FAUST XX. The chemical structure and temperature profile of the IRAS 4A2 hot corino at 20-50 au

Young low-mass protostars often possess hot corinos, compact, hot and dense regions bright in interstellar Complex Organic Molecules (iCOMs). Besides of their prebiotic role, iCOMs can be used as a powerful tool to characterize the chemical and physical properties of hot corinos. Using ALMA/FAUST data we aim to explore the iCOMs emission at < 50 au scale around the Class 0 prototypical hot corino IRAS 4A2. We imaged IRAS 4A2 in six abundant, common iCOMs (CH$_3$OH, HCOOCH$_3$, CH$_3$CHO, CH$_3$CH$_2$OH, CH$_2$OHCHO, and NH$_2$CHO), and derived their emitting size. The column density and gas temperature for each species were derived at 1$\sigma$ from a multi-line analysis by applying a non-LTE approach for CH$_3$OH, and LTE population or rotational diagram analysis for the other iCOMs. Thanks to the unique estimates of the absorption from foreground millimeter dust toward IRAS 4A2, we derived for the first time unbiased gas temperatures and column densities. We resolved the IRAS 4A2 hot corino finding evidence for a chemical spatial distribution in the inner 50 au, with the outer emitting radius increasing from ~ 22-23 au for NH$_2$CHO and CH$_2$OHCHO, followed by CH$_3$CH$_2$OH (~ 27 au), CH$_3$CHO (~ 28 au), HCOOCH$_3$ (~ 36 au), and out to ~ 40 au for CH$_3$OH. Combining our estimate of the gas temperature probed by each iCOM with their beam-deconvolved emission sizes, we inferred the gas temperature profile of the hot corino on scales of 20-50 au in radius, finding a power-law index $q$ of approximately -1. We observed, for the first time, a chemical segregation in iCOMs of the IRAS 4A2 hot corino, and derived the gas temperature profile of its inner envelope. The derived profile is steeper than when considering a simple spherical collapsing and optically-thin envelope, hinting at a partially optically-thick envelope or a gravitationally unstable disk-like structure.