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As an archaeologist it was always my job to back fill the 2m x 2m holes which was usually our unit of excavation. And you are right, there is never enough to put back in, even though you put in the same amount you took out.
Of course the reason is there really is that much air in soil that has not been disturbed. When you take it out, the air joins the atmosphere, and you are left short when you backfill.
And . . . that is the reason for no dig gardening. The air stays in. The soil doesn't have to rebuild itself to recover from the disturbance.
I would look no further than your neighbor next door, it seems odd that your dirt is disappearing while his stays the same. The solution to this problem is to wait until they leave on vacation and bring all that dirt back where it came from. Or you could bring it to me as my yard seems to be sinking too!!!