This article presents a novel, training-free method for predicting critical transitions in complex systems—specifically epileptic seizures and earthquakes—using a single mathematical framework grounded in non-associative algebra. The researchers demonstrate that by encoding multivariate time series (like EEG or seismic data) into octonions (8-dimensional numbers), they can measure a quantity called the associator. Because the associator is mathematically guaranteed to be zero for linear or stationary signals, it serves as a highly sensitive "instability metric" for detecting the subtle nonlinear coupling that precedes a system's collapse. Key Findings: Domain Independence: The exact same algorithm and parameter configuration were used for both brain activity and crustal displacement, suggesting a universal mathematical signature for "pre-ictal" states. Seizure Prediction: Validated on the full CHB-MIT dataset (24 patients), the method achieved a 34.4% pre-onset sensitivity with an average lead time of 38.6 minutes, all without any patient-specific training or AI model tuning. Earthquake Precursors: In a retrospective analysis of the 2010 M7.2 Baja California earthquake, the metric identified a "pre-ictal" crustal state 11.9 days before the mainshock, aligning with independently recorded foreshock activity. Interpretability & Efficiency: Unlike "black-box" deep learning, the method relies on a single, mathematically defined algebraic feature. It is computationally lightweight enough to run in real-time on a Raspberry Pi, making it suitable for wearable medical devices or remote seismic stations. The study concludes that non-associative algebraic structures provide a unique "lens" for identifying the universal precursors of failure in dynamical systems, offering a path toward reliable, zero-training early warning systems for both clinical and geophysical applications.