We develop a mathematically controlled operator-theoretic foundation for the global stability operator of Symplectic Active Geometrodynamics (SAGD). Working on a Sobolev-completed configuration space of metric--affine--symplectic fields, we construct a Hilbert space formulation of the theory and define a Laplace-type realization of the SAGD operator on a dense cylinder domain. Under explicitly stated structural assumptions---including coefficient regularity, coercivity of the effective potential, and relative boundedness of lower-order terms---we establish symmetry, semiboundedness, and essential self-adjointness of the operator using a Kato--Rellich framework. Assuming a Dirac-type factorization with non-vanishing Fredholm index, we prove the existence of non-trivial physical states defined as elements of the kernel. We further show elliptic regularity of such states on all finite-dimensional cylinder projections and demonstrate stability of the physical kernel under sufficiently small bounded perturbations, provided zero remains an isolated spectral point. These results provide a rigorous functional-analytic basis for the SAGD framework and clarify the mathematical conditions under which its operator formulation yields a well-defined and stable physical state space.