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This article constructs the moduli stack of torsion-free 
$G$-jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any 
$\infty$-topos equipped with any stable factorization systems.
In the intended applications of this theory, the factorization systems are given by the deRham-Stack construction. Homotopy type theory allows a formulation of this abstract theory with surprisingly low complexity. This is witnessed by the accompanying formalization of large parts of this work.
