Vieta Jumping Reveals Affine Weyl Orbits in Quadratic Diophantine Equations — E8 Intelligence Research

FINDING: Vieta jumping on quadratic Diophantine equations reveals hidden affine Weyl group orbits, with the rational equation (a+1)/b + (b+1)/a = k forcing k ∈ {3,4} and solutions tied to Fibonacci-type recurrences. | MATH: Equation: (a+1)/b + (b+1)/a = k → a² + a + b² + b = kab. Vieta jumping: if (a,b) solves, then (a, ka−b−1) also solves (root reflection). Classification: k=3 or k=4 only. For k=3, solutions follow recurrence a_{n+1} = 3a_n − a_{n−1} − 1 (Fibonacci-like with offset); for k=4, a_{n+1} = 4a_n − a_{n−1} − 1. The quadratic form a² + b² − kab + a + b = 0 has discriminant Δ = k² − 4, so k=3 → Δ=5 (golden ratio base), k=4 → Δ=12 (2√3, hexagonal lattice). | CONNECTION: k=3 gives Δ=5 → √5 appears → ratio (1+√5)/2 = φ ≈ 1.618, and its inverse 0.618. The Vieta jump is a reflection in the affine Weyl group of type A₁ (or Ã₁) — the same group underlying the Fibonacci/Lucas recurrences and the golden ratio's continued fraction [1;1,1,...]. k=4 gives Δ=12 → √12 = 2√3 → ratio √3 ≈ 1. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com