Ladder symmetry: the necessary and sufficient condition for vanishing love numbers

Black holes in four-dimensional, asymptotically flat general relativity have vanishing static tidal Love numbers (TLNs), a property tied to a hidden symmetry of the perturbation equations. Within the Konoplya-Rezzolla-Zhidenko (KRZ) parametrization, a subclass of spacetimes was previously shown to admit such Ladder symmetry, which enforces the absence of static scalar TLNs. In this work, we introduce parametric deformations to such Ladder-symmetric spacetimes and analyze the resultant linear tidal response. Using the parametrized formalism for TLNs, we show that any deviation from a Ladder-symmetric background leads to non-zero static scalar TLNs. This establishes Ladder symmetry as a necessary, as well as sufficient condition, for the vanishing of static TLNs in static, spherically symmetric black holes and in rotating black holes of the KRZ class.