Generalized Dynamics and Generation Mapping for Collatz-type Sequences

Let an odd integer X be represented as ₌ ₌ 2M + 2ᵐ - 1 for m 1, where 2ᵐ - 1 is called the Governor. The general dynamics is such that applying the Collatz-type function iteratively lead to ₍ ₁ 2N + 2^T - 1, where 2^T - 1 is referred to as the Trivial Governor. For the 3Z + 1 sequence, the Trivial Governor is 2¹ - 1, while for the 5Z + 1 sequence, the Trivial Governors are 2² - 1 and 2¹ - 1. It is demonstrated that for X to reappear in a Collatz-type sequence, its Governor must be the Trivial Governor. Specifically, in the 3Z + 1 sequence, ancestor mapping shows that odd integers in a repeating cycle are separated by two even integers. Successor mapping further indicates that there are no auxiliary cycles, as the Trivial Governor is transformed into a Governor with a different index. Similarly, in the 5Z + 1 sequence, successor mapping reveals that the smallest odd integers that form an auxiliary cycle are found between 2² and 2⁵. Finally, attempts to construct integers that diverge in the 3Z + 1 sequence suggest that no such integers exist.