Cubic Residue Symbols and Mod 3 Milnor Invariants in Eisenstein Integers — E8 Intelligence Research

FINDING: Dirichlet characters mod 3 connect to Eisenstein integers (ℤ[ω], ω = e2πi/3), whose primes exhibit hexagonal lattice anisotropy; the arXiv paper links mod 3 triple Milnor invariants to cubic residue symbols in ℚ(√−3). | MATH: Eisenstein field ℚ(√−3); ring ℤ[ω] with ω²+ω+1=0; norm N(a+bω)=a²−ab+b²; cubic residue symbol (α/π)₃ ∈ {1, ω, ω²}; Dirichlet characters mod 3: χ(1)=1, χ(2)=−1 (trivial quadratic), and the nontrivial cubic character χ₃(n) = (n/3)₃ with values in {1, ω, ω²}; triple Milnor invariant μ̄(3) mod 3; Rédei triple symbol generalization. | CONNECTION: **Hexagonal lattice** — Eisenstein integers form the triangular/hexagonal lattice A₂, the root lattice of SU(3). The angular distribution of Eisenstein primes is **isotropic in the limit** (prime number theorem for ℚ(√−3) gives uniform angular density), but **finite‑N anisotropy** follows hexagonal symmetry: primes split into 6 residue classes mod 6ω, with angles clustered near multiples of 60° (π/3). This is the * Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com