Universal geometric framework for black hole phase transitions: From multivaluedness to classification

Recent studies have revealed synchronized multivalued behavior in thermodynamic, dynamical, and geometric quantities during the black hole first-order phase transition, which enables a diagnosis from different perspectives, yet its fundamental origin has remained poorly understood. By constructing a unified geometric framework integrating real analysis and covering space theory, we reveal the universal mathematical mechanism behind this phenomenon. We prove that this multivaluedness originates from two non-degenerate critical points in the temperature function $T(r_+)$, where $r_+$ is the horizon radius, which fold the parameter space into a three-sheeted covering structure. As a direct application, we propose that a black hole undergoes a first-order phase transition if and only if its $T(r_+)$ curve has two extrema. Accordingly, we establish a classification scheme, denoted $A1$, $A2$, and $B$ for black holes. This scheme offers a complementary perspective to classifications based on global topological invariants. Our work provides a theoretical foundation for diagnosing phase transitions via multivaluedness and establishes a unified geometric perspective on black hole thermodynamics, chaotic dynamics, and spacetime structure during first-order phase transitions.