From Micro-Details to Global Closure: A Unified Framework for Collatz Trajectories via Two-Group Modular Partitioning

Abstract

Moving from local micro-structural details to global analytical closure, this paper establishes a comprehensive framework for Collatz (3n+1) trajectory analysis. By bridging foundational two-group modular partitioning (G_1 and G_2 modulo 8 classes) with non-iterative meta-optimization, Building upon this detailed parity decomposition (A_1--A_3) and a universal O(1) shortcut operator, the architecture culminates in an A_4 macro-distribution conservation law, demonstrating that all trajectories are permanently bounded across general scales.