As far as I can tell, they are unrelated beyond the relation all definitions of real numbers would have with it. There is a superficial Left and Right boundary defining an interval that contains the number, but it does not seem like there are a family of intervals for a given real number in this approach.

I do not get the feeling that there is a narrowing down to a given number in the surreal numbers, but I know very little about them.

I also do not immediately see how my construction could be extended to infinitesimals or the infinite numbers.