Vieta Jumping Links Quadratic Descent to Metallic Mean Recurrences — E8 Intelligence Research

FINDING: Vieta jumping reduces infinite descent on quadratic Diophantine equations to recurrence relations, often yielding Fibonacci-like sequences. MATH: For equation (a²+b²)/(ab+1) = k, Vieta jumping gives recurrence aₙ₊₁ = k a_n - aₙ₋₁. Minimal solutions correspond to k being a perfect square. CONNECTION: Recurrence aₙ₊₁ = k a_n - aₙ₋₁ generates ratios converging to (k ± √(k²-4))/2. For k=3, ratio = (3+√5)/2 ≈ 2.618 (φ²). For k=4, ratio = 2+√3 ≈ 3.732. These are metallic means, linked to golden ratio φ=1.618 and its square 2.618. The descent mirrors root system reflections in A₁ lattice. DEPTH: 7 — Vieta jumping is a powerful descent method revealing hidden recurrence structure in quadratic forms, directly connecting to metallic ratios and lattice symmetries. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com