Branch points of harmonic quasiregular mappings in three dimensions

We prove that the branch set of a nonconstant sense-preserving harmonic quasiregular mapping $f:\Omega\subset\R^3\to\R^3$ is empty. Equivalently, such a mapping has $J_f>0$ everywhere and is locally a real-analytic diffeomorphism. This gives a three-dimensional bounded-distortion form of Lewy's theorem: although Lewy's planar nonvanishing theorem fails for arbitrary harmonic homeomorphisms in higher dimensions, quasiregular distortion rules out the corresponding critical phenomenon in three-space. The proof is based on a homogeneous blow-up argument. A hypothetical zero of $J_f$ produces a nonconstant homogeneous harmonic polynomial quasiregular map $P:\R^3\to\R^3$ of degree $m>1$. We exclude such maps by a second-order trace identity for the spherical Jacobian $J_P|_{S^2}$: after normalizing the first jet at a positive minimum, the identity gives a negative spherical trace, contradicting the maximum principle. We also prove a companion topological obstruction, namely that no homogeneous harmonic polynomial map $\R^3\to\R^3$ of degree $m>1$ is one-to-one, and derive an affine Liouville theorem for entire harmonic quasiregular mappings in $\R^3$. The strictness of the homogeneous obstruction is sharp: we construct an explicit harmonic cubic with nonnegative Jacobian whose zero set on $S^2$ is the vertex set of a regular icosahedron. Finally, we classify the $O(n-1)$-equivariant harmonic cubic models in higher dimensions; this gives borderline examples in every even dimension and proves that no strict positive-Jacobian example exists in that natural equivariant class.