From Ringdown to Lensing: Analytic Eikonal Modes of Quasi-Topological Regular Black Holes

We develop an analytic eikonal description of perturbations for four-dimensional regular black holes in quasi-topological gravity. Using first-order Schutz–Will WKB together with a small-coupling expansion and a large-$\mathrm{\ell }$ expansion, we obtain closed quasinormal-mode formulas with explicit dependence on the black-hole parameters $(M,\mu ,\nu ,\alpha )$. We then map the same geodesic invariants $({\mathrm{\Omega }}_{\text{ph}},{\lambda }_{\text{ph}})$ to shadow and strong-lensing observables, deriving an explicit QNM–shadow–lensing correspondence. In this way, ringdown frequencies, shadow scale, and strong-deflection observables are unified in one analytic scheme for this quasi-topological family.