The structure theorem for sets of length for numerical semigroups

For sufficiently nice families of semigroups and monoids, the structure theorem for sets of length states that the length set of any sufficiently large element is an arithmetic sequence with some values omitted near the ends. In this paper, we prove a specialized version of the structure theorem that holds for any numerical semigroup $S$. Our description utilizes two other numerical semigroups $S_{\mathsf M}$ and $S_{\mathsf m}$, derived from the generators of $S$: for sufficiently large $n \in S$, the Ap\'ery sets of $S_{\mathsf M}$ and $S_{\mathsf m}$ specify precisely which lengths appear in the length set of $n$, and their gaps specify which lengths are "missing". We also provide an explicit bound on which elements satisfy the structure theorem.