  A few years ago we identified a recursion that works directly with the gaps
among the generators in each stage of Eratosthenes sieve. This recursion
provides explicit enumerations of sequences of gaps among the generators, which
sequences are known as constellations.
  Over the last year we identified a discrete linear system that exactly models
the population of any gap across all stages of the sieve. In August 2014 we
summarized our results from analyzing this discrete model on populations of
single gaps. This paper extends the discrete system to model the populations of
constellations of gaps.
  The most remarkable result is a strong Polignac result on arithmetic
progressions. We had previously established that the equivalent of Polignac's
conjecture holds for Eratosthenes sieve -- that every even number arises as a
gap in the sieve, and its population converges toward the ratio implied by
Hardy and Littlewood's Conjecture B. Extending that work to constellations, we
here establish that for any even gap $g$, if $p$ is the maximum prime such that
$p\# \; | g$ and $P$ is the next prime larger than $p$, then for every $2 \le
j_1 < P-1$, the constellation $g,g,\ldots,g$ of length $j_1$ arises in
Eratosthenes sieve. This constellation corresponds to an arithmetic progression
of $j_1+1$ consecutive candidate primes.
