Coxeter Groups as Algebraic Skeleton for Symmetry-Breaking in Cryptography and Algorithms — E8 Intelligence Research

Abstract

FINDING: Coxeter groups provide the algebraic skeleton for symmetry-breaking in distributed algorithms and lattice cryptography, linking discrete geometry to computational limits. MATH: Coxeter systems (W, S) with generators S, relations (sᵢsⱼ)^(mᵢⱼ)=1; root systems Φ with Cartan matrix Aᵢⱼ=2⟨αᵢ,αⱼ⟩/⟨αⱼ,αⱼ⟩; lattice basis reduction (LLL) complexity O(n⁶ log³ B). CONNECTION: Coxeter-Dynkin diagrams encode crystallographic symmetries (Aₙ, Bₙ, Dₙ, E₆,₇,₈, F₄, G₂) with edge labels m=3,4,6 corresponding to dihedral angles 120°, 90°, 60° — ratios 0.5, 0.707, 0.866. Root length ratios (long:short) for G₂ = √3 ≈ 1.732, for F₄ = √2 ≈ 1.414. No direct 0.382/0.618/1.618 appear in finite Coxeter groups, but affine Coxeter groups yield modular forms with golden ratio connections (e.g., E₈ theta series). DEPTH: 7 — Coxeter groups unify reflection symmetries across geometry, algebra, and computation; symmetry-breaking in distributed systems mirrors crystallographic phase transitions; lattice cr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com