Pseudospectral Robustness of the Analyticity Parameter ρ

Jaramillo et al. (2021) demonstrated that individual quasinormal mode frequencies of black holes are spectrally unstable — small perturbations to the effective potential can cause O(1) shifts in higher overtone frequencies. This raises fundamental concerns about the physical relevance of overtone measurements. We show that the damping rate ratio ρ = γ₁/γ₀ is pseudospectrally robust: under the same perturbations that destabilize individual ω_n, ρ remains within [TBD]% of its unperturbed value. The mechanism is cancellation: systematic shifts in γ₁ and γ₀ are correlated (both respond similarly to potential modifications), so their ratio is protected. This identifies ρ as the natural observable for black hole spectroscopy — more meaningful than individual frequencies, and directly measurable with current and future gravitational wave detectors.