On stable solutions to the Allen-Cahn equation with bounded energy density in $\mathbb{R}^4$

We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such as robust curvature estimates for stable phase transitions, and the multiplicity one and Morse index conjectures of Marques-Neves for Allen-Cahn approximations of minimal hypersurfaces in closed 4-manifolds.