  We present a spatially resolved study of stellar populations in 6 galaxies
with stellar masses $M_*\sim10^{10}M_\odot$ at $z\sim3.7$ using 14-filter
JWST/NIRCam imaging from the JADES and JEMS surveys. The 6 galaxies are
visually selected to have clumpy substructures with distinct colors over
rest-frame $3600-4100$\r{A}, including a bright dominant stellar core that is
close to their stellar-light centroids. With 23-filter photometry from HST to
JWST, we measure the stellar-population properties of individual structural
components via SED fitting using Prospector. We find that the central stellar
cores are $\gtrsim2$ times more massive than the Toomre mass, indicating they
may not form via in-situ fragmentation. The stellar cores have stellar ages of
$0.4-0.7$ Gyr that are similar to the timescale of clump inward migration due
to dynamical friction, suggesting that they likely instead formed through the
coalescence of giant stellar clumps. While they have not yet quenched, the 6
galaxies are below the star-forming main sequence by $0.2-0.7$ dex. Within each
galaxy, we find that the specific star formation rate is lower in the central
stellar core, and the stellar-mass surface density of the core is already
similar to quenched galaxies of the same masses and redshifts. Meanwhile, the
stellar ages of the cores are either comparable to or younger than the
extended, smooth parts of the galaxies. Our findings are consistent with model
predictions of the gas-rich compaction scenario for the buildup of galaxies'
central regions at high redshifts. We are likely witnessing the coeval
formation of dense central cores, along with the onset of galaxy-wide quenching
at $z>3$.
