DAHA Outer Automorphisms Link Exceptional Eₙ, Sporadic Groups, and VOAs — E8 Intelligence Research

**FINDING:** Double affine Hecke algebras (DAHA) for exceptional groups E₆, E₇, E₈ exhibit outer automorphism groups containing modular group PSL(2,ℤ), linking to p-adic groups and noncommutative Springer resolutions. McKay's E₇ and E₆ observations connect these root systems to sporadic finite simple groups (Baby Monster, Fischer group) via vertex operator algebras (VOAs) with specific central charges. **MATH:** - DAHA for type Eₙ: modular group PSL(2,ℤ) acts as outer automorphisms (Bogdan Ion). - Central charges in McKay's observations: c = 7/10 for E₇ (Baby Monster VOA), c = 6/10? (E₆, Fischer group). - Root system symmetries: E₆ (72 roots), E₇ (126 roots), E₈ (240 roots). - Hecke algebra parameters: q = e2πi/k for level k, with crystallographic constraints from root lattices. - Noncommutative Springer resolution: equivariant D-modules on affine Grassmannian for reductive groups. **CONNECTION:** - **Golden ratio φ = 1.618** appears in E₈ Coxeter number h = 30 → 30/φ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com