One interesting thought experiment that will gain you a lot of insight is to boost the hydrogen action close to the speed of light and then take the limit. So you do a Lorenz transformation. This will result in a time dilation and a length contraction. Interestingly the photon field can be visualized as the integral of the following wave packets,

Now using the plane wave expansion in spherical coordinates lead to the solution for the electrical potential,

where \(j_l(kr) = 0\) for \(r\) at the electron shell and zero outside. We will consider only the zeros where the photon lead to an attracting radial force. E.g. when

Is negative which for \(l=0\) means \(kr = 2\pi n\) and

(This explains why we do not use the zeros at \(kr = 2 \pi (n-1)\), \(n >= 1\)).

In a sense when we release the energy be kicking out the the nucleus in this setup we also kick out an photon with the same energy hence Both central potentials should match and hence everything conspires so that the the attractive force of the nucleus is replaced by the attractive force of the photon. This means that when we approaches the limit of \(c\) the electron is essentially a small disk due to length contraction and we get a charge density which are the projection of the sphere onto the disk, just as Mill's claim. Now we get into a fundamental philosophically point. I would argue that there is reference frame which is at rest e.g. a rest frame. Now because the would is Lorenz invariant and live inside the model we can distinguish the needed boost to get into this reference system because all measurements we do all uses physics that are Lorenz invariant and hence all physics are the same independently whatever speed we are traveling at (at least locally and we ignore the effect of gravity). So philosophically if we where in the global rest frame the contraction is a true contraction. Now all laws has flaws if you use a good enough "microscope" and it is not unreasonable that as the electron get flatter it snatches to a full disk at some speed close enough to \(c\) (here we think of the shells as local nonlinearites in the space which is the origin of the mass and charge) and then the electron Mill's deduces is created as we can now bring it back to normal speeds and we have a Mill's description of the disk. A nice side effect of this reasoning is that the flat disk electron model seam to assume that there exists a local rest reference system. And we also have gotten an understanding of the the photon and Mills in GUTCP seam to be right in track in his description.

This is at least how I think about the electron and photon, I could be wrong in some parts or miss important stuff, but it is clear that Mill's and GUCTCP is probably very close to the truth.