Published June 10, 2026 | Version v1

Re-Leveling and the Logarithmic Forbidden Interval in Scale Calculus

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I show that the forbidden zone κ ∈ (e⁻¹, e⁻¹/²) of Scale Calculus corresponds, under the natural logarithmic reparametrization ξ = ln(r₂/r₁), to the half-unit interval ξ ∈ (1/2, 1). This interval is the unique half-unit interval in ξ-space that contains ln 2, the logarithm of the natural harmonic overtone ratio, while remaining strictly below ln(2π), the logarithm of the full rotational period of the sphere. Numerical computation of the acoustic eigenmode spectrum for all five Platonic solid spherical shell cavities confirms that the fundamental harmonic ratio F₂/F₁ ranges from 1.865 to 1.992 across the five cavity types, with ln(F₂/F₁) falling inside (1/2, 1) in every case. I introduce the Re-Leveling Principle: no element of a self-similar resonant hierarchy can be identified with the hierarchy itself without losing its place within it. The forbidden zone is the interval over which this re-leveling occurs; traversal of it is what Scale Calculus calls a Directed Resonance Crossing. Conjecture G1 (Revised) states that these properties jointly determine the forbidden zone uniquely, promoting it from an axiom to a derived consequence of harmonic geometry and the exponential structure of the κ-map.

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