Although red giants deplete lithium on their surfaces, some giants are Li-rich. Intermediate-mass asymptotic giant branch (AGB) stars can generate Li through the Cameron-Fowler conveyor, but the existence of Li-rich, low-mass red giant branch (RGB) stars is puzzling. Globular clusters are the best sites to examine this phenomenon because it is straightforward to determine membership in the cluster and to identify the evolutionary state of each star. In 72 hours of Keck/DEIMOS exposures in 25 clusters, we found four Li-rich RGB and two Li-rich AGB stars. There were 1696 RGB and 125 AGB stars with measurements or upper limits consistent with normal abundances of Li. Hence, the frequency of Li-richness in globular clusters is (0.2 +/- 0.1)% for the RGB, (1.6 +/- 1.1)% for the AGB, and (0.3 +/- 0.1)% for all giants. Because the Li-rich RGB stars are on the lower RGB, Li self-generation mechanisms proposed to occur at the luminosity function bump or He core flash cannot explain these four lower RGB stars. We propose the following origin for Li enrichment: (1) All luminous giants experience a brief phase of Li enrichment at the He core flash. (2) All post-RGB stars with binary companions on the lower RGB will engage in mass transfer. This scenario predicts that 0.1% of lower RGB stars will appear Li-rich due to mass transfer from a recently Li-enhanced companion. This frequency is at the lower end of our confidence interval.
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