Moduli of sheaves on fourfolds as derived Lagrangian intersections

We show that any (-2) -shifted symplectic derived scheme X (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a (-1) -shifted symplectic derived scheme which is the (-1) -shifted cotangent stack of a smooth classical scheme. This leads to the possibility of the following viewpoint that is, at least to us, new: any n-shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme. We also give a separate proof of our main result in the case where the local Darboux atlas cdga for X has an even number of generators in degree (-1) ; in this case we strengthen the result by showing that X is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is C⁴.