Disentangling Metallicity Effects in Hot Jupiter Occurrence Across Galactic Birth Radius and Phase-Space Density

We explore how the correlation between host star metallicity and giant planets shapes hot Jupiter occurrence as a function of Galactic birth radius (\rbirth) and phase-space density in the Milky Way disk. Using the GALAH and APOGEE surveys and a galaxy from the NIHAO simulation suite, we inject hot Jupiters around stars based on metallicity power laws, reflecting the trend that giant planets preferentially form around metal-rich stars. For \rbirth\ $\geq 5$ kpc, hot Jupiter occurrence decreases with \rbirth\ by $\sim -0.1%$ per kpc; this is driven by the Galaxy's chemical evolution, where the inner regions of the disk are more metal-rich. Differences in GALAH occurrence rates versus APOGEE's and the simulation's at \rbirth\ $< 5$ kpc arise from survey selection effects. APOGEE and the NIHAO simulation have more high-$\alpha$ sequence stars than GALAH, resulting in average differences in metallicity (0.2--0.4 dex), $\alpha$-process element enrichment (0.2 dex), and vertical velocities (7--14 km/s) at each \rbirth\ bin. Additionally, we replicate the result of \cite{Winter20}, which showed that over 92% of hot Jupiters are associated with stars in phase-space overdensities, or "clustered environments." However, our findings suggest that this clustering effect is primarily driven by chemical and kinematic differences between low and high-$\alpha$ sequence star properties. Our results support stellar characteristics, particularly metallicity, being the primary drivers of hot Jupiter formation, which serves as the "null hypothesis" for interpreting planet demographics. This underscores the need to disentangle planetary and stellar properties from Galactic-scale effects in future planet demographics studies.