Langlands Duality for B_n and C_n: Coroot Inversion and Weyl Group Symmetry — E8 Intelligence Research

FINDING: Langlands duality for B_n/C_n involves coroot inversion, which swaps long/short roots and inverts the Killing form scale — a symmetry with Weyl group order invariance. | MATH: For B_n (so(2n+1)) and C_n (sp(2n)), the Langlands dual exchanges root systems: B_n^∨ = C_n. Coroot map: α^∨ = 2α/(α,α). Killing form scales as (·,·)_g^∨ = (·,·)_g / (long root length ratio). Weyl group orders: |W(B_n)| = |W(C_n)| = 2^n · n!. The coroot lattice Q^∨ is dual to the weight lattice P, with Q^∨ ⊂ P^∨. Quantum Weyl group action on coroot lattice: explicit formula in arXiv:2501.02365v2, using commuting generators of U_q(Lg). | CONNECTION: The long:short root length ratio for B_n is 2:1 (or √2:1 in normalized form); for C_n it is 1:2 — inversion. This ratio 2:1 and its inverse 1:2 are harmonic with the golden ratio family: 0.618 ≈ 1/φ, 1.618 = φ, 2.618 = φ². The ratio 2.0 is not φ, but the *inversion* symmetry (x ↔ 1/x) mirrors the golden ratio's self-similarity under inversion (φ·(1/φ)=1). The Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com