An Explicit Near-Conjugacy Between the Collatz Map and a Circle Rotation

We introduce an explicit logarithmic transformation $T(x)=\{{\log }_6(x+1\mathrm{/}5)\}$ under which the Collatz map becomes a rigid circle rotation by the irrational angle $\alpha ={\log }_{6}3$, perturbed by a uniformly bounded error term. We prove that for all positive integers $x$, $T(C(x))=T(x)+\alpha +\epsilon (x)(\mathrm{m}\mathrm{o}\mathrm{d}1)$, where $\mathrm{\mid }\epsilon (x)\mathrm{\mid }\le 0.2749$ and $\epsilon (x)=O(1\mathrm{/}x)$ as $x\to \mathrm{\infty }$. We derive the transformation from an exact functional equation linking the even and odd branches of the Collatz map, explain the arithmetic origin of the parameters $6$ and $1\mathrm{/}5$, and analyse the structure of the resulting error term. Extensive numerical computations up to ${10}^{12}$ confirm the sharpness of the bounds and show that cumulative errors remain uniformly bounded along all tested trajectories. While this near-conjugacy does not by itself resolve the Collatz conjecture, it provides a concrete and quantitative dynamical framework that clarifies the geometric structure underlying the Collatz iteration and may be useful in further analytical or experimental investigations of Collatz-type systems.