  The two known ``hyper-metal-poor'' (HMP) stars, HE0107-5240 and HE1327-2326,
have extremely high enhancements of the light elements C, N, and O relative to
Fe and appear to represent a statistically significant excess population
relative to the halo metallicity distribution extrapolated from [Fe/H] > -3.
This study weighs the available evidence for and against three hypothetical
origins for these stars: (1) that they formed from gas enriched by a primordial
``faint supernova'', (2) that they formed from gas enriched by core-collapse
supernovae and C-rich gas ejected in rotation-driven winds from massive stars,
and (3) that they formed as the low-mass secondaries in binary systems at Z ~
10^{-5.5} Zsun and acquired their light-element enhancements from an
intermediate-mass companion as it passed through an AGB phase. The observations
interpreted here, especially the depletion of lithium seen in HE1327-2326,
favor the binary mass-transfer hypothesis. If HE0107-5240 and HE1327-2326
formed in binary systems, the statistically significant absence of isolated
and/or C-normal stars at similar [Fe/H] implies that low-mass stars could form
at that metallicity, but that masses M ~< 1.4 Msun were disfavored in the IMF.
This result is also explained if the abundance-derived top-heavy IMF for
primordial stars persists to [Fe/H] ~ -5.5. This finding indicates that
low-mass star formation was possible at extremely low metallicity, and that the
typical stellar mass may have had a complex dependence on metallicity rather
than a sharp transition driven solely by gas cooling.
