Golden-Ratio BBP Formulas for Pi and Arctangents — E8 Intelligence Research

FINDING: The arxiv paper (1603.06307v1) is the substantive result — it derives novel BBP-type digit-extraction formulas for arctangents of odd powers of the golden ratio, and a golden-ratio-base BBP formula for π itself, linking Fibonacci/Lucas numbers to base-φ digit extraction. | MATH: Let φ = (1+√5)/2 ≈ 1.6180339887. The paper derives arctangent identities of the form: arctan(φ^(−k)) = Σ_{n=0}^∞ [a_n / (b^n)] where b is a power of 2 (binary BBP) or φ itself (golden-ratio-base BBP). Specifically, for odd k, arctan(φ^(−k)) admits BBP-type series: arctan(φ^(−k)) = (1/φ^k) Σ_{n=0}^∞ [F_{2n+1} / (φ^{2n+1} · 2^{m n})] — with F_n Fibonacci numbers, m depending on k. The golden-ratio-base BBP for π: π = Σ_{n=0}^∞ [1/(φ^n) · (A_n / (8n+1) + B_n / (8n+2) + ...)] where A_n, B_n are integer sequences derived from Lucas numbers L_n = φ^n + (−φ)^(−n). Key constants: φ, 1/φ = φ−1 ≈ 0.6180339887, φ² = φ+1 ≈ 2.6180339887, and the identity φ^n = F_n φ + F_{n−1}. | CONNECTION: The golden ratio φ is th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com