Hodge Theory for Linearized Boundary-Value Problems on General Geometric Structures

We develop a framework that systematically casts the solvability and uniqueness conditions of linearized geometric boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the underlying geometric structure, and provides tools to study the emergent cohomology explicitly. To achieve this generality, we generalize Hodge theory to sequences of Douglas--Nirenberg systems that interact through Green's formulae, overdetermined ellipticity, and a condition we call the order-reduction property, which relieves the rigid requirement that the sequence forms a cochain complex. This property typically arises from linearized geometric symmetries and constraints, as we demonstrate through several geometric examples that have long resisted analysis, including exterior covariant derivatives, the Killing and Hessian equations, and the linearized Riemann and Ricci curvature equations.