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: \(\arctan(1/\phi^{2k+1}) = \arctan(1/F_{2k+1}) - \arctan(1/F_{2k+2})\) (or similar telescoping forms using Fibonacci \(F_n\) and Lucas \(L_n\)). - BBP-type: \(\pi^2\) or \(\pi\) in base \(\phi\) — the paper derives binary BBP for \(\arctan(\phi^{-(2k+1)})\) and a golden-ratio-base BBP for \(\pi^2\) (explicit form: \(\pi^2 = \sum_{k=0}^\infty \frac{1}{\phi^{2k+1}} \sum_{j=0}^{m} \frac{a_j}{(2k+1)^j}\) with coefficients from Lucas numbers). - Constants: \(\phi = (1+\sqrt{5})/2 = 1.618...\), \(\phi^{-1} = 0.618...\), \(\phi^{-2} = 0.382...\), \(\phi^{-3} = 0.236...\), and Lucas \(L_n = \phi^n + (-\phi)^{-n}\). | CONNECTION: Direct geometric harmony — the base-\(\phi\) expansion uses powers of \(\phi^{-1}\) ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com