The smallest piece of matter possible

I was playing around with my spreadsheet and wondered what the rational approximation was for the best approximation of pi that would create a bubble trap.  1.5 * ln(phi), naturally. But how many bubble of space would be trapped? The basic idea is that the bubbles of space play a Whythoff game and once the most efficient position were reached, at the white and red barrier, then the game was won by the cold positions in the center, and mass formed.

That point, reached by Fibonacci numbers,  F17/F16 (or F61/F15) I am note sure yet, but that is about 1,000 bubbles.  Inside the barrier there are twice as many cold positions as hot, and just outside, twice as many hot as cold.

At the barrier, the divergence of moves is optimum and most moves take place inside the perimeter in circular fashion.  Pi is the curvature, computed, and held in position when Lucas swaps the positions in the center.  In that white and red ring, the system has locked onto the ratio as the tangent, and Lucas angle keep the paring straight, there are two quants, 1 inside the ring and 2 outside.  Feynman plays around in the barrier.

The blue are  hots trying to get cold.  Outside the hots try to get in; forcing cold [position out,   but the extreme accuracy of the barrier make it impossible to penetrate.

I know this little guy as to exist, the Lagrange mathematicions say we cannot make bigger things until we trap all the 'bad' phi approximations.   Any way, I call this the next biggest thing above the vacuum bubbles. These have to be the composites that make a lepton.

I know the point is likely F16 because I have run the numbers so many different ways, and I get phi^16 very close to (3/2)^19.  After splitting up the possible cold bubbles, I get (3/2)^18  for inside and outside, so I know this is close.

This is more accurate than the fine structure, and I can put a bunch of these things together and make a lepton using the 1+2^1/2 ratio. The effect is to flatten out tanh'' so the sweet spot ios not so sweet. But like the mathematicians say, with the bad phi approximations trapped in a well, I should have no problem adding spectral modes up all the way through quarkland.