The Geometric Meaning of Hyperbolic Spheres and Helices: A New Perspective in Quantum Field Theory

We establish a unified geometric framework based on a universal spiral generator defined on a hyperbolic sphere. Starting from the Bohr quantization condition, we show that the Fermat spiral naturally emerges as the geometric foundation of quantum orbital structure. The universal spiral generator unifies all spiral families, including the hyperbolic spiral (quantum regime) and logarithmic spiral (macroscopic regime). We derive the exact geometric expression for the fine structure constant: α = lnφ/(21π), where φ is the golden ratio. The theory also contains a binary quantization rule and a fine-structure splitting formula directly derived from spiral geometry. A natural radial vector field is introduced, whose divergence corresponds to the nuclear source and whose z-component of curl corresponds to quantum angular momentum, spin, and orbital magnetic moment. This framework reveals a deep geometric origin for fundamental constants and quantum structure.