{
  "abstract": "Harmonic analysis on the global Adèle Class Space \\mathbb{X} \\equiv \\mathbb{A}_\\mathbb{Q}/\\mathbb{Q}^\\times occupies a central position at the intersection of automorphic representation theory, analytic number theory, and noncommutative spectral geometry. The infinitesimal generator of multiplicative dilation, H = -i(x\\frac{d}{dx} + \\frac{1}{2}) \\cong -i\\frac{d}{du}, possesses a continuous, unbounded spectrum \\sigma(H) = \\mathbb{R} when acting on the automorphic Hilbert space \\mathcal{H}_\\mathbb{A} \\equiv L^2_0(\\mathbb{X}, d\\mu). Extending this unbounded scaling flow across the infinite product of non-Archimedean valuation trees produces severe domain instabilities, phase divergence, and ultraviolet singularities.\n\nIn this paper, we construct an unconditional, non-perturbative operator-theoretic framework for the two-sided regularized dilation operator: \\mathcal{D}_\\theta \\equiv H \\exp\\left(-\\theta |H|\\right) = H \\exp\\left(-\\theta \\sqrt{H^2}\\right). We establish that the spectral cutoff \\theta is an intrinsic mathematical invariant uniquely determined by a Three-Phase fixed-point hierarchy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "August Tudor"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0sf47q8wfbpv1.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23514877",
  "fulltext_url": null,
  "id": "0sf47q8wfbpv1",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23514877",
  "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": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0sf47q8wfbpv1.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:28Z",
      "native_id": "23514877",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23514877"
    }
  ],
  "subjects": [],
  "title": "Spectral Regulation of Scaling Flows on the Adèle Class Space: Intrinsic Regulator Derivation, Coercive Resolvent Reduction, and Dual Trace Factorization",
  "title_text": "Spectral Regulation of Scaling Flows on the Adèle Class Space: Intrinsic Regulator Derivation, Coercive Resolvent Reduction, and Dual Trace Factorization",
  "type": "preprint",
  "updated_at": "2026-11-03T12:20:28Z",
  "url": "https://research.draconitic.com/paper/0sf47q8wfbpv1",
  "venue": "Zenodo",
  "versions": []
}