Compression–Response Transition Index (CRTI): A Mechanism-Specific Early Warning Signal for Fold-Type Critical Transitions

This preprint introduces the Compression–Response Transition Index (CRTI), a bivariate early warning signal designed for detecting fold-type critical transitions in multivariate dynamical systems. The index is defined as T = R̂ / Φ, coupling a recovery-rate proxy derived from the autocorrelation structure (R̂) with a spectral concentration measure Φ = λ₁ / \sum_i λ_i, representing the dominance of the leading covariance mode.   Unlike classical early warning indicators based on variance or autocorrelation alone, CRTI explicitly integrates structural and dynamical information and is equipped with a validity gate via the Structural–Dynamic Separability (SDS) condition. The framework is mechanism-specific, with explicit boundary conditions covering Hopf bifurcations, noise-induced transitions, projection-induced distortion, and reflexive systems.   Simulation results demonstrate that CRTI provides earlier and more robust detection of fold bifurcations compared to AR(1) and variance-based indicators, while correctly failing outside its domain of validity. An empirical evaluation on the Peter Lake ecosystem dataset, based on a pre-registered protocol, supports the theoretical predictions.   CRTI is presented as a diagnostic instrument with explicitly defined scope, not as a universal early warning signal.     CRTI, early warning signals, critical transitions, fold bifurcation, multivariate time series, covariance structure, autocorrelation, spectral concentration, complex systems, nonlinear dynamics