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We study integrable spin chains and quantum and classical cellular automata with interaction range ℓ≥3. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest-neighbor interacting chains. This leads to a new integrability condition for medium-range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including U(1)-symmetric three-site interacting models, and
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