Base Scaling from First Principles in Geometric Thermodynamics and its Prediction for the Muon - 2 Anomaly

We establish a first-principles framework--Geometric Thermodynamics--from two postulates: (1) \textbf{Length Primacy}: length is the fundamental physical unit, with time, mass, and energy derived; (2) \textbf{Information Optimization}: physical structures organize to optimize information capacity. We prove that constructing stable, self-similar structures across discrete scales necessitates an optimal scale growth factor given by the golden ratio \phititle{\phic}, forming a ``\phititle{\phic}-base'' system. This naturally explains empirical relations like \phititle{M_{\Planck}/m_{\electron} = \phic^{-36}} and \phititle{\alpha = (2\pi/7)\phic^{10}}. Crucially, such a system's dynamics automatically satisfy the optimal non-resonance condition of the Kolmogorov-Arnold-Moser (KAM) theorem, ensuring maximal long-term stability. As the first falsifiable prediction, we derive a new-physics contribution to the muon anomalous magnetic moment: \phititle{\Delta a_{\muon}^{\GT} = \alpha \phic^{11} / (16\pi^2) \approx 2.59 \times 10^{-9}}, which aligns with the current experimental anomaly of \phititle{(2.51 \pm 0.59) \times 10^{-9}}. This work provides a deductive explanation for mathematical constants in physics and elevates Geometric Thermodynamics to a predictive theory.