{
  "abstract": "Every finite nonempty incomplete uniquely decipherable code with maximum word length $k$ has an uncompletable word of length at most $4k^2-3k$. The bound is independent of the number of codewords and their total length. Deleting a complete codeword cycle gives a finite path-counting identity; Kraft equality then supplies a short word of deficient compressed mass. Cyclic averaging and padding turn it into an uncompletable word. Conditional expectation makes the construction polynomial-time and also decides completeness. First-return words extend the bound to mortal families of nonnegative integer $n\\times n$ matrices with joint spectral radius at most one, provided every strongly connected component has a vertex meeting every cycle. Such a family has a zero product of length at most $4n^2-3n$. A binary partial deterministic family with $2k-1$ states has shortest zero product of length $k^2+k-1$, establishing the optimal quadratic order. The bounds and the explicit-code algorithm, including its polynomial work bound, are proved in Lean.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30817",
  "author_total": 2,
  "authors": [
    {
      "name": "Rahul Chandelkar"
    },
    {
      "name": "Samrath Singh Chadha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/edrbrfzcaj35s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30817",
  "fulltext_url": null,
  "id": "edrbrfzcaj35s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30817",
  "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",
      "cs.FL"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7214639431",
  "pdf_url": "https://arxiv.org/pdf/2609.30817",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/edrbrfzcaj35s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.30817",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30817"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Quadratic bounds for uncompletable words and matrix mortality",
  "title_text": "Quadratic bounds for uncompletable words and matrix mortality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/edrbrfzcaj35s",
  "venue": "arXiv",
  "versions": []
}