Mechanism-Dependent Sensitivity in Early Warning Signals: Boundary Conditions of Ratio-Based Composite Indicators

This preprint investigates the behavior of ratio-based composite early warning indicators of the form T(t) = R(t)/Φ(t) in complex dynamical systems. Here, R(t) represents adaptive response capacity, while Φ(t) captures structural compression in the covariance geometry of system fluctuations. Using multivariate Ornstein–Uhlenbeck (OU) processes as a canonical linear stochastic testbed, the analysis shows that ratio-based indicators do not outperform classical early warning signals such as variance and lag-1 autocorrelation in terms of early detection timing. This result is interpreted not as a failure, but as a boundary condition: classical indicators are amplitude-sensitive, whereas ratio-based indicators are structure-sensitive, responding to changes in covariance geometry and effective system dimensionality. The paper introduces the concept of sign non-invariance, demonstrating that ratio-based indicators can exhibit non-monotonic or directionally inconsistent behavior depending on the relative dynamics of numerator and denominator components. This property has direct implications for the interpretation and application of composite indicators in empirical settings. The findings support a mechanistic classification of early warning signals into amplitude-sensitive and structure-sensitive classes, providing a principled framework for indicator selection based on the underlying transition mechanism. The work contributes to clarifying the scope, limitations, and appropriate use of composite early warning indicators in transition monitoring. early warning signalscritical transitionsOrnstein-Uhlenbeck processcomposite indicatorscovariance structurestructural compressionsign non-invariancecritical slowing downcomplex systems