Are you contesting it, or just curious? You already know that my observation explains the "3 followed by 9" bias. You already know that the mathematicians call the conjecture "a much more refined model of randomness in primes" which is similar to how I described what I was doing. In addition, MathWorld's article on the k-Tuple Conjecture talks about residues mod q, which is similar to what I'm doing when I look at primes mod 10*3. All these elements point at some connection between the k-tuple conjecture and my observation.
Well 'points to a connection' is not the same as 'is a special case of', I'm not an expert on this subject, but looking at the conjecture is about the asymptotic distribution of certain patterns in prime numbers. I don't think that an example like the one you are giving is 'related' except in a hand-wavy vague way that anything dealing with prime numbers and patterns is related to everything else dealing with prime numbers and patterns of primes.
All I meant by "special case" was that "mod 30" isn't the whole story -- more like the most significant correction on top of what the OP said, with other smaller corrections possible, and the entire set of corrections being described by the k-tuple conjecture.
It's amazing how people can be picky and negative on HN. Someone positive would instead congratulate me for making the gist of what the prime k-tuple conjecture says about the biases easily understandable. Oh well.
So you are making comments about pure mathematics. If you want to use imprecise language and not be corrected, you should probably go write a book review or something. In math, precise language and correcting someone or forcing someone to give justification for something is expected and completely usual. It would be bizarre when talking to a mathematician about mathematics if they didn't immediately correct or demand clarification and justification when you say something vague or incorrect or unjustified.