If you cannot answer simple questions about your ATM idea then you are unable to defend it.

You assert that an equation is the covariant form of the Lorentz force. But all you do is algebraic manipulation that gives adifferent equationfrom the covariant form of the Lorentz force! What is worse is that your manipulation has no electromagnetism in it. You have no electric field. You have no magnetic field. You have no classical charge q.

I asked you some simple equations.

If you have an Lorentz force equation then use it to derive the electric field around a point charge.

If you have the same equation then the left hand side of your equation must be the same as the left hand side of that equation. But anyone can read that they are not the same set of symbols ! You need to show that they are the same.

If you have the same equation then where is the classical charge q that appears in the covariant form of the Lorentz force? That is not your "q is for elementary charge" (which is usually written as e to avoid confusion with the classical charge q), it is any value of charge at all.

If you have the same equation then your has to be the contravariant electromagnetic tensor (within a factor of q but see above!). Writing the same symbol does mow make your equation into the contravariant electromagnetic tensor so you need to show that the components match.