  The nature of the first Pop III stars is still a mystery and the energy
distribution of the first supernovae is completely unexplored. For the first
time we account simultaneously for the unknown initial mass function (IMF),
stellar mixing, and energy distribution function (EDF) of Pop III stars in the
context of a cosmological model for the formation of a MW-analogue. Our
data-calibrated semi-analytic model is based on a N-body simulation and follows
the formation and evolution of both Pop III and Pop II/I stars in their proper
timescales. We discover degeneracies between the adopted Pop III unknowns, in
the predicted metallicity and carbonicity distribution functions and the
fraction of C-enhanced stars. Nonetheless, we are able to provide the first
available constraints on the EDF, $dN/dE_\star \propto E_{\star}^{-\alpha_e}$
with $1\leq \alpha_e \leq2.5$. In addition, the characteristic mass of the Pop
III IMF should be $m_{\rm ch}<100\:{\rm M_\odot}$, assuming a mass range
consistent with hydrodynamical simulations (0.1-1000$\:{\rm M_\odot}$).
Independent of the assumed Pop III properties, we find that all [C/Fe]>+0.7
stars (with [Fe/H]<-2.8) have been enriched by Pop III supernovae at a $>20\%$
level, and all [C/Fe]>+2 stars at a $>95\%$ level. All very metal-poor stars
with $\rm [C/Fe]<0$ are predicted to be predominantly enriched by Pop III
hypernovae and/or pair instabillity supernovae. To better constrain the
primordial EDF, it is absolutely crucial to have a complete and accurate
determination of the metallicity distribution function, and the properties of
C-enhanced metal-poor stars (frequency and [C/Fe]) in the Galactic halo.
