@user1952009 These patterns are not random at all. These are the result of edge detecting filters and other features useful for image recognition.

I had trouble understanding if and why floor(0.999...) = 10.

Your link is dead, but there is another tool available for doing that, which is better: mrob.com/pub/ries

@DouglasZare see last comment.

@Wojowu Sorry, I have no education in number theory. I'm reading these papers because of my interest in AI and automated conjecture making. I edited my post to include more examples from the paper, which are perhaps more interesting.

If you can prove that it's unprovable, that proves it is true (because if a counterexample exists, that would make it provable, therefore no counterexample can exist.) Which is a contradiction. So it might be unprovable, and impossible to prove it is unprovable. --- Also interesting idea that the distribution of primes could be Turing complete. That would complicate things a lot, and is not at all impossible. Lots of simple systems like cellular automata are Turing complete.