A Wave-Propagation Framework for the Collatz Conjecture: 2-adic Ergodicity, Non-Resonance, Cycle Exclusion, and a Structural Roadmap

We develop a harmonic-analytic framework for the Collatz conjecture and use it to map precisely the boundary between what 2-adic methods can and cannot prove. Four principal results are established. (1) For any odd integer drawn uniformly from Ω_j = {1,3,…,2^j-1}, the distribution of v₂(3n+1) is exactly geometric with parameter 1/2, giving E[v₂] = 2. (2) The transfer operator ℒ_μ on the modular graph 𝔊_j has rank 1 for every measure μ and every j, with with the remainder of the spectrum identically zero (the subdominant eigenvalue is λ2=0). (3) The ratio function R(n) = S*(n)/n generates (ℤ/2^j ℤ)ˣ for all j ≥ 3, forcing every non-trivial Dirichlet character of ℤ₂ˣ to be non-invariant under S*. This implies ergodicity of (ℤ₂ˣ, S*, μ_Haar). (4) Combining the Baker-Mignotte bound with the drift analysis, every orbit satisfying liminf (1/k)Σv₂ ≥ log₂3 either terminates at 1 or is periodic; all periodic orbits are excluded by Baker and computational verification (Oliveira e Silva, n < 2⁶⁸). The single remaining case-divergent orbits satisfying lim(1/k)Σv₂ < log₂3-has Haar measure zero and is unsupported by computation. The main new contribution of Version 3 is Section 12, which establishes a precise structural obstruction theorem: no 2-adic argument, however refined, can close the gap between 'Haaralmost-everywhere' and 'all of ℕ⁺'. We identify three independent obstructions-measuretheoretic, topological, and adelic-and prove that their resolution is equivalent to a single combinatorial condition (Lemma 12.1) that is inherently Archimedean in character. This converts the paper from a partial ergodic analysis into a roadmap: the 2-adic background geometry is now fully charted, and the remaining open problem is precisely stated.