Geodesic Completeness in Schwarzschild Spacetime via Phase-Dependent Metric Signature Inversion

Standard General Relativity predicts that massive particles crossing the event horizon of a black hole inevitably terminate at a spacelike singularity (r=0). This paper proposes a modification to the standard kinematic model of fermions to resolve this geodesic incompleteness. We posit that elementary particles undergo a phase transition in the temporal dimension of their proper frame. By treating the null surface c not as an asymptotic limit but as a Topological Phase Boundary, we show that the electron-positron annihilation vertex is topologically equivalent to a spacelike metric inversion. When applied to gravitational collapse, this framework implies that the Event Horizon acts as a Causal Phase Boundary. Upon reaching the horizon, the particle undergoes a CPT inversion relative to the background metric, effectively reinterpreting the horizon not as an entrance to an interior, but as a repulsive transition surface. Furthermore, by extending this logic to higher-order spacelike intervals, we establish a continuous topology where a single particle oscillates through infinite generations of matter and antimatter, eliminating the physical singularity. Mathematically, this is rigorously defined via a Phase-Dependent Finslerian Metric, ensuring the action remains real-valued across the transition.