What is the limiting factor for Moon visibility at the pole, unlike mid-latitudes?

The local horizon, not the day/night cycle alone.

At mid-latitudes, such as $40^ ext{circ}$ North, the Moon always rises and sets daily, meaning the primary factor preventing observation during the night is whether the Moon is actually present in the sky during those hours (e.g., being in the New Moon phase and up during the day). However, at the exact North or South Pole, the geometry changes drastically. The limiting factor becomes the local horizon itself. If the Moon's declination is too far below the celestial equator relative to the pole's orientation, the celestial sphere appears to rotate parallel to the ground, keeping the Moon entirely below that local horizon line, irrespective of whether it is day or night above ground.