{
  "abstract": "The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\\mathbb{R}^3$. This note extends the underlying mechanism---the rep-$2^N$ chair $C_N = [0,2]^N \\setminus (1,2]^N$ with corner/socket markings---to $\\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00916",
  "author_total": 1,
  "authors": [
    {
      "name": "Dmitry Kamenetsky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/58qgpzba303b0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00916",
  "fulltext_url": null,
  "id": "58qgpzba303b0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00916",
  "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.CG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.MG"
    ],
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      "openalex",
      "2505|Quasicrystal Structures and Properties"
    ]
  ],
  "openalex": "W7216511749",
  "pdf_url": "https://arxiv.org/pdf/2610.00916",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/58qgpzba303b0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00916",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00916"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Towards Strongly Aperiodic Monotiles in Higher Dimensions",
  "title_text": "Towards Strongly Aperiodic Monotiles in Higher Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/58qgpzba303b0",
  "venue": "arXiv",
  "versions": []
}