A Bit-Length and Branch-Based Proof of the Collatz Conjecture

Astructured framework for analyzing the Collatz Con-
jecture using explicit branch formulas derived from the 4x and 4x + 2
decomposition, binary decomposition of odd numbers as R 0 1x, and a
bit-length quasi-invariant with a formal G/N partition. Each positive
integer is associated with a branch sequence, and the evolution of its
binary length under even and odd steps is rigorously bounded. By
organizing integers into recursively defined layers and proving finite
convergence within each layer, every sequence is shown to reach the
base layer. This approach formalizes the controlled reduction of num-
bers through the Collatz map and demonstrates, via layer induction
and bit-length analysis, that every positive integer ultimately reaches