Sato-Tate Conjecture for Weight 12 Cusp Forms and SU(2) Symmetry — E8 Intelligence Research

FINDING: Sato-Tate distribution for weight 12 cusp forms (e.g., Ramanujan tau function) converges to SU(2) Plancherel measure, linking modular forms to compact Lie group symmetries. MATH: For a weight \(k=12\) cusp form \(f(z)=∑ₙ₌₁^∞ a(n) e2π i n z\), normalized coefficients \(a(p)/p11/2\) for primes \(p\) lie in \([-2,2]\) and their distribution matches the SU(2) Plancherel measure \(1/2π√4-x^2\,dx\) on \([-2,2]\). The Sato-Tate group is SU(2). Fourier coefficients satisfy \(a(p) ≠ 0\) for all primes \(p > p_0\) (proved via Deligne's bound). CONNECTION: The measure \(√4-x^2\) is the semicircle law, with mean 0 and variance 1. The endpoints \(± 2\) correspond to eigenvalues of SU(2) matrices. The golden ratio \(φ=1.618\) appears indirectly: the trace of a matrix in SU(2) is \(2cosθ\), and the measure in \(θ\) is \(2/πsin^2θ\,dθ\), whose moments involve Catalan numbers (related to \(φ\) via continued Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com