{
  "abstract": "For the power mean $P_t$, $t<1$, let \\[ Λ_N(t)=\\sup_{x_k>0} \\frac{\\sum_{n=1}^N P_t(x_1,\\ldots,x_n)}{\\sum_{n=1}^N x_n} \\] be its finite Hardy constant. Negative powers, the geometric mean, and the usual positive-exponent Hardy inequality correspond respectively to $t<0$, $t=0$, and $0<t<1$. With the analytic parameter $q=t/(1-t)\\in(-1,\\infty)$, and writing $λ_N(q)=Λ_N(q/(1+q))$, we derive one exact scalar shooting map for all three regimes; its apparent singularity at $q=0$ is removable and gives the finite Carleman problem exactly. The map has a common terminal condition and its limiting vector field has a quadratic critical bottleneck at \\[ y_*(q)=1+q,\\qquad Λ_*(q)=(1+q)^{(1+q)/q}. \\] We prove that there is a unique real-analytic phase $κ(q)$ for which the finite-defect expansion has zero cubic coefficient and, uniformly for $q$ in compact subsets of $(-1,\\infty)$, for every fixed $L\\ge2$, \\[ Λ_*(q)-λ_N(q) =\\sum_{j=2}^{L}\\frac{A_j(q)}{(\\log N+κ(q))^j} +O\\!\\left((\\log N)^{-L-1}\\right). \\] In this phase-normalized scale, \\[ A_2=2π^2(1+q)Λ_*,\\qquad A_3=0,\\qquad A_4=-\\frac{π^2}{3}(2q^2+5q+5)A_2. \\] Thus the classical Carleman and positive-Hardy leading corrections are projections of a single analytic family. At the Carleman parameter we obtain $A_2(0)=2eπ^2$ and $A_4(0)=-(10/3)eπ^4$; the latter is also recovered by a direct local residue calculation. The phase itself contains the global discrete defect. After phase normalization, the first coefficient involving a nonconstant discrete response is $A_5$, at order $(\\log N+κ(q))^{-5}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10598",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianchi Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22hqq5h84p5ys.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10598",
  "fulltext_url": null,
  "id": "22hqq5h84p5ys",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10598",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7212229613",
  "pdf_url": "https://arxiv.org/pdf/2609.10598",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22hqq5h84p5ys.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.10598",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10598"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Finite Hardy Phase Theory for Power Means: An exact analytic shooting family through the Carleman parameter",
  "title_text": "Finite Hardy Phase Theory for Power Means: An exact analytic shooting family through the Carleman parameter",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/22hqq5h84p5ys",
  "venue": "arXiv",
  "versions": []
}