So we can count iinteger spheres in the proton

We got the sphere packing thing going.
The proton counts integers 3,2,1.  So its got six unit spheres going on. Lets count, 3 quarks, an electron and the quark bag. It gets a free pass on the 1 digit.

So I can guess the electron two root system, which makes the bag a conjugate root of the electron, in radius? The proton polynomial and electron polynomial separated (except for bounding error squared). That puts the entire third order polynomial inside the bag (except fractional error), with bounding error cubed.

The exponent quant have modifiers, the value log(2) for threes and log(2) for the twos, which is a finite p/q.