Universal near-resonance structure in quadratic quasinormal modes of Kerr black holes

We study the ratio Q= |∆ωR|/γ3 between the real detuning and the daughter-mode damping rate in quadratic quasinormal-mode (QNM) couplings of Kerr black holes. Numerically, several fundamental-daughter channels exhibit a weak spin dependence across 0 < a/M < 1, with the channel (2,2,0) ×(2,2,1) →(4,4,0) giving Q= 1.058 ±0.033. Using the near-extremal expansion ωℓmn ≃mΩH + cℓmn κ+ dℓmn κ2, we extract the leading coefficient cℓmn via a quadratic fit and derive the asymptotic estimate Q∞ = |∆cR|/|c3,I |. For channels with fundamental daughters, this estimate agrees with near-extremal numerics at a= 0.999 to ≲ 2% and remains compatible with the spin-averaged data at ≲ 0.74 σ. We interpret the result as evidence for an approximate near-resonance regularity. The Breit–Wigner structure implies that quadratic QNM amplitudes scale as 1/√1−a2 near extremality, potentially enhancing their detectability by a factor 3–7× for LISA sources, contingent on the quadratic coupling coefficients.