The JNL Framework: An Upgraded Map of the Syracuse Problem

This paper introduces a unique method for the statistical and structural analysis of the 3x + 1 problem. By transforming odd integers with the function f(x) = (x+1)/4, the framework organizes all positive integer sequences into a single linear number line called the JNL line. The system sorts values into distinct groups to track their movement. Group1 sequences create half-integer decimal drops, while Group 2 sequences result in positive integer increases. Rather than a traditional proof, this data-driven approach reveals a “wavy slide” pattern. Every path follows a universal trajectory toward a unified sink at 0.5, offering a clear model for studying how these number sequences behave.