Electromagnetic mass disproof I've been reading the Feynman's lectures lately and found the section of electromagnetic mass in this article: https://www.feynmanlectures.caltech.edu/II_28.html (section 3 at this page). Here Feynman is talking about the mechanical and the electromagnetic momentum of mass, and the "possibility that the mechanical piece is not there at all—that the mass is all electromagnetic". Lower in the text follows an argumentation pointing against this assumption:

for an arbitrary velocity $v$, the momentum is altered by the factor $1/\sqrt{1−v^2/c^2}$:
$$ p=\frac{2}{3}  \frac{e^2}{(ac^2)}\frac{v} {\sqrt{1−v^2/c^2}}$$

and

the field of the electron should have the mass
$$m′_{elec}=\frac{U_{elec}}{c^2}=\frac{1}{2}\frac{e^2}{(ac^2)}$$
which is not the same as the electromagnetic mass.

but I cannot quite understand how the given formulae actually prove the existence of "the mechanical piece" of mass, simply based on twisting of mathematical formulae. Can somebody put it into simpler words for me and help me understand it?
So, Q1: What is it that proves that the "mechanical mass" is there in deed?
And Q2: What part of the entire mass of an electron is non-electromagnetic, based on these formuli - is it 1/3, or 3/4 of the entire mass, and the rest is EMmass?
And on top of this - Q3:Why do the factors in the mentioned formuli change all the time - from $\frac{1}{2}$, to $\frac{2}{3}$, to $\frac{3}{4}$... (Excuse this too poor level of understanding of mine, but I will strongly appreciate some help!)
 A: Feynman is using the notation $a$ for the radius of the electron. The first point he makes, in section 28-1, is that this is infinite for a point charge. You could basically stop reading there. Current theory treats electrons as pointlike particles, so this infinite mass is a problem. For example, we see processes that create electron-positron pairs, which would be impossible if they had infinite mass-energy. The problem is really a problem with classical physics. We need quantum mechanics instead.
But Feynman is developing the absurd consequences that you arrive at when you know relativity, don't know about quantum mechanics, and try to develop a theory of electrons that have finite radius. This is the sort of thing that people would have tried to do ca. 1905-1927. The fact that you're confused by all the fractions like 3/4, etc., is because he's showing how silly it gets when you try to develop the theory in this way, and also because he follows the kind of complications that people tried to introduce to salvage the idea of point charges in classical physics. None of it actually ends up working out. E.g., you get equations for the motion of point charges (28-5) that have unphysical properties like the fact that charges accelerate before the time when a force is applied to them.
A: Feynman does not say that there must be mechanical mass, he explains that mass of a charged body cannot be purely electromagnetic, because something has to keep the charge concentrated on the body, and this something isn't electromagnetic force, so it has to be some non-electromagnetic force. This means there are non-electromagnetic contributions to energy and thus to mass.  The non-electromagnetic contribution can be negative or positive; in the case of the charged sphere held by non-EM forces, we expect it to be negative (attractive forces, negative energy in compact state).
Nobody knows how much of electron's mass is electromagnetic mass, because we do not know whether the electron is a charged sphere, and even if we assume it is, we do not know what the radius $a$ should be. Based on measurements of the electron g-factor there are estimates that electron radius is at most something like 1e-18 m and possibly much smaller.
This maximum possible radius combined with the charged sphere model imply much greater positive EM mass than observed, which then implies a negative non-electromagnetic mass that is much greater in magnitude than the observed mass. So in this "charged sphere model", two big contributions of opposite sign have to partially cancel each other and what remains is the observed mass. It is important model in EM theory, but its application to elementary particles brings mostly problems, and does not elucidate much. Electron is often thought of as point particle without structure.
The factors in different formulae are different because the formulae are for a different thing. Feynman's electromagnetic mass of a charged sphere is a different thing from the mass of EM energy of the charged sphere.
