This slide is going to eat my life.
Here is tabulated the increase in ln(M) upon increasing denominator or numerator of corresponding (unique) Farey representation.  The asymptotes for increasing just denominator and for increasing both numerator and denominator together, for the Fibonacci Sequence and for the Pell sequence, respectively, are quickly calculated at the bottom.  If both asymptotes are approached quickly enough “en route” to each distinct target ln (M), there emerges no way to get same sum going to different target sites, due to relative irrationality of asymptotes.  This fast approach (up) to applying asymptote when increasing numerator by j and denominator by k+j might be demonstrable with the Chebyshev representations:zn = z0Un(3ym/3) - xlUn-1(3ym/3), precisely, on each branch, where ym is the host and z0 is the base of the branch, and xl is the Markov number “mirrored up” to z0 via z0 = 3ymym-1 - xl #MarkovNumbers #Unicity