Spacetime Snap-back Theory: A Topological Resolution of Black Hole Singularities, Dark Matter, and the Stochastic Gravitational Wave Background

Update (Version 2.0): This version includes refined mathematical derivations and updated conclusions regarding the spacetime snap-back model. propose the Spacetime Snap-back model, a modified gravity framework in which black hole evaporation terminates via a topological phase transition rather than complete evaporation. Using a modified Einstein-Hilbert action with a differentiable Gauss-Bonnet coupling, we derive the topological stress-energy tensor T00 proportional to 1/M³ and the Geometric Snap-back Index n(a) for Schwarzschild and Kerr geometries. The model predicts: 1. Dark matter abundance from Planck-scale topological remnants 2. Unitary resolution of the Information Paradox via the Island Formula 3. Three falsifiable observational signatures: - GRB spectral cutoff (Fermi-LAT) - Stochastic gravitational wave background (LISA / NANOGrav) - Planck-mass pile-up in microlensing surveys (Roman / LSST)