To the long established idea of bounding turbulent convective heat transport by a variational method based on energetic constraints, we now add a richer class of ‘z-constraints’ with the hope of tightening bounds considerably. We establish that only certain moments of the governing equations are effective for this purpose. We explore the initial consequences of groups of such constraints by use of perturbation theory, which clarifies the need that a given set of elements be mutually congruent.