Hierarchy of Angular Instabilities in Scalarized Black Holes

We investigate the stability of scalarized black holes in Einstein-scalar-Gauss-Bonnet-Ricci theory along their fundamental branches. We show that initially stable solutions first lose nonspherical stability in the eikonal regime, while lower multipoles remain stable. As the branch is continued, instability extends systematically toward lower multipoles, forming an ordered hierarchy of deformation instabilities extending down to the quadrupole mode, while the dipole sector remains stable. The instability thresholds obey a common scaling law and approach finite eikonal limits, defining the boundary of the angularly stable region. We demonstrate that the previously identified quadrupole and angular-Laplacian instabilities are connected by a continuous hierarchy of instability thresholds spanning the angular sectors of the theory. This hierarchy is distinct from radial stability, which changes only at branch turning points, and reveals a previously unexplored angular organization of instabilities in scalarized black holes.