Remark: Family 099. The sharp distortion of edit distance into 1

Determines the least distortion of embedding edit distance on words of length at most $d$ into real ${\ell }_1$: 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.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.