{
  "abstract": "χ‑entropy is introduced as a rigorous operator–spectral invariant quantifying the internal differentiation of χ‑structures on the χ‑Boundary. The χ‑entropy functional S_chi = -Tr(Delta_chi * ln(Delta_chi)), defined via the χ‑time operator Delta_chi = exp(-T_chi) and the χ‑tension operator T_chi, arises directly from the non-local spectral action of the χ‑boundary derivation B = D * P_chi - P_chi * D, where D is an unbounded self-adjoint operator on a Hilbert space H and P_chi is an orthogonal projection onto the χ‑Boundary. Hyperbolic spectral curvature K(k) < 0 induces non-compact su(1,1) symmetry on spectral layers L(k) = P_k(H) and determines the non-equilibrium statistical distribution of χ‑vacuum modes. A complete operator–spectral reconstruction of χ‑entropy is derived, establishing its invariance under χ‑stable trace-class perturbations [V, P_chi] in S_1, its exact scaling under unitary representations, and its governing role in χ‑collapse dynamics, sub-harmonic χ‑fluctuations, and emergent χ‑noise. These results demonstrate that χ‑entropy forms the statistical backbone of χ‑reality, unifying non-commutative spectral geometry, non-compact representation theory, and quantum statistical mechanics within the non-perturbative ZEBTS‑SUPER 9.0 architecture.",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, retired person"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Anatolii Mukha",
      "orcid": "0009-0005-4614-6667"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/81zpfxykg5ddg.bib",
  "cited_by": 31,
  "doi": "10.33774/coe-2026-8x38k",
  "fulltext_url": null,
  "id": "81zpfxykg5ddg",
  "issns": [],
  "landing_url": "https://doi.org/10.33774/coe-2026-8x38k",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7196927748",
  "pdf_url": "https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6a797f26d1922e37d53e64ce/original/operator-spectral-entropy-and-hyperbolic-vacuum-structuring-statistical-foundations-of-reality.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/81zpfxykg5ddg.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T13:20:59Z",
      "native_id": "W7196927748",
      "source": "openalex",
      "url": "https://doi.org/10.33774/coe-2026-8x38k"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Advanced Operator Algebra Research",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Operator–Spectral χ‑Entropy and Hyperbolic Vacuum Structuring: Statistical Foundations of χ‑Reality",
  "title_text": "Operator–Spectral χ‑Entropy and Hyperbolic Vacuum Structuring: Statistical Foundations of χ‑Reality",
  "type": "preprint",
  "updated_at": "2026-11-09T13:20:59Z",
  "url": "https://research.draconitic.com/paper/81zpfxykg5ddg",
  "venue": null,
  "versions": []
}