describe a recursive, binary-based framework for the Collatz Con-jecture. I show that every positive integer can be decomposed into aprefix and trailing blocks of ones and zeros. Using a systematic 2n + 1recursion, 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 theFundamental Theorem of Arithmetic (FTA). Finally, I use inductionon layers to show that all numbers eventually reach 1.