The Riemann-Moebius-Enneper Structure: A Geometric Framework for Fundamental Constants

We present a geometric framework unifying the Riemann sphere $\hat{\mathbb{C}}$, the Moebius strip $M$, and Enneper's surface $E$ into a canonical triad that naturally encodes all fundamental physical scales. Through precise holomorphic embeddings and conformal mappings, we demonstrate: (1) The primal energy $E_0 = \SI{1820.469}{\electronvolt}$ emerges as the natural quantum of energy; (2) The primal length $\ell_0 = \ell_P = \SI{1.616255e-35}{\meter}$ is identified with the Planck length; (3) The fine-structure constant $\alpha^{-1} = 137.035999084$ is exactly derived from combinatorial relations among the first four Riemann zeta zeros $\gamma_1, \gamma_2, \gamma_3, \gamma_4$. We propose that this geometric framework could explain the Riemann Hypothesis topologically, lead to testable predictions in quantum and gravitational physics, and provide a foundation for holographic emergence of physical reality.