{
  "abstract": "The χ‑operator geometry provides a new spectral framework for describing extremal black holes, whose zero‑temperature states exhibit non‑zero entropy and a collapse of surface gravity. In this work, we reconstruct the extremal horizon as a χ‑spectral degeneracy manifold and show that the vanishing of temperature corresponds to a structural reduction of the χ‑spectrum rather than thermodynamic triviality. The χ‑operator encodes the geometric, quantum, and gravitational degrees of freedom in a unified spectral form, allowing extremal black holes to be treated as zero‑temperature χ‑states with finite horizon complexity. This approach reveals a new mechanism for entropy retention in extremal configurations and establishes a purely operator‑geometric origin of extremal horizon structure.",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher , retired person Kharkiv , Ukraine"
  ],
  "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/8a5jh1n9bzvkj.bib",
  "cited_by": 12,
  "doi": "10.33774/coe-2026-qc6jw",
  "fulltext_url": null,
  "id": "8a5jh1n9bzvkj",
  "issns": [],
  "landing_url": "https://doi.org/10.33774/coe-2026-qc6jw",
  "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",
      "3106|Black Holes and Theoretical Physics"
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  "openalex": "W7201856184",
  "pdf_url": "https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6a774e58810b9dcc82e17808/original/operator-geometry-of-extremal-black-holes-spectral-degeneracy-zero-temperature-and-non-zero-entropy.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8a5jh1n9bzvkj.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T13:20:42Z",
      "native_id": "W7201856184",
      "source": "openalex",
      "url": "https://doi.org/10.33774/coe-2026-qc6jw"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "openalex:3106|Black Holes and Theoretical Physics",
      "subfield": "high-energy"
    }
  ],
  "title": "χ‑Operator Geometry of Extremal Black Holes: Spectral Degeneracy, Zero Temperature, and Non‑Zero Entropy",
  "title_text": "χ‑Operator Geometry of Extremal Black Holes: Spectral Degeneracy, Zero Temperature, and Non‑Zero Entropy",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:23Z",
  "url": "https://research.draconitic.com/paper/8a5jh1n9bzvkj",
  "venue": null,
  "versions": []
}