The Resolution of the Collatz Conjecture: A Unified Arithmetic and Dynamical Framework

This paper presents a full resolution of the Collatz Conjecture.

The approach combines both a local view of how individual numbers behave under the rules of the problem, and a global view of how all numbers together form a complete structure.

From the local side, the paper shows how every number can be traced backwards in a predictable way using a new arithmetic method. This guarantees that each number comes from only one possible “parent,” creating a sort of family tree that never splits or loops.

From the global side, the paper builds a recursive framework that spreads out to cover all possible numbers through consistent patterns. This means that no number is left out and everything fits.

The core finding is that when both perspectives are put together, the Collatz process becomes fully closed. There’s no way for numbers to break off into infinite loops or escape upward forever. Every sequence must eventually settle at 1.

The paper walks through the logic, the patterns, and the consequences, providing a comprehensive explanation backed by arithmetic proofs and structural analysis.