Compression–Response Transition Index (CRTI): A Mechanism-Specific Early Warning Signal for Fold-Type Critical Transitions

This preprint introduces the Compression–Response Transition Index (CRTI), a bivariate early warning signal designed for detecting fold-type critical transitions in multivariate dynamical systems. The index is defined as T = R̂ / Φ, coupling a recovery-rate proxy derived from the autocorrelation structure (R̂) with a spectral concentration measure Φ = λ₁ / \sum_i λ_i, representing the dominance of the leading covariance mode.   Unlike classical early warning indicators based on variance or autocorrelation alone, CRTI explicitly integrates structural and dynamical information and is equipped with a validity gate via the Structural–Dynamic Separability (SDS) condition. The framework is mechanism-specific, with explicit boundary conditions covering Hopf bifurcations, noise-induced transitions, projection-induced distortion, and reflexive systems.   Simulation results demonstrate that CRTI provides earlier and more robust detection of fold bifurcations compared to AR(1) and variance-based indicators, while correctly failing outside its domain of validity. An empirical evaluation on the Peter Lake ecosystem dataset, based on a pre-registered protocol, supports the theoretical predictions.   CRTI is presented as a diagnostic instrument with explicitly defined scope, not as a universal early warning signal.     CRTI, early warning signals, critical transitions, fold bifurcation, multivariate time series, covariance structure, autocorrelation, spectral concentration, complex systems, nonlinear dynamics

Compression–Resonance Thermodynamic Index (CRTI): A Bounded Dynamic Phase Model for Structural Over-Compression in Complex Adaptive Systems

This paper introduces the Compression–Resonance Thermodynamic Index (CRTI) as a bounded dynamic phase model for diagnosing structural over-compression in complex adaptive systems (CAS). Moving beyond static efficiency-based indicators, the CRTI formalizes the interaction between Exploitation (E), Exploration (X), and Resonance Integrity (R) within a continuous-time dynamical framework.   The model is defined on a constrained phase space \Omega = [0,1]^3 and specifies coupled differential equations governing the evolution of E, X, and R. A formal bifurcation condition E^* = \beta / \kappa is derived, identifying the critical threshold at which exploration collapses and resonance decays toward a structurally muted attractor. The analysis includes Jacobian-based stability considerations, nullcline geometry in reduced phase space (E–X), hysteresis behavior, and sensitivity to stochastic shocks representing high-frequency environmental disruptions (e.g., technological change).   Unlike descriptive institutional diagnostics, the CRTI framework is explicitly falsifiable. The model specifies observable trajectories that would disconfirm its structural claims, including cases where high exploitation and low exploration coexist with sustained resonance under prolonged external shocks.   An empirical operationalization blueprint is provided, including:   Indicator mapping for E, X, and R Time-series data requirements State-space estimation logic Minimal viable dataset specification Monitoring database architecture for institutional implementation     The CRTI is proposed as a structural early-warning mechanism for detecting systemic brittleness before visible institutional collapse occurs. The framework is applicable across institutional domains (education, healthcare, corporate systems, public administration) and is designed to enable cross-system comparative analysis once calibrated with longitudinal data.   This publication represents a transition from a static compression ratio toward a mathematically bounded, dynamic, and testable phase model suitable for peer-level theoretical evaluation and empirical extension.   Complex Adaptive Systems (CAS) Phase Transition Modeling Structural Brittleness Over-Compression Dynamics Exploitation–Exploration Tradeoff Dynamical Systems Theory Bifurcation Analysis Institutional Resilience Feedback Permeability State-Space Modeling Stochastic Shock Response Organizational Adaptation Systemic Risk Diagnostics Hysteresis in Social Systems Early-Warning Indicators  

Collapse through Hyper-Stability: A Slow–Fast Dynamical Executive Framework for Optimization-Driven Structural Rigidity

Collapse through Hyper-Stability: A Slow–Fast Dynamical Executive Framework for Optimization-Driven Structural Rigidity