize h p) h v)).length = n) ∧        Real.exp (lower * exponentScale d) ≤ setDistortion          (Set.range fun v : MaskState h p (circleSize h p) h =>            encode c (maskWord h p (h ^ (2 * h)) (circleSize h p) h v))def ExcludedWitness (a C ε lower : ℝ) (d : ℕ) : Prop :=  ∃ m : ℕ, 2 ≤ m ∧ ∃ p : ℕ → ℕ → ℕ, ExcludedPrimes m p ∧    ∃ (w : ℕ) (c : ExcludedAlphabet m (circleSize m p) m → Fin w → Bool),      SubstitutionBounds a C c ∧ AnchorLocal ε c ∧      ∃ n : ℕ, 0 < n ∧ n ≤ d ∧        (∀ v : ExcludedState m (circleSize m p) m,          (encode c (excludedWord m p (m ^ (2 * m)) (circleSize m p) m v)).length = n) ∧        Real.exp (lower * exponentScale d) ≤ setDistortion          (Set.range fun v : ExcludedState m (circleSize m p) m =>            encode c (excludedWord m p (m ^ (2 * m)) (circleSize m p) m v))def UniformUpper (C : ℝ) (d₀ : ℕ) : Prop :=  ∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α → ∀ d : ℕ, d₀ ≤ d →    ∃ (M : ℕ) (F : Words α d → Fin M → ℝ), ∀ x y : Words α d,      (editDistance x.val y.val : ℝ) ≤ ∑ i, |F x i - F y i| ∧      (∑ i, |F x i - F y i|) ≤        Real.exp (C * exponentScale d) * editDistance x.val y.valdef FullMain : Prop :=  ∃ c C a Ccode δ ε : ℝ, ∃ d₀ : ℕ,    0 < c ∧ 0 < C ∧ 0 < a ∧ 0 < Ccode ∧ 0 < δ ∧ 0 < ε ∧    TwoCodes a Ccode δ ε ∧ UniformUpper C d₀ ∧    (∀ (α : Type) [Fintype α], 2 ≤ Fintype.card α → ∀ d : ℕ, d₀ ≤ d →      Real.exp (c * exponentScale d) ≤ E α d ∧      E α d ≤ Real.exp (C * exponentScale d)) ∧    (∀ d : ℕ, d₀ ≤ d →      MaskWitness a Ccode δ c d ∧ ExcludedWitness a Ccode ε c d ∧      Real.exp (c * exponentScale d) ≤ alphabetSupremum d ∧      alphabetSupremum d ≤ Real.exp (C * exponentScale d))end FiniteCircletheorem finiteCircle_fullMain : FiniteCircle.FullMain := by  sorryend 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).

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)

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

Configuration and file facts

File: lean/ComparatorChallenges/FiniteCircle.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.FiniteCircle.Main
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: 42 local vocabulary declarations
Lines of Lean, without comments or blank lines: 173
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.

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