A Geometric Theory of Water

Hydrology faces a fundamental divide: data-driven models lack causal insight, while physics-based models are rarely generalizable. This stems from applying deterministic mathematics to inherently stochastic and causally asymmetric water systems. We resolve this by establishing information geometry as the natural language for water causality. We demonstrate that causal direction is intrinsically encoded as a curvature asymmetry between statistical manifolds—formalized as the Causal Curvature Inequality. This transforms causation from a statistical inference into a measurable geometric property, shifting hydrology from site-specific calibration to principle-based understanding. Our framework thus provides a universal foundation for deciphering water system causality, essential for addressing global hydrological challenges.