That is a great paper. I came up with my own definition of real numbers [1] and using the simplified cut property of that paper was a pleasant pathway to establishing completeness.
[1]: https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
[1]: https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
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.