Four-Dimensional Spinorial Geometry and Chiral Spectral Duality via Temporal Spin Connections on Deformed Backgrounds

We construct a self-consistent four-dimensional spinorial framework to evaluate the first-order spectral deformations of the unreduced, massless Dirac operator ˆ/Dg = γae μa ∇μ re-stricted to an axisymmetrically deformed Lorentzian manifold. Utilizing the localized framebundle via explicit orthogonal tetrad and inverse dual coframe field decompositions, wepreserve the unreduced structural properties of the background geometry. By solving Car-tan’s first structural equation under the constraint of vanishing torsion, we demonstratethat the cross-term connection component ω02 induces a non-vanishing temporal couplingfield ω02t proportional to the angular gradient of the quadrupolar deformation polynomialY ′20(θ). This temporal connection generates a local, non-differential spatial Clifford potentialstructured via the γ2 matrix generator. We evaluate the full perturbed system within a self-adjoint Hilbert space, proving that the anti-commutation of the total Dirac operator withthe chiral volume element γ5 holds strictly up to the first order in the deformation parameterτ , enforcing a first-order chiral spectral duality ±λ within the linear perturbation regime.Finally, we formulate the relativistic field action and the corresponding Lagrangian density,establishing a natural geometric realization that is compatible with the standard particle-antiparticle interpretation after canonical secondary quantization within the unreduced 4DClifford space.