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Scaling K2. I. Revised Parameters for 222,088 K2 Stars and a K2 Planet Radius Valley at 1.9 $R_{\oplus}$

Previous measurements of stellar properties for K2 stars in the Ecliptic Plane Input Catalog (EPIC; Huber et al. 2016) largely relied on photometry and proper motion measurements, with some added information from available spectra and parallaxes. Combining Gaia DR2 distances with spectroscopic measurements of effective temperatures, surface gravities, and metallicities from the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) DR5, we computed updated stellar radii and masses for 26,838 K2 stars. For 195,250 targets without a LAMOST spectrum, we derived stellar parameters using random forest regression on photometric colors trained on the LAMOST sample. In total, we measured spectral types, effective temperatures, surface gravities, metallicities, radii, and masses for 222,088 A, F, G, K, and M-type K2 stars. With these new stellar radii, we performed a simple reanalysis of 299 confirmed and 517 candidate K2 planet radii from Campaigns 1--13, elucidating a distinct planet radius valley around 1.9 $R_{\oplus}$, a feature thus far only conclusively identified with Kepler planets, and tentatively identified with K2 planets. These updated stellar parameters are a crucial step in the process toward computing K2 planet occurrence rates.