Why Systems Fail Before They Look Unstable: Structural Compression as a Complementary Early Warning Signal in Multivariate Systems

This paper introduces structural compression as a conceptual framework and candidate observable for early warning of critical transitions in multivariate systems.   Classical early warning signals (EWS), such as rising variance and increasing autocorrelation, are based on the assumption that instability becomes visible through amplified fluctuations. We argue that this assumption is incomplete. In multivariate settings, systems may approach critical transitions not by expanding their dynamics, but by progressively losing effective degrees of freedom.   We formalize structural compression via the spectral entropy of the rolling covariance matrix, yielding an effective rank measure Φ. A sustained decline in Φ reflects increasing concentration of variance along fewer dimensions, consistent with a collapse of covariance geometry. We propose that Φ may serve as a complementary observable dimension to classical EWS, particularly in transition regimes where amplitude-based indicators remain uninformative.   To clarify scope and limitations, we introduce three conceptual mechanism classes: (M1) fold-type transitions with critical slowing down, (M2) structural compression without amplitude increase, and (M3) noise-driven dynamics. Structural compression is hypothesized to be informative primarily in M2-type regimes.   This work is explicitly framed as a conceptual contribution. No claims of empirical universality are made. The interpretation of Φ as an effective number of degrees of freedom is heuristic and depends on covariance estimation, projection, and noise assumptions. The framework is intended as a complementary extension to existing early warning signal approaches and as a basis for future empirical and theoretical investigation.     Primary keywords:   early warning signals critical transitions structural compression multivariate systems covariance structure     Secondary keywords:   spectral entropy effective rank tipping points complex systems nonlinear dynamics     Extended / discovery keywords:   critical slowing down system stability high-dimensional dynamics phase transitions resilience