Near-conjugacy of Collatz map to circle rotation

Abstract

We introduce a logarithmic transformation T(x) = log₆(x + 1/5) under which the Collatz map becomes a circle rotation by the irrational angle α = log₆3 plus a bounded error term. We prove that |ε(x)| ≤ 0.2749 for all positive integers x, with ε(x) = O(1/x) as x → ∞. Numerical verification up to 10¹² supports bounded cumulative error. This near-conjugacy reveals the geometric structure underlying Collatz dynamics without resolving the conjecture itself.