No Direct Result on B_n ↔ C_n Dual Coxeter Number Swap — E8 Intelligence Research

FINDING: Search returned mostly noise; only one relevant hit (coroots/Killing form/Langlands dual) plus a tangential Coxeter-invariant paper. No direct result on the B_n ↔ C_n dual Coxeter number swap. MATH: - Langlands dual: (G, T) → (G^∨, T^∨) with root datum (X, Φ, X^∨, Φ^∨) swapped: Φ ↔ Φ^∨, so long↔short roots exchange. - B_n: Φ = {±e_i ± e_j, ±e_i}; C_n: Φ = {±e_i ± e_j, ±2e_i}. Under duality B_n^∨ = C_n, C_n^∨ = B_n. - Dual Coxeter number h^∨ = (1/2) Σ_{α>0} ⟨α, α^∨⟩ = (1/2) Σ_{α>0} 2(α,α)/(α,α) — invariant under Langlands duality: h^∨(B_n) = h^∨(C_n) = 2n. (Check: B_n: 2n−1 long + 2n short... h^∨ = 2n; C_n: 2n short + 2n long... h^∨ = 2n. Equal.) - Killing form normalization: B(X,Y) = tr(ad X ∘ ad Y). For B_n vs C_n, the invariant form scales by the ratio of long-to-short root squared lengths: (α_long, α_long)/(α_short, α_short) = 2 for B_n, = 2 for C_n (inverse convention). This is the "swap" — the form is not literally identical, but the dual Coxeter number is. - Coxeter num Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com