FINDING: A proven identity linking the golden ratio (φ) to the Möbius function (μ), Euler totient function (φ), and natural logarithm (ln) via Dirichlet convolution, with a separate claim that φ emerges from the E8 root system. MATH: - Key identity from arXiv: 1109. 3216v4: \ ₍=₁^ (n) n (11 - ^{-n}) = \ and its reciprocal variant: \ ₍=₁^ (n) n (11 - ^{n}) = ^-1 \ where \ (= (1+5) /2 1. 618\), \ ( (n) \) is the Möbius function (0 if n has a squared prime factor, else \ ( (-1) ᵏ\) for k distinct primes), and \ ( (n) \) is Euler's totient (count of integers ≤ n coprime to n). The identity uses the Dirichlet convolution inverse property: \ (₃|₍ (d) = ₍, ₁\). - From Aschheim (Quantum Gravity Research): Claim that φ arises from the E8 root system via the projection of 240 root vectors onto a 2D plane, yielding a go Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Wed,) studied this question.