COS45 presents a geometric framework for representing high-dimensional dynamical systems as trajectories on the Grassmann manifold (Gr (r, n) ). The system state is encoded as ( (t) = (P (t), Ṗ (t) ) ), where (P (t) ) is a projection operator describing the dominant subspace structure of observed data. A geometric velocity (vG (t) = |Ṗ (t) |F) is defined to quantify the temporal evolution of the system’s dominant organization. Across heterogeneous datasets from electrochemical, mechanical, and biological systems, this quantity shows statistically consistent changes preceding observable macroscopic transitions. These results indicate a potential geometric lead-time structure in which variations in manifold dynamics may precede conventional observable indicators. The framework is model-agnostic and relies solely on spectral properties of the data without system-specific tuning. COS45 provides a unified geometric representation for analyzing transition dynamics in complex systems and for quantifying precursor behavior in high-dimensional time series.
Louis Morissette (Tue,) studied this question.