Phi as a natural sampling ratio in self sampled systems

We ave introduced the pit boss, defined as having access to the total bandwidth of the pit in the sense of being able to intervene after each bet by an independent trader with round roin access.

In sand box the two boundary conditions are simple: market share goes to zero and the system is gone, or; one or both queues tend to get ever larger and the system jams.  The pit boss always seeks to maximize its liquidity, but it splits the finite channel between a stable deposit queue and a stabler loan queue.

We have some conditions at the optimum, the channel partition is stable, and the bandwidth of the pit boss is equal to the sum of the other bandwidths. The channel partition is measured in quants, one quant being equal to the pit boss uncertainty. So each all bits at maximum entropy are equal constant power spectra, and the bits can be divided. At stability, that division leaves the round off error which is maintained by the pit boss and has one unit of power spectra, an extra bit.

The pit boss first looks at the deposit queue, without knowledge of the loan queue. It wants to put the deposit queue in cannonical form, remove queue in the structured queue so that all paths are equal length. He is, in essence loaning money for the depositors, and wants to do so under wht conditions? His error in doing will be higher than allowed by contract, but unobservable. He still is on the board and will be resructuring the loan queue.  By what principle can the pit boss act as lender, temporarily, to the depositors?  It must take the shortest path toward the optimum ration.  The pit boss is working a temporarily expanded bandwidth model to center the deposits aginst  a hypothetical loan queue.

The pit boss restructures queues then issues the interest swap that minimizes market risk.  But the pit boss always works from the same ratio, assumed to be optimum. One ends up with the equation for Phi, there is a Golden packing ratio that leaves the pit boss always finding the no arbitrage point in a two color sampler, Phi is a sampling constant, like Nyquist. Call it the Nash sampling constant, it appears when all parties in a two color market agree to a bounded error.