Möbius Maps Preserve Continued Fractions: PSL(2,ℝ) and 60-Smooth Reciprocals — E8 Intelligence Research

FINDING: Möbius transformations (PSL(2,ℝ)) act on the projective line ℙ¹(ℝ) and map continued fractions to continued fractions, with bounded period growth — the natural algebraic home of Babylonian 60-smooth reciprocal pairs. | MATH: M(x) = (ax+b)/(cx+d), ad−bc = 1. Reciprocal pair (n, n⁻¹) in base-60 ⟺ n = 2^a·3^b·5^c (60-smooth). Continued fraction [a₀; a₁, a₂, …] ↔ matrix product ∏[[aᵢ,1],[1,0]] ∈ SL(2,ℤ). Period after Möbius map bounded by Starosta–Řada result. | CONNECTION: PSL(2,ℤ) ≅ ℤ/2 ∗ ℤ/3 — the (2,3,∞) triangle group, the modular group whose fundamental domain has the golden-ratio-adjacent structure; 60-smooth numbers are exactly the units of ℤ[1/2,1/3,1/5], the S-units whose logarithms form a rank-3 lattice in ℝ³. | DEPTH: 8 --- **1. Key mathematical insight** The Babylonian scribal corpus (YBC 7289, Plimpton 322, the reciprocal tables) is not merely arithmetic — it is a *projective* arithmetic. The sexagesimal system's preference for **regular numbers** (60-smooth: 2^a· Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com