Structural Proof of the Collatz Conjecture via Factor Convergence and Binary Block Modeling

We present a structural proof of the Collatz Conjecture by introducing a binary classification model based on 4-bit block patterns and a concept called the Collatz factor. Unlike traditional sequence-based approaches, our method analyzes the transformation behavior of binary blocks under the Collatz operation and demonstrates that all natural numbers converge to 1. The proof is supported by an algebraic relationship between Collatz factors and Mersenne numbers, showing that every even-power Mersenne number minus one is divisible by three. Furthermore, we prove that all binary block patterns eventually transform into Collatz factors, and that local convergence is preserved regardless of carry propagation. This framework offers a predictive, visual, and universal model for understanding Collatz dynamics.