A Binary Branch–Layer Structure and Explicit Decreasing Invariant for the Collatz Map

Abstract
I present a binary-based framework for the Collatz Conjecture that
organizes all positive integers into a hierarchical branch–layer struc-
ture. Every integer admits a unique decomposition
n = 2^y(2^x(2R + 1) − 1) ,
which determines its position within a finite branch. I prove that
under the Collatz map T , the local branch parameters (x, y) strictly
decrease until reaching a canonical endpoint C = 2(3R + 1).
Using the recursive 2n+1 construction, I show that all odd integers
are generated uniquely, and I define an explicit integer-valued layer
index b(n) determined directly from this decomposition. I prove that
b(n) is well-defined for all n ≥ 1 and satisfies
b(T (n)) < b(n) for all n > 1.
Since b is nonnegative and strictly decreasing under T , every positive
integer descends through finitely many layers to the base layer C0 =
{4^n}, which trivially maps to 1. This establishes convergence of all
positive integers under the Collatz map.