We weren’t sure of how exactly JPL models the Moon’s brightness in HORIZONS, i.e.
The following plot shows that JPL calculates the actual observatory-object distance in giving the Moon absolute magnitude, but does not use any albedo map of the Lunar surface — so it is symmetrical with phase on either side of new moon (say).
Lower panel : apparent magnitude of the Moon over a 12 month period (September 2010-2012) computed with Horizons, and shown in a narrow range of the illuminated fraction (40 to 50 percent). The scatter in the apparent magnitude around the trend is ~ 0.1 mag.
Most of this scatter is due to the changing distance of the moon around its orbit (between ~0.019 and 0.023 light minutes). The middle panel shows the magnitude if we correct the photometry to a standard distance (0.0215 light minutes) — this reduces the scatter to 0.03 mag. (The distance provided by Horizons includes the position of the observatory on the Earth — this can make a difference of up to ~12,000 km in the Moon-Observatory distance).
The upper panel shows the residuals in a least squares fit to the middle panel, as a function of Julian day over the 1 year sample period. This clearly shows that most of the 0.03 mag scatter in the middle panel is due to the changing Sun-Earth distance during the course of the year. Accounting for this reduces the scatter to <0.01 mag. We interpret this to mean that no surface features of the Moon are being included in the Horizons’ apparent magnitude estimate, since we expect considerably more scatter than that (but we are working on this!).