{
  "abstract": "This work proposes a semantic environment for arithmetic geometry in which passage between presentations carries an explicit weight. Morphism costs induce a Lawvere generalized distance by taking infima over transports, and heights satisfy the corresponding transport bounds. A labeled quantitative $(2,1)$-category retains invertible $2$-morphisms as coherent comparisons between transports. Its vertical groupoids admit a transport interpretation inspired by the identity types of Voevodsky's C-systems. If a height factors through the underlying object, then a vertical comparison has equal endpoints, whatever its transport cost. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and we relate a bound of this form to the $abc$ conjecture. The defect remains visible between non-minimal and normalized presentations, while conductor complexity stays attached to the underlying curve. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, using strong point-count asymptotics with an explicit convergence requirement. The refined constant retains separately typed arithmetic factors. A finite differential correction makes the period independent of the chosen differential and links its normalization to model transport. Within the declared finite-descent theory, Tate--Shafarevich finiteness reduces to finitely many closed prime columns of Selmer shadows and a uniform cutoff. For $y^2=x^3-q^2x$ with $q\\equiv3\\pmod8$ prime, two-isogeny descent closes the two-primary column uniformly. The statements admit expression in $\\mathsf{ACA}_0$ relative to certified descent and Mordell--Weil data, with certified Cauchy names for the period and regulator.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.13609",
  "author_total": 1,
  "authors": [
    {
      "name": "R. Laniewski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55gm68keantqd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.13609",
  "fulltext_url": null,
  "id": "55gm68keantqd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.13609",
  "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.CT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7203604151",
  "pdf_url": "https://arxiv.org/pdf/2608.13609",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55gm68keantqd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.13609",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.13609"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Intensional semantics for comparison problems in arithmetic geometry",
  "title_text": "Intensional semantics for comparison problems in arithmetic geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/55gm68keantqd",
  "venue": "arXiv",
  "versions": []
}