{
  "abstract": "Quantifying change in a single covariance, coherence or spectral connection requires a distance that respects the positive-definiteness constraint and holds the surrounding entries fixed. We characterize the Riemannian geometry induced by the affine-invariant metric when one conjugate off-diagonal pair of a Hermitian positive-definite matrix varies under this constraint. A fixed Schur congruence maps the admissible set to an open complex disk and reduces its metric to a two-dimensional form independent of the ambient matrix dimension. The disk is complete, with Gaussian curvature in $(-1/4,0]$, and has unique minimizing geodesics and globally defined logarithms. Its intrinsic distance strictly exceeds the ambient affine-invariant distance between distinct points. Conservation of angular momentum reduces the geodesic boundary-value problem to a single scalar equation with elementary angular and length integrals, enabling distance and logarithm evaluation without shooting. For a declared reference distribution with a finite second intrinsic moment, we establish a unique Fréchet centre and consistency of its empirical estimate, together with a sandwich central limit theorem under the stated regularity conditions. Tangent covariance standardization yields moment identities, and exchangeable rank calibration supplies finite-sample score guarantees. These results provide a geometric and statistical framework for isolated connections on a fixed complement, with explicit limits on extension to varying complements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01202",
  "author_total": 2,
  "authors": [
    {
      "name": "Ronald García Reyes"
    },
    {
      "name": "Pedro Antonio Valdés-Sosa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6s4z6w2sg78y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01202",
  "fulltext_url": null,
  "id": "d6s4z6w2sg78y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01202",
  "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",
      "stat.ME"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6s4z6w2sg78y.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:21Z",
      "native_id": "2611.01202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "Geodesics and intrinsic statistics of isolated connections in Hermitian positive-definite matrices",
  "title_text": "Geodesics and intrinsic statistics of isolated connections in Hermitian positive-definite matrices",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/d6s4z6w2sg78y",
  "venue": "arXiv",
  "versions": []
}