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

Abstract

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 \(T_{n-1}\), right child \(T_{n-2}\). The closed form: \(F_n = \sum_{\text{paths }p \in T_n} \prod_{\text{edges }e \in p} w(e)\) with edge weights \(w=1\) for left, \(w=1\) for right — trivially counts paths, but the paper (arXiv:1302.6583) derives \(F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}\) where \(\varphi = \frac{1+\sqrt{5}}{2} \approx 1.618\), \(\psi = \frac{1-\sqrt{5}}{2} \approx -0.618\). The tree path sum reduces to the Binet form via generating functions: \(\sum_{n\ge0} F_n x^n = \frac{x}{1-x-x^2}\). | CONNECTION: The ratio \(\varphi = 1.618\) and its inverse \(\varphi^{-1} = 0.618\) 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