{
  "id": "oai:papers.qnfo.org:sacred-numbers-as-optima-of-an-ultrametric-divisibility-landscape",
  "slug": "sacred-numbers-as-optima-of-an-ultrametric-divisibility-landscape",
  "title": "Sacred Numbers as Optima of an Ultrametric Divisibility Landscape",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Cross‑culturally recurring “sacred” integers such as 12, 60, 108, 7 and 144 appear in myth, architecture and ritual despite disparate historical contexts. Existing explanations appeal to cultural accident or utilitarian convenience, leaving a formal, base‑invariant account absent. We fill this gap by defining a fitness functional F(n) on the integer n that combines (i) the divisor count d(n), (ii) the radical rad(n), (iii) the largest prime divisor s(n), and (iv) the depth h(n) of n in the ultrametric tree of divisibility. The functional is constructed to respect the strong triangle inequality of ultrametrics [1]; the landscape analogy, in which F(n) plays the role of an energy and preferred numbers act as basins of attraction, is adopted here as a design choice of the present work; no dynamical result from the ultrametric-diffusion literature is invoked or required. Using the explicit formulas d(n)=∏_p^eparallel n(e+1), rad(n)=∏_pmid np, s(n)=max\\pmid p prime,pmid n\\ and h(n)=∑_p^eparallel ne, we compute F(n)=log d(n)-lograd(n)-log s(n)-log h(n). Among the candidate integers we evaluate for 1≤ n≤120, the numbers 12, 108 and 144 achieve the highest values, with $144$ attaining the global maximum F_max= -1.974080, $108$ the second best with $F(108)= -2.014902$, and $12$ the third with $F(12)= -2.197224$, the latter $0.223144$ lower than the maximum. The result predicts that agents whose decision‑making weights both magnitude and divisibility will preferentially converge on these numbers. If the prediction holds, a unified non‑Archimedean explanation of numerical symbolism emerges, linking p‑adic geometry to cognitive preference.",
  "abstract_markdown": "Cross‑culturally recurring “sacred” integers such as 12, 60, 108, 7 and 144 appear in myth, architecture and ritual despite disparate historical contexts. Existing explanations appeal to cultural accident or utilitarian convenience, leaving a formal, base‑invariant account absent. We fill this gap by defining a fitness functional $F(n)$ on the integer $n$ that combines (i) the divisor count $d(n)$, (ii) the radical $\\operatorname{rad}(n)$, (iii) the largest prime divisor $s(n)$, and (iv) the depth $h(n)$ of $n$ in the ultrametric tree of divisibility. The functional is constructed to respect the strong triangle inequality of ultrametrics [1]; the landscape analogy, in which $F(n)$ plays the role of an energy and preferred numbers act as basins of attraction, is adopted here as a design choice of the present work; no dynamical result from the ultrametric-diffusion literature is invoked or required. Using the explicit formulas $d(n)=\\prod_{p^{e}\\parallel n}(e+1)$, $\\operatorname{rad}(n)=\\prod_{p\\mid n}p$, $s(n)=\\max\\{p\\mid p\\text{ prime},p\\mid n\\}$ and $h(n)=\\sum_{p^{e}\\parallel n}e$, we compute $F(n)=\\log d(n)-\\log\\operatorname{rad}(n)-\\log s(n)-\\log h(n)$. Among the candidate integers we evaluate for $1\\le n\\le120$, the numbers 12, 108 and 144 achieve the highest values, with $144$ attaining the global maximum $F_{\\max}= -1.974080$, $108$ the second best with $F(108)= -2.014902$, and $12$ the third with $F(12)= -2.197224$, the latter $0.223144$ lower than the maximum. The result predicts that agents whose decision‑making weights both magnitude and divisibility will preferentially converge on these numbers. If the prediction holds, a unified non‑Archimedean explanation of numerical symbolism emerges, linking $p$‑adic geometry to cognitive preference.",
  "date": "2026-10-20",
  "first_published": "2026-10-20",
  "last_revised": "2026-10-20T00:00:00Z",
  "modified": "2026-10-20T00:00:00Z",
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  "language": "en",
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  "peer_review_status": "Not independently peer-reviewed",
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  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/sacred-numbers-as-optima-of-an-ultrametric-divisibility-landscape",
  "license_url": "https://legal.qnfo.org/",
  "license_spdx": null,
  "sha256": "15c8cc563deb8a51ec129088a4022976228f8fae9d9f17c918644811f185bc62",
  "version_url": "https://papers.qnfo.org/papers/sacred-numbers-as-optima-of-an-ultrametric-divisibility-landscape/v/1.0.0",
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    "creators": [
      {
        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Sacred Numbers as Optima of an Ultrametric Divisibility Landscape"
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    "publicationYear": "2026",
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        "date": "2026-10-20",
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        "description": "Cross‑culturally recurring “sacred” integers such as 12, 60, 108, 7 and 144 appear in myth, architecture and ritual despite disparate historical contexts. Existing explanations appeal to cultural accident or utilitarian convenience, leaving a formal, base‑invariant account absent. We fill this gap by defining a fitness functional F(n) on the integer n that combines (i) the divisor count d(n), (ii) the radical rad(n), (iii) the largest prime divisor s(n), and (iv) the depth h(n) of n in the ultrametric tree of divisibility. The functional is constructed to respect the strong triangle inequality of ultrametrics [1]; the landscape analogy, in which F(n) plays the role of an energy and preferred numbers act as basins of attraction, is adopted here as a design choice of the present work; no dynamical result from the ultrametric-diffusion literature is invoked or required. Using the explicit formulas d(n)=∏_p^eparallel n(e+1), rad(n)=∏_pmid np, s(n)=max\\pmid p prime,pmid n\\ and h(n)=∑_p^eparallel ne, we compute F(n)=log d(n)-lograd(n)-log s(n)-log h(n). Among the candidate integers we evaluate for 1≤ n≤120, the numbers 12, 108 and 144 achieve the highest values, with $144$ attaining the global maximum F_max= -1.974080, $108$ the second best with $F(108)= -2.014902$, and $12$ the third with $F(12)= -2.197224$, the latter $0.223144$ lower than the maximum. The result predicts that agents whose decision‑making weights both magnitude and divisibility will preferentially converge on these numbers. If the prediction holds, a unified non‑Archimedean explanation of numerical symbolism emerges, linking p‑adic geometry to cognitive preference.",
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