From Circle to Wedge Sum of Two Circles: A Topological Phase Transition Model for Black Hole Singularities – Curvature-Topological Extension of Einstein Field Equations

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 S^1 (fundamental group \mathbb{Z} ), while the black hole interior evolves into the wedge sum S^1 \vee S^1 (fundamental group \mathbb{Z} * \mathbb{Z} ). Using the R \wedge 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 \rho as the control parameter, strictly reducing to general relativity for \rho > \rho_c and activating topological effects for \rho < \rho_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 \mu^2 = \mu_0^2 (\rho / \rho_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.