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) → vG (t) → Ξ (t) → Ω (t) → Pᵢnf. Central empirical finding. Across electrochemical, mechanical, and biological systems, increases in geometric velocity vG (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 = τₘacro − τgeo > 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.
Louis Morissette (Tue,) studied this question.