Packing spheres of multiple sizes

Its a hot topic, and the three sphere problem is still being worked.  I was thinking about it regarding the spectra of two spheres exchanging position, a relatively important topic. Too much for my tiny brain.

Let call the volume of a large sphere r^3 and the area r^2. Then log base r gives us the information content of the volume and area as 3 and 2, the volume always carrying 1.5 times the information as the area. The Shannon condition requires P(x)Log(px) be less than one for each.


We also know that Phi = 1/2+sqrt(5)/2 will measure the bit ratio from the Phi^75, with 9e-3 accuracy to Phi^107 at the same accuracy, with a symmetrical peak at Phi^91, having a 9e-5 accuracy.

So,  the 1+S/R in this equation is optimally equal to P/Q, a rational fraction equal to Phi^91, and delivering Gaussian error, for the packed proton, with about 32 digits of accuracy. Set the one digit to the proton sphere. So, the highest 16 bits likely measures area best at the shell, and the lowest bit measures volume best in the center. The proton peak is where area and volume are equally measured. The shell is likely where p(x)log(p(x) for area is close to one, and visa versa for the center.

Is all this enough info to discover how many vacuum bubbles and their radius? I would think so. The variations in pLog(p) , from the proton shell out, or in, tell us how the combinatorics change. We should see two or three or four patterns changes as we move in and out of the sphere.