{
  "abstract": "Purple Mathematics is an expository, unifying framework that gives every positive integer a structured identity through one local operation, the *dig*: factor n² − 1 = (n − 1)(n + 1) and read its largest prime factor ρ(n), comparing it to a fixed wall W = 11. This sorts the integers into a Zero domain and an Infinity domain along a \"bridge\" between 0 and infinity, with the unit 1 as the fixed point of x ↦ 1/x. The Zero domain is finite (Størmer's theorem): exactly 49 integers, the largest being 19,601. The paper develops the resulting structure, the choice of the wall, a triangular wall whose heights are the prime gaps, an identity card separating identity fields from diagnostic measurements, two independent signed calibrations, a structural-stability interpretation layer, an exact on-line detection criterion, and one genuinely open analytic problem about an associated prime-driven walk. The framework is explicitly a lens over classical results (Størmer, Dickman, prime gaps, Pólya–Liouville oscillation, twin-smooth integers); no new theorem is claimed, and every component is mapped to its established counterpart.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Samir Hanna Safar",
      "orcid": "0000-0001-5997-2347"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6scz31e5kgphd.bib",
  "cited_by": 6,
  "doi": "10.5281/zenodo.20753251",
  "fulltext_url": null,
  "id": "6scz31e5kgphd",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.20753251",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7165200141",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-05-18",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/6scz31e5kgphd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7165200141",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.20753251"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Purple Mathematics: A Visual Framework for Numerical Identity, Determination, and the On-Line Detection of Primes",
  "title_text": "Purple Mathematics: A Visual Framework for Numerical Identity, Determination, and the On-Line Detection of Primes",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:22Z",
  "url": "https://research.draconitic.com/paper/6scz31e5kgphd",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}