The hyperbolic yield curve

I sort of done it on the fly.  Let's look where I was headed:

The outcome I wanted was: 1-tanh(n*a)^2 = balance ratio.

The balance ratio was principal amount deposits over loans. The n was the step size and a, if we use Lucas numbers, is ln(Phi), I think, without looking things up.
The sinh and cosh were rates plus unit balances, for one period. Tanh then was the ratio of deposit flows to loan flows.  1-tanh^2 is the first differential of the flow ratio.

So the idea was that as n stepped up, the flows go to one and the principal balances go to zero. The balance ratio is the ratio that results when a careful economists goes from the weiner equation to the equation above:

For p(x,t) = t*tanh(x), where t was the balance ratio.



Where I left off was wondering what Lucas polynomials divide to make p(x,t), and I have barely looked.  The Lucas numbers generate sinh and cosh alternatively with each step of a.

Why do we care? Because if this equation is true then we know the limits of an optimum monetary zone, that is the balance ratio when it is as close to zero as we can get, which is the fine structure constant in physics, naturally, because nature does sphere packing as an estimation method.

Do economists care if tanh is polynomial? Not at first, they should figure the rest out themselves, I am retiring from estimation theory. I am too lazy or need a vacation, one or the other.