Lemma 24. Let A be a nonempty multiset of positive integers with total N , and let a=(a₁,…,a_l) be a nonempty prefix of the column-height list of a partition of N . If (N,∑_v=1^ (i,l) a_v+(i-l)₊a_l) ≤P_A(i) (1≤i< A), then every …

Every unknown entry is at most a_l . The left side of Equation is therefore an upper bound for the corresponding prefix sum of any completion. For i≥ A , the right side is already N , so the omitted inequalities hold automatically. If the prefix includes a final zero, it has no nonzero extension and the upper bounds are exact.