This confusion arises mainly from the serious mistake of not having recognized the fundamental role of the vector potential,

No need to run ahead of the locomotive ... Before talking about this - "This confusion arises mainly from the serious mistake of not having recognized the fundamental role of the vector potential,...", start to deal with who and when he introduced the concept " vector potential "... Understand how physics developed ... In what year was it done? Why was this done?

And the story goes like this...

In the middle of the 19th century, G. Helmholtz proved a theorem from which it followed that in any material, elastic medium, only two classes of perturbations can exist.

This is a vortex and gradient - rotational and translational displacement of the particles of the medium.

The gradient field is defined as the gradient of the scalar potential - gradφ.

The vortex field is defined as the curl of the vector potential - rofA.

Further, he proved the main theorem of classical field theory - the uniqueness theorem for the solution of field problems - the "Helmholtz theorem" from which it follows that:

Direct problem

- if the spatial distributions of the gradient field sources and/or vortex field exciters are given, then the spatial distribution of the scalar potential φ or/and the vector potential A are uniquely determined as a solution of the Laplace equations and/or the double vector product of the nabla operator, respectively.

Inverse problem

- if the spatial distribution of the scalar potential φ or/and the vector potential - A is given, then the spatial distribution of the sources φ or/and the exciters of the vector potential - A are uniquely determined by solving the corresponding differential equations.

By substituting the corresponding boundary and initial conditions into the solutions, we obtain exact particular solutions of specific constructive problems.

Based on the general field theory, Helmholtz built the theory of electric gradient and vortex electromagnetic fields.

Can we physicists trust Helmholtz's mathematics? Can we unmistakably assume that such physical objects as the electron and the proton obey the mathematics of Helmholtz? The answer is unequivocal - no! We cannot trust Helmholtz's mathematics... And as it turns out, we cannot trust Maxwell's mathematics! Physics, as we see from the results of experiments, does not follow mathematics - physics has its own laws and we have not studied them yet - we are just trying to do it!

The locomotive is physics, not mathematics!