Oka complements in algebraic tori and their logarithmic covers

Suppose $n\ge 2$ and $K\subset {\text{C}}^n$ is compact and polynomially convex. Write $q(z)=(e^{-i{z}_1},\dots ,e^{-i{z}_n})$, $X_K=({\text{C}}^{\ast }{)}^n\setminus (K\cap ({\text{C}}^{\ast }{)}^n)$, and $Y_K={\text{C}}^n\setminus q^{-1}(K)$. We prove that $X_K$ and its logarithmic covering $Y_K$ are Oka. More strongly, $Y_K$ admits normalized holomorphic families of entire embeddings through prescribed holomorphic sections. The proof combines polynomial pair Hamiltonian transport in multiplicative coordinates, compatible with the coordinate hyperplanes, with a fixed Fatou–Bieberbach model. Applications include Oka slab crosses over arbitrary nonempty connected planar domains, complements of periodic equal-radius Euclidean balls centered at products of planar lattices, and complements of sufficiently small closed metric balls in products of elliptic curves.