That was a close one

Yes, the conditions are met to meet the geometry of the sphere and quantize the most irrational number.  Yes, I got lost and equated volume, as an integral, with volume as a count of three spheres, so accommodating all the factors, Higgs and Avogadro match. The optimum finite number is real.

It is hard for me to see how any process in nature which has to pack integer things would not notice the separation of groups, and would not adapt an irrational number somehow.  If they cannot get a three sphere, too bad, they will still use the same principle. But most likely, they will find a three sphere line of symmetry. If not a three sphere, they will try a five configuration, and do the best it can be done. The most efficient number of groupings will always be limited by 107 and 127 in the exponent, that is always (speculation) the second best precision, and due to the density of groups being highest near the multiplicative identity, there will be some group match with better precision, so the middle and the end points of groupings are always defined. This all result from the necessity of having a multiplier in the system to maintain group structure, or remove adverse groups, in other words.

So the count of things will always be Higgs or some number close by using the most irrational number and the Heisenberg equivalent ratio times the packing efficiency.