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 aᵢ = 2 per odd step. We show that under this balanced regime, the accumulation term C reduces identically to 4ᵐ - 3ᵐ, uniquely yielding the trivial integer fixpoint $u = 1$. Furthermore, for all higher division powers $n > 2m$, we prove that the strict inequality 0 < C/2ⁿ - 3ᵐ < 1 holds, ruling out the existence of non-trivial integer cycles.