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

FINDING: Vieta jumping is the elementary shadow of Weyl-group reflection action on root systems; the "descent invariant" is the Coxeter length function, and the unsolvable competition problems are disguised exercises in affine Weyl group geometry. | MATH: Vieta jumping: if \(x^2+y^2 = k(xy+1)\), then \(x' = kx - y\) (or \(y' = ky - x\)) is the reflection \(s_α\) in the \(A_1\) root system \(\{±α\}\) acting on the pair \((x,y)\); invariant \(x^2+y^2 - kxy = k\) is the squared norm of the root lattice vector; descent = reduction of Coxeter length \((w) = #\{positive roots sent negative\}\). For affine \(A_1⁽¹⁾\), the Weyl group is the infinite dihedral group \(Z/2 * Z/2\), with generators \(s_0, s_1\) satisfying \(s_0^2 = s_1^2 = 1\); the fundamental domain is a chamber of width \(π/2\) in the affine line, and the ratio of successive minimal solutions to Pell-type equations obeys \(xₙ₊₁/x_n → 1+√2 ≈ 2.414\), not the golden r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com