Topological Langlands Duality and Coxeter Involutions — E8 Intelligence Research

FINDING: Langlands duality is being reformulated topologically and symplectically, with an explicit role for involutions in Coxeter groups — but no direct golden-ratio or self-duality constant emerges from these abstracts. MATH: - Langlands dual group: \( G \leftrightarrow {}^L G \) (root datum exchange: \( X^\vee \leftrightarrow X \), coroots ↔ roots). - Coxeter group involution product: \( \mathcal{W} = \{ w = xy \mid x^2 = y^2 = 1 \} \); for finite \( W \), \( W = \mathcal{W} \). Minimal defect \( \ell(x) + \ell(y) - \ell(w) \) (arXiv:1405.3051). - Poisson–Lie duality: \( G \) semisimple → dual \( G^* \) via Bohr–Sommerfeld quantization (Alekseev). - Topological Langlands: 1-form symmetries ↔ boundary conditions in 3-manifold TQFT (Jordan). CONNECTION: - Coxeter groups encode crystallographic root systems (A_n, D_n, E_6, E_7, E_8) — the same lattices underlying quasicrystalline and icosahedral symmetries. The involution product \( w = xy \) with \( x^2 = y^2 = 1 \) is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com