> the critical score... separates the world of infinite rationals with merely a finite number of rationals
I'm not sure that I understand what's being said here. There are countably (i.e. infinitely) many rationals, so is this saying that there is some particular finite set of rationals that are particularly relevant to the critical score?
Consider π. For any S>0, you can construct an infinite number of rational approximations that have a score of less than S.
But for any quadratic irrational (surd), as the depth of the corresponding continued fraction increases, the score will converge (in an alternating manner) to a critical score, S.
This means that for any score S < S, there is only a finite set of rational approximations that have a score of less than S.
For example, in figure 3, for S=0.4 < 1/√5 ≃ 0.447, there is only one fraction that gives a score of less than S=0.4.
Hope that helps!