{
  "abstract": "Socolar and Taylor asked for a single, simply connected three-dimensional prototile that forces nonperiodicity by shape alone, admitting no weakly nonperiodic tiling; the Schmitt-Conway-Danzer biprism and the three-dimensional Socolar-Taylor tile admit screw motions or a periodic stacking direction. We exhibit a rational polyhedral $3$-ball $Q$, which we call Chair44 (R44): a seven-cube chair whose $24$ exposed unit panels carry tiny square-pyramid features, and prove, as a proof submission, that $Q$ admits tilings of $\\mathbb{R}^3$ by congruent copies, reflections allowed, and that every such tiling has no translational period and a symmetry group of order at most $24$; every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The solid was designed to a reading of the aperiodic-monotile phenomenon reached with the Six Birds emergence calculus (Section 3.3), and the construction turns on a single finite test, checked by machine: the tile's own contact rule survives coarsening, so that the decoded parent tiling obeys the tile's rule and no other. The proof combines a written geometric argument, that the features force every tiling onto a registered lattice, with exhaustive finite enumerations; the companion census is replayed by two independent implementations, every finite gate is kernel-checked in Lean 4 (modulo a named compiler hook per native_decide theorem), and the written geometric lemmas and the logical assembly are Lean theorems as well, so that the theorem is kernel-checked modulo the named compiler hooks; the written proofs remain as exposition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19214",
  "author_total": 1,
  "authors": [
    {
      "name": "Ioannis Tsiokos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12q7et6xn8wr0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19214",
  "fulltext_url": null,
  "id": "12q7et6xn8wr0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19214",
  "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.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2505|Quasicrystal Structures and Properties"
    ]
  ],
  "openalex": "W7213593821",
  "pdf_url": "https://arxiv.org/pdf/2609.19214",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12q7et6xn8wr0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19214",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19214"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Strongly Aperiodic Monotile in Three Dimensions",
  "title_text": "A Strongly Aperiodic Monotile in Three Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/12q7et6xn8wr0",
  "venue": "arXiv",
  "versions": []
}