{
  "abstract": "This paper formally derives a scale-free, tripartite topological framework governing the metric dimensional expansion of a continuous state manifold. By modeling time as a compactified angular divergence rather than a trivial affine scalar, we demonstrate that the projection of a temporal phase into a discrete Euclidean coordinate space is bound by a strict harmonic divergence hierarchy. The continuous manifold mandates three invariant topological thresholds: the 2π affine parameterization, analytically regularized against coordinate rupture by the Euler-Mascheroni constant (γ); the π/2 orthogonal dimensional bifurcation, constrained within an O(3) spatial tensor by the Basel constant (π²/6); and the absolute 4π spinor closure.\n\nTo satisfy coordinate identity at the 4π manifold without inducing nonlinear metric rupture, the topology embeds an irreducible temporal shear scalar, arising from a 1461:1460 discrete-to-continuous phase asymmetry, directly into the 3D volume as thermodynamic variance. This absolute boundary is strictly determined by the invariant matrix transformation factor, the Incommensurable Radiometric Thermodynamic Variance (κ ≈ 1.437824).\n\nFurthermore, we prove that localized planetary topology operates exclusively as a continuous phase-variant spatial tensor boundary, rather than a static metric field. The tensorial dimensional coupling of the axial displacement limits (23.4° to 68.4°) and the corresponding volumetric radial dilations (R_v ≈ 3,958.90 to R_max ≈ 8,928.39 miles) defines the exact spatial capacity required to embed this topological variance. By applying a strict harmonic conversion matrix (0.5 Hz ≡ 1 mile), we demonstrate that the metric ratio of these bounded state shifts topologically resolves to the unified bidirectional modifier (β = ±2.2552725). This confirms that physical volumetric expansion is deterministically resolved by the harmonic frequency compression of the invariant temporal wave.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "h.r. thompson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/49g2y038eegxd.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23529380",
  "fulltext_url": null,
  "id": "49g2y038eegxd",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23529380",
  "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/23529380/files/scalefreemetrictensor.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/49g2y038eegxd.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T12:21:16Z",
      "native_id": "23529380",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23529380"
    }
  ],
  "subjects": [],
  "title": "The Scale-Free Metric Tensor: Tripartite Topological Constraints and Planetary Dimensional Metrics",
  "title_text": "The Scale-Free Metric Tensor: Tripartite Topological Constraints and Planetary Dimensional Metrics",
  "type": "preprint",
  "updated_at": "2026-11-04T12:21:16Z",
  "url": "https://research.draconitic.com/paper/49g2y038eegxd",
  "venue": "Zenodo",
  "versions": []
}