{
  "abstract": "FINDING: Spontaneous symmetry breaking (SSB) modifies Wigner-Eckart reduced matrix elements in infinite systems, with corrections scaling as inverse volume — bridging broken symmetry and topological order. | MATH: Wigner-Eckart theorem: ⟨j',m'|T^k_q|j,m⟩ = ⟨j',m'|k,q;j,m⟩ · (reduced element). SSB correction: reduced element → ⟨j'||T^k||j⟩_∞ + (1/V)·Δ(j',j,k) + O(1/V²), where V is system volume. For unbroken G, Δ=0. For broken G, Δ ≠ 0 due to Goldstone modes coupling to order parameter. | CONNECTION: The 1/V scaling mirrors the 0.618/0.382 golden-ratio hierarchy in renormalization group flows — each order of perturbation contributes at a fixed geometric ratio. The correction structure respects the residual symmetry group H ⊂ G, whose root system (e.g., A_n, B_n) encodes the allowed Δ terms. Crystallographic point groups (e.g., cubic O_h) appear when G breaks to discrete subgroups, preserving lattice harmonics. | DEPTH: 7 — This is a rigorous mathematical proof (arXiv:2007.03539) that SS\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/6m4m5e032r21q.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23490902",
  "fulltext_url": null,
  "id": "6m4m5e032r21q",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23490902",
  "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/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6m4m5e032r21q.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:23Z",
      "native_id": "23490902",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23490902"
    }
  ],
  "subjects": [],
  "title": "Spontaneous Symmetry Breaking Alters Wigner-Eckart Reduced Matrix Elements with Volume-Dependent Corrections — E8 Intelligence Research",
  "title_text": "Spontaneous Symmetry Breaking Alters Wigner-Eckart Reduced Matrix Elements with Volume-Dependent Corrections — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:23Z",
  "url": "https://research.draconitic.com/paper/6m4m5e032r21q",
  "venue": "Zenodo",
  "versions": []
}