2022AJ....164..137K

We present a catalog of ∼100,000 periodic variable stars in Transiting Exoplanet Survey Satellite (TESS) full-frame image data among members of widely distributed moving groups identified with Gaia in the previous papers in the series. By combining the periods from our catalog attributable to rotation with previously derived rotation periods for benchmark open clusters, we develop an empirical gyrochronology relation of angular momentum evolution that is valid for stars with ages 10-1000 Myr. Excluding stars rotating faster than 2 days, which we find are predominantly binaries, we achieve a typical age precision of ≈0.2-0.3 dex and improving at older ages. Importantly, these empirical relations apply to not only FGK-type stars but also M-type stars, due to the angular momentum distribution being much smoother, simpler, continuous, and monotonic as compared to the rotation period distribution. As a result, we are also able to begin tracing in fine detail the nature of angular momentum loss in low-mass stars as functions of mass and age. We characterize the stellar variability amplitudes of the cool stars as functions of mass and age, which may correlate with the starspot covering fractions. We also identify pulsating variables among the hotter stars in the catalog, including δ Scuti, γ Dor, and slowly pulsating B-type variables. These data represent an important step forward in being able to estimate precise ages of FGK- and M-type stars in the field, starting as early as the pre-main-sequence phase of evolution.