The interval conjecture for uniqueness of limit models in stable AECs

Let $\mathcal{K}$ be an AEC with $\mu$-amalgamation and suppose that it is $\mu$-stable. For a limit ordinal $\alpha$, a $(\mu,\alpha)$-limit model is a model obtained as the union of a limit chain of length $\alpha$ of models of cardinality $\mu$; write $(\mu,\mu)$-limit models analogously. Consider the set

$$\{\alpha<\mu^+: \operatorname{cf}(\alpha)=\alpha\text{ and }(\mu,\alpha)\text{-limit models are isomorphic to }(\mu,\mu)\text{-limit models}\}.$$

Interval conjecture. This set is a non-trivial interval of regular cardinals. Moreover, its minimum is an important measure of the complexity of $\mathcal{K}$.

The conjecture refocuses uniqueness of limit models in a $\mu$-stable AEC from whether all limit lengths yield isomorphic models to which regular limit lengths yield models isomorphic to $(\mu,\mu)$-limit models. It is motivated by analogous first-order results, while the paper develops tower methods for studying the question in classes that are stable but not necessarily superstable.