Finite-Depth Translation Symmetries on Arithmetic Progressions for the 3n+1 Map

For any positive integer N and any finite depth m we construct explicit arithmetic progressions (N + (3^k *2^t)) on which the (3n+1) map exhibits exact translation symmetry for the first m iterations. This symmetry provides an algebraic characterization of the well-known periodicity of parity sequences and clarifies the relationship between modular constraints and dynamical regularity in the Collatz problem. We show that infinite-depth translation symmetry occurs precisely when N converges to the 1-2 cycle. All results are unconditional, constructive, and provide a clean dynamical reformulation of classical modular properties.