The Compression–Resonance–Tension Index (CRTI): A Minimal Dynamical Framework for Efficiency-Driven Collapse in Complex Adaptive Systems

This publication introduces the Compression–Resonance–Tension Index (CRTI), a minimal nonlinear dynamical framework designed to analyze systemic fragility and collapse in complex adaptive systems. Traditional explanations of systemic collapse often emphasize external shocks, stochastic disturbances, or the gradual accumulation of disorder. In contrast, the CRTI framework proposes an alternative endogenous mechanism: efficiency-driven collapse caused by structural over-optimization and depletion of adaptive reserves. The model formalizes the interaction between two fundamental dynamical variables: Structural Compression (Φ) Represents optimization pressure, efficiency, and structural rigidity within a system. High values of Φ correspond to strongly optimized systems with reduced internal degrees of freedom. Adaptive Reserve (Ω) Represents redundancy, variance capacity, and regenerative buffers that enable a system to absorb perturbations and adapt to environmental change. The interaction between these variables is modeled using a minimal system of coupled nonlinear differential equations: dΦ/dt = αΦ(G − Φ) − βΩ² dΩ/dt = η(K − Ω) − θΦΩ where the parameters represent structural growth, reserve regeneration, and depletion dynamics. Within this framework, systemic collapse emerges when structural efficiency persistently consumes adaptive reserves faster than they can regenerate. This process results in a dynamical transition toward a state termed Singularization, characterized by high structural compression and near-zero adaptive flexibility. The CRTI framework introduces the dimensionless diagnostic ratio: χ = Φ / Ω This indicator measures the relative dominance of structural pressure over adaptive reserve capacity. A rising χ indicates increasing systemic fragility and may serve as an early warning signal for approaching collapse. Mathematical analysis of the CRTI dynamical system reveals several key features: • existence of a stability basin referred to as the Resonant Corridor • a separatrix boundary separating stable and collapse trajectories • systemic collapse occurring through a Saddle-Node bifurcation • shrinking stability basins under increasing structural pressure Unlike classical models where collapse arises from disorder or chaos, the CRTI framework suggests that many modern systems fail due to excessive order and optimization, which eliminate the adaptive variance required for long-term stability. The paper also proposes a reproducible empirical validation framework. As an illustrative example, the study outlines how CRTI indicators can be derived from publicly available Federal Reserve Economic Data (FRED) surrounding the 2008 global financial crisis. In this application: Structural Compression (Φ) proxies include: • Total Assets of Commercial Banks (TABS) • Total Bank Credit (TOTBKCR) Adaptive Reserve (Ω) proxies include: • Total Reserves of Depository Institutions (TOTRESNS) • Cash Assets (CASHBLNS) Preliminary analysis suggests that the CRTI indicator may reveal a period of structural vulnerability prior to the liquidity collapse of 2008, potentially providing earlier warning signals than traditional statistical early warning metrics such as autocorrelation or variance. Because the CRTI framework is formulated as a minimal dynamical model, it can be applied across multiple domains, including: • financial systems • ecological systems • supply chains and logistics networks • neural systems • infrastructure networks By providing a formal mathematical representation of the efficiency–resilience trade-off, the CRTI framework offers a cross-domain diagnostic tool for analyzing systemic risk and designing more resilient complex systems. Future work will focus on: • numerical bifurcation analysis of the CRTI dynamical system • empirical validation across multiple domains • statistical comparison with established early warning indicators • identification of universal thresholds of systemic fragility. Keywords CRTI Systemic Collapse Nonlinear Dynamics Complex Adaptive Systems Resilience Theory Structural Compression Adaptive Reserve Singularization Bifurcation Theory Early Warning Signals Systemic Risk Complex Systems Suggested Scientific Categories (für Zenodo) Empfohlen: Complex Systems Nonlinear Dynamics Systems Theory Applied Mathematics Resilience Research