{
  "abstract": "Heavy-tailed $p$-value combination tests are attractive under unknown dependence because their null tails can be first-order robust even when the exact dependent null distribution is unavailable. That robustness does not resolve calibration: $\\Pr\\{T>q(α)\\}=α+o(α)$ neither quantifies the remaining size error nor determines its sign. We develop a second-order calibration theory for positive Half-Cauchy and reciprocal, or harmonic-mean, aggregation. After exact marginal standardization, extremal dependence is represented by an index-one exponent measure on the coordinate-face lattice. Its support separates axial, full-interior, proper-face, and mixed geometries. A Möbius decomposition shows how the noncompact weighted half-space probes every face, while quantitative face limits determine whether the correction is integrable, critically amplified, or nonintegrable. With common-factor inversion and a tail-to-calibration map, this yields machinery applicable beyond individual parametric copula families. The results include an integrable hidden-face transfer theorem, a sharp fixed-dimensional local-corner theorem with explicit higher-face control, a common-heavy-factor theorem, and size and critical-value expansions relative to independence. Gaussian, standard multivariate-$t$, positive Clayton, and max-linear pair-shock models exhibit distinct power, logarithmic, radial--angular, and proper-face mechanisms; the Gaussian weighted-half-space transfer is conditional on explicit face-boundary hypotheses. Thus dependence geometry determines the rate, coefficient, and direction of the calibration error left unresolved by first-order validity. Asymptotics use $t\\to\\infty$, equivalently vanishing significance levels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29658",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianle Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12fbavhggtkyk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29658",
  "fulltext_url": null,
  "id": "12fbavhggtkyk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29658",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2003|Financial Risk and Volatility Modeling"
    ]
  ],
  "openalex": "W7214303437",
  "pdf_url": "https://arxiv.org/pdf/2609.29658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12fbavhggtkyk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29658"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Copula Geometry and Second-Order Calibration of Heavily Right Aggregation",
  "title_text": "Copula Geometry and Second-Order Calibration of Heavily Right Aggregation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:21Z",
  "url": "https://research.draconitic.com/paper/12fbavhggtkyk",
  "venue": "arXiv",
  "versions": []
}