Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 099-01. Edit Distance in l1: Matching Bounds up to Constants in the Exponent
Manuscript map · lines 4365–4374
OpenAI@2026-10-07#rem:oai-p-099-01
Authors: OpenAI. Family 099.
Title: Edit Distance in l1: Matching Bounds up to Constants in the Exponent.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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