General Dynamics and Generation Mapping for Collatz-Type Sequences

Let an odd integer \(X\) be expressed as \∑M > m bM 2M\ + 2ᵐ - 1, where bM ∈ \0, 1\ and 2ᵐ - 1 is referred to as the Governor. In Collatz-type functions, a high index Governor is eventually reduced to 2¹ - 1. For the 3Z + 1 sequence, the Governor occurring in the Trivial cycle is 2¹ - 1, while for the 5Z + 1 sequence, the Trivial Governors are 2² - 1 and 2¹ - 1. Therefore, in these specific sequences, the Collatz function reduces the Governor 2ᵐ - 1 to the Trivial Governor 2T - 1. Once this Trivial Governor is reached, it can evolve to a higher index Governor through interactions with other terms. This feature allows X to reappear in a Collatz-type sequence, since 2ᵐ - 1 = 2m - 1 + ⋯ + 2T + 1 + 2T + (2T - 1). Thus, if X reappears, at least one odd ancestor of \∑M > m bM 2M\ + 2m - 1 + ⋯ + 2T + 1 + 2T + (2T - 1) must have the Governor 2ᵐ - 1. Ancestor mapping shows that all odd ancestors of X have the Trivial Governor for the respective Collatz sequence. This implies that odd integers that repeat in the 3Z + 1 sequence have the Governor 2¹ - 1, while those forming a repeating cycle in the 5Z + 1 sequence have either 2² - 1 or 2¹ - 1 as the Governor. Successor mapping for the 3Z + 1 sequence further indicates that there are no auxiliary cycles, as the Trivial Governor is always transformed into a different index Governor. Similarly, successor mapping for the 5Z + 1 sequence reveals that the smallest odd integers forming an auxiliary cycle are smaller than 2⁵. Finally, attempts to identify integers that diverge for the 3Z + 1 sequence suggest that no such integers exist.