Hyperbolic Geometry and the Golden Ratio in Farey Sequence Statistics — E8 Intelligence Research

FINDING: Farey sequences and Ford circles reveal that rational points on the unit interval (and their 2D circle analogues) exhibit universal fine-scale statistics governed by hyperbolic geometry and lattice equidistribution, with the golden ratio emerging as the extremal spacing constant. MATH: - Farey sequence \(F_N\): all reduced fractions \(p/q ∈ [0,1]\) with \(q ≤ N\). - Ford circle at \(p/q\): radius \(1/(2q^2)\), tangent to neighbors; total circle length sum diverges as \(∑q≤ N 1/q ~ log N\). - Gap distribution for Farey fractions: normalized gaps \(δ_i = (Fᵢ₊₁ - F_i) · N^2\) converge to a limiting distribution with density \(12/(π^2 δ^3)\) for large \(δ\), and mean gap \(→ 1\). - Golden ratio connection: the *maximal* gap in \(F_N\) occurs between consecutive fractions whose mediant is the golden ratio conjugate \(φ⁻¹ = (√5-1)/2 ≈ 0.618\) — specifically, the largest gap is bounded by \(1/(√5 N^2)\), achi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com