A Pattern-Based Structural Approach Towards a Potential Proof of the Collatz Conjecture

Abstract This paper presents a new structural observation about Collatz trajectories based on a horizontal organisation of values across fixed iteration levels. By placing consecutive starting values in a row and extending each trajectory vertically, the terms appearing at any fixed step form a sequence of repeating “growth patterns.” At iteration (n), exactly (2^n) distinct horizontal patterns appear, each corresponding to one parity sequence of length (n). These patterns repeat across the row at uniform intervals and remain permanently fixed in both order and behaviour. Every pattern grows linearly, and each has a well-defined “descent threshold”: a point at which all starting values belonging to that pattern produce a term smaller than their seed at exactly the same iteration. Empirical examination of rows up to large heights (including notable patterns such as those containing seeds 27, 262171, and highly composite large seeds) shows that each growth pattern eventually yields a descent, and that this behaviour repeats predictably for all seeds in the same pattern class. This horizontal predictability suggests a possible framework for a descent-based proof of the Collatz conjecture: if every growth pattern is shown to possess a finite descent threshold, then every integer must eventually pass through a pattern that forces its trajectory to fall below its initial value. The paper develops the structure of these repeating patterns, formalises their relationship with parity sequences, and outlines how their descent behaviour may contribute to a general proof strategy.