The Gram Structure of the Quantum Geometric Tensor from the Spectral Adjoint

In a companion paper we showed that the Berry connection of a smoothly varying family of Hermitian operators can be reconstructed intrinsically from the Moore-Penrose pseudoinverse of the adjoint action. In the present work we show that the same operator also determines the quantum geometric tensor. For a smooth non-degenerate family H(lambda), we define the spectrally normalised deformation operators X_mu as the pseudoinverse of the adjoint action applied to the parameter derivatives of H. We prove that the band quantum geometric tensor admits an intrinsic operator representation given by the inner product of these spectrally normalised Hamiltonian deformations. This demonstrates that the quantum geometric tensor is precisely the Gram matrix of these operator deformations. Consequently, positivity of the quantum geometric tensor follows immediately from Hilbert-space geometry rather than perturbation theory.