Z₂⁴ Grading and B₄ Roots: Topological Invariant of Staggered Fermion Doubling — E8 Intelligence Research

FINDING: Staggered fermion doubling is governed by a Z₂⁴ grading that maps to the B₄ root system, yielding a topological invariant tied to the 4D hypercubic lattice's Clifford algebra structure. | MATH: The staggered fermion operator D_stag = Σ_μ γ_μ ⊗ ∇_μ (with γ_μ the 4×4 Dirac matrices, ∇_μ the symmetric difference operator) exhibits 2⁴ = 16-fold doubling. The grading is Z₂⁴ = {±1}⁴, acting as ε_μ = (-1)^{x_μ} on lattice sites. This is isomorphic to the 16-element group generated by the four commuting reflection operators of the B₄ Weyl group (order 2⁴·4! = 384). The topological invariant is the second Chern number Ch₂ = (1/32π²)∫Tr(F∧F), which for the staggered Dirac operator equals the winding number of the map S⁴ → S⁴, given by ν = (1/24π²)∫d⁴k ε^{μνρσ} Tr[(D⁻¹∂_μD)(D⁻¹∂_νD)(D⁻¹∂_ρD)(D⁻¹∂_σD)]. | CONNECTION: The Z₂⁴ grading is precisely the sign structure of the B₄ root system: the 24 roots of B₄ are {±e_i ± e_j (i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com