Weird. We (normal people that is) accept that interactions with fields can add mass-energy to particles.
I guess you missed some physics bachelor class. Mass is only added as a consequence of work. If you could understand that a field never can add mass without doing so (work) would be great.
Unluckily interactions with a conservative field are symmetric you can temporarily add mass e.g. on an eccentric orbit but this mass is released again.
The only thing you can claim is that a particle with e.g. the fringe Higgs field does a mass like interaction = it feels a force that in average must be "0", not quite what they want to have....
Finally you end up where I said: Higgs replaces gravitation because SM is unable to explain mass...
I've no idea even what this means! Mechanical mass is made up of 80% energy 20% rest mass of particles.
I hope you passed bachelor exams without this question. Up to date mechanics postulates that inertial mass and gravitational mass are 100% identical. For particles Einstein found the correct extension to this rule...
The correct definition of rest-mass is as the term says: Mass of a particle at rest, what is not always quite exact as e.g. an electron is never 100% at rest. (See De Broglie formula.)
As i see it, the only successes you claim are all numerical alchemy - magic formulae from which fundamental numbers can be derived. There is a long and well known history of this stuff - possible because semi-classical approximations are sometimes exact, and sometimes close, and hand-waving "corrections" easy to generate.
I agree it (SO(4) physics) looks like number alchemy to somebody with a SM troubled mind. The only question that remains is: Why does it fit for everything. Why can it give the gamma energy of e.g. 6Li, the levels of 4He and many others? Why can it calculate the magnetic moments from charge radii? Why does the gravitation constant pop up after it has been predicted (based on the 4-He mass measurement ) to interact with the electron perturbative mass? Why do all the isotopes (gamma spectra) follow the SO(4) quantum structure ?? Why can we e.g. explain= calculate the exact(at measurement) mass of 4He, 2H & neutron from proton and electron with the same basic structure?
I think from anwers thus far your main issue with physics is that you don't like algebra, are suspicious of symmetry-based explanations of physics, particularly don't like broken symmetry as explanation for physics:
I repeat it once more: Energy in nature is magnetic flux = photons. SM is potential based =>> only a subset of nature. The brimborium of SM/QED/QFR/QCD has been developed by mathematicians (like Dirac, Hilbert) that have no basic clue of physics or reality.
Higgs's idea is the o called Münchhausen cheat of physics define something missing by the missing...
The symmetry SM claims to see is not suited to represent magnetic coupling as this needs at least SO(4). How creepy must these people be to finally not understand that there is no symmetry breaking, when there in fact is not the correct symmetry used!!
I like algebra, but even more I like logic something that is completely missing when you analyze an SM paper and know the history.
If you don't understand that magnetic coupling is missing in SM and cannot be handele by the SM group structure then any future discussion is fruitless.
May be you believe, as others do, that the usage of a 4 potential is enough to satisfy magnetism. But this is utmost wrong as magnetism/magnetic coupling is not following the guv metric due to the fact that the magnetic mass already is at light speed and can no longer follow a time like manipulation. May be this is what SM believes to be symmetry breaking.... ( Do not tell me that you can - time like - convert a magnetic mass in a current/charge..I'm talking of a static force, magnetic interaction at a well defined radius...)