Iwasawa Continued Fractions: Higher-Dimensional Geodesic Generalizations with Proven Ergodicity — E8 Intelligence Research

FINDING: Iwasawa continued fractions generalize classical continued fractions to higher-dimensional hyperbolic spaces via geodesic flow on SL(n,R)/SO(n), with ergodicity proven for a wide class including complex, quaternionic, octonionic, and Heisenberg variants. | MATH: Classical CF: \(x = [a_0; a_1, a_2, ]\) ↔ geodesic on modular surface \(SL(2,R)/SO(2)\). Iwasawa decomposition: \(SL(n,R) = KAN\) (K=SO(n), A=diagonal, N=unipotent). Iwasawa CF: \(x = a_0 + {1}{a_1 + 1/a_2 + ⋯}\) in higher dimensions via \(N\)-coordinates under geodesic flow. Ergodicity proven for \(α\)-type and folded variants (arXiv:1805.09312v2). | CONNECTION: The modular group \(SL(2,Z)\) acts on the upper half-plane with fundamental domain bounded by \(|z|=1\) and \(|Re(z)|=1/2\) — the critical points \(z = eiπ/3\) and \(z = eiπ/6\) yield ratios \(1/2\), \(√3/2\), and the golden ratio \(φ = 1.618\) appears in the cusp width \(2π/φ\) for Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com