COS45 introduces a deterministic geometric framework for analysing observable transitions in dynamical systems through Hankel embeddings, singular value decomposition (SVD), and Grassmann geometry. The framework projects heterogeneous time-series datasets into a common observable space where geometric structures can be measured independently of the underlying physical mechanisms. The frozen protocol combines two complementary observables derived from the same SVD decomposition: • Branch A (Spectral): distance to a rank-constrained manifold through spectral observables. • Branch B (Geometric): Grassmann geodesic displacement relative to a frozen reference subspace. The methodology performs no machine learning, no parameter fitting, no domain-specific modelling, and no physical classification. All quantities are computed directly from the observable matrix representation. Empirical results reported in this deposit include battery aging (NASA PCoE), bearing fault evolution (CWRU), and biological collapse datasets. Across these independent domains, the same frozen pipeline produces reproducible geometric observables and statistically significant transition structures. The deposit also includes the exploratory extensions D14 and D14-A, introducing the candidate observables μₜ, Rₘod* (t), and Rbalₛmooth (t). These extensions expand the observable space from geometric direction alone to a joint geometric–spectral representation. They remain under active validation and are not part of the frozen protocol. COS45 does not claim diagnosis, prediction, causal inference, or a universal physical law. The framework establishes a reproducible observable geometry and defines a structured validation programme for future cross-domain investigations. Current evidence supports the possibility that independent dynamical systems may exhibit reproducible observable geometric structures despite fundamentally different physical mechanisms. Determining the extent, limits, and generality of these structures remains an open scientific question.
Morissette Louis (Fri,) studied this question.