Paper 23 — The ZD Dirac Operator and Fermionic Sector S₄

Paper 23 in the series on zero-divisors in Cayley–Dickson algebras. We construct the ZD Dirac operator D = Σ γ_a ∂/∂φ_a on the 84-dimensional scalar field manifold of S₈₄, where γ_a = ρ(a) ∈ Cl(0,7) are the Clifford matrices of Paper 7. We derive D² = −I₈Δ_fib (the negative Fano-fibre Laplacian), with purely imaginary symbol spectrum ±i‖ξ̃‖, complementary to the real integer spectrum {−4,−2,0,+2,+4} of the adjacency matrix A.

Restricting to the 4_s Majorana sector (four modes with ε-indices {4,5,6,7}), we prove by three independent arguments — block distribution 0+2+2, non-invariance of ker(L_a) under σ₃, and a Sylow order obstruction from |Stab_{G₈₄}(a)| = 2⁵ — that the C₃ Fano-block cyclic symmetry does not induce a 3+1 decomposition of the 4_s modes [Derived-Negative]. Open Problem 18.8 (sterile neutrino via Z₃) is thus closed negatively for the Fano-block mechanism.

We introduce the fermionic extension S₈₄^(f) with Yukawa coupling Σ φ_{a_j} Ψ̄ γ_{s_j} Ψ, derive the modified Dirac equation, compute the condensate shift (±2iC₀, purely imaginary), and identify the schematic fermion-loop correction to β(λ) as negative relative to the one-loop term. The canonical Cl(0,7) basis is positioned relative to the Gourlay–Gresnigt Cl(8): C⊗Cl(0,7) ≅ M₈(C) embeds as the octonionic block of End_C(C⊗S) ≅ M₁₆(C).