Fano Superselection Sectors in Algebraic Quantum Field Theory: a possible topological origin of the Electron Charge

We formulate a superselection sector for the electron within the framework of Algebraic Quantum Field Theory (AQFT), deriving its topological charge and exchange statistics from the projective geometry of the Fano plane PG(2,2). The construction proceeds through a sequence of nested algebraic structures. First, the Fano plane defines a topological defect through its incidence matrix M in {0,1}^{7x7}, whose associated Dirac operator D = [[0, M], [M^T, 0]] has spectrum { ±3, ±√2 }. Second, the spinorial phase θ = π/6 generates a monodromy operator U = exp(iπ X/6) on the two-qubit space C^2 ⊗ C^2, where X = 2√2 (σ_z ⊗ I_2) - √7 (I_2 ⊗ σ_z). The characteristic polynomial of X is χ_X(x) = x^4 - 30x^2 + 1, whose root ΔS = 2√2 - √7 is the entanglement deficit governing the renormalization flow. Third, the Fano 2-22 (Mori-Mukai ID-69) provides the algebraic moduli space through its Minkowski period coefficients c_5 = 24×45, c_6 = (4/3×45)(45+64), c_7 = 24×(4/3×45)(45-10), whose factorization encodes the string tension T=8, the bond dimension D=45, and the hinge invariant I_h = 4/(3π). Fourth, a sequence of graphs derived from these coefficients converges in cut norm to the graphon W(x,y) = κ Φ_VEV^2 exp(-γ d(x,y)) on S^3. The spectral gap satisfies λ_1/λ_0 = 1.99765, tending to 2 as γR → ∞, reflecting the isotropy of the emergent FLRW spacetime. The sector is defined as an endomorphism ρ_Fano of the observable algebra A localized in a spacelike cone via the split property. The exchange statistics are governed by the R-matrix R = τ ∘ exp(iπ X/6), which satisfies the Yang-Baxter equation globally, yielding a phase ε_ρ = exp(iπ/6) that is neither bosonic nor fermionic. The construction contains no free parameters, providing a possible topological origin for the electron charge and a falsifiable extension of the DHR classification in 3+1 dimensions.