{
  "abstract": "Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the associated congruent-number elliptic curve. More precisely, $M(N)\\le C^{1+r_N}$ for an absolute constant $C>1$, where $r_N$ is the rank of $y^2=x^3-N^2x$. Using the established Paulsen-West rank obstruction and Smith's results on quadratic twists, we then prove \\[ \\lim_{X\\to\\infty}\\frac1X \\sum_{\\substack{N\\le X\\\\N\\text{ positive squarefree}}}M(N)^q=0 \\qquad\\text{for every fixed }q>0. \\] Thus the result counts cuboid classes with their full multiplicities at each $N$. A quantitative decay estimate is also obtained. The geometric step excludes every positive-dimensional real abelian coset through the positive part of the compatibility surface; quantitative Mordell-Lang then gives the required bound. The argument is unconditional and uses the cuboid correspondence and rank obstruction as prior results.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38650",
  "author_total": 1,
  "authors": [
    {
      "name": "Valery Asiryan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/54ref5q3t2f5n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38650",
  "fulltext_url": null,
  "id": "54ref5q3t2f5n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38650",
  "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.GM"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7215071422",
  "pdf_url": "https://arxiv.org/pdf/2609.38650",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/54ref5q3t2f5n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38650",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38650"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Finiteness in Square Classes and Weighted Zero Density of Perfect Cuboids",
  "title_text": "Finiteness in Square Classes and Weighted Zero Density of Perfect Cuboids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/54ref5q3t2f5n",
  "venue": "arXiv",
  "versions": []
}