{
  "id": "oai:papers.qnfo.org:positional-notation-as-an-intrinsic-ultrametric-valuation",
  "slug": "positional-notation-as-an-intrinsic-ultrametric-valuation",
  "title": "Positional Notation as an Intrinsic Ultrametric Valuation",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Positional numeral systems are taught and implemented as points on the Archimedean real line, yet the digit strings that encode integers form a rooted b‑ary tree. This work identifies the concrete gap that the native metric induced by digit‑string structure has never been formalized, leading to hidden assumptions about distance and clustering. We adopt the valuation framework of non‑Archimedean number theory, define the b‑adic valuation v_b, and prove that the associated distance d_b(x,y)=b^-v_b(x-y) satisfies all ultrametric axioms for every integer base b≥2. Two quantitative illustrations are derived analytically: (1) exact counts of ultrametric distances for all unordered pairs of decimal integers 0≤ x<y≤999, yielding 450 000 pairs at distance $1$, 45 000 at $0.1$, and 4 500 at $0.01$; (2) the mean b‑adic distance for the interval 0≤ x<y≤99 is (101)/(110)≈0.918, the diameter‑normalized Euclidean mean is (101)/(297)≈0.340, giving a ratio of exactly $2.70$, and all $855$ pairs with Euclidean distance ≤9 are maximally distant (d_b=1). Recognizing the intrinsic ultrametric reshapes curricula, data‑structure design, and cryptographic primitives that rely on positional notation.",
  "abstract_markdown": "Positional numeral systems are taught and implemented as points on the Archimedean real line, yet the digit strings that encode integers form a rooted $b$‑ary tree. This work identifies the concrete gap that the native metric induced by digit‑string structure has never been formalized, leading to hidden assumptions about distance and clustering. We adopt the valuation framework of non‑Archimedean number theory, define the $b$‑adic valuation $v_b$, and prove that the associated distance $d_b(x,y)=b^{-v_b(x-y)}$ satisfies all ultrametric axioms for every integer base $b\\ge2$. Two quantitative illustrations are derived analytically: (1) exact counts of ultrametric distances for all unordered pairs of decimal integers $0\\le x<y\\le999$, yielding $450\\,000$ pairs at distance $1$, $45\\,000$ at $0.1$, and $4\\,500$ at $0.01$; (2) the mean $b$‑adic distance for the interval $0\\le x<y\\le99$ is $\\frac{101}{110}\\approx0.918$, the diameter‑normalized Euclidean mean is $\\frac{101}{297}\\approx0.340$, giving a ratio of exactly $2.70$, and all $855$ pairs with Euclidean distance $\\le9$ are maximally distant ($d_b=1$). Recognizing the intrinsic ultrametric reshapes curricula, data‑structure design, and cryptographic primitives that rely on positional notation.",
  "date": "2026-10-20",
  "first_published": "2026-10-20",
  "last_revised": "2026-10-20T00:00:00Z",
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  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/positional-notation-as-an-intrinsic-ultrametric-valuation",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Positional Notation as an Intrinsic Ultrametric Valuation"
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    "publicationYear": "2026",
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        "description": "Positional numeral systems are taught and implemented as points on the Archimedean real line, yet the digit strings that encode integers form a rooted b‑ary tree. This work identifies the concrete gap that the native metric induced by digit‑string structure has never been formalized, leading to hidden assumptions about distance and clustering. We adopt the valuation framework of non‑Archimedean number theory, define the b‑adic valuation v_b, and prove that the associated distance d_b(x,y)=b^-v_b(x-y) satisfies all ultrametric axioms for every integer base b≥2. Two quantitative illustrations are derived analytically: (1) exact counts of ultrametric distances for all unordered pairs of decimal integers 0≤ x<y≤999, yielding 450 000 pairs at distance $1$, 45 000 at $0.1$, and 4 500 at $0.01$; (2) the mean b‑adic distance for the interval 0≤ x<y≤99 is (101)/(110)≈0.918, the diameter‑normalized Euclidean mean is (101)/(297)≈0.340, giving a ratio of exactly $2.70$, and all $855$ pairs with Euclidean distance ≤9 are maximally distant (d_b=1). Recognizing the intrinsic ultrametric reshapes curricula, data‑structure design, and cryptographic primitives that rely on positional notation.",
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