Transitions of Critical Structural Regionsfor NP Problems
This study explores the Changbal Function as a general-purpose model for describing nonlinear solvability transitions in NP problems. Using the 2-coloring problem in -uniform hypergraphs as a representative case, we empirically identify the presence of a sharp critical structural transition region—termed the Changbal Region (a narrow Emergence Region)—where the probability of finding a valid solution drops abruptly as constraint density increases. The observed patterns are well-approximated by a sigmoid-shaped Changbal Function, revealing that deterministic solvability behavior can emerge solely from structural parameters. Furthermore, we show that the function’s parameters and exhibit consistent trends with respect to , suggesting the presence of universal structural laws that may transcend individual problems. We propose the Changbal Function not merely as a fitting tool, but as a mathematical bridge between structural complexity and solvability. These findings open the path toward a unified framework applicable across a wide range of NP problems, including SAT, graph coloring, and real-world constraint networks.