{
  "abstract": "This note records a geometric reduction of the Riemann Hypothesis under Topological Vacuum Electrodynamics. The continuous sheet used for the particle spectrum (a stochastic electromagnetic vacuum cut into force-free Beltrami tubes, with the fine-structure mode count $\\alpha^{-1}=4\\pi^3+\\pi^2+\\pi$ and the minor-radius cutoff) is identified with the dynamical partition function of the Reeb flow on the adele class space.\n\nNative closed-orbit periods $\\ell(\\gamma)=\\ln p$, half-density weights $p^{-1/2}$, and the Archimedean digamma kernel arise as geometric identities of that flow. Symplectic volume preservation and the functional equation give real-line unitarity of the scattering matrix. Compression of the unitary scaling group recovers the closed critical strip $0\\le\\operatorname{Re}(\\rho)\\le1$. The Clay requirement then reduces to the single unsigned condition\n\n$\\operatorname{Im}\\Bigl(-\\frac{\\Xi'(z)}{\\Xi(z)}\\Bigr)>0\\quad\\text{for all }\\operatorname{Im}(z)>0,$\n\nor any of its classically equivalent forms (Hermite–Biehler, Laguerre, canonical Hamiltonian positivity, Weil positivity, or Li’s criterion).\n\n$|E(z)|>|E(\\overline{z})|\\quad(\\operatorname{Im}z>0),$\n\n$L_\\Xi(x)=[\\Xi'(x)]^2-\\Xi(x)\\Xi''(x)>0\\quad(x\\in\\mathbb{R}),$\n\n$\\det\\mathcal{H}(t)\\ge0\\quad\\text{(almost everywhere)},$\n\n$W(f,f^*)\\ge0\\quad\\text{(for all admissible \\(f\\))},$\n\n$\\lambda_n\\ge0\\quad\\text{(for all \\(n\\ge1\\))}.$\n\nThe kinematic layers and the strip bound sit on the same sheet already used for mass and the forces. The Millennium problem is one global positivity condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Truth Vector"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5kt8800vzs9tf.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23440294",
  "fulltext_url": null,
  "id": "5kt8800vzs9tf",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23440294",
  "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": "https://zenodo.org/api/records/23440294/files/TVE_%20Riemann%20Hypothesis%20Reduced%2020261017.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5kt8800vzs9tf.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T12:21:12Z",
      "native_id": "23440294",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23440294"
    }
  ],
  "subjects": [],
  "title": "Riemann Hypothesis Reduced to the Herglotz Inequality under Topological Vacuum Electrodynamics",
  "title_text": "Riemann Hypothesis Reduced to the Herglotz Inequality under Topological Vacuum Electrodynamics",
  "type": "preprint",
  "updated_at": "2026-10-31T12:21:12Z",
  "url": "https://research.draconitic.com/paper/5kt8800vzs9tf",
  "venue": "Zenodo",
  "versions": []
}