The Golomb–Welch conjecture for arbitrary centre sets

We prove that for all $n\ge3$ and $r\ge2$, the Lee ball $B_n(r)$ does not tile $\Z^n$ by translations, resolving the Golomb–Welch conjecture for arbitrary centre sets and completing the existence classification of perfect Lee codes in $\Z^n$. The proof combines translation-invariant averaging, boundary contact identities, and integer moment inequalities with coefficient comparisons in stable symmetric functions. Consequently, we obtain finite local obstructions, strict Banach density gaps for packings and coverings, and the nonexistence of exact measurable Lee-ball towers. In the lattice specialization, our result classifies the attainability of the Lee bound for even-degree abelian Cayley graphs, excludes normal topological coverings and Galois graph coverings for Lee-ball completions, and gives a strict node bound for lattice cubature as well as the nonexistence of subgroup-indexed orthogonal exponential bases on $B_n(r)$.