Preperiodic integers for $x^d+c$ in large degree

Given a number field $K$, we completely classify the preperiodic portraits of the maps $x^d+c$ where $c\in K$ is an algebraic integer and $d$ is sufficiently large depending on the degree of $K$. Specifically, we show that there are exactly thirteen such portraits up to the natural action of roots of unity. In particular, we obtain some of the main results of recent work of the authors unconditionally for algebraic integers by replacing the use of the abc-conjecture with bounds on linear forms in logarithms. We then include applications of this work to several problems in semigroup dynamics, including the construction of irreducible polynomials and the classification of post-critically finite sets.