Number Theory

    

A Structural Proof of the Collatz Conjecture Via Non-Repeating Trajectory and Recursive Decay

Authors: Younghwan Yun

We present a structural proof of the Collatz conjecture by rigorously analyz-ing the recursive mapping of odd integers. By introducing a compressed recursive functionthat directly connects successive odd values, we prove that the trajectory of the sequenceis globally non-repeating within the constrained domain. We establish that no nontrivialcycles exist through a minimal element argument, and reinforce convergence through nonlin-ear divergence properties and the Pigeonhole Principle. Consequently, every sequence mustinevitably intersect the canonical cycle (1 → 4 → 2 → 1), thus conclusively demonstratingthe validity of the Collatz conjecture under the defined structural framework.

Comments: 11 Pages.

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Submission history

[v1] 2025-05-01 17:36:14
[v2] 2025-05-21 20:10:28
[v3] 2025-06-04 22:02:18

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