  We consider the Glauber dynamics of a ferromagnetic Ising-Kac model on a
three-dimensional periodic lattice of size $(2N + 1)3$, in which the flipping
rate of each spin depends on an average field in a large neighborhood of radius
$\frac1\gamma << N$. We study the random fluctuations of a suitably rescaled
coarse-grained spin field as $N \to \infty$ and $\gamma \to 0$; we show that
near the mean-field value of the critical temperature, the process converges in
distribution to the solution of the dynamical $\Phi^4_3$ model on a torus. Our
result settles a conjectured from Giacomin, Lebowitz and Presutti
(DOI:10.1090/SURV/064/03). The dynamical $\Phi^4_3$ model is given by a
non-linear stochastic partial differential equation (SPDE) which is driven by
an additive space-time white noise and which requires renormalisation of the
non-linearity. A rigorous notion of solution for this SPDE and its
renormalisation is provided by the framework of regularity structures
arXiv:1303.5113. As in the two-dimensional case arXiv:1410.1179, the
renormalisation corresponds to a small shift of the inverse temperature of the
discrete system away from its mean-field value.
