The Hausdorff Dimension of Arithmetically Constrained Collatz Orbits is Zero

The Collatz conjecture’s notorious difficulty stems from the tension between its deter- ministic arithmetic definition and the apparently random behavior of its orbits. Previous statistical approaches, while successful in proving density-one results, fail to exclude zero- measure sets of counterexamples because they operate in a purely symbolic space, ignoring crucial arithmetic constraints. This paper introduces a new framework that reintegrates these constraints. We define an ”arithmetically exceptional” set E within the 2-adic sym- bolic space, consisting of trajectories that are not only multiplicatively expansive but also satisfy a scale-dependent arithmetic condition necessary for true Archimedean divergence. We then prove that the topological entropy of this set is zero. By applying Bowen’s formula for the Hausdorff dimension of subshifts in the 2-adic metric, we conclude that the Haus- dorff dimension of E is zero. This result provides the strongest possible statement of the ”smallness” of the set of potential integer counterexamples within this dynamical framework.