Langlands Duality: Hidden Structural Identity of B_n and C_n Weyl Groups — E8 Intelligence Research

FINDING: Langlands duality between SO(2n+1) and Sp(2n) is a root-system duality where the ratio of long to short root lengths is 2:1, and the invariant theory of the corresponding Weyl groups yields fundamental degrees 2,4,...,2n for both — a structural identity hidden beneath the exchange of root systems. | MATH: Root systems: B_n (SO(2n+1)) has long roots of length √2, short roots of length 1; C_n (Sp(2n)) has long roots of length 2, short roots of length √2 (normalized so short roots = 1 in B_n, long roots = 2 in C_n). Ratio long:short = 2:1 in both, but the roles are swapped under duality. Weyl group invariant degrees: For B_n and C_n, both have degrees 2, 4, 6, ..., 2n (identical sequence). This is the classical "degree coincidence" that makes the Langlands L-group of SO(2n+1) equal to Sp(2n) and vice versa. The dual Coxeter number: h^∨(SO(2n+1)) = 2n−1; h^∨(Sp(2n)) = n+1. The ratio (2n−1)/(n+1) → 2 as n→∞, echoing the 2:1 root length ratio. | CONNECTION: The 2:1 ratio is the same Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com