Uncertainty, bandwidth and two period planning?

Everyone assumes we plan two periods ahead.  The reason is sample theory, optimally if the flow is band limited in variation, then the planner wants to sample at twice the bandwidth.  So the planner makes purchases at twice the rate in which he projects flow. This is standard in economics and signal processing. hence the hyperbolic condition cosh^2 - sinh^2 = 1, the result is a standardized constant unit flow.

And when there are three independent flows?

A good question which I have often pondered and never quite resolved. The quarks do it, I think. But that is where the Lucas polynomials come into to play.  Sample theory assume the bandwidth is known and fixed, information theory assumes bandwidth is symmetric, and redundancy theory says the system organizes to make bandwidth symmetric.  Those cosh and sinh functions are really Lucas polynomials in X, and n and make Lucas numbers only when X = 1.  Note that there must be two set of Lucas polynomials, the normal and the dual. I am not sure the mathematicians have written down the dual version.

But  n is an exponent, I have a hard time seeing that as a polynomial in time. Anyway, under minimum redundancy, we should meet the Brownian conditions. I am still pondering the issue, but I am sure the pros have already worked this. This is one of those issues which have always been above my head. 

My theory on this, unproven.
Brownian motion only works because the ensemble is locally connected and internally consistent.  To me that means the ensemble is a collection from the Wythoff array, the the transcendentals are finite approximations.  But, if this is true then I trust those Schramm-Loewner folks to figure it out.