  We give a complete classification of the Chern characters of constructive
exceptional vector bundles on $\mathbb{P}^3$ analogous to the work of Dr\'ezet
and Le Potier on $\mathbb{P}^2$, and using this classification prove that a
constructive exceptional bundle $E$ on $\mathbb{P}^3$ with $\mu(E) \geq 0$ is
globally generated.
