It's not surprising at all that you'd need modular arithmetic for natural scales. Modular arithmetic is still very positional vs our math now is in the same order but forgoes positional elements.

"like the inversion of the circle that all numerical relationships that lie between one and infinity also exist between one and zero..."

This is really interesting and it is interesting that you noticed a difference between how a circle is bounded internally and externally. By chance do you have any links/resources you can share/recommend that show this?

I think I may know what you are referring to but I haven't found a good example of this problem. Thanks!