Combinatorics and Graph Theory

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

Authors: Neel Alkoraishi
Comments: 13 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)

I present a binary structural framework for the Collatz map that organizes all positive integers into a hierarchical branch—layer system. Everyinteger admits a unique decomposition m = 2y (2x(2R + 1) − 1), which determines its position within a finite branch through the trailingbinary blocks of ones and zeros. Using the recursive 2n + 1 construction, I derive explicit linear branch formulas A(n, x) and B(n, x) that uniquely generate all odd integers. Eachbranch has a finite depth n and terminates at a branch endpoint C = 2 · 3nx + 4 ⌊(n−1)/2⌋ Xi=0 9i. Under the Collatz map, the local branch parameters (x, y) decrease deterministically until this endpoint is reached. At each branch endpoint the higher-order parameter satisfiesRu2032 = 3R + 1 2x+y, yielding the decreasing invariant Ru2032 < R (with a single boundary case).This provides an explicit numerical descent between successive branch endpoints.The reverse odd structure further shows that each branch contains a unique maximal node congruent to 3 (mod 6), which has no odd predecessor within the branch. This structure allows branches to be organized into a recursive hierarchy of endpoint sets C0, C1, C2, . . . , from which I define a global layer index L(m).I prove that every positive integer eventually reaches a branch endpoint belonging to a lower-numbered layer, yielding a strictly decreasinginvariant L(T k (m)) < L(m). Since L(m) is nonnegative, every trajectory must descend through finitely many layers to the base layer C0 = {2k }, which maps trivially to 1. This establishes convergence of all positive integers under the Collatz map within the branch—layer framework.
Category: Combinatorics and Graph Theory