  We prove that every bounded stable solution of \[ (-\Delta)^{1/2} u + f(u) =0
\qquad \mbox{in }\mathbb R^3\] is a 1D profile, i.e., $u(x)= \phi(e\cdot x)$
for some $e\in \mathbb S^2$, where $\phi:\mathbb R\to \mathbb R$ is a
nondecreasing bounded stable solution in dimension one. This proves the De
Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we
give a positive answer to the De Giorgi conjecture for boundary reactions in
$\mathbb R^{d+1}_+=\mathbb R^{d+1}\cap \{x_{d+1}\geq 0\}$ when $d = 4$, by
proving that all critical points of $$ \int_{\{x_{d+1\geq 0}\}} \frac12 |\nabla
U|^2 \,dx\, dx_{d+1} + \int_{\{x_{d+1}=0\}} \frac 1 4 (1-U^2)^2 \,dx $$ that
are monotone in $\mathbb R^d$ (that is, up to a rotation, $\partial_{x_d} U>0$)
are one dimensional.
  Our result is analogue to the fact that stable embedded minimal surfaces in
$\mathbb R^3$ are planes. Note that the corresponding result about stable
solutions to the classical Allen-Cahn equation (namely, when the half-Laplacian
is replaced by the classical Laplacian) is still open.
