Hexagonal Symmetry of Mod 3 Primes via Eisenstein Integers and Knot Invariants — E8 Intelligence Research

FINDING: Dirichlet characters mod 3 connect to Eisenstein primes (ring ℤ[ω], ω = e2πi/3), whose angular distribution in the complex plane exhibits hexagonal (6-fold) symmetry — the same lattice structure as the Eisenstein integers. The arXiv paper (1412.6894) explicitly links mod 3 triple Milnor invariants to cubic residue symbols in ℚ(√−3), revealing a knot-theoretic analog of prime distribution in this hexagonal lattice. MATH: - Eisenstein integers: ℤ[ω], ω = (−1 + i√3)/2, norm N(a + bω) = a² − ab + b². - Cubic residue symbol (χ₃(p)): for prime p ≡ 1 mod 3, χ₃(p) ∈ {1, ω, ω²}. - Dirichlet characters mod 3: trivial χ₀, and χ₁(n) = (n/3) with values {0, 1, −1} — the Legendre symbol mod 3. - Triple Milnor invariant μ̄(3) mod 3: generalizes Rédei triple symbol, defined via cubic residues in ℚ(√−3). - Key ratio: The hexagonal lattice has nearest-neighbor distance √3, and its Voronoi cell area = (3√3)/2 ≈ 2.598 — related to 2.618 (φ²) within 0.77% error. - Angular anisotrop Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com