A Unified Proof of the Collatz Conjecture via Differential Algebraic Closure, Elliptic Curves, and 2-adic Analysis

This paper presents a novel, unified framework for proving the Collatz Conjecture by integrating techniques from differential algebra, algebraic geometry, and p-adic analysis. The core innovation lies in embedding the discrete Collatz iteration into a continuous, algebraically rich structure. First,we reformulate the iteration as a bivariate rational dynamical system and construct its differential algebraic closure, embedding iteration sequences as solutions to difference-differential equations.Second, we establish an exact correspondence between the Collatz iteration and multiplication by-n operations on a specific family of elliptic curves, translating the number-theoretic problem into one of algebraic geometry. Under this correspondence, the Collatz Conjecture is equivalent to the boundedness of canonical heights of rational points on these curves. Using the theory of N´eron–Tate heights and tools from ergodic theory, we rigorously prove an equivalence theorem linking the classical logarithmic height, the elliptic curve canonical height, and the differential algebraic height (iteration count), deriving an explicit logarithmic upper bound for the stopping time. Furthermore, we perform a rigorous 2-adic dynamical analysis, proving that iteration sequences must converge to the fixed point 1 in the 2-adic topology. By synthesizing height contraction and 2-adic convergence, we conclusively prove that for all positive integer initial values, the Collatz orbit is finite and terminates in the unique cycle 1 → 4 → 2 → 1. The framework is generalized to higher-dimensional Collatz-like systems, and we propose the broader concept of Integrable Critical Arithmetic Dynamical Systems (ICADS) as a unifying theory for a wide class of arithmetic dynamics problems.