Thanks for going through this note. The meticulous checking by Horst and others has resulted in complete acceptance of ECE theory, and rejection of the standard model. In this case the integral is log sub e (mc / h bar) – log sub e (pi / a0) = loge sub e (mc a0 / h bar pi). The Bohr radius is a sub 0 = 4 pi eps0 h bar squared / (m e squared), so the integral is log sub e (4 eps0 h bar c / e squared) = loge sub e (1 / (alpha pi), where alpha is the fine structure constant. I checked this many times for past UFT papers, and it is based on a treatment of the Lamb shift given by googling "Lamb shift". Wikipedia is usually full of errors, which is why I checked the calculation. It was used for illustration only, in order to find the fluctuating m(r). The idea of using a charge density in multi electron states is a very good one, and leads into using computational quantum chemistry for the Lamb shift.
However there is only one log term in eq. (12), and this term contains constants that are not present in the limits of the integral. How can this be?
In eq. (13) the modulus of a single wave function psi(0) appears. This relates to a single state. In multi-electron systems we have for the charge density at r=0:
rho(0) = sum | psi_i (0) |^2
for all electrons i. Only s states contribute to psi_i(0). I guess that it would be allowed to use the total charge density rho(0) in eqs.(14/15). This would give an expression that is suited for quantumchemical calculations. The Lamb shift must be based on the total <Delta U>. It has to be added to the total electron potential. Thus it can be used in the self-consistency cycle.