The Evolution of Jupiter and Saturn as a function of the Parameter R$_{\rho}$

Computed using the APPLE planetary evolution code, we present updated evolutionary models for Jupiter and Saturn that incorporate helium rain, non-adiabatic thermal structures, and "fuzzy" extended heavy-element cores. Building on our previous Ledoux-stable models, we implement improved atmospheric boundary conditions that account for composition-dependent effective temperatures and systematically explore the impact of varying the parameter $R_{\rho}$, which allows one to explore in an approximate way the efficiency of semiconvection. For both Jupiter and Saturn, we construct models spanning from $R_{\rho}=1$ (Ledoux) to $R_{\rho}=0$ (Schwarzschild), and identify best-fit solutions that match each planet's effective temperature, equatorial radius, lower-order gravitational moments, and atmospheric composition at 4.56 Gyr. We find that lower $R_{\rho}$ values lead to stronger convective mixing, resulting in higher surface metallicities and lower deep interior temperatures, while requiring reduced heavy-element masses and lower initial entropies to stabilize the dilute inner cores. Our Saturn models also broadly agree with the observed brunt frequency profile inferred from Cassini ring seismology, with stable layers arising from both the helium rain region and the dilute core. These findings support the presence of complex, compositionally stratified interiors in both gas giants.