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Overview

The Collatz conjecture (also known as the 3x + 1 problem) asserts that for every positive integer n≥1,n \ge 1, the iterative process n↦{n2if n is even,3n+1if n is oddn \mapsto \begin{cases} \dfrac{n}{2} & \text{if } n \text{ is even}, \\ 3n + 1 & \text{if } n \text{ is odd} \end{cases} eventually reaches the trivial cycle (1,2)(1,2). Despite decades of study, extensive numerical verification, and many partial results, the conjecture remains open. This work presents a structural mathematical reduction of the problem.
Rather than studying individual trajectories, it analyzes the induced dynamics on residue classes modulo powers of two, allowing entire families of integers to be treated simultaneously.

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Main Result

The paper establishes a complete structural classification of all odd residue classes modulo 2k2^k. Each residue class falls into exactly one of the following regimes:
  • Structurally contractive
  • Reducible to a contractive class
  • Saturated (contraction certified but representation exhausts resolution)
  • Singular
Crucially,

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