Error bounding polynomials

If I start with 3/2 as an estimat of LogF(r), then my error is bound by 3/2.  If I have another degree of freedom then I can remover more error.

My equatin is r^2 - my previous error bounds = 1. The previous error bounds, in the new degree of freedom is (3/2)^2. So adding one more degree of freedom makes my new r a sqr funfion of 3/2 or 9/4. Compute the root and get the sqrt(2). Adding new root in jumps of one make the Lagrange set, I thi8nk. But we can make then in jumps of any amount. We always have to keep the hyperbolic condition: (T(q)^n -1) binds the available error spectrum.
So my 5-body equation has three digit types, 5,3,2.  Hence my quants count modulo 30. And I find my roots such that
LogF(5) / r5 = LogF(3)/r3 = LogF(2)/ r2; more or less, for each root.