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The interaction of fluid membranes with a scaffold, which can be a planar surface or a more complex structure, is intrinsic to a number of systems from artificial supported bilayers and vesicles to cellular membranes. In principle, these interactions can be either discrete and protein mediated, or continuous. In the latter case, they emerge from ubiquitous intrinsic surface interaction potentials as well as nature-designed steric contributions of the fluctuating membrane or from the polymers of the glycocalyx. Despite the fact that these nonspecific potentials are omnipresent, their description has been a major challenge from experimental and theoretical points of view. Here, we show that a full understanding of the implications of the continuous interactions can be achieved only by expanding the standard superposition models commonly used to treat these types of systems, beyond the usual harmonic level of description …