Why do small dents near a black hole's edge spell disaster?

Squeeze the geometry just outside a black hole's event horizon and you spawn a new purely imaginary oscillation mode. Push that deformation close enough to the horizon and the mode crosses into unstable territory — the black hole effectively rings itself apart. The work derives this threshold analytically and numerically, producing clean scaling relations between deformation strength and the critical distance, offering a spectral explanation for instabilities seen in recent time-domain simulations.