On the Impossibility of Non-Trivial Collatz Cycles

We formalize the parameterized master loop equation of the Collatz conjecture under the structural relation $n = 2m$ and $k = m$, corresponding to an average division power of $a_i = 2$ per odd step. We show that under this balanced regime, the accumulation term $C$ reduces identically to $4^m - 3^m$, uniquely yielding the trivial integer fixpoint $u = 1$. Furthermore, for all higher division powers $n > 2m$, we prove that the strict inequality $0 < \frac{C}{2^n - 3^m} < 1$ holds, ruling out the existence of non-trivial integer cycles.