My favorite sequence

Sum of (2/3)^n and sum of (1/3)^N

These power series sums make 2 and 1/2, so we have:

cosh^2-sinh^2 = 3/2  Spare capacity

I call this the optimum queuing model, it keep two queues. One queue has 1 or 2 in line at a time, the other 0 or 1. Phi^16 = (2/3)^19. Npw Phi is just a numberical tool, in this case it stands for the ratio of two Fibonacci numbers.

Anyway, to my thinking, 2/3 and 1/3 are the Boson and Fermion exchange rates whne there is no motion.  These are the conversion ratios when the bubbles are separated into the cold and hot positions of Wythoff.  For Fermions, the cold spots out number hot spots 2 to 1. Two times as many bubbles have no desire to go overlapping.

Now the 13th Fibonacci number is 377, about where the spare capacity is near the fine structure.  That should not surprise us, the Hyperbolics as simply a useful too to manage power series.  And Phi is just an estimation tool. So any reduced physics model will have lots of matches to some power series tool. But the adapted bandwidth requires sinh = cosh', and visa verso so it supports the impedance model. The tool is built around bandwidth adapted systems, two period constraint.

How does the system make (2/3)^n match (1/3)^n?  The 1/3 and 2/3 series shift, that makes charge, and they  become an additive series.