{
  "abstract": "On 7 and 8 September 2026 programmatic search produced singularities: a forced Navier-Stokes singularity at every fixed viscosity, statements (C) and (D) of the Clay problem, and two Euler singularities. The unforced problem, statements (A) and (B), stands open, and a search for it needs a target. This paper fixes one: the positive defect problem, that a Leray-Hopf solution from smooth data on the periodic cube loses energy on a finite window, at fixed viscosity, beyond what viscosity removes. A positive defect implies blowup and so a negative answer to statement (B); the converse is not known. The paper proves the target equivalent to a floor on the energy flux through the Littlewood-Paley shells, the Fourier-side form of the coarse-grained flux of the Onsager theory of turbulence, averaged over the window; states necessary conditions on a candidate: a singular time of Type II in velocity, energy concentrating on a set of zero length, a pressure outside L^2, a velocity outside the Onsager-critical class L^3_t B^{1/3}_{3,c_0}, an obstruction to collapse onto a fixed steady Euler profile; and states what cannot certify one: no finite computation witnesses a Galerkin-uniform ceiling, and selection and forcing return the question to a positive defect. A pseudo-spectral search at 128^3 and 256^3 shows which condition of the reduction binds: the fine-shell flux floor holds to within 1 to 6 percent of the ceiling for a third of a turnover time, and fails in scale at the Kolmogorov wavenumber, so a candidate must differ from generic turbulence in the depth of its cascade, not in its timing. Every implication not marked otherwise is a theorem in Lean 4 over Mathlib; the library contains no Navier-Stokes object, and the equation enters only through hypotheses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23868",
  "author_total": 2,
  "authors": [
    {
      "name": "Jarret Petrillo"
    },
    {
      "name": "James Glimm"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8da7409w6t6re.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23868",
  "fulltext_url": null,
  "id": "8da7409w6t6re",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23868",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "physics.flu-dyn"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7214001604",
  "pdf_url": "https://arxiv.org/pdf/2609.23868",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8da7409w6t6re.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.23868",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23868"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.flu-dyn",
      "subfield": "fluid-dynamics"
    }
  ],
  "title": "The Positive Defect Problem: Target and Admissibility Criteria for a Programmatic Search for Unforced Navier-Stokes Blowup",
  "title_text": "The Positive Defect Problem: Target and Admissibility Criteria for a Programmatic Search for Unforced Navier-Stokes Blowup",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8da7409w6t6re",
  "venue": "arXiv",
  "versions": []
}