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  "id": "oai:papers.qnfo.org:the-q-generalized-diameter-scaling-exponents-d-q-for-arbitrary-ultrametric-ratio",
  "slug": "the-q-generalized-diameter-scaling-exponents-d-q-for-arbitrary-ultrametric-ratio",
  "title": "The q-Generalized α-Diameter: Scaling Exponents D_q^α for Arbitrary Ultrametric Ratios",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "In the QNFO ultrametric physics program, the fine-structure constant α has been reframed as a cross-ratio of two measurable electron length scales, and adelic extensions on Bruhat–Tits buildings have been proposed. A natural but unresolved question is how the characteristic \"depth\" associated with the α-gap behaves when the underlying ultrametric hierarchy is re-parameterized to an arbitrary scaling ratio q rather than the privileged ratio q = α⁻¹. This paper adopts the convention D_q^α = ln(1/α)/ln q: the number of hierarchy levels, measured in units of the ratio q, needed to span the interval between the classical electron radius and the reduced Compton wavelength. (Two alternative conventions for the symbol D_q^α — a fractional q-derivative and a Rényi multifractal exponent — were proposed by independent drafts; per the reconciliation policy, they are documented in Appendix A rather than silently merged.) We derive D_q^α in closed form, prove its monotonicity, base covariance, and multiplicativity, and evaluate it numerically for q ∈ {2, 3, φ, 10, α⁻¹}: D₂^α = 7.09841, D₃^α = 4.47860, D_φ^α = 10.22469, D₁₀^α = 2.13683, and D_{α⁻¹}^α = 1 exactly. [...]",
  "abstract_markdown": "In the QNFO ultrametric physics program, the fine-structure constant α has been reframed as a cross-ratio of two measurable electron length scales, and adelic extensions on Bruhat–Tits buildings have been proposed. A natural but unresolved question is how the characteristic \"depth\" associated with the α-gap behaves when the underlying ultrametric hierarchy is re-parameterized to an arbitrary scaling ratio q rather than the privileged ratio q = α⁻¹. This paper adopts the convention D_q^α = ln(1/α)/ln q: the number of hierarchy levels, measured in units of the ratio q, needed to span the interval between the classical electron radius and the reduced Compton wavelength. (Two alternative conventions for the symbol D_q^α — a fractional q-derivative and a Rényi multifractal exponent — were proposed by independent drafts; per the reconciliation policy, they are documented in Appendix A rather than silently merged.) We derive D_q^α in closed form, prove its monotonicity, base covariance, and multiplicativity, and evaluate it numerically for q ∈ {2, 3, φ, 10, α⁻¹}: D₂^α = 7.09841, D₃^α = 4.47860, D_φ^α = 10.22469, D₁₀^α = 2.13683, and D_{α⁻¹}^α = 1 exactly. [...]",
  "date": "2026-10-02",
  "first_published": "2026-10-02",
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  "url": "https://papers.qnfo.org/papers/the-q-generalized-diameter-scaling-exponents-d-q-for-arbitrary-ultrametric-ratio",
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    "titles": [
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        "title": "The q-Generalized α-Diameter: Scaling Exponents D_q^α for Arbitrary Ultrametric Ratios"
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        "description": "In the QNFO ultrametric physics program, the fine-structure constant α has been reframed as a cross-ratio of two measurable electron length scales, and adelic extensions on Bruhat–Tits buildings have been proposed. A natural but unresolved question is how the characteristic \"depth\" associated with the α-gap behaves when the underlying ultrametric hierarchy is re-parameterized to an arbitrary scaling ratio q rather than the privileged ratio q = α⁻¹. This paper adopts the convention D_q^α = ln(1/α)/ln q: the number of hierarchy levels, measured in units of the ratio q, needed to span the interval between the classical electron radius and the reduced Compton wavelength. (Two alternative conventions for the symbol D_q^α — a fractional q-derivative and a Rényi multifractal exponent — were proposed by independent drafts; per the reconciliation policy, they are documented in Appendix A rather than silently merged.) We derive D_q^α in closed form, prove its monotonicity, base covariance, and multiplicativity, and evaluate it numerically for q ∈ {2, 3, φ, 10, α⁻¹}: D₂^α = 7.09841, D₃^α = 4.47860, D_φ^α = 10.22469, D₁₀^α = 2.13683, and D_{α⁻¹}^α = 1 exactly. [...]",
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