A microwave SQUID multiplexer ($mu$MUX) has been optimized for coupling to large arrays of superconducting transition-edge sensor (TES) bolometers. We present the scalable cryogenic multiplexer chip design in a 1820-channel multiplexer configuration for the 4-8 GHz rf band. The key metrics of yield, sensitivity, and crosstalk are determined through measurements of 455 readout channels, which span 4-5 GHz. The median white-noise level is 45 pA/$sqrt{textrm{Hz}}$, evaluated at 2 Hz, with a 1/f knee $leq$ 20 mHz after common-mode subtraction. The white-noise level decreases the sensitivity of a TES bolometer optimized for detection of the cosmic microwave background at 150 GHz by only 3%. The measured crosstalk between any channel pair is $leq$ 0.3%.
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