{
  "abstract": "We derive whole-space distribution results from the established compact, smoothly forced Navier--Stokes blowup construction of OpenAI. For every fixed positive viscosity and deadline, smooth forces causing classical breakdown from rest on $\\mathbb R^3$ are dense in the relative $L^1_tH^s_x$ topology exactly when $s<1/2$, and in the relative $L^2_tH^s_x$ topology exactly when $s<-1/2$. The positive results preserve every fixed initial velocity in $H^\\infty_σ(\\R^3)$. They follow from a compact divergence-free removal of a regular background, followed by exact insertion of a single whole-space singular packet. Separate critical estimates prove non-density; low spatial frequencies are controlled explicitly, without a periodic mean or a Poincaré inequality. The conclusions hold for spacetime-compact forces and for the rapidly decaying data class in the Clay formulation. We also obtain density in the usual completed energy-force spaces, strong energy-and-dissipation approximation of regular trajectories, and exact equality of cell observations on a prescribed finite family of whole-space grids. Initial-state spaces, forcing spaces and numerical error regularity are distinguished throughout. No claim of fixed-force instability, loss of weak existence or new formal verification is made.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10269",
  "author_total": 2,
  "authors": [
    {
      "name": "Shaozhen Cao"
    },
    {
      "name": "Zhuoni Chi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jqb801941c06.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10269",
  "fulltext_url": null,
  "id": "1jqb801941c06",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10269",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Navier-Stokes equation solutions"
    ]
  ],
  "openalex": "W7212130444",
  "pdf_url": "https://arxiv.org/pdf/2609.10269",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jqb801941c06.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.10269",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10269"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation",
  "title_text": "Singular Forces on the Whole Space: Sobolev Density Thresholds and Energy Approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/1jqb801941c06",
  "venue": "arXiv",
  "versions": []
}