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  "slug": "critical-values-on-a-non-uniform-number-line-bifurcation-markers-normalization-d",
  "title": "Critical Values on a Non-Uniform Number Line: Bifurcation Markers, Normalization Dependence, and the Conjectured Specialness of Physical Constants",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "We examine the conjecture that the real number line, as a model of physical state space, is misleadingly uniform: that certain parameter values act as topological phase boundaries rather than smooth increments, and that fundamental constants such as the fine-structure constant α ≈ 1/137 may mark bifurcation points between qualitatively different dynamical regimes. We formalize a number-to-phase-space map Φ that assigns to each parameter value the topological equivalence class of the associated dynamics, making precise in what sense a parameter value can be a \"boundary.\" We work two canonical examples in full arithmetic: the simple pendulum, whose separatrix sits at normalized energy ε = 2 exactly, and a central-force problem with a 1/r² correction, whose fall-to-center threshold sits at dimensionless coupling β = -1 exactly. Both thresholds land on small integers only because normalization conventions absorb dimensional constants into the critical value; the same physics yields $1$, 4/3, or $4$ under different normalizations of the double-pendulum flip energy. [...]",
  "abstract_markdown": "We examine the conjecture that the real number line, as a model of physical state space, is misleadingly uniform: that certain parameter values act as topological phase boundaries rather than smooth increments, and that fundamental constants such as the fine-structure constant $\\alpha \\approx 1/137$ may mark bifurcation points between qualitatively different dynamical regimes. We formalize a number-to-phase-space map $\\Phi$ that assigns to each parameter value the topological equivalence class of the associated dynamics, making precise in what sense a parameter value can be a \"boundary.\" We work two canonical examples in full arithmetic: the simple pendulum, whose separatrix sits at normalized energy $\\varepsilon = 2$ exactly, and a central-force problem with a $1/r^{2}$ correction, whose fall-to-center threshold sits at dimensionless coupling $\\beta = -1$ exactly. Both thresholds land on small integers only because normalization conventions absorb dimensional constants into the critical value; the same physics yields $1$, $4/3$, or $4$ under different normalizations of the double-pendulum flip energy. [...]",
  "date": "2026-10-11",
  "first_published": "2026-10-11",
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  "url": "https://papers.qnfo.org/papers/critical-values-on-a-non-uniform-number-line-bifurcation-markers-normalization-d",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
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        "title": "Critical Values on a Non-Uniform Number Line: Bifurcation Markers, Normalization Dependence, and the Conjectured Specialness of Physical Constants"
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        "description": "We examine the conjecture that the real number line, as a model of physical state space, is misleadingly uniform: that certain parameter values act as topological phase boundaries rather than smooth increments, and that fundamental constants such as the fine-structure constant α ≈ 1/137 may mark bifurcation points between qualitatively different dynamical regimes. We formalize a number-to-phase-space map Φ that assigns to each parameter value the topological equivalence class of the associated dynamics, making precise in what sense a parameter value can be a \"boundary.\" We work two canonical examples in full arithmetic: the simple pendulum, whose separatrix sits at normalized energy ε = 2 exactly, and a central-force problem with a 1/r² correction, whose fall-to-center threshold sits at dimensionless coupling β = -1 exactly. Both thresholds land on small integers only because normalization conventions absorb dimensional constants into the critical value; the same physics yields $1$, 4/3, or $4$ under different normalizations of the double-pendulum flip energy. [...]",
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