What declination condition ($\delta$) causes an object to remain permanently below the horizon at latitude $\phi$?

${\delta}$ must always be less than ${\phi} - 90^\circ$.

The criterion determining if a celestial object remains permanently below the observer's horizon involves comparing its declination ($\delta$) to a value derived from the observer's latitude ($\phi$). For an object to be continuously visible (circumpolar), its declination must always exceed $90^ ext{circ} - \phi$. Conversely, the condition ensuring that an object *never* rises above the horizon is precisely when its declination is consistently lower than the latitude minus $90$ degrees ($\delta < \phi - 90^ ext{circ}$). If the Moon's entire range of declination falls within this lower boundary based on the observer's latitude, that observer will never witness the Moon rise above their horizon.