Golden Ratio as Extremal Case in Farey Sequence Gap Structure — E8 Intelligence Research

FINDING: Farey sequences and Ford circles encode maximal-spacing rational approximations, with the golden ratio as the extremal case for gap structure. | MATH: Farey sequence \(F_n\): all reduced fractions \(a/b ∈ [0,1]\) with \(b ≤ n\), ordered. Adjacent terms satisfy \(bc - ad = 1\) (determinant condition). Maximal gap between consecutive Farey fractions is \(1/n\) (at endpoints), but the *largest minimal gap* across all intervals occurs near \(φ = (1+√5)/2 ≈ 1.618\), where the continued fraction \([1;1,1,]\) gives the slowest convergence — hence the largest "hole" in the Stern–Brocot tree. Ford circles: tangent circles of radius \(1/(2q^2)\) at \((p/q, 1/(2q^2))\); tangency iff \(|ps - qr| = 1\). The golden ratio's convergents \(Fₖ₊₁/F_k\) (Fibonacci ratios) produce the *maximal* gap in the Farey tessellation of the hyperbolic plane. | CONNECTION: Golden ratio \(φ\) and its inverse \(φ⁻¹ = φ - 1 = 0.618\) appear as the *unique* real number whos Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com