{
  "abstract": "Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, $\\textit{canonically}$ stems from a $\\textit{single}$ definitional primitive $-$ the ($\\mathbb R^{\\vcenter{\\hbox{$\\scriptscriptstyle+$}}}\\!$-quantitative) invariant $\\mathcal Q$-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared $p = |\\mathfrak a|^2$ $-$ the quantum Born rule $-$ and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the $\\textit{well-defined}$ interpretable turns out to be, up to gauge $\\mathcal Q {\\,\\to\\,} \\mathrm{const} {\\,\\vcenter{\\hbox{$\\scriptstyle\\times$}}\\,} \\mathcal Q$, the unique gauge-invariant measure $\\mathcal Q=|\\hspace{-0.18em}| {\\cdot}{\\cdot}{\\cdot} |\\hspace{-0.18em}|^2$; independently of the field $\\mathbb R$ or $\\mathbb C$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24943",
  "author_total": 1,
  "authors": [
    {
      "name": "Yu. V. Brezhnev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5q0gqr1x0fk9k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24943",
  "fulltext_url": null,
  "id": "5q0gqr1x0fk9k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24943",
  "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|Mathematics and Applications"
    ]
  ],
  "openalex": "W7204461476",
  "pdf_url": "https://arxiv.org/pdf/2608.24943",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5q0gqr1x0fk9k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2608.24943",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24943"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Mathematics and Applications",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Note on the Measure of Vector and Pythagorean Theorem",
  "title_text": "A Note on the Measure of Vector and Pythagorean Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/5q0gqr1x0fk9k",
  "venue": "arXiv",
  "versions": []
}