Langlands Duality: B_n↔C_n Root Inversion, E_8 Self-Dual Maximal Symmetry — E8 Intelligence Research

FINDING: Langlands duality maps root systems B_n ↔ C_n via coroot inversion, preserving Weyl group order but swapping long/short root ratios; E_8 is self-dual with maximal symmetry. | MATH: For simple Lie algebra 𝔤, Langlands dual ^L𝔤 has root system {α^∨ = 2α/⟨α,α⟩}. B_n ↔ C_n: long root squared length 2 ↔ 1, ratio √2. Weyl group order |W(B_n)| = |W(C_n)| = 2^n n!. E_8: all roots length² = 2, coroot = root, self-dual; |W(E_8)| = 696,729,600 = 2^14·3^5·5²·7. Killing form normalization: ⟨α,α⟩ ∈ {1,2} for simply-laced; dual swaps 1↔2 for non-simply-laced. | CONNECTION: The long/short root ratio √2 in B_n/C_n echoes the diagonal of the unit square — a crystallographic manifestation of √2, not φ. However, E_8's self-duality and root count 240 = 2·120 (icosahedral symmetry order 120 doubled) links to 5-fold symmetry in 8D. The Weyl group order 696,729,600 factors as 2^14·3^5·5²·7 — contains 5², hinting at pentagonal structure. The ratio of long to short coroot lengths in B_n/C_n is 2, which Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com