Rational approximation

Let us consider this equation.

(r+e)^2 - (r-e)^2 = 1

Solving this we find that r = 1/e, this is the hyperbolic system.  It is really a condition on indexing, the area between the two concentric circles must be on, we are indexing circular rings.

The solutions r with asymmetric error, 1/r are also satisfied with the rational fibonacci ratios. And the sequence of fibonacci ratios makes the 'estimate' of r become increasingly irrational until r become, in the limit, Phi, the most irrational number of all.

What does this have to do with banking?

The banker is receiving two sequences, overs and unders; moments when agents realize they have enough, or not enough and cause liquidity events, mainly dumping digits back to the banker of taking them from the banker.  I think the banker needs only do the subtraction, two independent sequences are squared products because the agents have been packing sphere.

The banker wants his window of  deposits to be about 1.5 times his window of loans. Window meaning number of transactions before the expected repeat sequence of some rational approximation.  In  other words, the pit boss is using index arithmetic, just divying up the spots in the generator that balances loans and deposits.  Once the banker has the proper window sizes, the banker can enter market maker, small loans or deposits, to fill in space, equivalent to making their generators isomorphic. Then, the interest charges should force the closest match that generates irrational bit error sequence.