General Dynamics and Generation Mapping for Collatz-type Sequences

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General Dynamics and Generation Mapping for Collatz-type Sequences

Let an odd integer \(\mathcal{X}\) be expressed as $\mathcal{X} = \left\{\sum\limits_{M > m} 2^M\right\} + 2^m - 1,$ where $2^m - 1$ is referred to as the Governor. In Collatz-type functions, a high index Governor is eventually reduced to $2^1 - 1$. For the $3\mathcal{Z} + 1$ sequence, the Governor occurring in the trivial cycle is $2^1 - 1$, while for the $5\mathcal{Z} + 1$ sequence, the Trivial Governors are $2^2 - 1$ and $2^1 - 1$. Therefore,…