Golden Ratio as the Dominant Eigenvalue of the Fibonacci Recurrence — E8 Intelligence Research

FINDING: Derivation of the golden ratio φ from the Fibonacci recurrence relation as the dominant eigenvalue of its characteristic equation. | MATH: Fibonacci recurrence: F_n = Fₙ₋₁ + Fₙ₋₂. Characteristic equation: r² = r + 1 → r = (1 ± √5)/2. Dominant root: φ = (1 + √5)/2 ≈ 1.618034. The other root: ψ = (1 - √5)/2 ≈ -0.618034. Binet formula: F_n = (φ^n - ψ^n)/√5. Ratio of successive terms: limn→∞ Fₙ₊₁/F_n = φ. | CONNECTION: φ is the fundamental ratio of geometric harmony — directly yields 1/φ = 0.618, φ² = 2.618, φ⁻² = 0.382. The recurrence structure mirrors self-similar scaling in pentagonal symmetry (5-fold) and quasicrystal diffraction patterns. The bifurcation constant δ (Feigenbaum) is not derived here; this is the linear case, not chaotic period-doubling. | DEPTH: 7 — foundational link between linear recurrence, irrational ratio, and natural growth patterns; essential for understanding scaling in phyllotaxis, spiral geometries, and golden-angle-based lattice structure Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com