Affine Weyl Group Dynamics on Markov Triples via Root Lattice Orbits — E8 Intelligence Research

FINDING: Affine Weyl groups (A₂ root lattice) generate discrete integrable dynamics via Markov triples, with orbit classification tied to the affine Weyl group's action on the root lattice. | MATH: Markov triple equation: x² + y² + z² = 3xyz. The affine Weyl group of type A₂ (denoted Ã₂) acts on the root lattice Q(A₂) ≅ ℤ² with Cartan matrix [[2,-1],[-1,2]]. The group is generated by simple reflections s₁, s₂ with (s₁s₂)³ = 1, extended by the translation part (coroot lattice). Orbits of Markov triples under the Vieta involutions (x,y,z) → (x, y, 3xy−z) correspond to Weyl group orbits. The Markov constant: μ = 3√(1 − 1/9) = 3√(8/9) = 2√2 ≈ 2.828. | CONNECTION: The A₂ root lattice has hexagonal symmetry (6-fold). The Markov triples' orbit space is parametrized by the fundamental domain of the affine Weyl group — a 60° sector (equilateral triangle). The ratio of successive Markov numbers along a branch approaches 9 (not golden ratio), but the *geometric* structure of the root lattice yiel Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com