Result 099, Convex and metric geometry

The sharp exponential scale of edit-distance distortion

Determines the least distortion of embedding edit distance on words of length at most d into real ℓ1: it is exp⁡(Θ(log⁡d log⁡log⁡d))\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.

Lean formalization New or sharp bound

The bigger picture

Why it matters

Edit distance counts the fewest unit-cost insertions, deletions and substitutions needed to transform one string into another. The manuscripts claim sharp limits on how faithfully this distance can be represented using a familiar geometry of real vectors.

What changes?

An l1 embedding represents strings by real vectors, measuring distance by summed absolute coordinate differences. Distortion measures the worst multiplicative mismatch after rescaling. For all sufficiently large d, the reported optimal distortion lies between exp(c sqrt(log d log log d)) and exp(C sqrt(log d log log d)). The positive constants c and C are absolute, uniform over finite alphabets with at least two symbols, even when growing with d. The domain contains all strings of length at most d, including the empty string.

What does that help mathematicians do?

The lower bound already arises among binary strings of a single common length. Researchers therefore cannot blame the obstruction on large alphabets or comparisons between unequal lengths. Together with the reported histogram embedding upper bound, this identifies the best possible order of the logarithm of distortion. It rules out substantially more faithful l1 representations, while leaving room to improve constants in the exponent.

Are there practical applications?

The immediate value is foundational: the result quantifies an unavoidable loss when translating edit-distance questions into l1 geometry. It also gives a limit for any computational approach that relies on preserving all these distances through such an embedding. The abstracts describe an embedding construction, but do not establish practical speedups or performance on real-world string data.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

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

September 27, 2026 15 pages

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⁡(clog⁡d log⁡log⁡d)\exp(c\sqrt{\log d\,\log\log d}) and exp⁡(Clog⁡d log⁡log⁡d)\exp(C\sqrt{\log d\,\log\log d}) for absolute constants c,C>0c,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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026,
  author = {{OpenAI}},
  title = {{Edit Distance in $\ell_1$: Matching Bounds up to Constants in the Exponent}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026/paper.pdf}{OAI:Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026}},
  year = {2026}
}

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

September 27, 2026 33 pages

We give two finite-circle constructions of binary strings whose least ℓ1 distortion is exp⁡(Ω(log⁡d log⁡log⁡d))\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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026,
  author = {{OpenAI}},
  title = {{Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf}{OAI:Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026}},
  year = {2026}
}

Tree Constructions for the l1 Distortion of Binary Edit Distance

September 27, 2026 23 pages

We give two independent constructions of binary words of one length at most d whose ordinary edit-distance metrics require ℓ1 distortion exp⁡(Ω(log⁡d log⁡log⁡d))\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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026,
  author = {{OpenAI}},
  title = {{Tree Constructions for the $\ell_1$ Distortion of Binary Edit Distance}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026/paper.pdf}{OAI:Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/099.md.

The sharp exponential scale of edit-distance distortion

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization determines the exponential scale of the least ℓ1\ell_1 distortion of unit-cost edit distance on strings of length at most dd. For every sufficiently large dd, uniformly over finite alphabets with at least two symbols, the distortion lies between exp⁡(clog⁡d log⁡log⁡d)\exp(c\sqrt{\log d\,\log\log d}) and exp⁡(Clog⁡d log⁡log⁡d)\exp(C\sqrt{\log d\,\log\log d}) for absolute constants c,C>0c,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 ℓ1\ell_1 distortion is at least exp⁡(clog⁡d log⁡log⁡d)\exp(c\sqrt{\log d\,\log\log d}) for all sufficiently large length caps dd. 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⁡(Clog⁡d log⁡log⁡d)\exp(C\sqrt{\log d\,\log\log d}) for all strings of length at most dd, 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 ℓ1\ell_1 distortion of ordinary edit distance on binary words. For every sufficiently large length cap dd, each construction supplies a finite set of at least two binary words of one common length at most dd whose least ℓ1\ell_1 distortion is at least exp⁡(clog⁡d log⁡log⁡d)\exp(c\sqrt{\log d\,\log\log d}) for an absolute c>0c>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.

Comparator links

Result Comparator statement
Matching exponential-scale bounds for edit-distance distortion EditDistance.lean
Finite-circle obstructions and histogram embeddings for edit distance FiniteCircle.lean
Binary edit-distance distortion lower bound BinaryEditLower.lean
Tree-based binary edit-distance distortion lower bound TreeEdit.lean

Posts about this result

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.