Loop Structure of Collatz-Type Functions 3x+n: A Conjugacy Theorem and Powers of Three

We record two elementary structural facts about the loop behavior of the Collatz-type maps x ↦ 3x+n for odd n, working with the odd-only reduced map. First, the net displacements around any cycle sum to zero; this is a pure arithmetic identity, and we note that inverse pairs — odd inputs whose displacements are equal in magnitude and opposite in sign — are one mechanism by which a cycle can realize the required cancellation. Second, and the main point, we prove a conjugacy identity f_n(nx) = n · f_1(x) valid for all odd n and odd x, and show that when n = 3^k a 3-adic confinement forces every orbit eventually onto the sublattice on which this identity is an exact rescaling of 3x+1. Consequently 3x+3^k is dynamically subordinate to 3x+1, which explains why the powers of three are exactly the odd values n > 1 that do not generate new nontrivial cycles. The argument is self-contained, and the conjugacy result is unconditional; statements about the absence of cycles are conditional on the corresponding fact for 3x+1.