{
  "abstract": "Pate proved ordinary irreducible-immanant permanental dominance through order $13$ and identified $(4,4,3,3)$ as the sole remaining order-$14$ case, with $(5,4,3,3)$ and $(3^5)$ forming the order-$15$ frontier. These three cases are settled here; consequently $d_λ(A)/f^λ\\le \\operatorname{per}(A)$ for every partition $λ\\vdash n$ with $n\\le15$ and every complex Hermitian positive-semidefinite matrix $A$. The argument also yields results beyond this finite frontier: an exact four-term bridge for $(4,4,3,3)$, the uniform family $(m,4,3,3)$, a two-parameter family $(a,b,3,3)$ for $a\\ge b\\ge4$ and $5a\\ge8b$, and a long-first-row criterion for arbitrary fixed tails. These results arise from explicit specializations of Pate's $W$-function positivity framework using partial swaps, Young projectors, Pieri--content identities, and branching data. For $(3^5)$, an exact Farkas certificate shows that the central-projector partial-swap cone is insufficient; a branching-refined one-swap construction escapes this obstruction and yields a positive $106+19$-witness certificate. Boundary-compression and node-moving results further describe the reach and limitations of the local-filter method. All finite certificates are checked by exact integer or rational arithmetic and are supplied as ancillary material. The order-$14$ bridge is additionally formalized and kernel-checked in Lean 4 for all complex Hermitian positive-semidefinite matrices, including the exact coefficient normalization and the deduction of $(4,4,3,3)$ permanental dominance from four explicitly stated Pate inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13412",
  "author_total": 1,
  "authors": [
    {
      "name": "Yinjie Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9c53f4nxaccs3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13412",
  "fulltext_url": null,
  "id": "9c53f4nxaccs3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13412",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213348971",
  "pdf_url": "https://arxiv.org/pdf/2609.13412",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9c53f4nxaccs3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13412",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13412"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Lieb's Permanental Dominance Conjecture for Ordinary Immanants through Order Fifteen",
  "title_text": "Lieb's Permanental Dominance Conjecture for Ordinary Immanants through Order Fifteen",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9c53f4nxaccs3",
  "venue": "arXiv",
  "versions": []
}