I think I found the real problem with how the Moller generator was initially set up.
This one affects the generated distribution a LOT more than the RNG precision probably did, and explains the remaining strangeness in the generated distribution.
Looking at the egs5 Moller procedure, the angular cut was defined as:
abs(v(np)) > 0.010 radians
where v(np) is supposed to be theta.
However, the variables u, v, and w in egs are actually directional cosines, p_x = p*u, p_y = p*v, and p_z = p*w.
So this means that in reality, the generator was saving Moller events such that
abs [ sin(theta)*sin(phi) ] > 0.010
which has a periodic nature to it. Plot this equation for some value of phi (or just think about it) and you'll see what was likely making these strange hills and gaps in the energy distribution - full-wave rectified Mollers!
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