Sub-Neptune Memories I: Implications of Inefficient Mantle Cooling and Silicate Rain

We explore the evolution of sub-Neptune (radii between $\sim$1.5 and 4 R$_\oplus$) exoplanet interior structures using our upgraded planetary evolution code, \texttt{APPLE}, which self-consistently couples the thermal and compositional evolution of the whole structure. We incorporate stably stratified regions with convective mixing and, for the first time, ab initio results on the phase separation of silicate-hydrogen mixtures to model silicate rain in sub-Neptune envelopes. We demonstrate that inefficient mantle cooling can retain sufficient heat to Gyr ages: inefficient heat transport from mantle to envelope alone keeps radii $\sim$10\% larger than predicted by adiabatic models at late times. Silicate rain can contribute an additional $\sim$5\% to the radius, depending on envelope mass and initial metal abundance. The silicate-hydrogen immiscibility region may lie in the middle or even upper envelope, far above the envelope-mantle boundary layer, and bifurcates the envelope into two an upper, hydrogen-rich region and a lower, metal-rich region above the mantle. If silicate rain occurs, atmospheres should appear depleted of silicates while radii remain inflated at late ages. To demonstrate this, we present interior evolution models for GJ 1214 b, K2-18 b, TOI-270 d, and TOI-1801 b, showing that hot, liquid silicate mantles with thin envelopes reproduce their radii and mean densities, providing an alternative to water-world interpretations. These results imply that bulk compositions inferred from mean density must account for mantle thermal state and envelope mixing/phase separation history; such thermal ``memories'' may constrain formation entropies and temperatures when metallicities are better measured.