  We prove a result for the spin-$1/2$ quantum Heisenberg ferromagnet on
$d$-dimensional boxes $\{1,\dots,L\}^d \subset \mathbb{Z}^d$. For any $n$, if
$L$ is large enough, the Hamiltonian satisfies: among all vectors whose total
spin is at most $(L^d/2)-n$, the minimum energy is attained by a vector whose
total spin is exactly $(L^d/2)-n$.
