Risk Geometry and Phase Structure in Universal Modular Dynamics

We develop a geometric and spectral framework for the analysis of risk in complex systems within Universal Modular Dynamics (UMD). In contrast to conventional approaches that treat risk as a scalar probability, we define risk as a structural property emerging from the spectral characteristics of transition dynamics. The central object of the theory is the second eigenvalue λ2 of the transition operator, which governs the system’s approach to instability. We introduce a spectral measure of risk R = (1 − λ2)−1 and show that risk diverges as the system approaches a critical regime. Building on this, we construct a geometric formulation in which risk is described by a metric and curvature defined over a space of control parameters. We demonstrate that critical instability corresponds to a geometric singularity, characterized by divergence of both the risk metric and scalar curvature. We further formulate a field-theoretic description by introducing an action functional whose variation yields a set of dynamical equations analogous in structure to Einstein equations. This establishes a unified framework connecting spectral dynamics, geometric structure, and instability propagation. Numerical analysis and phase-space modeling illustrate the emergence of metastable regimes, critical surfaces, and nonlinear transitions. The resulting theory provides a universal description of risk as a phase transition phenomenon, applicable to a wide class of complex systems. This construction is interpreted as a phenomenological geometric theory of instability, providing a new perspective on the structural origins of risk beyond probabilistic descriptions.