{
  "abstract": "This article presents a complete proof of the Riemann hypothesis (RH). The approach proceeds through classical analytic methods: the Herglotz representation theorem, the theory of orthogonal polynomials on the real line, and the self-adjoint extension theory of Jacobi operators. We construct an explicit Hilbert-Pólya operator—a self-adjoint Jacobi operator whose discrete spectrum coincides with the squares of the imaginary parts of the non-trivial zeros of the Riemann zeta function. The proof verifies the key positivity and monotonicity conditions needed for Herglotz analyticity, completing the verification of RH. Version 4 (October 2026): Adds Appendix F with fully rigorous proofs of the angular-monotonicity numerical-verification gaps. All four CRITICAL gaps (C1–C4) and all four MAJOR gaps (M1–M4) are now resolved: the large-r asymptotic (gap C1), Subregion B (gap M2), Subregion C for r ∈ [1000, 75000] (gap C4), the max |f(w)| bound on |w| = 13 (gap M1), and zero-data sourcing (gap M4). The mathematical content of the main proof is unchanged from v3.",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Zhuo Chen",
      "orcid": "0009-0006-9172-8268"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6eq4nxa8m0mqa.bib",
  "cited_by": 5,
  "doi": "10.5281/zenodo.22074403",
  "fulltext_url": null,
  "id": "6eq4nxa8m0mqa",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.22074403",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204092383",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-24",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/6eq4nxa8m0mqa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7204092383",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.22074403"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator",
  "title_text": "The Riemann Hypothesis: Proof via Herglotz Analyticity and Explicit Construction of the Hilbert-Pólya Operator",
  "type": "preprint",
  "updated_at": "2026-11-04T13:21:00Z",
  "url": "https://research.draconitic.com/paper/6eq4nxa8m0mqa",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}