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.
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