{
  "abstract": "Let $F$ be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass \\[ M_*(F)=\\inf_{covol_g(N)=1}M_g(F) \\] and prove the two-sided comparison \\[ \\sqrt{\\frac83}\\,m(F)\\leq M_*(F)\\leq 3\\sqrt{2\\,q(F)}. \\] Here $q(F)=deg(F\\cdot F)$ is stable self-intersection and $m(F)$ is the least degree of $F$ under a primitive rank-two lattice quotient. For a surjection $A$, one has $deg(A_*F)=deg(F\\cdot[\\ker A_R])$; hence $m(F)$ is equivalently the least stable intersection degree with a complete weight-one rational two-plane. In particular, \\[ q(F)\\geq\\frac4{27}m(F)^2. \\] The proof combines an exact Gaussian--kinematic formula for mixed stable intersection with an affine normalization principle for stationary spherical links. In dimension four, the identification $Gr^+(2,4)\\simeq S^2\\times S^2$ converts ambient isotropy into a sharp second-moment transversality estimate. We prove affine covariance for $M_*$, derive equality and stability conditions for the transversality estimate, and give examples where ambient isotropy coexists with zero mixed intersection, as well as a three-fan in six-space with zero self-intersection. Determining the optimal numerical constant remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28498",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikita Kalinin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bj2d11mfxq80m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28498",
  "fulltext_url": null,
  "id": "bj2d11mfxq80m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28498",
  "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.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214367047",
  "pdf_url": "https://arxiv.org/pdf/2609.28498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bj2d11mfxq80m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans",
  "title_text": "Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/bj2d11mfxq80m",
  "venue": "arXiv",
  "versions": []
}