General 2^{k}-Division Shortcut Operator and Modular G_{1}/G_{2} Decomposition for Hardware-Level O(1) Throughput Enhancement in Supercomputers for the Collatz Problem

Abstract

In this study, a general algebraic shortcut operator is derived to eliminate iterative steps and intermediate parity checks in Collatz (3n+1) trajectory simulations. By integrating G_{1}/G_{2} modulo 8 classification, the framework achieves 100% coverage over all positive odd integers. Through branchless execution and instruction fusion, the proposed method reduces simulation time complexity to O(1) constant time in High-Performance Computing (HPC) environments.