The article is too vague to assess how interesting the claims are, sadly. By my reading, the article seems to state the key import of the finding quite clearly

The article is too vague to assess how interesting the claims are, sadly. By my reading, the article seems to state the key import of the finding quite clearly: "This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers." However this statement: If you simulate a bunch of random 'next trains', 1 is much more likely than 9 because P(9) approx = !P(1,3,5,7). This is true for all bases. I'm afraid I don't follow at all. (Do you really mean we should expect that P(1|1) > P(1|9), for either random trains or for subsequent primes? Say wha?) That said, perhaps you might want to skip straight to the arxiv article itself, or perhaps do some experiments on your own. It's definitely not hard to generate a non-"minimal" amount of data (out to the first few million primes or so) on one's laptop, these days.