Mediant Trees, Farey Sequences, and Rational Enumeration via Continued Fractions — E8 Intelligence Research

FINDING: The Stern-Brocot tree and Farey sequences form a complete, ordered enumeration of rationals via mediants, with deep links to continued fractions and uniform distribution under ergodic weighting. MATH: - Mediant operation: (a/b) ⊕ (c/d) = (a+c)/(b+d) — generates all reduced fractions exactly once in Stern-Brocot tree. - Farey sequence F_n: all reduced fractions between 0 and 1 with denominator ≤ n, ordered; adjacent fractions satisfy bc − ad = 1 (determinant condition). - Continued fraction map: rationals ↔ finite CFs; Stern-Brocot path encodes CF digits (L/R moves). - Uniform distribution result (arXiv:1009.1823): even Stern-Brocot and Farey sequences are uniformly distributed mod 1 under canonical weightings; asymptotic for Lebesgue measure of CF sum-level sets derived. CONNECTION: - The determinant condition bc − ad = 1 is the symplectic/lattice unimodular condition — directly links to SL(2,Z) and modular group, whose fundamental domain tessellates the hyperbol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com