Beyond Variance: Distinguishing Critical Instability from Active Amplification — A Boundary Condition for the Universal Resonance Model (URM)

This paper clarifies the mathematical boundary conditions of the Universal Resonance Model (URM). It distinguishes bifurcation-driven instability from non-normal transient amplification in active non-equilibrium systems. While classical early-warning frameworks associate increasing variance and autocorrelation with proximity to critical transition, this work shows that fluctuation amplification can arise without loss of stability in non-reciprocal or energy-driven systems. The article therefore introduces an explicit discrimination criterion based on recovery dynamics, separating genuine eigenvalue softening from amplified variability without stability loss. This contribution strengthens the theoretical rigor of URM and extends its foundational scope by defining the conditions under which early-warning interpretation is valid. This paper is part of the foundational scope-defining work of the Universal Resonance Model (URM). It clarifies the mathematical boundary conditions of URM by distinguishing bifurcation-driven instability from non-normal transient amplification in active non-equilibrium systems. While classical early-warning interpretations often associate increasing variance with proximity to critical transition, this work demonstrates that fluctuation amplification can arise without loss of stability in non-reciprocal systems. The article introduces an explicit discrimination criterion based on recovery dynamics, thereby refining the stability logic of URM and strengthening its theoretical foundations.