Open Quantum Systems as Regular Holonomic mathcal{D}-Modules: The Mixed Hodge Structure of Spectral Singularities

The geometric description of open quantum systems via the Quantum Geometric Tensor (QGT) traditionally relies on the assumption that the physical states form a differentiable vector bundle over the parameter manifold. This framework becomes ill-posed at spectral singularities, such as Exceptional Points, where the eigen-bundle admits no local trivialization due to dimension reduction. In this work, we resolve this obstruction by demonstrating that the family of Liouvillian superoperators mathcal{L}(k) over a complex parameter manifold X canonically defines a regular holonomic mathcal{D}_X-module mathcal{M}. By identifying the physical coherence order with the Hodge filtration and the decay rate hierarchy with the Kashiwara filtration, we show that the open quantum system underlies a Mixed Hodge Module (MHM) structure in the sense of Saito. This identification allows us to apply the Grothendieck six-functor formalism rigorously to dissipative dynamics. We prove that the divergence corresponds to a non-trivial cohomology class in Ext¹_{mathcal{D}_X}, thereby regularizing the Quantum Geometric Tensor without ad-hoc cutoffs. Specifically, the "singular component" of the Complete QGT arises as the residue of the connection on the Brieskorn lattice associated with the vanishing cycles functor.