I for one didn''t know that this is true, and I doubt there is a proof stating exactly that (it seems a rather unmathematical thing).

The main problem in your post is that there has been well established pattern in the primes, but it does not stick absolutely to it. Once you start looking at a localized area (which you are neatly avoiding by scrolling fast), you always find small errors. Furthermore, having such a pattern does not mean there is ultimately a formula. (E.g. the constant omega has 50% 1s and 50% zeros in binary, but it is uncomputable i.e. the sequence will never be known).

Are you sure about your result. The links given by other posters serve to remind me that the primes get more spaced out the higher, hence, even with the digit shifing occasionally, it shouldn''t smoothly scroll. The time to increase by one should be shorter for the higher numbers (8 or 9).