But this bound is almost always too strong. In fact, just considering the n-sticks alone you find there is no possible dissection, except ...

To further clarify, the bound thus obtained is only *just* too strong - you get a pair of constraints Q<=p and Q>p. So the next boundary-point down is the one you want. I wrongly hoped that this would be the other element in the min(), but as the s(5,7) case shows this isn't remotely true.