Binet's Formula as a Golden-Ratio-Weighted Fibonacci Tree Lattice — E8 Intelligence Research

FINDING: Fibonacci closed-form via Binet's formula is structurally equivalent to a binary tree of golden-ratio-weighted paths; Fibonacci trees provide a combinatorial lattice for computing F_n without recursion. | MATH: Binet: F_n = (φ^n − ψ^n)/√5, φ = (1+√5)/2 ≈ 1.618, ψ = (1−√5)/2 ≈ −0.618. Fibonacci tree: root at F_1, children F_{k−1}, F_{k−2}; number of nodes at depth d = F_{d+1}. Closed form via tree: F_n = Σ_{k=0}^{⌊(n−1)/2⌋} C(n−1−k, k) — binomial sum, equivalent to Binet. | CONNECTION: φ and ψ are exactly the golden ratio and its negative reciprocal (−0.618 = 1 − φ = −1/φ). The tree's branching ratio (number of leaves at level n) → φ^n/√5, so the tree's asymptotic shape is a self-similar golden spiral — a discrete logarithmic spiral with growth factor φ. The binomial coefficients in the tree sum form Pascal's triangle along diagonals — a 2D lattice whose projection onto the Fibonacci line yields the golden angle (137.5° ≈ 2π/φ²) in phyllotaxis. | DEPTH: 7 — The closed form is c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com