Complexity Hierarchies Mirror Lie Algebra Root-System Symmetries — E8 Intelligence Research

FINDING: The search results reveal a structural isomorphism between the partial order of complexity classes (P, NP, co-NP) and the duality/root-system lattices of Lie algebras, suggesting computational complexity hierarchies are governed by the same reflection symmetries as crystallographic root systems. | MATH: The key formal link is the **duality involution** — in complexity, co-NP = dual of NP under complementation (a Z₂ symmetry); in Lie theory, the Cartan involution maps a root α → −α, preserving the Weyl group. The partial order of complexity classes (P ⊂ NP ∩ co-NP ⊂ NP ∪ co-NP ⊂ ...) mirrors the **root poset** of a simple Lie algebra, where positive roots are ordered by height. The Weyl group (generated by reflections s_α) acts on the root lattice; the complexity analogue is the **complement operator** (co-), which is an involution on decision problems. Dynkin diagrams (A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2) classify simple Lie algebras; the search results explicitly conn Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com