{
  "abstract": "We study one-dimensional binary cellular automata with a five-slot radius-two local rule of exact algebraic-normal-form degree two, acting on periodic rings of length n. We prove that every such rule is non-injective whenever 4 | n and n >= 8. The proof begins with the four-cell collapse, where the two extreme formal slots coincide. A structural classification of the resulting four-variable maps separates the 65,472 exactly quadratic rules into 63,456 rules with an immediate ring-four collision and 2,016 exceptional lifts. The latter split into layers of sizes 480 and 1,536. Their remaining finite obligations are represented by 136 parameter-region constructions and 768 per-lift records, respectively. Each certificate supplies differentiating closed walks of lengths 8 and 12 with a common pair-graph base vertex. Concatenation then gives lengths 8a + 12b, which are exactly the multiples of four from eight onward. The load-bearing finite certificate core therefore contains 904 = 136 + 768 independently replayable objects checked by standalone, non-searching programs. The complete checker CLIs additionally reconstruct expected populations and execute coverage, complement, and regression/guard checks; 904 is not a count of total checker operations. Periodic extension also yields full-shift non-injectivity; that consequence is used here only as a corollary.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35921",
  "author_total": 1,
  "authors": [
    {
      "name": "Houqiao Fu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11zqftvyp5h2t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35921",
  "fulltext_url": null,
  "id": "11zqftvyp5h2t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35921",
  "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"
    ],
    [
      "arxiv",
      "nlin.CG"
    ],
    [
      "openalex",
      "1703|Cellular Automata and Applications"
    ]
  ],
  "openalex": "W7214987319",
  "pdf_url": "https://arxiv.org/pdf/2609.35921",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11zqftvyp5h2t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:31Z",
      "native_id": "2609.35921",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35921"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.CG",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Finite-ring obstructions for quadratic binary radius-two cellular automata",
  "title_text": "Finite-ring obstructions for quadratic binary radius-two cellular automata",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/11zqftvyp5h2t",
  "venue": "arXiv",
  "versions": []
}