{
  "abstract": "For a finite real alphabet $T$, let $g(T)$ be the supremum of the lengths of words over $T$ containing no two adjacent nonempty blocks of equal length and equal sum. We classify all four-element alphabets with $g(T)<139$. Up to affine equivalence, these are the known endpoint-sum family and seven isolated classes. Exhaustive finite searches give the isolated values, while 77 explicit words prove that every remaining real alphabet admits a word of length 139. The latter argument uses exact intersections of finite unions of lines, so it covers irrational parameters as well. The cutoff is sharp because $g(\\{0,2,3,7\\})=139$. We also obtain a sharp cutoff of 167 within the family $\\{0,1,x,x+2\\}$, $x>1$. The required supplement prints the word certificates and specifies the complete-search evidence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02215",
  "author_total": 1,
  "authors": [
    {
      "name": "Enkai Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fqy4j7bq38k6z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02215",
  "fulltext_url": null,
  "id": "fqy4j7bq38k6z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02215",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7220442903",
  "pdf_url": "https://arxiv.org/pdf/2610.02215",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fqy4j7bq38k6z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02215",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02215"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The additive-square-free spectrum below 139",
  "title_text": "The additive-square-free spectrum below 139",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/fqy4j7bq38k6z",
  "venue": "arXiv",
  "versions": []
}