The Collatz map as a (2,3)-adic bridge: a framework for characterizing the canonical transfer between incompatible completions of ℚ

We develop a structural framework in which the Collatz map is interpreted as a canonical transfer between incompatible completions of ℚ, specifically linking the 2-adic and 3-adic regimes. In this perspective, the Collatz dynamics is not treated as a purely combinatorial process, but as a bridge between arithmetic structures that are otherwise incompatible. We construct a multi-layer correspondence between arithmetic residue dynamics, orbit structure, spectral representations, and operator-theoretic behavior. Within this framework, the system decomposes into a resolved sector, where symmetry and averaging mechanisms yield strong statistical control, and an obstruction sector consisting of points whose dynamics fails to stabilize at any finite depth. We prove that the obstruction sector is dynamically invariant under the Collatz return map and is thin, with density O (2^-m) among residue classes modulo 2ᵐ, and therefore zero in the profinite limit. This sector admits consistent realizations across multiple structures, including arithmetic towers, unresolved residue classes, bounded cocycles, and sparse spectral components. The Collatz map thus functions as a canonical interface between distinct arithmetic completions, with the obstruction sector representing the precise locus where compatibility fails. This reframes the Collatz conjecture as a structural problem of extending compatibility across these regimes.