A Modular Structure for the Collatz Dynamics: Upper Bound and Uniqueness of the 4–2–1 Cycle

Abstract

This work introduces a modular mechanism for controlling Collatz orbits through two operators: the projection function S(n) and the elevator E(n)=n·64^k. The function S(n) is defined through a complete classification of residues modulo 18 and maps every natural number directly into the stable class 16 (mod 36). This projection is fully compatible with the closed modular structure of the even Collatz dynamics, where all even values eventually fall inside the system {4, 16, 22, 34} (mod 36). Within this structure, the classes 22 and 34 act as corridors leading to 16, while the classes 10 and 28 disappear after the first cycle. Although S(n) always lands in the contractive class 16, its value may lie below the true peak of the orbit. To resolve this, the elevator operator E(n)=n·64^k is introduced. Multiplication by powers of 2 preserves the tail of the Collatz orbit, and for every n there exists a k ≤ 2 such that E(S(n)) becomes its own stable peak while remaining in the class 16 (mod 36). Consequently, for any natural number n, the composite E(S(n)):– belongs to the stable class 16 (mod 36),– is its own peak,– and shares exactly the same tail as n. This mechanism provides an explicit upper bound for every Collatz orbit and complements the modular structure developed for the even dynamics.