{
  "abstract": "FINDING: Tiling decidability collapses from algorithmic certainty to undecidability at dimension thresholds, while aperiodic single-tile constructions (Einstein tile) reveal that local rules can force global non-periodicity — a deep link between computation limits and crystallographic order. MATH: - Translational tiling: finite \\(F \\subset \\mathbb{Z}^d\\) tiles \\(\\mathbb{Z}^d\\) iff \\(\\exists T \\subset \\mathbb{Z}^d\\) such that \\(\\mathbb{Z}^d = \\bigoplus_{t \\in T} (F+t)\\) (direct sum, disjoint translates). - Decidability: For \\(d=1,2\\) — decidable (Berger 1966, later simplified). For \\(d \\geq 3\\) — undecidable (Greenfeld & Tao, 2021; also Greenfeld's work). - Finite tiling problem: undecidable in Euclidean plane (Kari 1994) and hyperbolic plane (Margenstern, 2007). - Domino problem (Wang tiles): undecidable in \\(\\mathbb{Z}^2\\) (Berger 1966). - Einstein tile: single aperiodic tile (Smith et al., 2023) — a 13-sided polygon (hat) with no periodic tiling of the plane. CONNECTION\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
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  "affiliations": [
    "Independent Researcher, United Kingdom"
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  "article_type": null,
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  "author_total": 1,
  "authors": [
    {
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      ],
      "name": "Andrew Stewart Caldin"
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  "bibtex": "https://research.draconitic.com/paper/ejzggxn2hdkyt.bib",
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  "doi": "10.5281/zenodo.23635485",
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  "published": "2026-10-25",
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  "ris": "https://research.draconitic.com/paper/ejzggxn2hdkyt.ris",
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      "first_seen": "2026-11-08T12:21:54Z",
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  "title": "Tiling Decidability Thresholds and the Einstein Tile's Aperiodic Order — E8 Intelligence Research",
  "title_text": "Tiling Decidability Thresholds and the Einstein Tile's Aperiodic Order — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:54Z",
  "url": "https://research.draconitic.com/paper/ejzggxn2hdkyt",
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