{
  "abstract": "This paper settles the first two problems of the research program posed in Purple Numbers: Foundations (DOI 10.5281/zenodo.21168196). Problem 5 (continuity at the Zero pole) is solved by a dichotomy theorem: the determination weight T — being locally constant — extends uniquely to a continuous function on the profinite integers, taking the value 0 at the pole itself (the pole is undetermined: perfectly balanced) and the extreme values ±ln 15015 at ±1 (beside the full spectrum, determination is maximal); whereas the reach ρ, the dig κ, and the magnitude log n admit no continuous extension, being unbounded on every profinite neighbourhood. Residue functionals cross to the pole; magnitude functionals do not: the framework's Zero/Infinity division is a topological theorem. Problem 3 (joint determination) is resolved empirically over all 455 million primes below 10^10: the archimedean determination A(p) and the finite determination T(p) are correlated, but the correlation obeys the clean two-term law corr(A,T) ≈ −0.51/ln x, vanishing at exactly the rate the independence postulate (Axiom 6 of the Foundations) requires; the archimedean reading alone leaves the direction of the reach essentially undecided (within ±2% of even, against the finite reading's 8%–92% range); and conditionally on T the residual effect of A is below one percent. The measured coupling constant −0.51 joins the two constants found earlier in the program, giving three independent instances of the same second-order structure: a limit constant plus a 1/ln x correction. The classical neighbours of these results — the Lemke Oliver–Soundararajan biases and standard profinite topology — are attributed throughout.Notes field: Third paper of the Purple Mathematics series. Solves Problem 5 (continuity dichotomy theorem at the Zero pole, with elementary proof) and empirically resolves Problem 3 (joint law of archimedean and finite determination over 455 million primes to 10^10) of the research program stated in Purple Numbers: Foundations. Classical ingredients (profinite topology; Chinese Remainder Theorem; the Lemke Oliver–Soundararajan consecutive-prime biases, arXiv:1603.03720) are attributed in the text; the dichotomy statement, the extension values at the pole, the measured coupling constant C_AT ≈ −0.51, and the separability finding are original to this paper.",
  "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/6cdw6ghd48jws.bib",
  "cited_by": 5,
  "doi": "10.5281/zenodo.21194306",
  "fulltext_url": null,
  "id": "6cdw6ghd48jws",
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  "landing_url": "https://doi.org/10.5281/zenodo.21194306",
  "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"
    ]
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  "openalex": "W7167369363",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-04",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/6cdw6ghd48jws.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7167369363",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.21194306"
    }
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "First Results on the Purple Program: The Continuity Dichotomy at the Zero Pole, and the Joint Law of the Two Determinations",
  "title_text": "First Results on the Purple Program: The Continuity Dichotomy at the Zero Pole, and the Joint Law of the Two Determinations",
  "type": "preprint",
  "updated_at": "2026-11-04T13:21:00Z",
  "url": "https://research.draconitic.com/paper/6cdw6ghd48jws",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}