{
  "abstract": "Let $U$ be Haar-uniform on $\\mathbb S^{p-1}$, let $a_1,\\ldots,a_k$ be arbitrary nonzero vectors, and let $w_1,\\ldots,w_k$ be simplex weights. Define \\[ g(U)=\\sum_{j=1}^k w_j\\frac{a_j}{a_j^\\top U}, \\qquad N(U)=\\frac{g(U)}{\\|g(U)\\|}. \\] We prove the universal incidence law \\[ \\{U^\\top N(U)\\}^2\\sim\\operatorname{Beta}\\!\\left(\\frac12,\\frac{p-1}{2}\\right), \\] independently of the number, arrangement, rank, or overcompleteness of the directions and of the weights. Thus a deterministic, generally non-Haar function of $U$ has the same squared-cosine law as an independent Haar direction. One proof combines a Herglotz--Cauchy boundary principle, a Haar-random two-plane with one common phase, and an exact Beta--Cauchy tangent-projection equivalence. A second proof specializes the positive-semidefinite Pillai--Meng identity. The planar structure leads to converses: plane-conditional Cauchy laws recover positivity, while for signed measures an exact phase-cancellation deficit equals twice the hidden negative mass. This yields local-to-global rigidity under a phase-norming condition strictly weaker than injectivity and an unconditional exclusion of negative atoms. The law extends to probability measures under almost-sure reciprocal integrability. We characterize this condition by an exact Wiener--Dini belt series, prove finite Shannon entropy to be the sharp universal criterion for countable weights, and give an entropy--geometry extension for clustered measures. Every compact carrier of zero one-dimensional Hausdorff measure is admissible, whereas a nonzero rectifiable arc component forces divergence on a set of positive Haar measure. In orthogonal coordinates, the theorem also gives a weight-free scaled $F$ law for Pearson divergence from a fixed simplex vector to a $\\operatorname{Dirichlet}(1/2,\\ldots,1/2)$ vector.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00603",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianle Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6psr9bypeg8wm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00603",
  "fulltext_url": null,
  "id": "6psr9bypeg8wm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00603",
  "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",
      "math.CV"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7206180521",
  "pdf_url": "https://arxiv.org/pdf/2609.00603",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6psr9bypeg8wm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00603",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00603"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements",
  "title_text": "Universal Beta Incidence Angles: Cauchy Rigidity and Infinite Arrangements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/6psr9bypeg8wm",
  "venue": "arXiv",
  "versions": []
}