Chauve--Zhang condensed-neighborhood unary minimality

Chauve and Zhang, On the size of the neighborhoods of a word, arXiv:2505.13796v2, Section 6, printed page 7, ask:

It was also shown in [6] that unary words have the smallest neighborhoods among all words of the same length over a given alphabet, thus leading to lower bounds for the size of neighborhoods. It is thus natural to ask if a similar property holds for condensed and super condensed neighborhoods, namely that unary words have the smallest condensed or super condensed neighborhoods.

This dossier tracks the condensed assertion. Equations (2.1)--(2.2) define N(w,d) as the words at Levenshtein distance at most d and CN(w,d) as the members of N(w,d) with no proper prefix in N(w,d). The assertion asks whether, for every finite alphabet, length n, radius d, letter a, and word w of length n,

|CN(a^n,d)| <= |CN(w,d)|.

The paper proves exact cardinality formulas for condensed neighborhoods of unary words. The closing question asks whether those values are lower bounds for arbitrary words, extending the known minimality of unary words for whole Levenshtein neighborhoods.

Version 2 retains the closing question. The bounded literature check recorded in issue #8618 covered the arXiv version history, exact topic searches, the paper's cited whole-neighborhood result, and the available Semantic Scholar citation data. It found no proof or refutation of the condensed assertion in that scope. Google Scholar and exhaustive worldwide literature were not checked.

Use the binary alphabet. At length three and radius two, CN(000,2) = {0,10,110} while CN(001,2) = {0,1}. Hence the proposed condensed inequality specializes to 3 <= 2.

The source-form distance is the infimum of alignment costs. The public identity levenshtein_eq_dlev proves that the executable Wagner--Fischer recurrence computes this distance, so the finite condensed-neighborhood calculation applies to the paper's stated metric.

The route would fail if the displayed neighborhood enumeration were incorrect under the minimum-alignment-cost definition. A failure of levenshtein_eq_dlev, or a proof that the displayed three-element set has cardinality at most the corresponding two-element set, would also invalidate the refutation.

• Frozen identity: D5/S1/Words/ChauveZhangUnaryNeighborhoodRefutation.levenshtein_eq_dlev.
• Frozen condensed refutation: D5/S1/Words/ChauveZhangUnaryNeighborhoodRefutation.resultCondensed.
• The formal claim quantifies the alphabet, length, radius, unary letter, and comparison word, and retains the source's unrestricted radius.

theorem; resolution refuted. The identity and condensed refutation have proof_shape: content. The identity has admission_basis: escape-witness; the published-question refutation has admission_basis: open-problem-resolution under preregistration #8618.

The literature review is bounded and does not establish exhaustive absence of corrections, unpublished work, paywalled citing papers, or unindexed resolutions. No worldwide novelty or publication-priority claim is made.