The proposed solution to this dichotomy between the near and far sides of the moon is that during the explosive moon-forming impact, two moons formed instead of one. Under most circumstances, this is highly unlikely. However, if the second moon were to form at a point of orbital stability with the Earth and the other moon, called a Trojan point, it would be possible to have a stable configuration with two moons for tens of millions of years. Eventually the second moon's orbit would be disturbed, sending it on a collision course with the larger moon.
Jutzi and Asphaug simulated this type of collision and varied the size, composition and velocities of the two different moons. The larger of the two moons would still have a magma ocean based off of thermal evolution models. While their simulations cannot predict specific features of the lunar surface, they can predict the overall composition of the surface and look at the near and far side discrepancy.
The primary findings are that a smaller moon moving at speeds lower than sound speed in silicates will not produce a crate, but will eventually re-create the lunar surface we see today. The smaller moon ends up displacing the magma ocean from one side of the moon and pushes it to the other side. The magma ocean becomes the near side of the moon we have today, and the smaller moon gets spread out over the other hemisphere, which becomes the far side of the moon we have today.
There are other plausible ways to create the surface features we see on the moon, so this model can be viewed as one more option in a the pile of possibilities. Thankfully there are experiments that can be done to either confirm or refute this theory. Going to the moon and looking for some sort of compositional difference that could arise from having two initial moons is a possibility, but the differences would likely be small considering that the two moons would have been made of the same material. Another intriguing possibility is using lunar seismology to get more detailed measurements of the crust, or using gravitational measurements. If materials were concentrated only on one half of the moon, this should be detectable by future work by NASA.
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