An Extended Action for Generalized Supergravity NS–NS Equations with Exact Recovery of the Solution Set

We construct an extended variational formulation of the NS–NS sector of the generalized supergravity equations (GSE). Starting from subdivision, Taylor approximation and recon- struction from prescribed partial derivatives, we examine discrete multipliers and continuum compatibility conditions before introducing independent adjoint fields. Successive integra- tion of the six equation groups gives a local continuum action. We prove that the projection of its full stationary set onto the original fields is exactly the solution set of the original equations. The continuum projection result has been verified in Lean.