You are not serious, you are asking DaviD to prove something with math? Did you look at his feedback?
I bet that you don't understand about spins half, that understand I. And to prove it is very hard, becouse magnetism is somewhow related with mass in units, very stupid, this is not my foult, that magnetism is mikxed with mass!!

edit: here is atom model of spins and charges. Electrons have quantized orbits, becouse charge distance working... and electrons orbits like you can see from schrodinger equation not so quantized

So all electron quantizeation and orbits is possible to explain with charges and spins interactinos.
BTW, why then electrons don't fuse with protons (nuclears)? Who say that they don't fuse? WE don't know they size and they structure, waves and so on so electrons maybe fuse... Over answer can be that all spin froce between nuclears and electrons taking all mass of electron and force don't enough of spn to change somthing of electron and when electron flying around nuclear then nuclear spin is over after 180^o and electron is pushed instead atracted. And for lussers: electron don't radiating magnetic waves, becouse electron flying in each orbit with constant speed! There everyhting is so unclear, that I wonder how schrodinger equation can somthing explain and somthing to determine... And since schrodinger equation explaining so don't much, it can be cosidence...
Why for example neutron don't atracting electron? But who say, that it don't atracting? Spin must atract! Thus electron mush fusse with neutron and to produce somthing, but since imposible to izolate them both then by thermal noise and spins of over particles electron won't fuse with neutron. And neutron is product of proton and neutron fusion... Spins and charges. Charge force atraction energy is production of antineutrino. And neutrons can decay, becouse thermal noise of over particles spins and charges disbalancing spin balnace between electron and proton spins and thus spins becoming acidently oposit and thus neutron decay into proton and electron and charge force products antineutrino. As we know proton don't decay if there no positrons and probably over thricks maybe from cosmic rays...