Bernd von Mallinckrodt 🦋Singularization Framework v4: ...
Singularization Framework v4: Bifurcation Structure, Stochastic Collapse, and Early Warning Signals in the CRTI Minimal Model
This paper presents Version 4 of the Singularization Framework, a minimal dynamical theory of structural collapse in complex adaptive systems. The framework introduces three state variables — Structural Compression C(t), Adaptive Resonance R(t), and Systemic Tension T(t) — forming the Compression–Resonance–Tension Index (CRTI).The central contribution is the analytic derivation of the collapse region from system dynamics. The C–R subsystem is shown to undergo a saddle-node bifurcation at the critical parameter κ = αγ/(δ_C β) = 1, at which the healthy equilibrium annihilates with the separatrix saddle, leaving the collapse attractor P₃ = (1, 0) as the sole stable fixed point. The collapse basin Ω_c is defined as the basin of attraction of P₃, bounded by the stable manifold of the saddle — replacing the ad-hoc threshold definitions of previous versions.A stochastic Langevin extension connects the framework to Kramers escape rate theory, providing quantitative collapse probability estimates. Near the bifurcation, classical critical slowing down signatures are analytically guaranteed: rising variance, rising lag-1 autocorrelation, and diverging recovery time. A fragility index φ = |κ − 1| provides an operationalisable systemic vulnerability criterion.The framework is positioned relative to May’s fold catastrophe model, Scheffer–Dakos early warning signal theory, Freidlin–Wentzell quasi-potential theory, and Ashby’s Law of Requisite Variety. Empirical proxies for C, R, and T are proposed across organisational, ecological, financial, and physiological domains. complex adaptive systems · structural collapse · bifurcation theory · saddle-node bifurcation · critical transitions · early warning signals · critical slowing down · stochastic dynamics · Kramers escape rate · resilience theory · CRTI · Singularization Framework · nonlinear dynamics · quasi-potential · Lyapunov function · adaptive resonance · structural compression · systemic tension · complexity science · panarchy
Prof. R. I. Sujith
Andy Scollick