A Binary Decomposition Approach to the Collatz Conjecture

 describe a recursive, binary-based framework for the Collatz Con-
jecture. I show that every positive integer can be decomposed into a
prefix and trailing blocks of ones and zeros. Using a systematic 2n + 1
recursion, I define branch formulas A, B for all odd integers and asso-
ciate each branch with a target number C, forming layers C0, C1, . . . .
I formally prove that every integer is generated uniquely using the
Fundamental Theorem of Arithmetic (FTA). Finally, I use induction
on layers to show that all numbers eventually reach 1.