The LNL Project (LSR): A Structural Framework for the Collatz Conjecture

Abstract This preprint presents the LNL (Lange Number Line) and LSR (Lange Structural Research) framework, a structural research program for investigating the Collatz Conjecture. Instead of studying individual trajectories, this work introduces a structural representation based on the Ironing Board Model, the Lange Pillar Graph (LSG), First Activation, the empirical Lange Constant, and a block-based DNA representation of Collatz trajectories. The manuscript clearly distinguishes between: mathematical definitions, rigorously proven local lemmas, empirical computational observations, and open research hypotheses. The objective of this work is not to claim a completed proof of the Collatz Conjecture. Instead, it proposes a new mathematical framework intended to simplify the dynamics and identify structural properties that may contribute to a future proof. The manuscript is released as a research preprint to invite mathematical discussion, criticism, and independent verification.Source code and supplementary material:GitHub Repository – LNL / LZR Collatz