Transient Amplification and Capacity-Induced Collapse in Physical and Complex Systems

Abstract We develop a unified framework for instability in physical and complex systems based on transient amplification in non-normal operators. Classical stability criteria based on eigenvalues fail to capture the dynamics of real systems where amplification dominates. We introduce a capacity-based instability condition: K = \frac{G_{\max} \cdot \|\delta x_0\|}{S}, and show that instability occurs when K \ge 1. We demonstrate that this criterion applies across domains including quantum systems, geophysical flows, and networked systems. The results establish a universal mechanism for collapse driven by internal amplification rather than external forcing.