Langlands Duality as Root Data Symmetry Exchanging Root and Coroot Lattices — E8 Intelligence Research

FINDING: Langlands duality is a symmetry of root data (R → R∨) that exchanges the root lattice and coroot lattice, preserving the Weyl group and Cartan matrix structure while inverting the Killing-form normalization. | MATH: For semisimple Lie algebra 𝔤 with root system Φ, coroots α∨ = 2α/⟨α,α⟩; Cartan integers A_ij = ⟨α_j, α_i∨⟩ = 2(α_i,α_j)/(α_j,α_j) are invariant under duality. Killing form B(x,y) = tr(ad x · ad y); dual group Ĝ has root data (X*(T), Φ∨, X_*(T), Φ). Weyl group order |W| = ∏(m_i + 1) where m_i are exponents; for A_n: |W| = (n+1)!, B_n/C_n: 2^n·n!, D_n: 2^(n-1)·n!, E_8: 696729600 = 2^14·3^5·5^2·7. | CONNECTION: Root systems ARE the crystallographic symmetry groups — the 4D/6D/8D lattices (A_n, D_n, E_6, E_7, E_8) are the only even unimodular lattices in their dimensions. E_8 lattice has kissing number 240 = 2·120, packing density π^4/384. Angles between simple roots: 90°, 120°, 135°, 150° — cosines are 0, −1/2, −√2/2, −√3/2. The golden ratio φ = 1.618 appears in H_3/H Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com