The approach is correct, my skills are lacking

The proton is not solving cubic roots. It is arranging a counting system for maximum entropy and that means optimum separation so rational remainders  are bounded.  And maximal group separation almost certainly means prime roots. It can really only manipulate two variables, quant separation and cycle length, all integer. Dimensionality is fixed to the prime number, and no primes above 17, they are most likely 1,2,3 and maybe 5. The electron likely does not have a root system three, and its unit sphere odd shaped.  Beyond that, the theory is incomplete and above my head at the moment.