I still don't think you're conserving energy. Start with two particles far apart and approaching each other at some speed; define this state as the zero energy. Let them approach each other, slowing down all the while, and eventually heading back out. When they reach their initial separation, they have kinetic energy from two sources: One source is popping back the forces they experienced during their approach, the other is the forces they experienced as they separated. Since they are at their initial separation again, the stack is empty, so there is zero potential energy; and there's no reason the kinetic energy should be what it was initially. So energy has been added or subtracted.
The idea of having only "magnetic" forces seems to work, yes. But, as you say, we then lose special relativity, and that imposes a preferred frame of reference, which in turn means that the laws are no longer invariant under translation. So then you lose conservation of momentum, if I remember my Noether correctly.
One source is popping back the forces they experienced during their approach, the other is the forces they experienced as they separated. Since they are at their initial separation again, the stack is empty, so there is zero potential energy; and there's no reason the kinetic energy should be what it was initially. So energy has been added or subtracted.
You got it backwards. The stack reads in from infinity, not from 0 separation. As they approach, they're pushing, not popping. Plus, the contents of the stack are included in the potential energy, so eit...
This is a thread where people can ask questions that they would ordinarily feel embarrassed for not knowing the answer to. The previous thread is at close to 500 comments.