Dear Neal,
Your essay conjoins two topics I would not have immediately thought of as being closely related in an interesting way. A few comments:
1.Your eqn. 2 reminded me a little of Ehrenfest's argument already over 100 years ago that one way to answer the question about the dimensionality of space is to consider that in space dimensions other than 3 orbits become unstable. Now, I realize your argument is very different, even drawing from a different theory (EM vs. CM) but it still has a similar flavor.
I tend to be skeptical of such arguments because the takeaway I get from them is not that space had to be 3D, but that for other dimensionalities, the "stuff" that would be the analog of mass in spacetime would have to have different properties and obey different laws.
2.Your discussion of the 4/3 problem reminded me that there have been over the last few years claims of having solved it. One name that comes to mind is Fritz Rohrlich. He also wrote a book "Classical charged particles" which you may enjoy, given your interest(I found it extremely readable, which in the EM literature is not always the case).
3.Regarding your discussion of whether mathematics has the capacity of formally expressing conceptually intuitive ontological distinctions I agree with you that in its present form it does not. However, I also believe that compared to what it could express, the current form is very impoverished, and that the only reason most of us don't see this is because we are too beholden to its present state. To better understand what I mean, I invite you to peruse any reference on various non-classical logics. You will find a large menagerie of species, mostly developed by philosophers for comparatively narrow purposes, many of which an offer the possibility for serving as a foundation of mathematics with expressive powers beyond what you might have thought possible. Indeed, my own main area of research is in this area, and in fact my entry in this contest is concerned precisely with introducing the distinction between actuality and potentiality into mathematics.
Overall, your paper offers several interesting ideas, I hope that some experts in the area will take the time to examine your dimensionality argument in depth. From my perspective, its correctness would be interesting not so much for the reasons you give but because it might have the potential to illuminate other foundational questions in that area.
Best wishes,
Armin