Hanf number of the first stability cardinal in AECs

We show that ℶ(2LS(K))+ is the lower bound to Hanf numbers for length of order property and stability in stable abstract elementary classes (AECs). Our examples satisfy joint embedding property, no maximal model, (<ℵ0)-tameness but not necessarily amalgamation property. also define variations on syntactic properties insofar as it allows index set be linearly ordered rather than well-ordered. Combining with Shelah's theorem, we deduce our can have up any μ<ℶ(2LS(K))+. Boney conjectured needed two type-counting lemmas. solved conjecture by showing independent ZFC.