Searching for GEMS: The Occurrence of Giant Planets orbiting M-dwarfs within 100 pc

We present results from a systematic search for transiting short-period Giant Exoplanets around M-dwarf Stars (GEMS; $P < 10$ days, $R_p \gtrsim 8~R_\oplus$) within a distance-limited 100 pc sample of $149,316$ M-dwarfs using TESS-Gaia Light Curve (TGLC) data. We describe the development and application of the \textit{TESS-miner} package and associated vetting procedures used in this analysis. To assess detection completeness, we conducted $\sim$72 million injection-recovery tests across $\sim$26,000 stars with an average of $\sim$3 sectors of data per star, subdivided into early-type (M0--M2.5), mid-type (M2.5--M4), and late-type (M4 or later) M-dwarfs. Our pipeline demonstrates high sensitivity across all subtypes within the injection bounds. We estimate the occurrence rates of short-period GEMS as a function of stellar mass, and combine our measured rates with those derived for FGK stars, fitting an exponential trend with stellar mass, consistent with core-accretion theory predictions. We find GEMS occurrence rates of $0.118\% \pm 0.068\%$ for early-type M-dwarfs, $0.153\% \pm 0.069\%$ for mid-type, and $0.036\% \pm 0.024\%$ for late-type M-dwarfs, with a mean rate of $0.068\%\pm0.024\%$ across the full sample. While our search spanned $1.0~\mathrm{days} < P < 10.0$ days, these rates were calculated using planets orbiting with $1.0~\mathrm{days} < P < 5.0$ days. This work establishes the basis for future occurrence rate studies of transiting GEMS.