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 degeneracy of its projective differential data. A global intersection-theoretic computation determines this number exactly, while a local analysis along the degeneracy locus yields a strictly larger lower bound for the same quantity. This contradiction excludes degree six and proves the conjectured bound.