Magnetic Moment of the Proton #3: RMS Current?

Using RMS for the current for case 1:

$$\bar{M}={\sqrt{2}\over2}4\mu_N$$

$$\bar{M}=2.8284271\mu_N$$

Which is closer.  Have to re-check current calculation.  

More later...
This is not justifiable in the derivation, as a DC circular loop current was assumed, however, it's fairly close to measured! More serious attempts to follow...