When the power series do not converge evenly

If we take the series sum: (1/3)^n, n = 1 to some large N, then we get 1/2, or in the game of wythoff, we get one hot move per cold move, I guess.
And the series sum of (2/3)^n, where n goes to the same N, gets 2, and that is 2 hot moves per cold move.

The first is the fermion the second is the boson. Now that is fine when we can add energy into the system. We get this situation when we do fixed Shannon channels in engineering.  In the vacuum the bubbles do not have that choice, they have to adjust chemical potential, to match both the vacuum at the edge while avoiding the winning position in the center.  So the fermion gains a few hot moves and the boson loses a few.  On net, the cold positions in the fermion get shared, so they move around relative to the hot positions. Physicists call this relativity, I call it making Ito's calculus work.

Hence, in my original spectrum, from sheer dumb luck, I was finding the solution by matching the series sum (Phi)^k to the boson (3/2)^j.  It turns out that is the solution. It is stable because it is the connected solution, always locally additive. The math is hyperbolic at discrete Lucas angles. So the silly metaphor becomes that in the adapted system, adiabasis occurs when the hot players and the cold players choose not to win the game, but keep on playing. What happens if one of the fermion bubbles wins, it lands in the center, the abyss. I dunno, ask Professor Higgs. But is seems to me we get empty space if that happens, but that is impossible, there is no such thing.