No infinite zero-dimensional fields in NIP expansions by closed sets

Let ${\mathscr{R}}$ be an NIP expansion of $(\mathbb{R},<,+)$ by closed subsets of Euclidean space. A definable set has dimension zero when its dimension in the paper's dimension theory is zero. Zero-dimensional field conjecture. The structure ${\mathscr{R}}$ cannot define a field $(Z,\oplus,\otimes)$ such that $Z$ is infinite and zero-dimensional. This is a stronger formulation from which the field-type conjecture follows via the paper's theorem that a definable infinite field of positive dimension yields field-type behavior; the positive-characteristic case is proved separately.