Fluctuation-stable generalized entropy probes of spectral heterogeneity

Generalized entropy measures are widely used to characterize localization and multifractality, and the regime $q>1$ is often empirically found to exhibit improved numerical stability and cleaner scaling behavior. Here, we develop a fluctuation-stability framework for generalized entropy diagnostics and show that weak-amplitude spectral fluctuations are amplified for $q<1$ and suppressed for $q>1$, thereby providing a theoretical basis for the physically robust $q>1$ regime. A thermodynamic scaling analysis further identifies an asymptotically stable regime beyond a critical threshold. As an application, we introduce the entropy-gradient susceptibility ${\chi }_q$ as a coarse-grained probe of spectral heterogeneity. Using the Aubry-André and generalized Aubry-André models, we demonstrate that ${\chi }_q$ sharply distinguishes homogeneous localization transitions from mobility-edge coexistence regimes. Our results establish fluctuation stability as a guiding principle for generalized entropy diagnostics in quasiperiodic systems.