{
  "abstract": "FINDING: Phase transitions may occur without hidden symmetry breaking, challenging the Landau paradigm; spontaneous symmetry breaking (SSB) modifies Wigner-Eckart relations in infinite systems via Goldstone-mode corrections. | MATH: Landau free-energy functional \\(F[\\phi] = \\int d^dx (|\\nabla\\phi|^2 + r\\phi^2 + u\\phi^4)\\); SSB correction to Wigner-Eckart: \\(\\langle j'||T^k||j\\rangle_{\\text{broken}} = \\langle j'||T^k||j\\rangle_{\\text{sym}} + \\sum_{n} \\frac{\\langle j'||V||n\\rangle\\langle n||T^k||j\\rangle}{E_j - E_n + i\\epsilon}\\) (Goldstone pole dominates as \\(q\\to 0\\)); unbroken group \\(G\\), broken subgroup \\(H\\), coset dim = # Goldstone modes = dim(G/H). | CONNECTION: Goldstone modes correspond to generators of \\(G/H\\) — root system weights of Lie algebras (e.g., \\(SU(2)\\to U(1)\\) gives ratio 1:2 in magnon dispersion); crystallographic point groups (e.g., \\(D_6\\) for graphene) constrain SSB patterns; no explicit golden ratios emerge, but the coset structure echoes root lattice geometry\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/8ft1t45hs2nn7.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23553800",
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  "id": "8ft1t45hs2nn7",
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  "landing_url": "https://zenodo.org/records/23553800",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ft1t45hs2nn7.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:21:16Z",
      "native_id": "23553800",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23553800"
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  ],
  "subjects": [],
  "title": "Beyond Landau: Symmetry Breaking and Goldstone Corrections to Wigner-Eckart Relations — E8 Intelligence Research",
  "title_text": "Beyond Landau: Symmetry Breaking and Goldstone Corrections to Wigner-Eckart Relations — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-05T12:21:16Z",
  "url": "https://research.draconitic.com/paper/8ft1t45hs2nn7",
  "venue": "Zenodo",
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}