Microlocal classification of invariant order-two metric Hessians, with spectral applications

We prove a microlocal classification theorem for principal symbols of classical pseudodifferential operators of order two acting on symmetric two-tensor fields and arising as Hessians of diffeomorphism-invariant metric functionals on Riemannian manifolds. Assuming orthogonal equivariance of the principal symbol, we show that the leading symbol is necessarily of universal transverse–traceless plus trace type: it annihilates the longitudinal gauge sector and, on the microlocal gauge quotient, acts by scalar multiplication on the transverse–traceless and transverse-trace summands. We further identify the precise additional condition required for a pure transverse–traceless reduction, formulated as a quotient-space trace Ward identity on the microlocal gauge quotient. Under this hypothesis, the transverse-trace mode is eliminated and the principal symbol reduces to the pure transverse–traceless form. As a motivating application, we discuss renormalized spectral functionals associated with Laplace-type operators and admissible spectral deformations. Using standard pseudodifferential functional calculus, we show that such deformations preserve the scalar principal symbol and improve resolvent differences by two orders, thereby explaining the microlocal mechanism behind the spectral examples. The resulting spectral applications are stated conditionally, via an explicit analytic realization hypothesis for the renormalized spectral Hessian.