A Symbolic Prefix Decomposition of the Collatz Map and Its Empirical Determination of Per-Class Statistics

This paper establishes properties of the prefix decomposition and presents empirical evidence that the decomposition's terminal value a_final determines per-class statistics of the total stopping time σ on odd integers in residue class r mod 2^k. The contribution has two parts — an algebraic component (the decomposition itself) and an empirical component (the per-class statistical regularities it predicts).