COS45 — Geometric Resolution, Coherence, and the Interpretation of Randomness

Abstract COS45 is a geometric preprint introducing a reproducible framework for studying how structural organization becomes observable at different geometric resolutions. Starting from a raw signal x(t), the framework constructs a trajectory Γ on the Grassmann manifold G(2,256) through a deterministic Hankel-SVD pipeline. All observables are derived from this single trajectory and are interpreted as progressively higher-resolution observations of the same geometric object. The framework introduces a coherence-resolution threshold, σ*, representing the geometric resolution at which stable organization becomes observable under a frozen protocol. Geodesic displacement, curvature variability, geometric activity, and morphological stability are treated as complementary descriptions of the same trajectory rather than independent quantities. A central hypothesis of COS45 is that apparent randomness may correspond, in part, to geometric coherence that remains unresolved at the current observational scale. The objective is not to classify systems as ordered or random, but to identify the geometric resolution at which their organization becomes most observable. Preliminary results across independent domains suggest that distinct systems may exhibit reproducible coherence-scale neighborhoods while retaining different underlying dynamics. The framework remains open to independent validation, refinement, and falsification. COS45 is released as an open mathematical resource for exploring the geometry of observability across scales, representations, manifolds, and domains.