Chirality of the Seeley-DeWitt coefficients and quartic Weyl structure in the spectral action

We prove that the traced Seeley-DeWitt coefficient tr(a_8) for the Dirac operator on a Ricci-flat four-manifold has the form c(p² + q²) with zero cross-term pq, where p = |C⁺|² and q = |C⁻|² are the norms of the self-dual and anti-self-dual Weyl tensors. The proof rests on a chirality theorem: the generators σ^{rs} = (1/4)[γ^r, γ^s] commute with γ₅ in d = 4, rendering the spin connection curvature Ω_{μν} block-diagonal in the chiral basis. Consequently the heat kernel e^{−tD²} splits into left- and right-handed sectors, each coupled to only one half of the Weyl tensor (crossed chirality assignment). This result applies to the full Standard Model Dirac operator D on the product geometry M × F and holds at all…