Paper 39 — The Majorana–Dirac Operator and the Fano–Cayley–Dickson Doubling

We study the restriction of the ZD Dirac operator 𝒟 = Σ_{p∈ℓ̄_a} γ_p ∂p to the Majorana sector 4_s of the sedenion scalar field theory S₈₄, where γ_p = L{e_p}^{𝒜₃} are the Clifford generators (octonion left-multiplications) and ℓ̄_a is the set of four Fano points non-incident to the line ℓ_a associated with a ZD-active element a.

We first prove [Derived-Negative, GAP 4.15.1] that 𝒟|_{4_s} does not map ker(L_a) = V_ni(a) to itself: every Clifford generator γ_p with p ∈ ℓ̄_a sends V_ni(a) into the complementary on-line subspace ℓ_a, not back into V_ni(a).

The correct closure holds at second order: we prove [Derived, GAP 4.15.1] that 𝒟² maps ker(L_a) to itself. The algebraic mechanism is the Cayley–Dickson doubling 𝒜₂→𝒜₃: the decomposition Im(𝒜₃) = ℓ_a ⊕ V_ni(a) identifies ℓ_a = Im(ℍ) as the imaginary quaternions (pre-doubling sector) and V_ni(a) = ℍ·e₄ as the post-doubling sector isomorphic to ker(L_a). The complete bipartition table — V_ni × V_ni → ℓ_a, ℓ_a × V_ni → V_ni, V_ni × ℓ_a → V_ni, ℓ_a × ℓ_a → ℓ_a — is verified by GAP 4.15.1 for all seven Fano lines simultaneously and follows analytically from the Fano XOR rule without invoking associativity.

As a consequence, 𝒟²|_{ker(L_a)} is a well-defined real 4×4 operator whose spectrum provides the algebraic input for the boundary-condition parameter β in BC(0,β) (Paper 38, Conj. 5.1), reducing Open Problem 37.4 and OP 12.1 to an explicit spectral computation (OP 39.1). Two GAP 4.15.1 scripts are deposited as supplementary files. (on Zenodo)