Proof of the Necessity of the Odd-Even-Even-Odd Structure in the Collatz Conjecture

This paper proves that for any positive odd integer n, in the Collatz iteration, a parity structure of the form “even → even → odd” (EEO) must necessarily appear. The proof is based on modulo 4 classification and a parameter recursion method. We first address the case n ≡ 1 (mod 4). Then, for numbers n ≡ 3 (mod 4), by introducing parameters A, W, t, … and employing the method of infinite descent, we prove that such numbers will eventually transform into either n ≡ 1 (mod 4) numbers or directly enter a trajectory of multiples of 4. This result provides a structural foundation for the eventual proof of the Collatz Conjecture.