Adelic Spectral Approach to the Collatz Conjecture via Hecke Operators — E8 Intelligence Research

FINDING: The Collatz map is reframed as a dynamical system on the adelic ring (product of all p-adic integers), with spectral analysis via Hecke operators and trace formulas, aiming to isolate a spectral gap proving global convergence. | MATH: Adelic ring \(Ẑ = ∏_p Z_p\); diagonal embedding \(Z Ẑ\); Collatz accelerated map \(T(n) = (3n+1)/2v_2(3n+1)\); spectral zeta function \(ζ_T(s) = ∑λ ∈ spec(T) λ⁻ˢ\) (formal); Euler product structure \(ζadelic(s) = ∏_p (1 - p⁻ˢ)⁻¹\) (classical, referenced); Hecke operators \(T_p\) acting on adelic functions; trace formula \(Tr(T_p^N) = ∑periodic orbits weight/stability\) (Selberg-type). | CONNECTION: The adelic product structure naturally encodes all prime moduli simultaneously — a lattice/root-system-like symmetry across primes. The spectral gap sought (if \(λₘₐₓ < 1\)) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com