Complete quasinormal modes of Type-D black holes. II. Extremal Kerr black holes

We develop a double-confluent Heun (HeunD) method that extends our construction of complete quasinormal-mode (QNM) spectra to extremal Kerr black holes. Matching the horizon-ingoing and infinity-outgoing radial solutions together with angular regularity yields the frequencies directly at extremality, without extrapolation from nonextremal spins. For gravitational perturbations, we obtain an expanded spectrum on both conventional and unconventional Riemann sheets, including additional QNMs revealed by analytic continuation across frequency branch cuts. Continuation in spin connects the extremal modes to our complete Schwarzschild spectrum, including its unconventional branches. We show for the first time that unconventional Class I sequences with $m<0$, continued from unconventional Schwarzschild modes, approach negative superradiant bounds with vanishing damping in the extremal limit. We also identify finite-damping modes interspersed among sequences approaching zero damping, showing that a single overtone cutoff cannot describe the transition to extremality. Independent scattering-amplitude calculations verify representative frequencies and their sheet assignments, while a contour count establishes spectral completeness within a bounded frequency domain. Uniform matching near the superradiant bound further distinguishes critical accumulation from isolated extremal QNMs and resolves the associated Riemann-sheet structure. These results bring extremal Kerr into our unified treatment of black-hole QNM spectra and reveal how the Schwarzschild mode families reorganize in the extremal limit.