{
  "abstract": "FINDING: Reciprocal lattice duality between FCC and BCC is a foundational Fourier-space symmetry, but the search results are pedagogical videos and one lattice QCD paper — no new mathematical discovery is presented. The core structural fact: the reciprocal of an FCC lattice is BCC, and vice versa, with a scale factor of \\(2\\pi\\) and a volume ratio of \\(2\\).\n\nMATH: For a direct lattice with primitive vectors \\(\\mathbf{a}_1, \\mathbf{a}_2, \\mathbf{a}_3\\), reciprocal vectors are \\[ \\mathbf{b}_1 = 2\\pi \\frac{\\mathbf{a}_2 \\times \\mathbf{a}_3}{\\mathbf{a}_1 \\cdot (\\mathbf{a}_2 \\times \\mathbf{a}_3)} \\] (cyclic permutations). For FCC with conventional cubic side \\(a\\), primitive vectors: \\[ \\mathbf{a}_1 = \\frac{a}{2}(\\hat{y}+\\hat{z}),\\quad \\mathbf{a}_2 = \\frac{a}{2}(\\hat{z}+\\hat{x}),\\quad \\mathbf{a}_3 = \\frac{a}{2}(\\hat{x}+\\hat{y}) \\] Reciprocal lattice is BCC with cubic side \\(4\\pi/a\\). Conversely, BCC reciprocal is FCC. Volume ratio: \\(V_{\\text{recip}} = (2\\pi)^3 / V_{\\text{cell}}\\).\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aqx8vk4qsvr9m.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23610273",
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  "id": "aqx8vk4qsvr9m",
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  "landing_url": "https://zenodo.org/records/23610273",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aqx8vk4qsvr9m.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T12:21:35Z",
      "native_id": "23610273",
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      "url": "https://zenodo.org/records/23610273"
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  "subjects": [],
  "title": "FCC-BCC Reciprocal Lattice Duality: A Foundational Fourier-Space Symmetry — E8 Intelligence Research",
  "title_text": "FCC-BCC Reciprocal Lattice Duality: A Foundational Fourier-Space Symmetry — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-07T12:21:35Z",
  "url": "https://research.draconitic.com/paper/aqx8vk4qsvr9m",
  "venue": "Zenodo",
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}