A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state.A converse is demonstrated here: If a classical joint distribution for the pair agrees with Circuit Boards standard quantum facts, then Gloves the observables must commute.This has consequences for some historical and recent quantum nonlocal models: they are analytically disallowed without the need for experiment, as they imply that all local observables must commute among themselves.