  We report observations and modeling of the stellar remnant and presumed
double-degenerate merger of Type~Iax supernova Pa30, which is the probable
remnant of SN~1181~AD. It is the only known bound stellar SN remnant and the
only star with Wolf-Rayet features that is neither a planetary nebula central
star nor a massive Pop I progenitor. We model the unique emission-line spectrum
with broad, strong O~{\sc vi} and O~{\sc viii} lines as a fast stellar wind and
shocked, hot gas. Non-LTE wind modeling indicates a mass-loss rate of $\sim
10^{-6}\,\rm M_\odot\,yr^{-1}$ and a terminal velocity of $
\sim$15,000~km\,s$^{-1}$, consistent with earlier results. O~{\sc viii} lines
indicate shocked gas temperatures of $T \simeq 4\,$MK. We derive a magnetic
field upper limit of $B<2.5\,$MG, below earlier suggestions. The luminosity
indicates a remnant mass of 1.0--1.65\,\rm M$_\odot$ with ejecta mass
$0.15\pm0.05\,\rm M_\odot$. Archival photometry suggests the stellar remnant
has dimmed by $\sim$0.5 magnitudes over 100 years. A low Ne/O$\,<0.15$ argues
against a O-Ne white dwarf in the merger. A cold dust shell is only the second
detection of dust in a SN Iax and the first of cold dust. Our ejecta mass and
kinetic energy estimates of the remnant are consistent with Type Iax
extragalactic sources.
