The sharp exponential scale of edit-distance distortion
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is \(\exp(\Theta(\sqrt{\log d\,\log\log d}))\). Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
The other possible matches are Theorem 1.1 in Edit Distance in l1 () and Theorem 1.1(Binary lower bound) in Tree Constructions for the l1 Distortion of Binary Edit Distance ().
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,
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.
Reported proof verification
Four Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We determine the exponential scale of the least ℓ1 distortion of unit-cost edit distance on all strings of length at most d. For every sufficiently large d, uniformly over finite alphabets of size at least two, the distortion lies between \(\exp(c\sqrt{\log d\,\log\log d})\) and \(\exp(C\sqrt{\log d\,\log\log d})\) for absolute constants \(c,C\gt 0\). The lower bound already holds on binary strings of one common length. Thus the order of logarithmic distortion is sharp up to absolute constants.
We give two finite-circle constructions of binary strings whose least ℓ1 distortion is \(\exp(\Omega(\sqrt{\log d\,\log\log d}))\), where d bounds their length. Both constructions supply words of one common length for every sufficiently large cap. Two direct binary coding arguments transfer the constructions with absolute distortion and logarithmic block width. We also develop the overlapping-substring method of Ostrovsky and Rabani into a complete finite histogram embedding at the same exponential scale, uniformly over all finite alphabets and all words of length at most d, including the empty word.
We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ1 distortion \(\exp(\Omega(\sqrt{\log d\,\log\log d}))\) for every sufficiently large d. We also prove a constant-distortion binary conversion for one prescribed input length. Together with the companion upper embedding theorem, these lower bounds determine the order of logarithmic distortion uniformly over finite alphabets with at least two symbols.
Four Lean statements in this family
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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
The Lean source’s comment reads: /-- Introduction, thm:binary: actual equal-length binary witnesses for every
sufficiently large cap, from the finite-tree family and quadratic block code. -/
The Lean statement whose source cites this theorem
The Lean source cites this theorem. No review of whether the statements mean the same thing is recorded.
BinaryEditLower.lean, the release lists it as a supporting statement.
theorembinary_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 := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
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).
The words they share: witnessed, common, length, one, binary, bound, sqrt, alphabet, exp, constants, log, finite, integer, exponent, edit, distance, strings, absolute.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
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.
The words they share: two, length, upper, lower, distortion, words, map, bound, sigma, sqrt, alphabet, exp, log, finite, integer, codes, circle, edit, distance, constants, absolute.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
FiniteCircle.lean, the release lists it as a supporting statement.
defFullMain : 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))⋮theoremfiniteCircle_fullMain : FiniteCircle.FullMain := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
TreeEdit.lean, the release lists it as a supporting statement.
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
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
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Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:None.
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)
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
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 $c,C>0$. The lower bound has a witness consisting of binary strings of one common length, and the same scale controls the supremum over finite alphabets.
The formalization gives two finite-circle constructions of binary strings whose least $\ell_1$ distortion is at least $\exp(c\sqrt{\log d\,\log\log d})$ for all sufficiently large length caps $d$. The witnesses use strings of one common length and include the binary coding transfers with their stated uniform bounds.
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.
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 $c>0$.
The selected statements are the binary lower bounds. The paper’s constant-distortion binary conversion and the companion upper embedding theorem are outside them.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the tree-constructions paper: for every large length cap d, a set of binary words of one common length with ℓ1 distortion at least exp(c√(log d log log d)). The paper’s binary conversion and upper embedding are outside it.
Matching exponential-scale bounds for edit-distance distortionEditDistance
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the least ℓ1 distortion of edit distance on strings of length at most d lies between exp(c√(log d log log d)) and exp(C√(log d log log d)), uniformly over finite alphabets.
Finite-circle obstructions and histogram embeddings for edit distanceFiniteCircle
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the finite-circle paper: finite-circle lower-bound witnesses and a histogram embedding together give matching exponential scales for edit-distance distortion.
Unreviewed AI draft: states a paper’s theorem as the paper does. States the same binary lower bound as BinaryEditLower, the main theorem of the tree-constructions paper, in a separately written statement file.
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What these papers cite in the release, and what cites them
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Address a question raised in an independent assessment
Atlas records no review confirming that the two statements mean the same thing.
Independent assessment
MathVet
Their term
“full”
What their term means
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 69 s; peak memory 6.2 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 114 s; peak memory 6.3 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 108 s; peak memory 6.3 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
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.
Matching exponential-scale bounds for edit-distance distortion
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
Finite-circle obstructions and histogram embeddings for edit distance
Role
the release lists it as a supporting statement
Paired with
Theorem 1.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.
No paper theorem is paired with this Lean statement.
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.