On-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries

We study a generalization of the Holst action where we admit nonmetricity and torsion in manifolds with timelike boundaries (both metric tetrad formalism). prove that its space solutions is equal to one Palatini action. Therefore, conclude sector fact identical general relativity (GR), which defined by Einstein-Hilbert further that, despite defining same solutions, (the generalized) Lagrangians are not cohomologically equal. Thus, presymplectic structure charges provided covariant phase method might differ. However, using relative bicomplex framework, show spaces both theories equivalent (and GR), as well their charges, clarifying some open problems regarding dual equivalence different formulations.