State Topology and Trajectory Storage: A Geometric Framework for Monitoring Complex Dynamic Systems

Abstract Contemporary monitoring of complex dynamic systems relies on a storage and query model inherited from relational database theory: system state is represented as discrete snapshots, and alerts are generated by threshold violation. This model is structurally mismatched to systems in which state evolves continuously, failure is a trajectory, and the operationally relevant question is not "what is the current state" but "what does the current trajectory resemble." We propose State Topology and Trajectory Storage (STTS), a framework in which systems are represented as continuous trajectories through an n-dimensional embedding space, stored in a vector-queryable index, and monitored by geometric similarity search against a corpus of historical trajectories with known outcomes. The primary monitoring query is a nearest-neighbor search: how similar is the current trajectory to trajectories that preceded known failure states. Under stated conditions, this query fires before any individual parameter threshold is crossed, recovering an intervention window that threshold monitoring cannot see. We identify three applicability conditions under which the framework provides meaningful value, and argue that eight domains satisfy them: aerospace, launch systems, marine transport, clinical medicine, power grid stability, financial systems, epidemiology, and structural integrity. Cross-validated empirical results on the NASA C-MAPSS turbofan benchmark — four sub-datasets spanning single and multi-condition operation — yield F1 scores of 0.88–0.97, exceeding the closest domain-specific prior art on three of four held-out evaluations. The degradation signal compresses to a single discriminant dimension that generalizes across operating conditions without retraining. Validation on NASA battery degradation data confirms the framework's geometric structure in a third physical domain — electrochemical capacity fade — with identical pipeline architecture: V1 separation of 320.9x (p < 10⁻⁸³) and detection of all 10 run-to-failure batteries before end-of-life. Validation on near-Earth asteroid close approach trajectories — orbital elements computed by JPL Horizons from DE441 numerical integration — extends the framework to a fourth physical domain: 973 confirmed Earth close approaches, V1 = 3.8x (p ≈ 0), V2 ρ = 0.631, F1 = 1.000 95% CI: 0.998–1.000 on 795 held-out test objects. With 1,825-day trajectory histories, mean detection lead reaches 1,693 days (4.6 years) before close approach; 57.6% of objects are detected within 90 days of any point in their tracked history, and no object requires more than 665 days of history to trigger detection. The distribution is right-truncated at the 1,825-day window — the signal precedes the available data. Applied out-of-sample to asteroid 99942 Apophis, the framework produces a triage signal from 45 days of observational arc, 24.4 years before the 2029 flyby, using a corpus that contained no Apophis observations. Across four physically distinct domains — turbofan engines, bearings, batteries, and orbital mechanics — the framework's geometric structure (V1) holds universally regardless of corpus size, while detection performance (V2, F1) tracks corpus sufficiency monotonically, providing empirical evidence that the stated applicability condition P1 is a binding constraint. We further present illustrative analyses of two historical events — STS-107 Columbia and STS-51-L Challenger — tracing precursor trajectory signatures in published forensic records prior to threshold violation.