{
  "abstract": "FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner-Eckart reduced matrix elements with inverse-volume scaling corrections in infinite systems. | MATH: Wigner-Eckart theorem: \\(\\langle j' m' | T^k_q | j m \\rangle = \\langle j k; m q | j' m' \\rangle \\cdot \\langle j' || T^k || j \\rangle / \\sqrt{2j'+1}\\). SSB correction: \\(\\langle j' || T^k || j \\rangle_{\\text{broken}} = \\langle j' || T^k || j \\rangle_{\\text{symmetric}} + O(1/V)\\), where \\(V\\) is system volume. The correction term is proportional to \\(V^{-1}\\) times a Goldstone-mode overlap integral, scaling as \\(\\sim \\int d^dx \\, \\langle \\Omega | [Q_a, T^k_q] | \\Omega \\rangle / V\\). | CONNECTION: The \\(1/V\\) scaling is the topological inverse-volume factor — in 3D, \\(V \\sim L^3\\), so corrections vanish as \\(L^{-3}\\), a cubic lattice harmonic. The Goldstone bosons are the zero-momentum modes; their count equals the dimension of the coset space \\(G/H\\), whose rank relates to root-system degeneracies (e.g., \\(SU(2)\\to U(1)\\) gives 2\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/bjs4kwcv6sy6j.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23490659",
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  "id": "bjs4kwcv6sy6j",
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  "landing_url": "https://zenodo.org/records/23490659",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjs4kwcv6sy6j.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:27Z",
      "native_id": "23490659",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23490659"
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  "subjects": [],
  "title": "Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume Corrections — E8 Intelligence Research",
  "title_text": "Spontaneous Symmetry Breaking Alters Wigner-Eckart Matrix Elements with Volume Corrections — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:27Z",
  "url": "https://research.draconitic.com/paper/bjs4kwcv6sy6j",
  "venue": "Zenodo",
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}