I think the S tool reward rate changes after the first one:

I have 2 accounts...both forged at the same time everyday...the first account forges twice...the second account forges once.

As expected, the first account gets their first S tool after 15 days (30 forges)
also as expected, about 2 weeks later, the 2nd account gets their first S tool (again 30 forges)

however, if the 30-forge thing were a constant, then the 1st account would gotten their 2nd S tool on the same day time that 2nd account got their first S tool

(since the first account would have 60 forges by that time)

however, after a week of waiting (about 15 forges later)..the 1st account still hasn't gotten their 2nd S tool

maybe it's 30 forges for the 1st S tool...and a higher number (or chance) for additional S tools?