The sandbox S/L really targets

 The number of coin tosses.  

Coin tosses, market size, are actually the uncertainty.  In essence the pit boss at the S/L is maintaining the rank of the structured queues to be stable. That number appears in an exponent, and he is actually logbinomial, a discrete version of lognormal. The pit boss is trying to match the probability moments between savings and lendings. He is trying to make them look bell curved, because then the pit boss does the minimal amount of market intervention.  Quantum mechanical, closed color operator, maximizing entropy, finding the best set of relative primes to handle market size. All the same idea.

The currency banker can do this because transaction costs are trivial, like a tort judge relative to damages and liability. The the best pit boss technology are bearer assets in a secure pit. A public art auction, the boss wants all bidders prequalified so the check handed is mostly secure, no need to halt the auction and check the account. Whenever the pit boss needs to intervene that is a lost coin toss, like the velocity of the tax dollar collapsing under all the interventions.

As a pit boss, the central banker has a simple role.  It is government that must manage its own scofflaw queue, the default rate and mechanism. Then that happens outside the pit, the transaction cost od S/L trades drops dramatically, velocity rises, banking marlet in tax dollar dominate shadow banking.  If government manages its own losses then we not hounding the banks with the tax and regulations. I think it is pretty clear what Due Process banking is, and bankers almost certainly agree.

The currency banker wants to be x in the Markov triple, like spin, rarely make a bet. That means the rank of the structured queue is binomial with a very small variance such the the rank is most stable.  The savings and lendings are kept moment matched by the traders themselves.

Define rank as being mostly stable but can drop one level.  Anything beyond that and the memory is erased, it is called a collapse and requant, in the three tuples.

The packed public auction is a good example. You would like the distance from pit boss to buy as binomial.  Interactions are minimized, the crowd is a circle. But then the total number of traders is logbinomial. If k, the number of coin tosses, stable then I think it reverts to Poisson, but don;t quote me, yet.