{
  "abstract": "We classify the polyominoes obtained by adjoining four straight arms to a single square, allowing zero arm lengths, according to their ability to tile rectangles, half-strips, bent strips, quadrants, strips, half-planes, and the plane. We also classify their ability to tile an integer enlargement of themselves. Tiles occupy whole square-grid cells; translations, rotations, and reflections are permitted. Exactly five capability profiles occur. For the family $P(n,1,1,0)$, the rectangle profile holds for $n\\le3$ and the bent-strip profile, with no half-strip or rep-tiling, for every $n\\ge4$. A cross with four positive arms tiles the plane precisely when two opposite arms have length one; it never tiles a half-plane. Explicit periodic constructions and geometric obstructions are combined with finite symbolic case certificates. A Lean 4 development verifies the full classification for every natural four-tuple, including the interpretation of the certificates as statements about arbitrary infinite tilings. The account incorporates the author's 2020--2021 L- and T-polyomino work, reconstructs Dahlke's gun argument, and documents the subsequent AI-assisted proof development and formalization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13641",
  "author_total": 1,
  "authors": [
    {
      "name": "Angel Ivanov Raychev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/02mzs7cb2x3qq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13641",
  "fulltext_url": null,
  "id": "02mzs7cb2x3qq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13641",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213290532",
  "pdf_url": "https://arxiv.org/pdf/2609.13641",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/02mzs7cb2x3qq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13641",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13641"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Four-arm polyominoes in Golomb's hierarchy: A complete classification with Lean verification",
  "title_text": "Four-arm polyominoes in Golomb's hierarchy: A complete classification with Lean verification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/02mzs7cb2x3qq",
  "venue": "arXiv",
  "versions": []
}