All of this gets much easier if you notice that the smaller eigenvalue is less than one. So its contribution becomes exponentially smaller at higher F_i. Hence,

All of this gets much easier if you notice that the smaller eigenvalue is less than one. So its contribution becomes exponentially smaller at higher F_i. Hence, you can just take the corresponding power of phi, multiply by some constant I forgot, and round to the nearest integer. No need for an exact formula.