  Quasi-normal modes (QNMs) uniquely describe the gravitational-wave ringdown
of post-merger black holes. While the linear QNM regime has been extensively
studied, recent work has highlighted the importance of
second-perturbative-order, quadratic QNMs (QQNMs) arising from the nonlinear
coupling of linear QNMs. Previous attempts to quantify the magnitude of these
QQNMs have shown discrepant results. Using a new hyperboloidal framework, we
resolve the discrepancy by showing that the QQNM/QNM ratio is a function not
only of the black hole parameters but also of the ratio between even- and
odd-parity linear QNMs: the ratio QQNM/QNM depends on what created the ringing
black hole, but only through this ratio of even- to odd-parity linear
perturbations.
