Analysis of the Collatz Conjecture: Excluding Infinite Growth.

Abstract

Part 2 of a two‑paper program toward an analytical proof of the Collatz conjecture. Using the modulo‑3 “One‑Way Gate”, the 3‑adic contraction of T(n)=(3n+1)/2^{v2(3n+1)}, and an exact Diophantine reduction with a telescoping identity, we exclude infinite growth of Collatz trajectories. Every trajectory must enter the target set S={n | 3n+1=2^k} and thus reach 1. The integer fixed‑point condition is D|S(σ) with n=S(σ)/D, where D=2^b−3^a and S(σ)=2^b C(σ). Orders with any p_i≥3 are excluded by inequalities; for mixtures p∈{1,2} a gcd argument forces the trivial class p_i=2, yielding n=1. Modular invariants (Bang–Zsigmondy) and a finite congruence check close remaining borderline cases. This paper complements Part 1 (Smooth Model & Diophantine Bridge), which eliminates nontrivial cycles.