The Quantum Fracture Limit: Resolution of the Schwarzschild Singularity via Structural Topology

Abstract

This paper formalizes the structural degradation of a continuous, compact manifold under the singular tidal dynam- ics of an interior Schwarzschild spacetime. We demonstrate that introducing a quantum-geometric lower bound, specifically the Planck length, mathematically halts the unbounded frag- mentation cascade predicted by classical general relativity. By evaluating proper dimensions within a co moving frame, this spatial cutoff induces a discrete topological phase transition that maps the smooth geometric manifold onto a finite, connected, weighted combinatorial graph. By utilizing the spectral prop- erties of Weighted Graph Laplacian operators, this relational framework preserves the foundational mass-energy and infor- mation states of the system, effectively resolving the classical infinite-density singularity paradox without violating causality constraints or established topology-change horizons.