I'll have a try:
Power is proportional to the ion density squared.
Now a polywell confines the ions using an electron cloud, which has nearly the same charge density as the ions (the deviation procudes the ion confining E-field).
So the fusion power (=P_fusion) is proportional to the electron density (=n) squared:
P_fusion ~ n^2
The B-field is used to confine the electrons, but the electrons produce a B-field too, which works against the external B-field. At some point those fields cancel each other. This happens when the kinetic energy density of the electrons (=n*e*V, V: acceleration/grid voltage) is equal to the energy density of the external B-field (this is called beta=1 condition).
(In a polywell this is intended to happen inside of the grids: with increasing density the B-field gets 'forced' further outside, were the external field is stronger.
At some point however the external field gets weaker again. The density to cancel the field there is the maximal density possible, at this density the well will 'blow out'.)
So magnetic field energy density (proportional B^2) is equal to n*e*V, thus:
n ~ B^2 => P_fusion ~ B^4
Still, this is just the power not the power gain.
The scaling bases on the assumption that the major losses are electrons hitting unshielded surfaces.
Since there should only be unshielded surfaces outside of the wiffleball the losses are proportional to the outside electron density, which is the inside density divided by a trapping factor.
According to the valencia paper this trapping factor scales with B^2, so:
P_loss ~ n / B^2 ~ B^2 / B^2 = 1
so the losses are constant regarding the B-field and gain too scales with B^4.
My understanding is, that there are only small areas around the cusp axis through which the electrons can escape. The radius of these 'holes' is propotional to the gyro radius of the electrons, which again is proportional to B. Thus the area through which the electrons can escape is ~ B^2.
For anybody who doesn't like this 'explanation' of the trapping factor, the B^2 scaling might be relatively easy to test via simulation and I'm sure that would be appreciated around here