Unified Theory of Transient Stability in Non-Normal Systems: A Norm-Based Criterion via the Kaupp Number

Abstract: This paper introduces a fundamental paradigm shift in stability analysis for dynamical systems, moving beyond the classical asymptotic constraints of Lyapunov theory. For over a century, spectral analysis has been the primary tool for determining system safety; however, for non-normal operators, eigenvalues fail to predict transient energy growth, leading to catastrophic failures in "spectrally stable" systems. We present the Kaupp Stability Framework (KSF), a norm-based criterion designed for finite-time operational environments. The core of this framework is the Kaupp Number (\bm{K}), a dimensionless metric that quantifies the maximum transient amplification (\bm{G_{max}}) relative to the system's structural capacity (\bm{S}). By integrating pseudospectral analysis and the Kreiss inequality into an engineering-ready formulation, the Kaupp Number provides a universal threshold (\bm{K < 1}) for system survival. This work reconciles abstract operator theory with physical reality, offering a robust diagnostic tool for fields ranging from fluid dynamics and power grids to neural networks and structural mechanics. Keywords: Non-normal operators, Transient growth, Kaupp Number, Pseudospectra, Structural stability, Kreiss constant.