{
  "abstract": "A binary word is a shuffle square if its positions can be partitioned into two identical subsequences. A finite word over $\\{1,2\\}$ is smooth if repeated finite run-length differentiation remains binary until the empty word is reached. This paper proves that every smooth binary word in which both letters have even multiplicity is a shuffle square once its length is at least $18$. There are exactly $34$ exceptions, all of length at most $16$. The infinite part of the proof is a parity-return atom induction: a smooth even-Parikh word factors uniquely into smooth parity atoms, and every such atom has length at most $8$. A finite, independently checkable certificate handles lengths $18$ through $40$. As a consequence, a nonempty prefix of the classical Kolakoski sequence is a shuffle square if and only if both symbols occur an even number of times and its length is not $4$ or $8$. There are infinitely many such prefixes. In fact, the entire infinite Kolakoski sequence can be partitioned into two identical infinite subsequences with monochromatic blocks of uniformly bounded length.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02262",
  "author_total": 1,
  "authors": [
    {
      "name": "Kory Kilpatrick"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b8wwqqahm6btz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02262",
  "fulltext_url": null,
  "id": "b8wwqqahm6btz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02262",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.FL"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7220429226",
  "pdf_url": "https://arxiv.org/pdf/2610.02262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b8wwqqahm6btz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02262"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Shuffle squares in smooth binary words and the Kolakoski sequence",
  "title_text": "Shuffle squares in smooth binary words and the Kolakoski sequence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/b8wwqqahm6btz",
  "venue": "arXiv",
  "versions": []
}