The least distortion of embedding edit distance into l1

Theorem 1.1: there are absolute $c,C>0$ and $d_0$ with $\exp(c\sqrt{\log d\log\log d})\le E_\Sigma(d)\le\exp(C\sqrt{\log d\log\log d})$ for all $d\ge d_0$ and all finite $|\Sigma|\ge2$ , so the Ostrovsky-Rabani scale is sharp up to constants in the exponent. The upper bound is Ostrovsky-Rabani's method made uniform over alphabets and lengths; the new content is the matching lower bound, given by three constructions across the family. The constants $c,C$ are not determined, and nothing is said about computational efficiency.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 27, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The least distortion of embedding edit distance into l1 · Edit distance into l1: distortion

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed record: The least distortion of embedding edit distance into l1

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question: absolute $c,C$ with $\exp(c\sqrt{\log d\log\log d})\le E_\Sigma(d)\le\exp(C\sqrt{\log d\log\log d})$ for all large $d$ , uniformly over finite alphabets of size at least two, the lower bound witnessed by binary strings of one length. The challenge is not in lean/formalization.yaml; lean/ComparatorChallenges/EditDistance.json exists with solution_module OAI.Combinatorics.EditDistance.Main, whose file exists at the pinned commit. The statement EditDistance.lean was read here; not rebuilt here. It defines edit distance by unrestricted unit-step scripts with substitutions, distortion as the product of the two extreme ratios over injective maps into $\ell^1(\mathbb N)$ , and states the two-sided bounds for every finite alphabet type, the binary witness and the alphabet supremum. This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.