The Compression–Resonance–Tension Index (CRTI): A Spectral Fragility Framework for Complex Adaptive Systems

This paper introduces the Compression–Resonance–Tension Index (CRTI), a composite spectral parameter designed to quantify systemic fragility in complex adaptive systems. The framework models systemic collapse as a phase transition occurring when constraint-induced stress exceeds the system’s structural capacity for relaxation and reorganization. The CRTI is formulated as a dimensionless ratio that combines structural constraint density and stress variance with spectral measures of network connectivity and configurational entropy. In its spectral representation, the index relates the variance of internal load distributions to the algebraic connectivity of the system’s graph Laplacian and the entropy of its accessible configuration space. The proposed canonical formulation is: chi = (Phi * sigma_Delta^2) / (lambda_2(L) * H_conf) where: Phi represents constraint density within the system structure sigma_Delta^2 represents stress variance across system components lambda_2(L) is the algebraic connectivity (spectral gap) of the graph Laplacian H_conf denotes configurational entropy representing the system’s adaptive reserve Within this framework, systemic instability arises when stress heterogeneity grows faster than the network’s spectral capacity to redistribute perturbations. The CRTI therefore functions as a spectral fragility indicator, identifying critical regimes in which small perturbations can trigger cascading failures or structural collapse. Unlike traditional resilience metrics that rely solely on topology or load conditions, the CRTI integrates structural and dynamical information into a single dimensionless control parameter. This allows early detection of fragility across multiple domains. Potential applications include: cascading failure prediction in electrical power grids representation collapse detection in neural networks fragility monitoring in financial and logistic networks rigidity transitions in physical and biological systems The CRTI provides a unified mathematical framework linking spectral graph theory, statistical physics, and complex systems research. By combining constraint density, stress variance, spectral connectivity, and configurational entropy, the framework offers a candidate general metric for quantifying fragility in complex adaptive systems.