This paper elaborates on a new approach of the question of the proof-theoretic study of concurrent interaction called "proofs as schedules". Observing that proof theory is well suited to the description of confluent systems while concurrency has non-determinism as a fundamental feature, we develop a correspondence where proofs provide what is needed to make concurrent systems confluent, namely scheduling. In our core logical system (a variant of multiplicative linear logic), processes and schedulers appear explicitly as proofs in different fragments of the proof language and cut elimination between them does correspond to execution of a concurrent system. This separation of roles suggests new insights for the denotational semantics of processes and new methods for the translation of π-calculi into prefix-less formalisms (like solos), as the operational counterpart of a translation between logical systems.