Langlands Duality: B_n ↔ C_n via Coroot Inversion with √2 Root Ratio — E8 Intelligence Research

FINDING: Langlands duality maps B_n ↔ C_n via coroot inversion, exchanging root systems while preserving Weyl group order 2^n n! and revealing a root-length ratio of √2 between long and short roots. MATH: - Weyl group order: |W(B_n)| = |W(C_n)| = 2^n · n! - Coroot inversion: α∨ = 2α/(α,α) → for B_n (long roots length √2, short length 1) and C_n (long roots length 2, short length √2), the duality swaps long↔short. - Root length ratio (B_n): long/short = √2; (C_n): long/short = 2/√2 = √2. - Killing form normalization: (α,α) = 2 for long roots in simply-laced; for B_n/C_n, the dual pairing inverts the Cartan matrix: A(B_n)ᵀ = A(C_n)⁻¹ (up to scaling). - Quantum coroot lattice action (arXiv:2501.02365): explicit formula for U_q(Lg) Weyl group action on coroot lattice Q, using commuting generators — exact expression involves q-shifted exponentials, e.g., ex_i acting as q±(λ,α_i∨). CONNECTION: - The √2 ratio is not a golden-ratio constant, but it is a crystallographic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com