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Microwave spectro-polarimetry of matter and radiation across space and timeAbstract: This paper discusses the science case for a sensitive spectro-polarimetric survey of the microwave sky. Such a survey would provide a tomographic and dynamic census of the three-dimensional distribution of hot gas, velocity flows, early metals, dust, and mass distribution in the entire Hubble volume, exploit CMB temperature and polarisation anisotropies down to fundamental limits, and track energy injection and absorption into the radiation background across cosmic times by measuring spectral distortions of the CMB blackbody emission. In addition to its exceptional capability for cosmology and fundamental physics, such a survey would provide an unprecedented view of microwave emissions at sub-arcminute to few-arcminute angular resolution in hundreds of frequency channels, a data set that would be of immense legacy value for many branches of astrophysics. We propose that this survey be carried out with a large space mission featuring a broad-band polarised imager and a moderate resolution spectro-imager at the focus of a 3.5 m aperture telescope actively cooled to about 8K, complemented with absolutely-calibrated Fourier Transform Spectrometer modules observing at degree-scale angular resolution in the 10–2000 GHz frequency range. We propose two observing modes: a survey mode to map the entire sky as well as a few selected wide fields, and an observatory mode for deeper observations of regions of specific interest.