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

Lemma 23. The first-four balance test and the band tests cover every target when 9≤M≤21 , r>0 , except possibly for (M,r)∈(10,8),…,(10,17) ∪(12,14),…,(12,20) ∪(14,22),(14,23).

Lemma 22. If M≥22 , a target satisfying Equation passes the band test in one of its two orientations.

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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.

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