Catastrophic transitions in complex systems, from engineered infrastructure to socio-economic networks, often occur with limited warning. Here, we identify a universal geometric precursor to these failures: the saturation of an effective geometric degree of freedom. We demonstrate that this instability threshold predicts critical events across seven orders of spatial magnitude—from laboratory-scale nuclear experiments to historical ffnancial market crashes—where event sizes follow a robust scale-free distribution (α ≈ 1.96). Beyond prediction, this geometric constraint offers a control strategy: real-time modulation of the governing parameter successfully steers systems away from collapse, establishing a uniffed framework for mitigating systemic risk.
栎洋 丁 (Thu,) studied this question.