Wow, I've been working all day on a similar project, and then found your notebook. Super cool. Have you considered a parsimony measurement, that factors in the size of the denominator? If you use d^2 for the complexity cost, then phi, the most irrational number, have local extrema in a horizontal line. You are already doing something like that with your dotted red line. Interesting that you are focusing on the fraction decomposition of irrational numbers. I started my project to look at calendar cycles, especially the Metonic cycle of the moon. It turns out, the lunar phase cycle restarts almost exactly every 19 years. There are about 12.36 lunar cycles per year, which is one reason we have 12 months. Since it isn't exactly 12, lunar calendars have to alternate between having 12 and 13 lunar months per year. It makes sense that if you increase the denominator, like to 19, you could get a better approximation. But I wanted to know if this was a specially good ratio, or if there was an even better one with a larger d. It turns out, if you apply the d^2 complexity cost, then the 19 year Metonic cycle is clearly the best rational approximation. Even if higher ds have more accuracy, they are only marginally better given how much more complex you have to make the fraction. Using parsimony, I found that the earth and Uranus opposition cycle is very close to 84/1. Fraction decomposition seems like a best kept secret for natural ratios. I always get hung up on coincidences that are caused from our base 10 numbers, because I know the choice of base is arbitrary and just a matter of convince. Fraction decomposition is universal.

For φ the fractional approximations are the ratios between subsequent numbers of the Fibonacci sequence, and the error indeed has a constant ratio compared to 1/d²: on a logarithmic graph this is visible as a constant distance under the red dotted line. The way these graphs are created is discussed in the linked blog post on lcamtuf’s thing.