A Constructive Proof of P = NP via Continuous Energy Minimization: Amanollahi Method

This article presents a novel, constructive, and verifiable proof of the P = NP conjecture based on a deterministic continuous optimization method, termed the Amanollahi Methodology. By formulating Boolean SAT instances as continuous energy landscapes with tunable sharpness, we demonstrate that both satisfiable and unsatisfiable cases can be resolved in provable polynomial time. The energy function guarantees zero minima for SAT and a strictly positive lower bound for UNSAT instances. The method successfully solves benchmark instances such as uf75, uf125, uuf50, and uuf250—historically hard 3-SAT challenges—without relying on external SAT solvers. The implementation is entirely independent and reproducible, and the proof includes rigorous analytical lemmas, complexity bounds, and empirical validation. For the first time in history, the instances uf125 and uuf250 have been solved without any external reinforcement. For interactive demonstrations and detailed results, readers may consult the following Google Colab notebooks: Colab: uf125 Results Colab: uuf250 Results