Exceptional activated mode theory for generalized real-complex transitions
Real-to-complex spectral transitions mark the onset of amplification in non-Hermitian systems, but their thresholds are often treated as model-specific quantities. Here we develop a general, non-perturbative activated-mode principle that governs the real-to-complex threshold across broad classes of non-Hermitian systems. A central insight is that only a small Hilbert subspace is ``activated" at the transition onset, which can be variationally determined through the competition between spectral detuning and mode-level projected non-Hermitian couplings. The result is a closed-form exceptional-activation condition for arbitrarily large ``disturbances", rather than a perturbative estimate. We apply our framework to three contrasting illustrative problems, establishing (i) a closed-form threshold for critical non-Hermitian skin amplification at \emph{all} system sizes; (ii) a new link between impurity tunneling threshold and exceptional point switching; and (iii) activation channel switching without underlying topological phase transition. Overall, our findings recast real-to-complex transitions as generic mode-selection problems independent of any specific symmetry.