BBP-Type Arctangent Formulas for Odd Powers of the Golden Ratio — E8 Intelligence Research

FINDING: BBP-type formulas exist for arctangents of odd powers of the golden ratio, expressible in binary and golden-ratio-base digit extraction, with Fibonacci/Lucas identities underpinning them. | MATH: Key identities from the paper (arXiv:1603.06307): - Arctangent identities: \((1/φ²ᵏ⁺¹) = (1/F₂ₖ₊₁) - (1/F₂ₖ₊₂)\) (or similar telescoping forms using Fibonacci \(F_n\) and Lucas \(L_n\)). - BBP-type: \(π^2\) or \(π\) in base \(φ\) — the paper derives binary BBP for \((φ⁻⁽²ᵏ⁺¹⁾)\) and a golden-ratio-base BBP for \(π^2\) (explicit form: \(π^2 = ∑ₖ₌₀^∞ {1}{φ²ᵏ⁺¹} ∑ⱼ₌₀ᵐ a_j/(2k+1)^j\) with coefficients from Lucas numbers). - Constants: \(φ = (1+√5)/2 = 1.618...\), \(φ⁻¹ = 0.618...\), \(φ⁻² = 0.382...\), \(φ⁻³ = 0.236...\), and Lucas \(L_n = φ^n + (-φ)⁻ⁿ\). | CONNECTION: Direct geometric harmony — the base-\(φ\) expansion uses powers of \(φ⁻¹\) ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com