{
  "abstract": "For every right half we determine the unique left half whose Rule 30 central trace is constant: an alternating tail selected by the leading one of the right half when the initial center is 0, and a single universal checkerboard when it is 1. Every nonzero member of either fiber carries infinitely many ones, so the zero row is the only finite configuration with a constant trace. For support radius w the sharp maximum constant-prefix length is 2*ceil(w/2)+1 when the initial center is 0 and 2*floor(w/2)+2 when it is 1, so the maximum over both is w+2, attained by exactly 2^w configurations for even w and 2^w - 1 for odd w. Hence no column of a nonzero finite Rule 30 orbit is eventually constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09431",
  "author_total": 1,
  "authors": [
    {
      "name": "David L. Condrey"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/729mxbtawyt4x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09431",
  "fulltext_url": null,
  "id": "729mxbtawyt4x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09431",
  "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.DS"
    ],
    [
      "arxiv",
      "nlin.CG"
    ],
    [
      "openalex",
      "1703|Cellular Automata and Applications"
    ]
  ],
  "openalex": "W7212108149",
  "pdf_url": "https://arxiv.org/pdf/2609.09431",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/729mxbtawyt4x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.09431",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09431"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.CG",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Finite Configurations Cannot Generate a Constant Trace in Rule 30",
  "title_text": "Finite Configurations Cannot Generate a Constant Trace in Rule 30",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/729mxbtawyt4x",
  "venue": "arXiv",
  "versions": []
}