Fenyman diagrams are Wythoff game moves

They look like this.The process on the left samples the blue line  as does the process on the right. Two fermion samples per boson in the middle, Nyquist rate.   In this game the players are trying to reach the hottest position by adding chips to their pile. That is a reverse Wythoff goal.

There should be another dual of the Feynman diagram in which two Bosons sample the fermion in the middle.  That would be the normal Wythoff game.  When mass fermions sample boson, the ratio of nulls (lowest state vacuum element, cosh(0)) to bosons is 2 to 1, still  Nyquist.  When the bosons sample the Nulls, ratio is still 2 to 1. In one case the effective sample rate is 2/3 and in the other 3/2.

The hyperbolics do this. cosh(theta)&2 - sinh(theta)^2 - 1. Thus, the one ssmpled element remains. reversing the process becomes cosh(-theta)&2 - sinh(-theta)^2 - 1. And there are some 2^50 games being played in the hydrogen ion.  Each game has some 2^16 moves available at any given time.

The whole key to understanding the adapted Weiner process seems to always be Nyquist/Shannon sample theorem. When ever I go back to that, issues clear up.  The Nyquist rate and Avogadros number are related, the first setting the second.