{
  "abstract": "FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner–Eckart (W-E) relations in infinite systems via finite-volume corrections, with Goldstone modes mediating the deviation. | MATH: W-E theorem: \\(\\langle j' m' | T^{(k)}_q | j m \\rangle = \\frac{1}{\\sqrt{2j+1}} \\langle j' || T^{(k)} || j \\rangle \\langle j m; k q | j' m' \\rangle\\) (Clebsch-Gordan coefficient). SSB correction: For unbroken group \\(G \\to H\\), the leading correction to W-E is **not** a constant reduced matrix element but acquires a **Goldstone pole** — scaling as \\(1/(p^2)\\) in momentum space, giving finite-volume correction \\(\\propto 1/L^2\\) (where \\(L\\) is system size) for the ratio of matrix elements. The paper (arXiv:2007.03539) proves corrections are **non-analytic** in symmetry-breaking parameter, proportional to \\(\\langle \\pi | J^\\mu | 0 \\rangle / f_\\pi\\) (Goldstone decay constant). | CONNECTION: The Clebsch-Gordan coefficients are intimately tied to \\(SU(2)\\) root systems — the same algebraic structure as cry\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/4pzz7d4g5mfv5.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23658383",
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  "id": "4pzz7d4g5mfv5",
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  "language": "eng",
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  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4pzz7d4g5mfv5.ris",
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    {
      "first_seen": "2026-11-09T12:21:36Z",
      "native_id": "23658383",
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      "url": "https://zenodo.org/records/23658383"
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  "subjects": [],
  "title": "Finite-Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research",
  "title_text": "Finite-Volume Corrections to Wigner–Eckart Relations via Spontaneous Symmetry Breaking — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-09T12:21:36Z",
  "url": "https://research.draconitic.com/paper/4pzz7d4g5mfv5",
  "venue": "Zenodo",
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