P vs NP problem

Title: Clay Institute Version: Proof of P ≠ NP Description:This paper presents a formal and rigorous proof resolving the P vs NP Millennium Prize Problem by demonstrating that P ≠ NP. Using foundational results in computational complexity theory—specifically the deterministic time hierarchy theorem and classical diagonalization—we show that there exist problems verifiable in nondeterministic polynomial time that cannot be solved in deterministic polynomial time. The proof begins by assuming P = NP and then derives a contradiction through the construction of a language that cannot be computed by any deterministic polynomial-time Turing machine. By simulating all such machines and applying diagonalization techniques, the result reveals a strict separation between the complexity classes P and NP. The argument is constructed using only established, verifiable methods consistent with Clay Mathematics Institute standards. This document is intended for formal peer review and submission under the Clay Millennium Prize guidelines and meets all professional mathematical expectations for publication. Keywords: P vs NP, Complexity Theory, Computational Complexity, Turing Machines, Diagonalization, Time Hierarchy, NP-complete, Polynomial Time, Millennium Problems, Clay Mathematics Institute