I hope I have written this question in an answerable form. Basically, I am assuming this theorem or conjecture exists, but I can't find it from my reading / searching (I'm not a mathematician). What I know:
A "balanced prime" is the middle prime number in a sequence of three consecutive primes in arithmetic progression. This would make the number "CPAP-3", and thus the minimum separation is 6 = 3# = 3 * 2.
What I've noticed from just analyzing balanced primes (see my post on GooglePlus: I am too new of a user to post an image) is that not only is their minimum separation 6 = 3# = 3 * 2 (see the lowest magenta "line" in picture). But greater separations must be a multiple of 6 (by observation)
So, my question is: is there a named theorem or conjecture that describes the isolation of balanced primes must be a multiple of 3#? I'm not a mathematician, but I'm guessing this exists and I just can't find it.
I'm also very interested to know if there's a theorem or conjecture about how the maximum isolation of a balanced prime grows. It appears to grow faster than log(Pn).
PS: I labeled the graph with terms like "prime loneliness" before I was pointed to better terminology. "swiss primes" are actually "balanced primes."
EDIT (again): From comments: this is a proven (and easily explained) theorem (much more elementary than the Green-Tao theorem), which covers the more general case of primes in arithmetic progression (AP-K; not necessarily consecutive primes). Thank you to all for pointing me to the obvious! :)