Evolution of the radius valley around low mass stars from $Kepler$ and $K2$

We present calculations of the occurrence rate of small close-in planets around low mass dwarf stars using the known planet populations from the $Kepler$ and $K2$ missions. Applying completeness corrections clearly reveals the radius valley in the maximum a-posteriori occurrence rates as a function of orbital separation and planet radius. We measure the slope of the valley to be $r_{p,\text{valley}} \propto F^{-0.060\pm 0.025}$ which bears the opposite sign from that measured around Sun-like stars thus suggesting that thermally driven atmospheric mass loss may not dominate the evolution of planets in the low stellar mass regime or that we are witnessing the emergence of a separate channel of planet formation. The latter notion is supported by the relative occurrence of rocky to non-rocky planets increasing from $0.5\pm 0.1$ around mid-K dwarfs to $8.5\pm 4.6$ around mid-M dwarfs. Furthermore, the center of the radius valley at $1.54\pm 0.16$ R$_{\oplus}$ is shown to shift to smaller sizes with decreasing stellar mass in agreement with physical models of photoevaporation, core-powered mass loss, and gas-poor formation. Although current measurements are insufficient to robustly identify the dominant formation pathway of the radius valley, such inferences may be obtained by $TESS$ with $\mathcal{O}(85,000)$ mid-to-late M dwarfs observed with 2-minute cadence. The measurements presented herein also precisely designate the subset of planetary orbital periods and radii that should be targeted in radial velocity surveys to resolve the rocky to non-rocky transition around low mass stars.