Just to clarify, Mills is not claiming that these rings actually exist, they are a mathematical tool used to describe the motion of the current on the 2-sphere.
From GUTCP latest edition starting on Page 67:
I agree that one can choose any mathematical tool you want, regardless of how ridiculous it is, if it makes precise predictions. But, if the predictions are better without an arbitrary construction, then one should throw away the construction.
This is the argument some of Mills' supporters use -- that a hypothesis can make any arbitrary construction one wants, provided it works. Particle-wave duality is pretty out-there, after all, but it works.
The reality is that no, you can't just create an arbitrary construction, without determining the implications of your construction. In Mills case of the rotating charge rings, he doesn't even get a stable construction -- remember he is trying to use classical physics. Then, his calculations don't even get correct results.
I totally understand why Mills created his charged rings. They take the radial dimension out of the equation (once he defines a stable radius), making the math a lot easier (2-D, not 3-D). He needs rings to prevent the radiation condition.
What he has done is created a 2-D projection of the 3-D wavefunction, for Hydrogen, onto a hard sphere. Yes, this makes math really easy. Too bad physicists never thought of it (sarcasm).
Except, it doesn't work for anything except Hydrogen, and it doesn't even work for Hydrogen, when you try to get high accuracy. QM does give the precise result. I went through Mills' fine structure constant derivation, and anything beyond the first term is not even derived. The first term is well known, and does have a classical analog. Mills invents the farther terms with no backup (almost like he invented them), and still gets an answer that is not close enough.
As for Mills' "Millsian" program, yes he has invented formulas that match empirical results reasonably well -- but they have no backup in his theory. No-one knows where the terms come from -- almost like he invented them to match experiment. It is well known that there are patterns in ionization energies that can be fit to curves, and it is difficult to calculate them using QM from first principles. It is much easier to look in the back of the textbook, and say "look, I can come up with a formula".
Now, Mills claims that his formulas only use fundamental constants. This is "partly true", by which I mean it is misleading. I can fit anything to anything using only fundamental constants. I can put them in exponents, denominators, multiplicative factors, and raise them to different powers. I bet with a bit of work, I could invent more accurate formulas than Mills, just by adding a few higher order terms. Of course, I'd have to update it, when new experimental results invalidated them.