Nucleic acid circuits using polymerases can maintain dynamic steady states by consuming nucleoside or deoxynucleoside triphosphates7,8, demonstrating diverse functions, such as multistate memories9 and nonlinear classification10. However, with a few exceptions11, they continuously consume energy even without input changes, leading to energy waste and limited operation times12. Enzyme-free nucleic acid circuits use only rationally designed components, making them more robust to environmental changes, such as temperature and salt conditions13, while performing complex information processing tasks, including Boolean logic14,15 and neural computation16,17.