{
  "abstract": "We study Eratosthenes sieve as a discrete dynamic system. At each stage of the sieve there is a cycle of gaps ${\\mathcal G}(p^\\#)$ of length $ϕ(p^\\#)$ and span $p^\\#$. There is a recursion ${\\mathcal G}(p_k^\\#)\\longrightarrow {\\mathcal G}(p_{k+1}^\\#)$ that creates the next cycle from the current one. If we take initial conditions from the cycle ${\\mathcal G}(p_0^\\#)$, then for all constellations of span $|s| < 2p_1$, including gaps $g < 2p_1$, the driving terms of various lengths form Markov chains. These yield {\\em exact} models for the populations $n_s(p_k^\\#)$ for all further stages of the sieve. If $s$ is an admissible constellation of length $J$, then its population $n_{s,J}(p^\\#)$ grows as $Θ\\left( \\prod (q-J-1)\\right)$. So we factor out the superexponential growth to obtain the exact model for the relative population $w_s(p_k^\\#)$ of the constellation $s$ across all further stages of the sieve. $$ w_{s,J}(p_k^\\#) \\; = \\; n_{s,J}(p_k^\\#) \\, / \\, \\prod_{J+1 < p \\le p_k} (p-J-1) $$ The asymptotic value of the relative population is a constant ${w_{s,J}(\\infty) \\ge 1}$ that depends only on the odd prime factors that divide a span in $s$. Assuming that the instances of a constellation $s$ are approximately uniformly distributed in ${\\mathcal G}(p_k^\\#)$, we develop initial estimates of the number of instances $s$ that would occur in the interval of survival $ΔH(p_k) = (p_k^2, p_{k+1}^2)$. We define a statistic $η_s(p_k)$, the quadratic density of the constellation $s$ over the interval $ΔH(p_k)$. We show that the initial estimates $\\widehat{η_g}(p)$ for prime gaps agree with samples up to $5.677\\,E14$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26384",
  "author_total": 1,
  "authors": [
    {
      "name": "Fred B. Holt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cyze17sdjxan6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26384",
  "fulltext_url": null,
  "id": "cyze17sdjxan6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26384",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204561695",
  "pdf_url": "https://arxiv.org/pdf/2608.26384",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cyze17sdjxan6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2608.26384",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26384"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Discrete dynamics of Eratosthenes sieve",
  "title_text": "Discrete dynamics of Eratosthenes sieve",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/cyze17sdjxan6",
  "venue": "arXiv",
  "versions": []
}