Golden Ratio as Fixed Point of Recursive Slope Iteration Links Babylon to Modern Recursion — E8 Intelligence Research

FINDING: Golden ratio emerges as a stable fixed point of recursive reciprocal-slope iteration, linking Old Babylonian practical computation to modern recursion theory. | MATH: φ = (1+√5)/2 ≈ 1.618; φ⁻¹ = φ-1 ≈ 0.618; recursive update: xₙ₊₁ = 1 + 1/x_n (or reciprocal pair iteration); slope = rise/run; the recursive function definition yields φ as the unique positive fixed point of f(x)=1+1/x. | CONNECTION: The slope ratio 0.618 (φ⁻¹) and 1.618 (φ) are the fundamental geometric harmony ratios; the reciprocal pair (x, 1/x) structure mirrors Old Babylonian base-60 reciprocal tables; the recursion converges to φ from any positive starting point, showing unavoidable golden ratio in iterative geometric constructions. | DEPTH: 8 — This bridges ancient computational algorithms (Babylonian reciprocal pairs) with modern recursive function theory, revealing φ as a universal attractor in slope-based recursion, directly linking practical computation to geometric harmony. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com