bibkey: chauve2025neighborhoods authors: Cedric Chauve, Louxin Zhang year: 2025 title: On the size of the neighborhoods of a word doi: 10.48550/arXiv.2505.13796 url: https://arxiv.org/abs/2505.13796v2 claim: The closing question asks whether unary words minimize condensed and super condensed neighborhoods. strata_touched:

• D5/S1/Words/ChauveZhangUnaryNeighborhoodRefutation license: citation-only triage: anchor

On the size of the neighborhoods of a word

On printed page 2, the paper defines Levenshtein distance first as the minimum number of insertion, deletion and substitution edits, and states that this equals the minimum cost of an alignment. Its two-row alignments have no gap-over-gap column; the cost counts unequal-letter or gap columns. Equations (2.1)–(2.3) on the same page define the Levenshtein neighborhood, the condensed neighborhood obtained by excluding words with a proper neighborhood prefix, and the super condensed neighborhood obtained by excluding words with a proper neighborhood subword.

The conclusion on printed page 7 asks whether unary words have the smallest condensed or super condensed neighborhoods among words of the same length over a fixed alphabet. Version 2, dated September 3, 2025, retains that question.

This note attests the definitions and the published closing question, not the repository’s refutations. Bounded arXiv and Semantic Scholar searches found no later settlement; that scope is not an exhaustive claim about all literature.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2505.13796
• URL: https://arxiv.org/abs/2505.13796v2
• Version and location: arXiv:2505.13796v2, printed pages 2 and 7, equations (2.1)–(2.3) and Section 6.