{
  "abstract": "We introduce farthest-cell triplet entropy, the conditional Shannon entropy of the farthest-prototype label given three random prototypes. For independent queries and prototypes, its estimator records only the farthest label, not coordinates or numerical distances. The statistic is bounded by $\\log 3$, is invariant under common strictly increasing transformations of the dissimilarities, and has an exact mutual-information interpretation. In high-dimensional isotropic radial models $X_d=R_dU_d$, the Euclidean ordering reduces to scores $λ_dξ_{i,d}-Z_i$, where $λ_d=\\sqrt d\\,\\operatorname{sd}(R_d)/\\mathbb{E} R_d$ and the $Z_i$ are independent standard Gaussian variables. This gives angular-dominated, intermediate, and radial-dominated entropy limits $\\log 3$, $H_{\\infty}(λ;F)$, and $0$. In hyperbolic space of curvature $-κ_d^2$, the same master curve appears at $λ_{d,\\mathbb H}=\\sqrt d\\,τ_d A(s_d)$, where $τ_d=\\operatorname{sd}(R_d)/\\mathbb{E} R_d$, $s_d=κ_d\\mathbb{E} R_d$, and $A(s)=s\\coth s$. With a calibrated radial law and $τ_d$, and a monotone operating interval, entropy inversion identifies the scale-invariant target $s_d^2$; absolute curvature requires an external length unit. CPU simulations give Euclidean and hyperbolic master-curve RMSEs of $0.0164$ and $0.0209$. Inversion from observed synthetic latent coordinates has a median relative error in $κ$ of $6.6\\%$, while angular anisotropy increases this error to $68.9\\%$. Thus the entropy statistic is comparison-based, whereas curvature recovery remains model-calibrated, is not graph-only, and is not robust to anisotropy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02362",
  "author_total": 1,
  "authors": [
    {
      "name": "Chongkun Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b25azx1302pke.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02362",
  "fulltext_url": null,
  "id": "b25azx1302pke",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02362",
  "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.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Applications"
    ]
  ],
  "openalex": "W7207803934",
  "pdf_url": "https://arxiv.org/pdf/2609.02362",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b25azx1302pke.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02362",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02362"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification",
  "title_text": "Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/b25azx1302pke",
  "venue": "arXiv",
  "versions": []
}