Isomorphic Rational Trees Link Farey, Stern-Brocot, and Markov Spectra — E8 Intelligence Research

FINDING: The Stern-Brocot tree, Farey sequences, Calkin-Wilf tree, and continued fractions are isomorphic representations of ℚ⁺, with the Lagrange spectrum and Markov triples emerging as the arithmetic shadow of badly-approximable irrationals — the same objects that generate the golden ratio and its metallic-mean family. MATH: Core structures (all bijective on ℚ⁺): - Stern-Brocot: mediant insertion. From 0/1, 1/0, insert (a+c)/(b+d). Every positive rational appears exactly once. - Farey F_n: fractions in [0,1] with denominator ≤ n; consecutive terms a/b, c/d satisfy bc − ad = 1. - Calkin-Wilf: breadth-first enumeration via f(a/b) = a/(a+b), (a+b)/b. - Continued fractions: [a₀; a₁, a₂, …] ↔ Stern-Brocot path (L/R runs of length aᵢ). Unimodular invariance: - det[[a, c],[b, d]] = ad − bc = ±1 for Farey neighbors → SL(2,ℤ) action. - Stern-Brocot = Cayley graph of PSL(2,ℤ) with generators L, R. Lagrange spectrum: - L(α) = liminfq→∞ q‖qα‖. - Lagrange's theorem: α badly approximable ⟺ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com