On the Sausage Catastrophe in 4 Dimensions

The Sausage Catastrophe of J. Wills (1983) is the observation that in $d=3$ and $d=4$, the densest packing of $n$ spheres in ${\mathbb{R}}^d$ is a sausage for small values of $n$ and jumps to a full-dimensional packing for large $n$ without passing through any intermediate dimensions. Let $n_d^{\ast }$ be the smallest value of $n$ for which the densest packing of $n$ spheres in ${\mathbb{R}}^d$ is full-dimensional and $N_d^{\ast }$ be the smallest value of $N$ for which the densest packing of $N$ spheres in ${\mathbb{R}}^d$ is full-dimensional for all $N\ge N_d^{\ast }$. We extend the work of Gandini and Zucco (1992) to obtain new upper bounds of $n_4^{\ast }\le 338,\text{\hspace{0.17em}⁣}196$ and $N_4^{\ast }\le 516,\text{\hspace{0.17em}⁣}946$. Some lengthy and repetitive components of the proof of the latter result were obtained using interval arithmetic.