Golden Ratio as the Fundamental Scale Factor in Geometric Thermodynamics:Predictions for the Tau Lepton Mass and Neutrino Mass Scale

We extend the Geometric Thermodynamics framework with $\phic$-base scaling to predict two additional fundamental quantities: the tau lepton mass and the neutrino mass scale. The $\phic$-base system, derived from first principles of Length Primacy and Information Optimization, provides a discrete scaling law $m_n = m_0 \phic^{n}$ for particle masses. For the tau lepton, we predict $m_\tau/m_e = \phic^{-17} \approx 3571.7$, corresponding to $m_\tau \approx 1824$ MeV, which compares favorably with the experimental value of 1776.86 MeV (2.7\% deviation). For the neutrino mass scale, we predict $m_\nu = m_e \phic^{32} \approx 0.0955$ eV, aligning with experimental constraints of $\sim 0.1$ eV (4.5\% deviation). Combined with our previous successful prediction of the muon $g-2$ anomaly ($\Delta a_\mu^{\GT} = \alpha\phic^{11}/(16\pi^2) \approx 2.59 \times 10^{-9}$, matching experiment within 3.2\%), these results demonstrate the remarkable predictive power and universality of the $\phic$-base scaling system across more than 38 orders of magnitude.