functions where the microscopic fluctuation fields, namely the density and the height, will be defined and we give the proper definition of these fields. Section \[s3\] contains all the rigorous definitions of solutions to the SPDEs that we obtain, namely the OU/SBE with Dirichlet boundary conditions, the KPZ equation with Neumann boundary conditions and the SBE with linear Robin boundary conditions, and we explain how these equations are linked. In Section \[s4\] we prove the convergence of the microscopic fields to the solutions of the respective SPDEs, namely Theorems \[theo:convergence\] and \[theo:convheight\]. In Section \[s5\] we prove the second order Boltzmann-Gibbs principle, which is the main technical result that we need at the microscopic level in order to recognize the macroscopic limit of the density fluctuation field as an energy solution to the SBE. Finally, in Section \[sec:energy-pr\] we give the proof of the uniqueness of solutions to the aforementioned SPDEs. The appendices contain some important aside results that are needed along the paper, but to facilitate the reading flow we removed them from the main body of the text. In particular, Appendix \[app:micro-CH\] sketches how one could prove that the microscopic Cole-Hopf transformation of the microscopic density fluctuation field 