Vieta Jumping as Weyl Group Reflection in Rank-1 Root System A₁ — E8 Intelligence Research

FINDING: Vieta jumping is structurally identical to the action of the Weyl group of a rank-1 root system (A₁), where the "descent" step is a reflection that preserves a quadratic invariant. | MATH: For A₁: roots {±α}, reflection s_α(x) = x − 2⟨x,α⟩/⟨α,α⟩·α. Vieta jump: given (a,b) solving a² + b² = k(ab+1), the other root is b' = kb − a. Invariant: a² + b² − kab − k = 0 (quadratic form). The jump (a,b) → (b, kb−a) is exactly s_α acting on the lattice Z² with Cartan integer ⟨α,α⟩=2. | CONNECTION: The ratio of consecutive terms in minimal Vieta-jump descent sequences approaches φ = (1+√5)/2 ≈ 1.618 (and its inverse 0.618) when k=1 (Fibonacci case); for general k, the ratio approaches (k+√(k²−4))/2, whose reciprocal is (k−√(k²−4))/2 — these are the eigenvalues of the Coxeter element of A₁. The descent terminates because the quadratic form is positive definite, mirroring the finite Weyl group property. | DEPTH: 9 --- FINDING: The "impossible" competition problem (IMO 1988 Problem 6) is a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com