Weak universality of $\Phi^4_3$: polynomial potential and general smoothing mechanism

We consider a class of stochastic reaction-diffusion equations on the three dimensional torus. The non-linearities are odd polynomials in the weakly non-linear regime, and the smoothing mechanisms are very general higher order perturbations of the Laplacian. The randomness is the space-time white noise without regularisation. We show that these processes converge to the dynamical $\Phi^4_3 (\lambda)$ model, where the coupling constant $\lambda$ has an explicit expression involving non-trivial interactions between all details of the smoothing mechanism and the non-linearity.