  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
are known as constellations.
  As the recursion proceeds, adjacent gaps within longer constellations are
added together to produce shorter constellations of the same sum. These
additions or closures correspond to removing composite numbers that are
divisible by the prime for that stage of Eratosthenes sieve. Although we don't
know where in the cycle of gaps a closure will occur, we can enumerate exactly
how many copies of various constellations will survive each stage.
  In this paper, we broaden our study of these systems of constellations of a
fixed sum. By generalizing our methods, we are able to demonstrate that for
every even number $2n$ the gap $g=2n$ occurs infinitely often through the
stages of Eratosthenes sieve. Moreover, we show that asymptotically the ratio
of the number of gaps $g=2n$ to the number of gaps $g=2$ at each stage of
Eratosthenes sieve converges to the estimates made for gaps among primes by
Hardy and Littlewood in Conjecture B of their 1923 paper.
