Planetary engulfment events have long been proposed as a lithium (Li) enrichment mechanism contributing to the population of Li-rich giants $(A(mathrm{Li}) geq 1.5$ dex). Using $GALAH$ survey data and MESA stellar models, we calculate the strength and duration of the Li enrichment signature produced in the convective envelope of a host star that has engulfed a hot Jupiter (HJ) companion. We consider solar-metallicity stars in the mass range of $1.0-2.0~mathrm{M_{odot}}$ and the Li supplied by a HJ of $1.0~mathrm{M_{J}}$. We explore engulfment events that occur near the main sequence turn-off (MSTO) and out to orbital separations of $R_{star}{sim}~0.1~mathrm{AU}= 22~mathrm{R_{odot}}$. We map our results onto the Hertzsprung-Russell (H-R) Diagram, revealing the parameter space where planetary engulfment events produce significant Li enrichment signatures. We also map the associated survival times of these signatures, which range across 9 orders of magnitude. Our calculations indicate that if the HJ engulfment event occurs near the MSTO or on the subgiant branch, Li enrichment can be measured at a $5sigma$ confidence level and with strengths that exceed meteoritic abundance measurements. Moreover, for stars of $1.4~mathrm{M_{odot}}$, these signatures are predicted to survive for up to 1 Gyr. We determine that Li enrichment beyond the subgiant branch must be produced by other mechanisms, such as the Cameron-Fowler process or accretion of material from an AGB companion.
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