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 Method. 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 `uf75-01.cnf`—a historically hard 3-SAT challenge—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. Included Files: - Main manuscript in LaTeX format (`P_equals_NP_proof.tex`) - Supplementary response to peer criticisms (`Reply_to_Criticisms_on_P_equals_NP_Proof.pdf`) - Python implementation of the SAT solver based on the Amanollahi Method (`amanollahi_proof_sat_solver_uf75.py`) Author: Mohammadreza Sarvamanollahi Affiliation: Independent Researcher Date: July 2025 DOI will be assigned by Zenodo upon submission.