{
  "abstract": "We prove a periodic long-time Boltzmann--Grad limit for hard spheres in every fixed spatial dimension $d\\ge4$. Under the assumptions of the main theorem, the rescaled $s$-particle correlations converge in $L^1$ to the corresponding tensor-product Boltzmann profile at rate $\\varepsilon^{1/(400d)}$, uniformly for $1\\le s\\le|\\log\\varepsilon|$ and $0\\le t\\le t_{\\rm fin}$. In particular, when the activity and weighted solution bounds are fixed, $t_{\\rm fin}=O(\\log|\\log\\varepsilon|)$. The componentwise long-bond estimate used in dimensions two and three is insufficient in higher dimension. We replace it by a joint estimate for two connected time sublayers, selecting the first two lower collisions as landing roots. Disjoint landing edges yield a direct tangential frame. When the edges overlap, the newly appearing particle line contains a collision-free chord whose radial relative speed produces a rank-$(d-1)$ positive Schur factor. A positive Jacobi-network argument prevents focusing, while bounded-degree finite-fibre coarea globalizes the estimate for arbitrary nonnegative joint kernels. A first-failure construction repairs periodic double overlaps, and the resulting packet bound closes the exceptional top-layer contribution. A paid epoch restart for sealed complete joint kernels, using the fixed-word multi-landing operator for every fixed integer $k\\ge1$, yields the stated time range. Within the regular connected full two-sublayer packet class, every packet with at least $2k-1$ physical lines admits a canonical $k$-birth flag and the operator factor $\\varepsilon^{(d-1)(k-1)}\\varepsilon_*^{-k(d-1)}$; $k=2$ is the first gain-producing case. A complementary full-chord estimate provides a coarse fallback.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29566",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhixu Hua"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dw80q5r5dw295.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29566",
  "fulltext_url": null,
  "id": "dw80q5r5dw295",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29566",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Gas Dynamics and Kinetic Theory"
    ]
  ],
  "openalex": "W7214651278",
  "pdf_url": "https://arxiv.org/pdf/2609.29566",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dw80q5r5dw295.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.29566",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29566"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension $d\\ge 4$",
  "title_text": "A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension d≥ 4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/dw80q5r5dw295",
  "venue": "arXiv",
  "versions": []
}