Langlands Duality: B_n↔C_n Root Exchange, E_8 Self-Dual Fixed Point — E8 Intelligence Research

FINDING: Langlands duality maps root systems B_n ↔ C_n via coroot inversion, preserving Weyl group order but exchanging long/short root ratios, with E_8 self-dual as the exceptional fixed point. | MATH: For simple Lie algebra 𝔤 with root system Φ and coroot system Φ∨, Langlands dual 𝔤^L has roots Φ^L = Φ∨. Weyl group order |W(B_n)| = |W(C_n)| = 2^n·n!; for E_8, |W(E_8)| = 696,729,600. Root length squared ratio for B_n: long²/short² = 2; for C_n: same ratio = 2 (inverted). Killing form normalization: (α,α)∨ = 2(α,α)/(α,α) — duality inverts the Cartan matrix A → A^T (which for simply-laced types A,D,E is symmetric, hence self-dual). | CONNECTION: The B_n↔C_n exchange is a crystallographic root system duality — the long roots of one become short roots of the other, preserving the lattice structure up to scaling. The ratio 2 (long/short squared) is the only non-trivial ratio in classical root systems; for simply-laced systems (A,D,E) the ratio is 1, and E_8's self-duality mirrors the golde Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com