{
  "abstract": "The Boltzmann equation is the fundamental kinetic bridge between molecular mechanics and fluid behaviour. It converts binary encounters into the evolution of a velocity distribution, underlies rigorous routes from hard-sphere dynamics to Euler and Navier--Stokes--Fourier equations, and remains indispensable when continuum descriptions fail, from hypersonic shock layers and atmospheric re-entry to rarefied multiscale flows. Yet its nonlinear collision term is conventionally represented by an infinite three-index tensor of collisional brackets, with apparently new coefficients at every polynomial order. Here we prove that this entire hard-sphere tensor contains only one primitive collision law: the exact rank-two stress-production vertex $S$. Every polynomial collision vertex is obtained from $S$ by a finite sequence of universal differential and tensor operations, and every conventional bracket follows by differentiation. Thus the interaction physics and its representation separate exactly: all hard-sphere-specific information is concentrated in $S$, whereas the hierarchy above it is universal algebra. A blind eighth-order test in a Mach-5 shock agrees with independent collision sampling and strongly amplifies a weak kinetic precursor. The hard-sphere hierarchy is infinite in dimension, but its primitive collision content consists of a single scalar law.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17123",
  "author_total": 1,
  "authors": [
    {
      "name": "I. Karlin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4kh188fzqa2b6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17123",
  "fulltext_url": null,
  "id": "4kh188fzqa2b6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17123",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "physics.flu-dyn"
    ],
    [
      "openalex",
      "2604|Gas Dynamics and Kinetic Theory"
    ]
  ],
  "openalex": "W7221797810",
  "pdf_url": "https://arxiv.org/pdf/2610.17123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4kh188fzqa2b6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.17123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17123"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.flu-dyn",
      "subfield": "fluid-dynamics"
    }
  ],
  "title": "One vertex generates the hard-sphere Boltzmann collision hierarchy",
  "title_text": "One vertex generates the hard-sphere Boltzmann collision hierarchy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:06Z",
  "url": "https://research.draconitic.com/paper/4kh188fzqa2b6",
  "venue": "arXiv",
  "versions": []
}