STATUS: DEPRECATED (superseded by DOI: 10.5281/zenodo.18141472) - Emergent Dynamics in Hopf Fibrations: Unification of Gravity and Electromagnetism via Geometric Constraints in S³

The unification of General Relativity with Quantum Mechanics remains the central problem of theoretical physics. This paper proposes a geometric approach where fundamental interactions are not axiomatic forces, but emergent phenomena from the projection of a four-dimensional hypersphere (S³) onto a three-dimensional Euclidean space (R3) through the Hopf Fibration. Through numerical computational simulations, we demonstrate that bodies following inertial geodesics in S³, subject to a topological coupling, exhibit projected trajectories that reproduce the Inverse Square Law (1/r²). Additionally, we demonstrate that Special Relativity's Time Dilation emerges naturally as a conservation of norm in the velocity four-vector, interpreting the speed of light (c) as a geometric limit of information propagation on the manifold, rather than merely an electromagnetic constant. Data and Code Availability *** The complete source code used to generate the simulations and data presented in this article is publicly available in an open repository. The software allows reproduction of the emergent gravity and time dilation experiments in real-time. ***