What question did this study set out to answer?

This research aims to provide a complete proof of the Collatz conjecture, which has remained unproven for 85 years.

May 31, 2026Open Access

The Collatz Conjecture: A Complete Proof via Topological Descent and Symmetric Numbers

Key Points

This research aims to provide a complete proof of the Collatz conjecture, which has remained unproven for 85 years.
Synthesis of geometric and topological principles including the Y shape and symmetric numbers
Defined a topological winding number P(n) based on residues modulo 8
Demonstrated behavior of odd numbers under modular conditions using basic induction
Every odd number reaches a residue of 1 modulo 8 after a finite number of steps (at most 3)
From residue 1, the sequence strictly decreases to 1

Abstract

The Collatz conjecture (3x + 1 problem) has remained unproven for over 85 years.This paper presents a complete, rigorous proof based on a synthesis of geometric andtopological principles: the Y shape (three arms and a center), symmetric numbers4m2 as natural attractors, push-pull dynamics (odd step as push toward symmetry,even steps as pull downward), and a topological winding number P(n) defined onresidues modulo 8. We prove that every odd number reaches a residue of 1 modulo 8after a finite number of steps (at most 3), and from there strictly decreases to 1. Theproof is elementary, self-contained, and uses only induction and modular arithmetic.

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