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 \(\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p\); diagonal embedding \(\mathbb{Z} \hookrightarrow \hat{\mathbb{Z}}\); Collatz accelerated map \(T(n) = (3n+1)/2^{v_2(3n+1)}\); spectral zeta function \(\zeta_T(s) = \sum_{\lambda \in \text{spec}(T)} \lambda^{-s}\) (formal); Euler product structure \(\zeta_{\text{adelic}}(s) = \prod_p (1 - p^{-s})^{-1}\) (classical, referenced); Hecke operators \(T_p\) acting on adelic functions; trace formula \(\text{Tr}(T_p^N) = \sum_{\text{periodic orbits}} \frac{\text{weight}}{\text{stability}}\) (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 \(\lambda_{\max} < 1\)) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com