Distribution-Metric Geometry: Phase Transitions as Information-Geometric Bifurcations

 We introduce Distribution–Metric Geometry (DMG), a geometric framework for analyz
ing phase transitions directly in the space of empirical probability distributions generated
 by microscopic models. Instead of focusing on model-specific order parameters, DMG con
structs a multi-metric embedding of each model into an information-geometric manifold.
 Phase transitions then appear as geometric events: sudden reorientation of the trajectory in
 metric space, peaks in geometric speed and curvature, and temporary expansion of intrinsic
 dimensionality. Across three classical 2D lattice models (Villain, XY, Ising), DMG reveals
 that their trajectories in metric space require, respectively, one, two, and three principal ge
ometric modes. These intrinsic dimensions are stable under the choice of metrics and system
 size, and behave as robust geometric invariants of each model. . The framework is model
agnostic and extends naturally to complex systems where traditional order parameters are
 unknown or purely topological.