In general relativity, black hole singularities represent a core difficulty where curvature and the stress-energy tensor diverge. Departing from traditional curvature regularization, this paper proposes a spacetime topological phase transition hypothesis: the singularity is a smooth interface between two classes of topological manifolds. Normal spacetime corresponds to S¹ (fundamental group Z), while the black hole interior evolves into the wedge sum S¹ S¹ (fundamental group Z * Z). Using the R R topological density of curvature two-forms, we construct an extended Einstein-Hilbert action with a density-dependent coupling function and derive modified field equations. The model uses an effective vacuum perturbation density as the control parameter, strictly reducing to general relativity for > c and activating topological effects for < c. Under the static spherically symmetric approximation, we achieve approximate smooth matching of interior and exterior metrics (using the Hayward regular black hole as a phenomenological interior example), establish a strict coupling with Memory Gravity (TMG) via ² = ₀² (/ c - 1), derive the black hole shadow shift from the effective potential, and provide quantitative predictions for gravitational wave echoes along with falsification criteria. This model offers a new topological interpretation of the singularity problem and yields testable predictions for next-generation experiments.
Ma et al. (Fri,) studied this question.