Structural Compression as an Early Warning Signal: The Compression–Response Transition Index (CRTI) for Mechanism-Specific Critical Transitions

This paper introduces a mechanism-specific early warning framework for critical transitions in complex systems based on the concept of structural compression. While classical early warning signals (EWS) rely on critical slowing down and corresponding increases in variance and autocorrelation, these indicators fail in systems where transitions are not accompanied by amplitude amplification.   We define structural compression as a reduction in the effective dimensionality of system dynamics, quantified via the spectral entropy of the rolling covariance matrix. This yields a structural measure Φ that captures the collapse of the covariance spectrum toward low-dimensional configurations. To incorporate dynamical information, we introduce a persistence-based measure R derived from the AR(1) coefficient of the leading principal component. The combined Compression–Response Transition Index (CRTI), defined as T = R / Φ, captures joint structural–dynamical degradation.   To ensure statistical robustness, the covariance spectrum is interpreted within the framework of Random Matrix Theory (RMT). Using the Marchenko–Pastur bounds, noise-induced eigenvalues are separated from statistically significant structural modes, yielding a noise-corrected dimensionality estimate Φ_sig. This ensures that CRTI measures statistically resolvable structure rather than sampling noise.   The framework is explicitly mechanism-specific and is particularly suited for transition classes characterized by structural compression without variance amplification. Simulation results demonstrate that Φ and CRTI provide early warning signals in regimes where classical amplitude-based indicators remain uninformative. A real-data illustration further shows the operational feasibility of the approach in multivariate financial systems.   The CRTI is not proposed as a universal indicator but as a complementary diagnostic tool that extends the applicability of early warning methodology to previously undetectable transition mechanisms. Its validity is bounded by explicit statistical and structural conditions, including sufficient sample size, spectral separability, and independence between structural and dynamical measures.     CRTI, structural compression, early warning signals, spectral entropy, covariance matrix, random matrix theory, Marchenko–Pastur, critical transitions, complex systems, multivariate time series, dimensionality reduction, system resilience, bifurcation analysis