  We investigate the current debate on the Mn valence in Ga$_{1-x}$Mn$_x$N, a
diluted magnetic semiconductor (DMSs) with a potentially high Curie
temperature. From a first-principles Wannier-function analysis, we
unambiguously find the Mn valence to be close to $2+$ ($d^5$), but in a mixed
spin configuration with average magnetic moments of 4$\mu_B$. By integrating
out high-energy degrees of freedom differently, we further derive for the first
time from first-principles two low-energy pictures that reflect the intrinsic
dual nature of the doped holes in the DMS: 1) an effective $d^4$ picture ideal
for local physics, and 2) an effective $d^5$ picture suitable for extended
properties. In the latter, our results further reveal a few novel physical
effects, and pave the way for future realistic studies of magnetism. Our study
not only resolves one of the outstanding key controversies of the field, but
also exemplifies the general need for multiple effective descriptions to
account for the rich low-energy physics in many-body systems in general.
