Topology of Vanishing Sums in the Collatz Problem:\\ 2-adic Reduction, 3-adic Scales, and Dynamic Incompatibility

A deterministic framework is developed for excluding the existence of non-trivialcycles in the Collatz map $3n+1$ on $\mathbb{N}$. The trajectory is interpreted not asa pseudo-random process, but as an evolution through a hierarchy of $3$-adic scalelevels: each odd step necessarily increases the $3$-adic denominator depth, forcing thesystem into a finer representational layer. This mechanism is combined with the local$2$-adic structure, which uniquely determines the sequence of divisions by $2$, and withthe global Diophantine constraint given by the cycle equation. A dynamic top-levelcontribution arising at each odd step is identified. Under normalization to depth $3^K$,the final transition produces a non-removable residue $2^{A_{K-1}} \bmod 3$, whereas allpreceding contributions are multiples of $3$ and vanish. This residue cannot be cancelledby the nodal term $n(2^M - 3^K)$, yielding a structural incompatibility between the$2$-adic determinism of the descent sequence and the topology of the $3$-adic scalehierarchy. As a consequence, no non-trivial cycles with $K>1$ can occur. The analysisplaces the Collatz problem within the framework of deterministic non-Archimedeanarithmetic dynamics, where the $3$-adic scale structure acts as a stable invariant of thetransformation.