Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 099-02. Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance
Manuscript map · lines 4375–4384
OpenAI@2026-10-07#rem:oai-p-099-02
Authors: OpenAI. Family 099.
Title: Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance.
Abstract: We give two finite-circle constructions of binary strings whose least ℓ1 distortion is ${}exp({}Omega({}sqrt{{}log d{},{}log{}log d}))$, 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.
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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