{
  "id": "oai:papers.qnfo.org:prime-valuation-depth",
  "slug": "prime-valuation-depth",
  "title": "Prime Valuation Depth: Multiplication as Branching, the Calculus of Indications, and the Structural No-Cloning Reading",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "This paper develops a structural reading of the p-adic valuation as a measure of depth along a prime branch of the integer factorization tree, rather than a measure of size. The reading, grounded in Ostrowski's classification of absolute values on the rationals, is proposed as a bridge between the calculus of indications (Spencer-Brown's theory of distinction) and number theory (the tree of prime divisors). The correspondence is stated at the level of shared structural laws: both structures satisfy a nesting/hierarchy condition, and the p-adic valuation is the unique measure of multiplicity along a labeled branch. The reading is then extended to quantum mechanics: the tensor product multiplies Hilbert-space dimensions, so prime factorization of dimension labels the branch types and the p-adic valuation of dimension counts tensor-branch depth. Within this vocabulary, the no-cloning theorem is expressed as the impossibility of linearly duplicating multiplicative branching: the diagonal map sending a state to its tensor square is nonlinear, and the image of such a map is not a linear subspace. The categorical content of this statement is established in the literature by Coecke and Duncan and by Coecke and Paquette; the contribution of the present paper is the p-adic-depth vocabulary and its branch-depth framing, which is explicitly labeled as an interpretive reading rather than new physics. Falsifiability conditions are stated for every non-empirical claim, and the limits of the reading are discussed, including the absence of a direct literature anchor for p-adic-specific no-cloning theorems.",
  "abstract_markdown": "This paper develops a structural reading of the p-adic valuation as a measure of depth along a prime branch of the integer factorization tree, rather than a measure of size. The reading, grounded in Ostrowski's classification of absolute values on the rationals, is proposed as a bridge between the calculus of indications (Spencer-Brown's theory of distinction) and number theory (the tree of prime divisors). The correspondence is stated at the level of shared structural laws: both structures satisfy a nesting/hierarchy condition, and the p-adic valuation is the unique measure of multiplicity along a labeled branch. The reading is then extended to quantum mechanics: the tensor product multiplies Hilbert-space dimensions, so prime factorization of dimension labels the branch types and the p-adic valuation of dimension counts tensor-branch depth. Within this vocabulary, the no-cloning theorem is expressed as the impossibility of linearly duplicating multiplicative branching: the diagonal map sending a state to its tensor square is nonlinear, and the image of such a map is not a linear subspace. The categorical content of this statement is established in the literature by Coecke and Duncan and by Coecke and Paquette; the contribution of the present paper is the p-adic-depth vocabulary and its branch-depth framing, which is explicitly labeled as an interpretive reading rather than new physics. Falsifiability conditions are stated for every non-empirical claim, and the limits of the reading are discussed, including the absence of a direct literature anchor for p-adic-specific no-cloning theorems.",
  "date": "2026-08-13",
  "first_published": "2026-08-13",
  "last_revised": "2026-08-13T00:00:00Z",
  "modified": "2026-08-13T00:00:00Z",
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  "language": "en",
  "keywords": [
    "p-adic valuation",
    "laws of form",
    "calculus of indications",
    "no-cloning theorem",
    "tensor product",
    "Ostrowski's theorem"
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  "url": "https://papers.qnfo.org/papers/prime-valuation-depth",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "Prime Valuation Depth: Multiplication as Branching, the Calculus of Indications, and the Structural No-Cloning Reading"
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        "subject": "laws of form"
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      {
        "subject": "calculus of indications"
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      {
        "subject": "no-cloning theorem"
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        "subject": "tensor product"
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        "subject": "Ostrowski's theorem"
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