Sharp degree bound for rational proper maps from $\mathbb B^2$ to $\mathbb B^4$

We prove D'Angelo's degree conjecture for rational proper holomorphic maps from $\mathbb{B}^2$ to $\mathbb{B}^4$, establishing the sharp degree bound of five. Suppose, to the contrary, that a rational proper map of degree six exists. We associate to the map a characteristic number measuring the dege