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Smoothness constraints have become central to the training of machine-learning interatomic potentials, whether imposed explicitly through regularization or implicitly through Sobolev objectives that couple energies with forces. Although this strategy improves interpolation, stabilizes learned potential energy surfaces, and embeds physically motivated local structure, its broader consequence is more restrictive than current practice typically acknowledges. This article shows that smoothness does not merely regularize learning within the training domain; it also limits the model’s capacity to extrapolate beyond that domain in a mathematically structured way. Working in a Sobolev-space framework, the analysis demonstrates that extrapolation error grows at least exponentially with distance from the training set, with the growth rate controlled by the force-weighting parameter λ. On that basis, the article formalizes a set of fundamental trade-offs linking interpolation accuracy to extrapolation distance, force accuracy to energy extrapolation, strong smoothness priors to representational flexibility, and data efficiency to extrapolation range. The argument further identifies the mechanisms through which these tensions arise, including prior dominance, spectral filtering, gradient-matching interference, and optimization bias toward overly smooth solutions. The result is a unified theoretical account of why Sobolev-trained potentials perform so well near known configurations yet remain fragile in genuinely novel regions of configuration space. These findings reposition smoothness from a default virtue to a task-dependent design choice and establish the need for adaptive regularization, extrapolation-aware evaluation, and hybrid modeling strategies when machine-learning potentials are deployed in materials discovery settings that cannot avoid out-of-distribution prediction.