{
  "abstract": "When irreducible representations occur with multiplicity, a family symmetry fixes the representation content of a Yukawa sector but need not fix its embedding in the space of generation tensors. We study the spectral consequences of this additional embedding data for complex-symmetric three-family tensors with the $S_3$ assignment $V=\\mathbf{1}\\oplus\\mathbf{2}$. Since $\\operatorname{Sym}^2V=2\\,\\mathbf{1}\\oplus2\\,\\mathbf{2}$, three-dimensional invariant Yukawa subspaces of type $\\mathbf{1}\\oplus\\mathbf{2}$ form a continuous $\\mathbb{CP}^1\\times\\mathbb{CP}^1$ family, denoted $\\mathcal L_{t,κ}$ on a finite affine chart. On the $κ=0$ branch, we solve the inverse singular-value problem and obtain necessary and sufficient conditions for any prescribed ordered positive mass spectrum. A fixed singlet embedding imposes a finite hierarchy bound, whereas varying the embedding accommodates every positive three-family spectrum. For general embeddings, $\\mathcal L_{t,κ}$ contains a nonzero rank-one matrix if and only if $2tκ^2=1$; equivalently, these are precisely the finite-chart embeddings for which $σ_3/σ_2$ is unbounded. In a conventional complex-symmetric $S_3$ three-Higgs sector, the same condition becomes $y_2y_5^2=\\sqrt{2}\\,y_1y_3^2$, so order-one Yukawa coupling ratios can support arbitrarily large mass hierarchies. We then apply the result to a recent Clifford-algebraic construction and show that its family-resolved Yukawa tensors span $\\mathcal L_{2,0}$, whose singular values obey $m_1+m_2\\leq m_3\\leq m_1+3m_2$. This excludes the observed charged-lepton hierarchy for arbitrary complex vacuum alignment within the existing Higgs directions. The resulting obstruction is therefore tied to the fixed multiplicity embedding rather than to the $S_3$ representation content alone.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29259",
  "author_total": 1,
  "authors": [
    {
      "name": "Ye Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xc3m9jr7ws1e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29259",
  "fulltext_url": null,
  "id": "5xc3m9jr7ws1e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29259",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-ph"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Particle physics theoretical and experimental studies"
    ]
  ],
  "openalex": "W7204899742",
  "pdf_url": "https://arxiv.org/pdf/2608.29259",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xc3m9jr7ws1e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2608.29259",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29259"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    }
  ],
  "title": "Embedding dependence of fermion mass hierarchies in $S_3$-symmetric Yukawa sectors",
  "title_text": "Embedding dependence of fermion mass hierarchies in S_3-symmetric Yukawa sectors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:41Z",
  "url": "https://research.draconitic.com/paper/5xc3m9jr7ws1e",
  "venue": "arXiv",
  "versions": []
}