P vs np

Title: A Constructive Resolution to the P vs NP Problem Description:This paper presents a novel and constructive approach to the P vs NP problem—one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute. The work rigorously investigates the computational complexity of decision problems and demonstrates, through logical and algorithmic frameworks, whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. Utilizing both classical proof methods and enhanced symbolic modeling via autonomous AI collaboration, this paper aims to deliver a definitive resolution to the P vs NP question. The implications of this resolution extend to cryptography, optimization, artificial intelligence, and computational theory at large. Keywords: P vs NP, polynomial time, computational complexity, verification, algorithm design, NP-complete, millennium problem, computer science, AI-assisted proof.