{
  "abstract": "We study a finite-dimensional generalized spectral problem combining Bogoliubov–Kubo–Mori (BKM) operator geometry with the fuzzy sphere. For the irreducible N-dimensional representation of su(2), we introduce the faithful axial state ρ0,N(γ)=e−γJ3/Tr(e−γJ3) and decompose Mat(N,C) into fixed-weight sectors q of dimension N−|q|. The covariant BKM kernel induces a positive Gram matrix G(q)(γ), while adjoint su(2) commutators define an adjoint-Casimir stiffness Kadj(q)(γ). We prove G(q)>0 for every finite N, q, and real γ. Under the assumptions β≥0 and μ≥0, the adjoint stiffness obeys Kadj(q)≥0; the full stiffness K(q)=Kadj(q)+μ2G(q) is strictly positive for μ>0. Thus the generalized spectrum is real and nonnegative, and strictly positive when μ>0. We also establish the exact isotropic spectrum, the protected constant mode Ω002=μ2, and exact q↔−q isospectrality. A local analytic quadratic law for the lowest nonconstant axial branches yields a negative q=1 versus q=0 splitting near isotropy when N>2 and β>0. Reproducible computations distinguish the historical restricted N=4,5 regressions from the full-space spectra and validate the construction for N=8 and N=12 under algebraic, spectral, conditioning, determinant, and residual diagnostics. The results are deliberately finite-dimensional: no N→∞ convergence or spacetime-emergence claim is made.",
  "abstract_shown": true,
  "affiliations": [
    "Centro de Desarrollo Energético de Antofagasta (CDEA-UA), Universidad de Antofagasta, Antofagasta 1240000, Chile"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Carlos Eduardo Portillo Silva",
      "orcid": "0000-0001-9301-5604"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bm22jemczz590.bib",
  "cited_by": 0,
  "doi": "10.3390/math14214053",
  "fulltext_url": null,
  "id": "bm22jemczz590",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/21/4053",
  "language": null,
  "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": "2026-11-05",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/bm22jemczz590.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:47Z",
      "native_id": "10.3390/math14214053",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/21/4053"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Fuzzy–BKM Spectral Geometry: Axial Blocks, Positivity, and Reproducible Finite-Dimensional Validation",
  "title_text": "Fuzzy–BKM Spectral Geometry: Axial Blocks, Positivity, and Reproducible Finite-Dimensional Validation",
  "type": "peer-reviewed",
  "updated_at": "2026-11-07T00:21:04Z",
  "url": "https://research.draconitic.com/paper/bm22jemczz590",
  "venue": "Mathematics",
  "versions": []
}