OpenAI · 099-02 · 2026-09-27 · Convex and metric geometry
Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance
preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026
Family 099. The sharp distortion of edit distance into .
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Determines the least distortion of embedding edit distance on words of length at most into real : it is . 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 ; binary words already force the lower bound.
We give two finite-circle constructions of binary strings whose least ℓ1 distortion is , 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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 099: The sharp exponential scale of edit-distance distortion. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.