Intrinsic Reconstruction of the Berry Connection from the Spectral Adjoint

The Berry connection of a smoothly varying family of Hermitian operators is conventionally defined through the differentiation of instantaneous eigenvectors, a construction that introduces local gauge choices and becomes singular at energy degeneracies. In this paper we show that the Berry connection admits an intrinsic, gauge-independent reconstruction through the Moore-Penrose pseudoinverse of the adjoint action. By defining the connection directly on the operator level via the spectral adjoint, we eliminate the need to track eigenbases or construct local coordinates. This provides a unified operator formulation that applies equally to non-degenerate and degenerate spectral subspaces.