  In the three-dimensional parameter space defined by velocity dispersion,
effective radius (R_e), and effective surface brightness (I_e), early-type
galaxies are observed to populate a two-dimensional fundamental plane (FP) with
finite thickness. In Paper III of this series, we showed that the thickness of
the FP is predominantly due to variations in the stellar mass surface density
(Sigma_*) inside the effective radius R_e. These variations represent
differences in the dark matter fraction inside R_e (or possibly differences in
the initial mass function) from galaxy to galaxy. This means that galaxies do
not wind up below the FP at lower surface brightness due to the passive fading
of their stellar populations; they are structurally different. Here, we show
that these variations in Sigma_* at fixed dynamical mass (M_dyn) are linked to
differences in the galaxy stellar populations, and therefore to differences in
their star formation histories. We demonstrate that the ensemble of stellar
population and Sigma_* variations through the FP thickness can be explained by
a model in which early-type galaxies at fixed M_dyn have their star formation
truncated at different times. The thickness of the FP can therefore be
interpreted as a sequence of truncation times. Galaxies below the FP have
earlier truncation times for a given M_dyn, resulting in lower Sigma_*, older
ages, lower metallicities in both [Fe/H] and [Mg/H], and higher [Mg/Fe]. We
show that this model is quantitatively consistent with simple expectations for
chemical enrichment in galaxies. We also present fitting functions for
luminosity-weighted age, [Fe/H], [Mg/H], and [Mg/Fe] as functions of the FP
parameters velocity dispersion, R_e, and I_e. These provide a new tool for
estimating the stellar population properties of quiescent early-type galaxies
for which high-quality spectra are not available.
