  The use of Type Ia Supernovae (SNe Ia) as cosmological tools has motivated
significant effort to: understand what drives the intrinsic scatter of SN Ia
distance modulus residuals after standardization, characterize the distribution
of SN Ia colors, and explain why properties of the host galaxies of the SNe
correlate with SN Ia distance modulus residuals. We use a compiled sample of
$\sim1450$ spectroscopically confirmed, photometric light-curves of SN Ia and
propose a solution to these three problems simultaneously that also explains an
empirical 11$\sigma$ detection of the dependence of Hubble residual scatter on
SN Ia color. We introduce a physical model of color where intrinsic SN Ia
colors with a relatively weak correlation with luminosity are combined with
extrinsic dust-like colors ($E(B-V)$) with a wide range of extinction parameter
values ($R_V$). This model captures the observed trends of Hubble residual
scatter and indicates that the dominant component of SN Ia intrinsic scatter is
from variation in $R_V$. We also find that the recovered $E(B-V)$ and $R_V$
distributions differ based on global host-galaxy stellar mass and this explains
the observed correlation ($\gamma$) between mass and Hubble residuals seen in
past analyses as well as an observed 4.5$\sigma$ dependence of $\gamma$ on SN
Ia color. This finding removes any need to prescribe different intrinsic
luminosities to different progenitor systems. Finally we measure biases in the
equation-of-state of dark energy ($w$) up to $|\Delta w|=0.04$ by replacing
previous models of SN color with our dust-based model; this bias is larger than
any systematic uncertainty in previous SN Ia cosmological analyses.
