  We measure how the properties of star-forming central galaxies correlate with
large-scale environment, $\delta$, measured on $10$Mpc/h scales. We use group
catalogs to isolate a robust sample of central galaxies with high purity and
completeness. The properties we investigate are star formation rate (SFR),
exponential disk scale length $R_{\rm exp}$, and Sersic index of the light
profile, $n$. We find that, at all stellar masses, there is an inverse
correlation between SFR and $\delta$, meaning that above-average star forming
centrals live in underdense regions. For $n$ and $R_{\rm exp}$, there is no
correlation with $\delta$ at $M_{\rm star}\lesssim 10^{10.5}$ $M_\odot$, but at
higher masses there are positive correlations; a weak correlation with $R_{\rm
exp}$ and a strong correlation with $n$. These data are evidence of assembly
bias within the star-forming population. The results for SFR are consistent
with a model in which SFR correlates with present-day halo accretion rate,
$\dot{M}_h$. In this model, galaxies are assigned to halos using the abundance
matching ansatz, which maps galaxy stellar mass onto halo mass. At fixed halo
mass, SFR is assigned to galaxies using the same approach,but $\dot{M}_h$ is
used to map onto SFR. The best-fit model requires some scatter in the
$\dot{M}_h$-SFR relation. The $R_{\rm exp}$ and $n$ measurements are consistent
with a model in which these quantities are correlated with the spin parameter
of the halo, $\lambda$. Halo spin does not correlate with $\delta$ at low halo
masses, but for higher mass halos, high-spin halos live in higher density
environments at fixed $M_h$. Put together with the earlier installments of this
series, these data demonstrate that quenching processes have limited
correlation with halo formation history, but the growth of active galaxies, as
well as other detailed properties, are influenced by the details of halo
assembly.
