STATUS: EXPLORATORY HYPOTHESIS (superseded by DOI: 10.5281/zenodo.18141753) - Emergent Coulomb Potentials and Orbital Quantization via Classical Dynamics on $S^3$-Hopf Fibrations

The reconciliation of General Relativity (geometric determinism) and Quantum Mechanics (probabilistic indeterminism) remains the central challenge of modern physics. This paper presents a computational framework where fundamental quantum phenomena emerge from strictly deterministic dynamics in a 4-dimensional hyperspherical manifold ($S^3$). We demonstrate that a classical harmonic oscillator confined to $S^3$ geometry and projected onto $\mathbb{R}^3$ via the Hopf Fibration naturally reproduces the dynamics of a $1/r$ Coulomb potential. Numerical validation using a 4th-order Runge-Kutta integrator reveals a Virial Ratio of $\langle T \rangle / \langle V \rangle \approx 0.500$, consistent with hydrogen-like systems. Furthermore, the introduction of an action-based phase condition (De Broglie relation) results in the emergence of discrete topological structures indistinguishable from Schrödinger’s probability orbitals. These findings suggest that the probabilistic nature of the wavefunction may be an artifact of projecting higher-dimensional geometric trajectories onto lower-dimensional space.