Unified Lattice Structure of Stern-Brocot, Farey Sequences, and Modular Group — E8 Intelligence Research

Abstract

FINDING: The Stern-Brocot tree, Farey sequences, and the modular group PSL(2,Z) form a unified bijective lattice structure — with uniform distribution properties under canonical weightings, linking continued fractions to ergodic theory. | MATH: Stern-Brocot tree: mediant operation (a/b ⊕ c/d = (a+c)/(b+d)); Farey sequence F_n = all reduced fractions p/q ∈ [0,1] with q ≤ n, ordered; bijection: Stern-Brocot tree ↔ PSL(2,Z) via matrices [[a,b],[c,d]] acting on rationals; continued fraction expansion: x = [a₀; a₁, a₂, ...] ↔ path in Stern-Brocot tree (L/R moves); key result (arXiv:1009.1823): even Stern-Brocot sequence and Farey sequence are uniformly distributed mod 1 under canonical weightings — asymptotic for Lebesgue measure of continued fraction sum-level sets: μ({x ∈ [0,1] : a₁+...+aₙ = k}) ~ (1/√(2πn)) · e^{-(k-2n)²/(2n)} (Gaussian-type, from infinite ergodic theory). | CONNECTION: The Stern-Brocot tree's mediant operation generates all rationals with a natural binary tree structure Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com