Paper 40 — Fano Trifrontality and the Spectrum of the Majorana–Dirac Operator

We prove that the Fano plane PG(2,F₂) plays three distinct algebraic roles in the Cayley–Dickson zero-divisor series — a triple we call the Fano trifrontality — all three now established at the [Derived, GAP 4.15.1] level:

(i) multiplication organiser in 𝒜₃ (incidence = nonzero product);

(ii) annihilation organiser in 𝒜₄ (non-incidence = ker(L_a));

(iii) Cayley–Dickson doubling organiser in Im(𝒜₃) (the bipartition (ℓ_a, V_ni(a)) is precisely the decomposition Im(ℍ) ⊕ ℍ·e₄ of the doubling 𝒜₂→𝒜₃, established in Paper 39).

As the primary spectral consequence of the trifrontality, we compute the matrix of 𝒟²|_{V_ni(a)} in the S₄-equivariant basis {e₁,e₃,e₅,e₇}: it equals J₄ − 5I₄ (all-ones minus 5·identity), with spectrum {−1, −5, −5, −5} [Derived, GAP 4.15.1].

Via Schur's lemma applied to ker(L_a)|_{S₄} = irr₁ ⊕ irr₃ (Paper 37), the operator 𝒟² acts as −1 on the gravitational singlet irr₁ and as −5 on the gauge triplet irr₃.

The spectral zeta function ζ_{𝒟²}(s) = 1 + 3·5^{−s} gives regularised determinant det_ζ(−𝒟²) = 5³ = 125, providing the algebraic input for the boundary-condition parameter β in BC(0,β) (Papers 37–38).

The portfolio of Fano roles (Paper 34 §7) is updated with four new entries and two upgrades. Two GAP 4.15.1 scripts are deposited as supplementary files. (on Zenodo)