Arithmetic Geometry of Planck Scale: Deriving Kg · C = 1 from Zeta Zeros

We present a comprehensive derivation of the geometric factor $K$ that establishes a mathematical bridge between the first four nontrivial Riemann zeta zeros and fundamental physical constants. Through high-precision computation (200+ digits) we demonstrate that $K$ decomposes into two exactly inverse components: the \textit{geometric seed} $K_g \approx 0.008353870129$ and the \textit{completion factor} $C \approx 119.700000000$, with $K_g \cdot C = 1$. This identity reveals that the Planck length $\ell_P = \sqrt{G\hbar/(c^3 K)}$ is intrinsically determined by arithmetic relationships among $\gamma_1, \gamma_2, \gamma_3, \gamma_4$. The framework provides first-principles derivations of $\ell_P = \SI{1.616255e-35}{\meter}$, $E_0 = \SI{1820.469}{\electronvolt}$, and $\alpha^{-1} = 137.035999084$, all emerging from the same geometric structure. The work resolves previous apparent inconsistencies and establishes a mathematical foundation for the geometric origin of physical scales.