Variability Properties of 4 Million Sources in the $TESS$ Input Catalog Observed with the Kilodegree Extremely Little Telescope Survey

The Kilodegree Extremely Little Telescope (KELT) has been surveying more than $70\%$ of the celestial sphere for nearly a decade. While the primary science goal of the survey is the discovery of transiting, large-radii planets around bright host stars, the survey has collected more than $10^6$ images, with a typical cadence between $10-30$ minutes, for more than $4$ million sources with apparent visual magnitudes in the approximate range $7<V<13$. Here we provide a catalog of 52,741 objects showing significant large-amplitude fluctuations likely caused by stellar variability and 62,229 objects identified with likely stellar rotation periods. The detected variability ranges in $rms$-amplitude from 3 mmag to 2.3 mag, and the detected periods range from $\sim$0.1 days to $\gtrsim$2000 days. We provide variability upper limits for all other $\sim$4 million sources. These upper limits are principally a function of stellar brightness, but we achieve typical 1$σ$ sensitivity on 30-minute timescales down to $\sim5$ mmag at $V\sim 8$, and down to $\sim43$ mmag at $V\sim 13$. We have matched our catalog to the $TESS$ Input catalog and the AAVSO Variable Star Index to precipitate the follow up and classification of each source. The catalog is maintained as a living database on the Filtergraph visualization portal at the URL https://filtergraph.com/kelt$\_$vars