Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s$ close to zero

We prove that stable nonlocal minimal cones in three dimensions are flat, for $s$ sufficiently close to zero. Our approach develops new structural properties of stable $s$-minimal surfaces in every dimension as $s \downarrow 0$, notably a compactness theory and pinching theorems near a half-space.