Lattice-Theoretic Order in Complexity Classes Mirrors Root System Symmetries — E8 Intelligence Research

Abstract

FINDING: Complexity class posets exhibit structural parallels to root system symmetries in Lie algebras, suggesting a deep lattice-theoretic order underlying computational hierarchy. MATH: Root systems (e.g., \(A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2\)) define Weyl groups with Coxeter numbers; complexity classes (P, NP, PSPACE, EXP, etc.) form a poset under polynomial-time reductions. Key constants: Coxeter number \(h\) for \(E_8 = 30\), for \(G_2 = 6\); golden ratio \(\phi = 1.618\) appears in \(E_8\) root system norms. CONNECTION: Root system \(E_8\) has 240 roots with angles governed by \(\phi\) (e.g., \(\cos 72^\circ = 0.309\), \(\cos 144^\circ = -0.809\), related to \(\phi/2 = 0.809\)). The poset of complexity classes may map to a subset of root lattice points, with reductions analogous to Weyl group reflections. DEPTH: 7 — The analogy is suggestive but not yet proven; it hints at a unified symmetry underlying computational limits and algebraic geometry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com