Graphite is one of the many presolar circumstellar condensate species found in primitive meteorites. While the isotopic compositions of low-density graphite grains indicate an origin in core-collapse supernovae, some high-density grains have extreme isotopic anomalies in C, Ca and Ti, which cannot be explained by envelope predictions of asymptotic giant branch (AGB) stars or theoretical supernova models. The Ca and Ti isotopic anomalies, however, match the predictions of He-shell abundances in AGB stars. In this study, we show that the C, Ca, and Ti isotopic anomalies are consistent with nucleosynthesis predictions of the H-ingestion phase during a very late thermal pulse (VLTP) event in post-AGB stars. The low $^{12}$C/$^{13}$C isotopic ratios in these grains are a result of abundant $^{12}$C efficiently capturing the protons that are being ingested during the VLTP. Very high neutron densities of $sim 10^{15}$ cm$^{-3}$, typical of the $i$-process, are achieved during this phase in post-AGB stars. The large $^{42,43,44}$Ca excesses in some graphite grains are indicative of neutron capture nucleosynthesis during VLTP. The comparison of VLTP nucleosynthesis calculations to the graphite data also indicate that apparent anomalies in the Ti isotopic ratios are due to large contributions from $^{46,48}$Ca, which cannot be resolved from the isobars $^{46,48}$Ti during the measurements. We conclude that presolar graphite grains with moderate to extreme Ca and Ti isotopic anomalies originate in post-AGB stars that suffer a very late thermal pulse.
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