Closed-Form Fibonacci via Path-Sums on Fibonacci Trees — E8 Intelligence Research

FINDING: Fibonacci numbers admit a closed-form via combinatorial path-sums on Fibonacci trees, independent of prior terms. | MATH: Let \(F_n\) be the \(n\)-th Fibonacci number. The Fibonacci tree \(T_n\) has root value \(F_n\), left child \(Tₙ₋₁\), right child \(Tₙ₋₂\). The closed form: \(F_n = ∑paths p ∈ T_n ∏edges e ∈ p w(e)\) with edge weights \(w=1\) for left, \(w=1\) for right — trivially counts paths, but the paper (arXiv:1302.6583) derives \(F_n = {φ^n - ψ^n}{√5}\) where \(φ = {1+√5}{2} ≈ 1.618\), \(ψ = {1-√5}{2} ≈ -0.618\). The tree path sum reduces to the Binet form via generating functions: \(∑n≥0 F_n x^n = x/1-x-x^2\). | CONNECTION: The ratio \(φ = 1.618\) and its inverse \(φ⁻¹ = 0.618\) appear directly. The tree's branching (left = \(n-1\), right = \(n-2\)) mirrors the Fibonacci recurrence — a self-similar, binary recursive structure. The closed Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com