Paper 34 — A Single Fano Plane: Zero-Divisors, Gauge Structure and the Massless Sector

The Fano plane PG(2,F₂), with its GL(3,F₂)-symmetry, is the universal local structure of the entire Cayley–Dickson zero-divisor tower: every ZD pair at every level belongs to exactly one Fano-plane block carrying the same standard action (Paper 4, Fano universality corollary). This single combinatorial fact cascades into seven structural results assembled here for the first time in a unified analysis.

We prove that the normalized residues s_k of the ZD zeta function Z(x) at the four Hurwitz poles encode the structural dimensions of the last normed division algebra A₃ = 𝕆: (s₀,s₁,s₂,s₃) = (1,7,14,8), equal respectively to dim(ℝ), |PG(2,F₂)|, dim(G₂) = dim(Aut(𝕆)), and dim(𝕆). This closes OP 33.3 at the [Derived] level for each individual identification. A new structural observation: the spectral invariant 𝒮 = Σ(2+μ)m(μ) = 168 of the quantum field theory S₈₄ equals the order of the Fano plane automorphism group |GL(3,F₂)| = 168.

The ZD zeta function Z(x) = 336x⁴·Z_Weil(PG(3,F₂),x) is identified as the mathematical bridge between the two faces of the Fano bifrontality (Papers 32–33): its numerator 336x⁴ encodes the annihilation face (ZD structure of A₄), its Z_Weil factor encodes the multiplication face (Frobenius structure of the Hurwitz tower), and the Schubert stratification of PG(3,F₂) maps four strata bijectively to the four Hurwitz algebras and their associated forces.

G₂ appears at the ℍ-pole of Z(x) with s₂ = dim(G₂) = 14 rather than dim(Aut(ℍ)) = 3, because G₂ = Aut(𝕆) is the first exceptional Lie group simultaneously containing the gauge groups of levels n=2 (SU(2)) and n=3 (SU(3)_C), via the chain S₄ ⊂ PSL(2,7) ⊂ SU(3) ⊂ G₂ ⊂ SO(7).

The Level-2 massless sector (μ = −2) is shown to be the unique eigenvalue for which the Fano bipartite identity Σ(2+μ)m(μ) = 168 implies zero contribution to the gravitational spectral sum, providing a new [Derived] Fano-combinatorial route to address OP 24.5. The decomposition 𝒮_eff = 164 = 168 − 4 is expressed explicitly as the difference of two Fano-combinatorial quantities: the bipartite spectral sum and the Fano non-incidence count dim(ker(L_a)) = 4. The cascade implications of OP 29.1 are mapped precisely in two branches.