{
  "abstract": "This work studies the three-dimensional incompressible Navier-Stokes equations in two parts. Part I uses structural decomposition and reconstruction to verify that the kinematic viscosity coefficient nu appearing in the initial viscous system is preserved through the energy, pressure, vorticity, two-point, midpoint, finite-cell, refinement, and reconstruction stages. Part II examines the Clay breakdown alternative on R^3 with a smooth rapidly decaying external force f. Under the conditional hypothesis of a finite maximal smooth time T_* < infinity, first-hitting intervals are constructed on which the enstrophy doubles, and each interval is normalized exactly from enstrophy 1 to 2. The rescaled force satisfies an integrated smallness estimate, yielding\n\nF_n -> 0,\n\nintegral from 0 to theta_n of (z_n - nu*y_n) ds -> 1/2.\n\nConsequently, every sufficiently deep normalized block contains a time s_n^# at which the nonlinear stretching exceeds viscous dissipation. The argument also identifies a genuine sufficient differential criterion for finite-time blow-up, namely a persistent pointwise lower bound of the form\n\nV - nuY + F >= deltaX^3\n\non a sufficiently long interval. The proved blockwise and one-time conclusions do not imply that persistent bound. Thus Part II reaches an exact stopping boundary: it derives necessary structure under the hypothesis T_* < infinity, but it neither constructs data and forcing that satisfy Clay Statement (C) nor proves finite-time breakdown.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Son Tuyet Tran",
      "orcid": "0009-0007-9871-9871"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cm4pjgntm7kms.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23493359",
  "fulltext_url": null,
  "id": "cm4pjgntm7kms",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23493359",
  "language": "eng",
  "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/23493359/files/Navier_Stokes.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cm4pjgntm7kms.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:19Z",
      "native_id": "23493359",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23493359"
    }
  ],
  "subjects": [],
  "title": "Solving the Three-Dimensional Incompressible Navier–Stokes Global Regularity Problem by Structural Decomposition and Normalized Growth Analysis",
  "title_text": "Solving the Three-Dimensional Incompressible Navier–Stokes Global Regularity Problem by Structural Decomposition and Normalized Growth Analysis",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:19Z",
  "url": "https://research.draconitic.com/paper/cm4pjgntm7kms",
  "venue": "Zenodo",
  "versions": []
}