{
  "abstract": "We prove thirteen results concerning three aspects of magnitude theory. The first four concern small-scale magnitude and microscopic weightings. We identify the derivative of the magnitude function at the origin with maximal energy, allowing the value $+\\infty$, and construct finite spaces for which small-scale magnitude diverges or for which the one-point property and finite concentration do not produce a microscopic weighting. A fifth result disproves the conjectured monotonicity of magnitude by giving a fourteen-point metric space of strict negative type whose magnitude function decreases. The remaining eight results concern continuity and dimension. Magnitude is Hausdorff-continuous at finite subsets of finite-dimensional positive definite normed spaces, but need not be continuous on compact subsets, even under convexity, geodesicity, bounded tractability, or simultaneous Hausdorff and Lipschitz convergence. The same constructions disprove an $\\ell_1$-intrinsic-volume formula and two tractability conjectures. Finally, we construct a countable compact space of negative type whose Minkowski dimension is zero and whose lower and upper magnitude dimensions are both infinite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13958",
  "author_total": 1,
  "authors": [
    {
      "name": "Venkata Siddharth Pendyala"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9dgf83eqv9scf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13958",
  "fulltext_url": null,
  "id": "9dgf83eqv9scf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13958",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7221781683",
  "pdf_url": "https://arxiv.org/pdf/2610.13958",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9dgf83eqv9scf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.13958",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13958"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Thirteen Results in Magnitude Theory",
  "title_text": "Thirteen Results in Magnitude Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/9dgf83eqv9scf",
  "venue": "arXiv",
  "versions": []
}