I'm pretty solidly convinced of the pattern at this point: I've checked the "reflect across the anti-diagonal" pattern for the first 10 million primes, expresse

I'm pretty solidly convinced of the pattern at this point: I've checked the "reflect across the anti-diagonal" pattern for the first 10 million primes, expressed in every base from 3 to 20, and it seems to hold up. (I haven't tried to establish any sort of bounds.) As I've expressed in another comment, my preferred way of thinking about this symmetric pattern goes something like this: The probability that (a prime congruent to x mod b is followed by a prime congruent to y mod b) seems to be equal to the probability that (a prime congruent to -y mod b is followed by a prime congruent to -x mod b). I still haven't figured out whether it ought to be obvious, though if it is then I expect the language I've used above to be relevant. It's definitely not trivially obvious, because it's not an exact equality: if the pattern only shows up clearly after you've accumulated thousands or millions of primes, then it doesn't seem that it could be enforced by any sort of exact transformation. (For example, if the symmetric entries were somehow just counting the same pairs in two different ways, the numbers ought to be precisely equal rather than just increasingly close.) There are a rather a lot of other patterns in the data; I expect that at least some of them must be accounted for in the original paper, but I haven't more than glanced through it yet.