Have been plugging away some more looking at patterns that develop in the candidate primes and k-tuples with progressive sieving to the n-th prime. Thought I would mention a couple of interesting observations.
The first is simply an observation of behaviour that may or may not extend to infinity. I suspect it does, and must, but can’t prove it:
After sieving to the n-th prime, Pn, the next candidate twin higher than Pn is always an actual twin. So, of the infinitude of remaining candidate twins after n rounds of sieving, the next candidate will not be removed until we reach, in successive rounds of sieving, the lower prime in the twin pair. Or in other words, there will be no lower candidate twin pairs that contain one or more composite numbers.
The other observation, which is I think an easy claim to defend, but I think has important implications in what I am trying to develop, is that in the (n+1)-th round of sieving, the only prime candidate between Pn^2 and P(n+1)^2 that can be removed as a candidate prime, not having been removed by prior rounds of sieving, is P(n+1)^2 itself. Therefore, at most one candidate twin pair can be removed between Pn^2 and P(n+1)^2, and at most two twin prime pairs less than or equal to P(n+1)^2 can be removed (i.e. those affected by P(n+1) and P(n+1)^2).