First-Order Spectral Deformations of the Helical Vorticity Operator on Curved Four-Dimensional Backgrounds

We construct a self-consistent geometric framework to evaluate the first-order spectral de-formations of a spinorial covariant extension of the linear helical vorticity operator restrictedto a compact, non-singular domain within a curved four-dimensional pseudo-Riemannianmanifold. Utilizing the axisymmetrically deformed interior metric tensor derived in ourprior horizon-averaged surface gravity framework, we formalize the local frame bundle viaexplicit orthogonal tetrad and inverse tetrad field decompositions. By solving Cartan’sfirst structural equation under the constraint of vanishing torsion, we evaluate the linearperturbations of the localized spin connection ωabμ (τ ) up to the first order in the scalar de-formation parameter τ , verifying the temporal sector explicitly. We demonstrate that thequadrupolar metric deformation breaks the flat-space operational symmetry, mapping theperturbation directly onto a non-local spinorial deformation operator ˆVspin. This covariantformulation parameterizes the first-order shifts of the discrete eigenvalue spectrum λn(τ )within a self-adjoint Hilbert space framework, providing a mathematically controlled base-line for analyzing structural wave transitions within a deterministic, deformed space-timemedium.