COS45: Geometric Observability and Precedence in Complex Dynamical Systems

This work presents the COS45 framework, a geometric approach for representing complex evolving systems through dominant spectral subspaces on the Grassmann manifold Gr(r, n). The framework introduces a unified geometric state and organizes its observables into a hierarchical structure: P(t) → v_G(t) → Ξ(t) → Ω(t) → P_inf. Central empirical finding. Across electrochemical, mechanical, and biological systems, increases in geometric velocity v_G(t) detected by a pre-registered CUSUM procedure with fixed parameters (k = 0.5, h = 5) are statistically observed to precede macroscopic transition thresholds under fixed detection parameters, with positive lead time Δt = τ_macro − τ_geo > 0. On NASA PCoE battery data (B0005-B0018, n = 636 discharge cycles), the CUSUM geometric change point is detected 29 to 92 cycles before the 80% capacity threshold, with relative lead-time ratios R_Δ ∈ [0.48, 0.75]. Block-bootstrap confidence intervals confirm Δt > 0 on B0005 (95% CI [4, 69]); wider intervals on B0006-B0018 reflect cohort heterogeneity and motivate independent replication. All findings are reported within the tested datasets and under fixed preprocessing assumptions; no claim of universality is made. The same geometric representation reveals statistically consistent ordering structures across the evaluated heterogeneous systems under identical preprocessing and detection protocols.