We propose a novel scheme for final muon ionization cooling with quadrupoledoublets followed by emittance exchange in vacuum to achieve the small beamsizes needed by a muon collider. A flat muon beam with a series of quadrupoledoublet half cells appears to provide the strong focusing required for finalcooling. Each quadrupole doublet has a low beta region occupied by a dense, lowZ absorber. After final cooling, normalized transverse, longitudinal, andangular momentum emittances of 0.100, 2.5, and 0.200 mm-rad are exchanged into0.025, 70, and 0.0 mm-rad. A skew quadrupole triplet transforms a round muonbunch with modest angular momentum into a flat bunch with no angular momentum.Thin electrostatic septa efficiently slice the flat bunch into 17 parts. The 17bunches are interleaved into a 3.7 meter long train with RF deflector cavities.Snap bunch coalescence combines the muon bunch train longitudinally in a 21 GeVring in 55 microseconds, one quarter of a synchrotron oscillation period. Alinear long wavelength RF bucket gives each bunch a different energy causingthe bunches to drift in the ring until they merge into one bunch and can becaptured in a short wavelength RF bucket with a 13% muon decay loss and apacking fraction as high as 87%.
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