<p>This paper investigates the global polynomial lag synchronization (GPLS) of neural networks (NNs) with proportional delays. By designing adaptive controllers as well as adaptive pinning controllers, building appropriate Lyapunov functionals, and exploiting the properties of proportional delay functions, GPLS criteria are derived. Introducing polynomial functions simultaneously in the controller and Lyapunov functional enables unconstrained determination of GPS. This approach directly handles the unboundedness of proportional delays, unifies global polynomial convergence with lag synchronization, and yields synchronization criteria that are either condition-free or involve only algebraic inequalities, avoiding the complexity of linear matrix inequalities. Moreover, the proposed adaptive controller contains no sign functions, which eliminates chattering near the equilibrium point. Finally, the effectiveness of the theoretical results is verified through a numerical example with three cases, along with simulations and image encryption/decryption.</p>