Endogenous Collapse in Nonlinear Systems: Structural Compression, Observability Limits, and the CRTI Index
This preprint introduces a structural framework for detecting collapse precursors in high-dimensional nonlinear systems. Classical early warning signals (EWS), such as rising variance and lag-1 autocorrelation, are grounded in the theory of critical slowing down (CSD) and detect collapse only as the dominant recovery rate approaches zero near a bifurcation. We show that, under conditions of slow endogenous parameter drift, collapse may be preceded by a distinct regime characterized by progressive loss of effective dimensionality. This structural compression is quantified via the effective dimensionality d_{\mathrm{eff}}(t), derived from the entropy or participation ratio of the covariance spectrum, and its normalized form \Phi(t) = d_{\mathrm{eff}}(t_0)/d_{\mathrm{eff}}(t). We define the Compression–Response Transition Index (CRTI), T(t) = R(t)/\Phi(t), where R(t) denotes adaptive capacity measured by the dominant recovery rate. A formal proposition (CL-1) establishes that, under explicit conditions (existence of a hyperbolic fixed point, slow drift, weak noise, and fold-class bifurcation), a dissociation interval can arise in which d_{\mathrm{eff}}(t) decreases while R(t) remains bounded. During this interval, T(t) declines monotonically while classical EWS remain stationary. A minimal analytical model demonstrates the existence of this dissociation regime and provides a closed-form expression for the lead time advantage of CRTI relative to classical EWS. We further formalize Projection-Induced Determinism (PID), showing that under partial observability, structural compression can remain undetected due to rank deficiency in the projected Fisher Information Matrix. The framework is falsifiable, explicitly bounded in scope, and applies to high-dimensional nonlinear systems undergoing slow endogenous drift toward fold bifurcations. It does not apply to shock-driven transitions, high-noise regimes, or non-fold bifurcations such as Hopf transitions. The results suggest that collapse in complex systems may be preceded not only by dynamical slowing but also by structural reduction of effective degrees of freedom, introducing a complementary pathway to early warning detection. early warning signals, critical transitions, critical slowing down, nonlinear systems,high-dimensional systems, structural compression, effective dimensionality,participation ratio, covariance spectrum, collapse dynamics,CRTI, compression-response transition index, projection-induced determinism,observability limits, Fisher information, endogenous collapse,fold bifurcation, saddle-node bifurcation, complex systems, system stability
Bernd von Mallinckrodt 🦋
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