Geometry, Observability, and the Limits of Early Warning Signals in Complex Systems: A Dissertation Summary

This dissertation summary presents a theoretical framework addressing a persistent limitation in the detection of critical transitions in complex systems. Classical early warning signals (EWS), such as rising variance and increasing lag-1 autocorrelation, are commonly interpreted as indicators of critical slowing down. However, their empirical reliability has remained inconsistent across domains.   This work argues that this inconsistency is not primarily due to noise, data limitations, or model misspecification, but reflects a more fundamental constraint: the geometry of observation. Specifically, the detectability of critical system dynamics depends on the alignment between the system’s critical mode and the observation space. When this alignment is weak or dynamically degraded, early warning signals may be suppressed or entirely absent, even as the system approaches a transition.   The framework is formalized through the Compression–Response Transition Index (CRTI), defined as T = R/Φ, where R denotes adaptive capacity and Φ represents structural compression. This formulation distinguishes between amplitude-based indicators and structure-sensitive measures, and introduces boundary conditions under which early warning signals are expected to succeed or fail. In particular, the concepts of Structural–Dynamic Separability and the Relaxation–Coupling Failure Mode define regimes of detectability.   The central contribution is the formulation of a detectability bound under projection: a general constraint arising from the dimensional mismatch between system dynamics and observation. This bound implies that the absence of early warning signals can be a necessary consequence of projection geometry, rather than a failure of measurement.   The results suggest a shift in perspective from improving statistical indicators toward understanding and designing observation systems that preserve alignment with critical system modes. More broadly, the work positions detectability limits as an intrinsic property of the observer–system relationship in high-dimensional dynamical systems.     early warning signals; critical transitions; complex systems; observability; projection; geometry; detectability; CRTI; structural compression; adaptive capacity; bifurcation; critical slowing down; system dynamics; high-dimensional systems