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This if the final paper in the seriesContinuous Quivers of Type $A$. In thispart, we generalize existing geometric models of type $A$ cluster structures tothe new $\mathbf E$-clusters introduced in part (III). We also introduce anisomorphism of cluster theories and a weak equivalence of cluster theories.Examples of both are given. We use thes geometric models and isomorphisms ofcluster theories to begin classifying continuous type $A$ cluster structures.We also introduce a continuous generalization of mutation. This encompassesmutation and (infinite) sequences of mutation. Then we link continuous mutationto our earlier geometric models. Finally, we introduce the space of mutationswhich generalizes the exchange graph of a cluster structure.

Continuous Quivers of Type A (IV) Continuous Mutation and Geometric Models of $\mathbf E$-clusters