Below are draft charts of spin addresses for numbers and their powers. The boxes at the bottom (or end) of each column (or row) show where the first ‘double is rolled’. I made the following graphics by hand, but hope to have a computer on the job soon for data runs for hundreds or thousands of numbers in any given range, which should suggest how uniquely pi and its approximations tend toward infinity and eliminate any errors I’ve made.

In the second chart, I simply look at numbers in regular increments without regard to special numbers such as pi or e. At this resolution, there is a peak near pi and a larger one near 2.8. Both may be random, but the larger one is suspiciously close to e. Note, e itself, in the chart above, is unremarkable. I will investigate this range more closely soon.

In a range tight around pi, those values nearest pi itself appear to have higher roll values.

When the values approach pi on the order of 0.000001, the roll values, on average, continue to increase.