Stern-Brocot Tree: Continued Fractions Unifying Quantum Hall and Anyon Statistics — E8 Intelligence Research

FINDING: The Stern-Brocot tree is the complete binary tree of all positive rationals generated by mediants, and its paths encode continued fraction expansions — the same structure that organizes quantum Hall filling fractions and anyon statistics. MATH: - Mediant operation: given a/b, c/d → (a+c)/(b+d). Starting from 0/1 and 1/0, iterating the mediant generates every positive rational exactly once. - Continued fraction: [a₀; a₁, a₂, …] ↔ path in Stern-Brocot tree via L/R runs of lengths a₁, a₂, … - Convergents pₙ/qₙ satisfy pₙqₙ₋₁ − pₙ₋₁qₙ = (−1)ⁿ⁻¹ (unimodularity / SL(2,ℤ)). - Farey sequence Fₙ: fractions in [0,1] with denominator ≤ n; adjacent Farey fractions a/b, c/d satisfy bc − ad = 1. - Quantum Hall filling fractions observed: ν = 1/3, 2/5, 3/7, 2/3, 5/2, 12/5 … — these are precisely Farey/Stern-Brocot mediants of neighboring plateaus (Haldane-Halperin hierarchy). - Fibonacci anyons: braiding phases e^{iπ/5}, quantum dimension φ = 1.618…; Jones polynomial at roots of unity. CON Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com