Final Cooling for a High-Energy High-Luminosity Lepton Collider

A high-energy muon collider scenario require a "final cooling" system that reduces transverse emittance by a factor of ~10 while allowing longitudinal emittance increase. The baseline approach has low-energy transverse cooling within high-field solenoids, with strong longitudinal heating. This approach and its recent simulation are discussed. Alternative approaches which more explicitly include emittance exchange are also presented. Round-to-flat beam transform, transverse slicing, and longitudinal bunch coalescence are possible components of an alternative approach. Wedge-based emittance exchange could provide much of the required transverse cooling with longitudinal heating. Li-lens and quadrupole focusing systems could also provide much of the required final cooling.