CRTI: A Mechanism-Specific Measurement Framework for Early Warning Signals Based on Structural Compression in Fold Bifurcations

The Compression–Response Transition Index (CRTI) is a mechanism-specific measurement framework for detecting early warning signals (EWS) in complex systems approaching fold (saddle-node) bifurcations.   Classical EWS—such as variance and lag-1 autocorrelation—capture changes in dynamic memory but do not resolve the geometric reorganisation of multivariate system states. CRTI addresses this limitation by introducing structural compression Φ(t), derived from the spectral entropy of the covariance matrix, as a scale-invariant measure of effective dimensionality. This structural component is combined with an adaptive response measure R(t), based on an AR(1) recovery proxy, into a composite index T(t) = R(t) / Φ(t).   Under explicitly stated domain-of-validity conditions—fold bifurcation dynamics, additive approximately isotropic noise, and multivariate observability (d ≥ 2)—CRTI yields a falsifiable prediction: the composite index T(t) decreases toward zero as the system approaches a critical transition.   A central methodological contribution is the introduction of the Structural–Dynamic Separability (SDS) condition, defined via the correlation between R(t) and Φ(t). If separability is violated (|ρ| ≥ θ), the composite index is declared invalid. The Relaxation–Coupling Failure Mode (RCFM) is identified as the primary mechanism underlying SDS failure.   CRTI is not proposed as a universal indicator but as a domain-restricted, validity-gated measurement instrument. Its applicability, assumptions, and limitations—including projection-induced distortion (PID), noise anisotropy, dimensionality constraints, and windowing artefacts—are explicitly defined.   This work provides a structured extension to the early warning signal framework by incorporating covariance geometry alongside classical dynamical indicators, enabling more specific detection of structural precursors in systems approaching fold-type critical transitions.   early warning signals critical transitions fold bifurcation structural compression spectral entropy covariance geometry AR(1) complex systems tipping points dynamical systems multivariate analysis system stability