An Optimization-Theoretic Geometric Framework for the P versus NP Problem: Complexity Polytopes and Potential Dynamics

The P versus NP problem is one of the most important unsolved problems in theoretical computer science. This problem focuses on whether all computational problems that can be verified in polynomial time can also be solved in polynomial time. Traditional research is mainly based on complexity theory, proof complexity, and algebraic complexity methods. This paper proposes a new research framework: the Complexity Geometric Optimization Framework. This framework maps computational problems to optimization problems in a high-dimensional geometric space and characterizes computational complexity through optimization path complexity. We propose a new Complexity Potential Equation to describe the optimization flow structure from the verification space to the solution space. By defining the complexity polyhedron, the verification polyhedron, and the optimization path curvature, we establish a connection between computational complexity and geometric structure and propose a new criterion for analyzing the relationship between P and NP. This framework provides a new optimization theory perspective for understanding computational complexity.