{
  "abstract": "Let $G=C_2\\times C_2$ and $A=\\{0,1\\}$. We characterize the submonoid of $\\text{CA}(G;A)$ generated by lazy and invertible cellular automata. A cellular automaton belongs to this submonoid if and only if, whenever it maps the two free $G$-orbits onto the same free orbit, it maps both constant configurations and some two-point orbit to the same constant. The proof gives a factorization using elementary orbit collapses and three two-cell lazy maps. The criterion yields the cardinality $59\\,712$ and identifies the maps missed by a weaker constant-separation invariant. Adjoining one idempotent generates the full monoid, and the generated submonoid is contained in exactly one maximal proper submonoid, of cardinality $65\\,024$. The submonoid generated by lazy maps alone has $4\\,007$ elements, and for both generating sets the maximum shortest-word length is six. The criterion and the cardinalities $59\\,712$ and $4\\,007$ are formally verified in Lean.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20329",
  "author_total": 4,
  "authors": [
    {
      "name": "Edgar Alcalá-Arroyo"
    },
    {
      "name": "Alonso Castillo-Ramirez"
    },
    {
      "name": "Haobo Ma"
    },
    {
      "name": "Wenlin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6q9tagxg330ym.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20329",
  "fulltext_url": null,
  "id": "6q9tagxg330ym",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20329",
  "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",
      "math.GR"
    ],
    [
      "openalex",
      "1703|Cellular Automata and Applications"
    ]
  ],
  "openalex": "W7222114429",
  "pdf_url": "https://arxiv.org/pdf/2610.20329",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6q9tagxg330ym.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20329",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20329"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generation by lazy and invertible cellular automata over the Klein four-group",
  "title_text": "Generation by lazy and invertible cellular automata over the Klein four-group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/6q9tagxg330ym",
  "venue": "arXiv",
  "versions": []
}