Extreme excesses of $^{13}C$ ($^{12}C$/$^{13}C$<10) and $^{15}N$ ($^{14}N$/$^{15}N$<20) in rare presolar SiC grains have been considered diagnostic of an origin in classical novae, though an origin in core collapse supernovae (CCSNe) has also been proposed. We report C, N, and Si isotope data for 14 submicron- to micron-sized $^{13}C$- and $^{15}N$-enriched presolar SiC grains ($^{12}C$/$^{13}C$<16 and $^{14}N$/$^{15}N$<~100) from Murchison, and their correlated Mg-Al, S, and Ca-Ti isotope data when available. These grains are enriched in $^{13}C$ and $^{15}N$, but with quite diverse Si isotopic signatures. Four grains with $^{29,30}Si$ excesses similar to those of type C SiC grains likely came from CCSNe, which experienced explosive H burning occurred during explosions. The independent coexistence of proton- and neutron-capture isotopic signatures in these grains strongly supports heterogeneous H ingestion into the He shell in pre-supernovae. Two of the seven putative nova grains with $^{30}Si$ excesses and $^{29}Si$ depletions show lower-than-solar $^{34}S$/$^{32}S$ ratios that cannot be explained by classical nova nucleosynthetic models. We discuss these signatures within the CCSN scenario. For the remaining five putative nova grains, both nova and supernova origins are viable because explosive H burning in the two stellar sites could result in quite similar proton-capture isotopic signatures. Three of the grains are sub-type AB grains that are also $^{13}C$ enriched, but have a range of higher $^{14}N$/$^{15}N$. We found that $^{15}N$-enriched AB grains (~50<$^{14}N$/$^{15}N$<~100) have distinctive isotopic signatures compared to putative nova grains, such as higher $^{14}N$/$^{15}N$, lower $^{26}Al$/$^{27}Al$, and lack of $^{30}Si$ excess, indicating weaker proton-capture nucleosynthetic environments.
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