Iii- The Pure-Halving Case of Collatz

We analyze the subset of integers under the Collatz map: for which every iteration proceeds purely by halving. We show that only powers of two possess this property: starting from , the trajectory: contains no odd branch, and conversely every non-power of two eventually encounters an odd step. Using the 2-adic valuation , we identify a precise geometric distribution for over odd , derive the resulting mean contraction rate , and examine a probabilistic “random-walk” heuristic for expected stopping times and excursion heights. This framework separates provable structure (the halving class and valuation law) from heuristic dynamics (statistical contraction and random-walk drift), illustrating why the Collatz conjecture remains resistant to full resolution despite its strong average-case convergence.