The gauge equation is a generalization of the conjugacy relation for Koszul connection to 1 bundle morphisms that are not isomorphisms. The existence of non-trivial solution to this equation, 2 especially when duality is imposed upon related connections, gives important information about the 3 geometry of the manifolds considered. In this article, we use the gauge equation to introduce spectral 4 sequences that are further specialized to Hessian structures. 5