A Σ₂–Pseudospectral Diagnostic for Ringdown Mode Legitimacy

The paper introduces an operator-theoretic diagnostic framework for assessing when quasinormal-mode (QNM) claims in black hole ringdown analyses are physically interpretable, rather than artifacts of early-time fitting in non-normal dynamical systems. Ringdown is treated as a generally non-normal linear relaxation process, for which eigenvalues alone do not control stability or mode separability. Building on pseudospectral theory, the work defines: an operator-based ringdown onset time determined by resolvent isolation in reduced angular sectors, and a Σ₂-motivated legitimacy criterion requiring that candidate modal reconstructions monotonically reduce a closure-energy functional measuring projection inconsistency between radiative and asymptotically stationary (Kerr-like) representations. Synthetic numerical-relativity–like demonstrations show that additional overtones can improve early-time waveform fits while violating the legitimacy criterion due to pseudospectral overlap and non-normal mixing, clarifying their role as reconstruction elements rather than independent physical relaxation channels prior to spectral isolation. The framework is diagnostic and interpretive: it does not modify General Relativity, Kerr geometry, or quasinormal-mode spectra. Instead, it constrains when and how modal descriptions acquire physical meaning, complementing mismatch-based and Bayesian pipelines—particularly in high–signal-to-noise ratio events, where systematic modeling ambiguities dominate statistical uncertainty. An appendix provides a concrete realization of the closure residual using Newman–Penrose curvature scalars and Kerr-consistent projections, along with a minimal toy non-normal matrix analogy for intuition. Limitations, noise robustness, and applicability to future numerical-relativity catalog studies are discussed explicitly.