Electron capture is made more mysterious, not less, in Mills's model. Mills's model does not fit what we already know in this instance, unless we are also to abandon our understanding of how the weak interaction works.
According to Mills logic there is no weak force, at least not in the sense of a primary (force) phenomen. Mills did a huge task and I will not blame him for not covering everything to the deepest possible level. K-Shell captures according Mills should be modeled as an effect of multi pol forces & energy transitions. (39.52 for transition possibilities).
For the alpha-decay he has some beefed up formulas in the 2016 version, showing that his first guess is in line with the expected decay rates. Whether it is better than the standard, has to be answered by the literature.
The dirac fields couples with electromagnets, I think you need to give me some background on why you don't think that a resonant coupling can't ork. What I mean with this is just as with resonant couplings you see nada when there is no resonanse e.g. it is only for special setups of the EM fields that you will get a measurable effect so it is quite possible that we missed it. Allso Mills model is an attractive idea due to it's simplicity, everything is electromagnetic theory and also the weak forces and strong forces are actually a special electromagnetic phenomena.
On nuclear level, we seem to have only resonances = ratios of energies/frequencies. There is a lot of ongoing work (- not using the classical standard assumption, that never had success). Mills theory ends (fails = needs extensions) as soon as we go to the first true nucleus. The adding of a neutron to a proton still can be calculated (=> correct deuterium mass) but for higher compounds the model breaks = less accurate. This is no surprise as also Maxwell needs some strict assumptions, which are no longer given in a soup of protons & neutrons. (E.g. We live once in a light-like field, where some masses move at light speed if we assume they are masses - e.g. the electron of a neutron and in other cases - proton it is - mechanically non-kinetic.) In Mills model you have to carefully separate the light-like frame parts from the others, because for each (nested) frame of reference you have to use different measures. There are nested relativistic frames.. and who knows where they exactly sit.
Regarding the proof of Mills convolution formulas: I did once a different approach based on a more logical reasoning: What we know is, that in a synchronized system waves must interfere in the turning point of the second derivation, if we want a perfect non resonance (No added curvature => no added momentum). This implies that the waves must meet orthogonally, in phase, in all intersection points. The two great circles are per definition orthogonal, different Legendre polynoms are orthogonal too, thus their convolution always holds this criteria. In the case of a locked in photon the third circle alway violates the first criteria. Thus the proof is only challenging for outer non S-orbits.
I would put it differently. It is hard to follow several volumes of word salad, because the individual details are disjoint and do not provide an actual mathematical argument.
Eric: Can you give us a specific example of word salad? I agree, if you are use to standard terminology, then you expect other words.
For an alpha particle scattering off of lead, at ~ 25 MeV the scattering angle starts to depart significantly from the Rutherford prediction as a result of the nuclear interaction. There are probably tens of thousands other such experimental phenomena that will also need to be examined anew if we're to set aside the nuclear and weak interactions and attempt to explain them as being derivative of the electromagnetic force.
Another example: if the electromagnetic force is infinite in range, and the weak interaction is derivative, why does the weak interaction work at only 0.1 percent of the diameter of a proton?
@Why is QUED not able to calculate the proton charge radius? The measured deviation is larger than your 1%...