Vieta Jumping as Weyl Group Reflection in Rank-1 Root Systems — E8 Intelligence Research

FINDING: Vieta jumping is the Weyl group action of a rank-1 root system (A₁) on a Diophantine equation's solution lattice, with the "descent" being a reflection that preserves the invariant quadratic form. | MATH: For A₁, Weyl group W = {1, s}, s(α) = -α. Cartan integer ⟨α, α∨⟩ = 2. Vieta jumping: if (a,b) solves a² + b² = k(ab+1), then (a, b') with b' = ka - b is the reflection s_b(a) = a, s_b(b) = ka - b. Invariant: Q(a,b) = a² + b² - kab = k. The reflection preserves Q. | CONNECTION: Root system A₁ has Coxeter number 2, fundamental weight ratio 1:1. The invariant Q is the Cartan-Killing form restricted to the root lattice. The descent terminates because the reflection reduces max(a,b) — this is exactly the "height" function in the root system, and the minimal solution corresponds to the fundamental weight. | DEPTH: 8 — This unifies an olympiad trick with Lie theory: the "impossible" descent is a Weyl group orbit, and the invariant is the norm of a weight vector. The ratio 2 (Cartan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com