Shelah's eventual categoricity conjecture for abstract elementary classes

Theorem 1.2: for every infinite cardinal $\lambda$ there is $\mu(\lambda)\ge\lambda$ such that every AEC $K$ with $LS(K)\le\lambda$ categorical in some $\kappa\ge\mu(\lambda)$ is categorical in every $\kappa'\ge\mu(\lambda)$; equivalently (Theorem 1.3) an AEC categorical in unboundedly many cardinals is categorical on a tail. It applies to $L_{\omega_1,\omega}$ sentences in countable languages. It gives no explicit value for $\mu(\lambda)$; the prescribed $\beth_{\omega_1}$ threshold for $L_{\omega_1,\omega}$ and the Hanf threshold for AECs are left separate (the companion refutes the latter under CH).

Theorem 1.2: for every infinite cardinal $\lambda$ there is $\mu(\lambda)\ge\lambda$ such that every AEC $K$ with $LS(K)\le\lambda$ categorical in some $\kappa\ge\mu(\lambda)$ is categorical in every $\kappa'\ge\mu(\lambda)$; equivalently (Theorem 1.3) an AEC categorical in unboundedly many cardinals is categorical on a tail. It applies to $L_{\omega_1,\omega}$ sentences in countable languages. It gives no explicit value for $\mu(\lambda)$; the prescribed $\beth_{\omega_1}$ threshold for $L_{\omega_1,\omega}$ and the Hanf threshold for AECs are left separate (the companion refutes the latter under CH).