The type and cardinality of minimal presentations of numerical semigroups with embedding dimension four

For a numerical semigroup $S$ with embedding dimension four, we study the relationship between its type $t(S)$ and the cardinality of its minimal presentations $η(S)$. Using an approach based on the geometry of the Apéry set, we prove that $4t(S) + 5 \geq η(S) \geq t(S)-11$. This resolves a problem of Moscariello and Sammartano asking whether $t(S)$ is bounded by a function of $η(S)$, and improves the previously known bound $9t(S) + 4 \geq η(S)$ due to Bresinsky.