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We have investigated a highly energetic H-ingestion event during shell Heburning leading to H-burning luminosities of $\log(L_\mathrm{H}/\mathrm{\Lsun})\sim 13$ in a $45 \Msun$ Pop III massive stellar model. In order to track thenucleosynthesis which may occur in such an event, we run a series ofsingle-zone nucleosynthesis models for typical conditions found in the stellarevolution model. Such nucleosynthesis conditions may lead to$\textit{i}-$process neutron densities of up to $ \sim 10^{13}\mathrm{cm}^{-3}$. The resulting simulation abundance pattern, where Mg comesfrom He burning and Ca from the $\textit{i}$ process, agrees with the generalobserved pattern of the most iron-poor star currently known, SMSSJ031300.36-670839.3. However, Na is also efficiently produced in these $i$process conditions, and the prediction exceeds observations by $\sim 2.5\unitstyle{dex}$. While this probably rules out this model for SMSSJ031300.36-670839.3, the typical $\textit{i}-$process signature of combined Heburning and $\textit{i}$ process of higher than solar [Na/Mg], [Mg/Al] and low[Ca/Mg] is reproducing abundance features of the two next most iron-poor starsHE\,1017-5240 and HE\,1327-2326 very well. The $i$ process does not reach Fewhich would have to come from a low level of additional enrichment.$\textit{i}$ process in hyper-metal poor or Pop III massive stars may be ableto explain certain abundance patterns observed in some of the most-metal poorCEMP-no stars.

Pop III $\textit{i}$-process Nucleosynthesis and the Elemental Abundances of SMSS J0313-6708 and the Most Iron-Poor Stars