Positive-characteristic validity of the singularity classification algorithm

Let $\overline{K}$ be an algebraically closed field of characteristic $p$, and let the paper's singularity-classification algorithm be denoted by Algorithm 1. Positive-characteristic hypothesis. The algorithm is valid over $\overline{K}$ for every characteristic satisfying

$$p\neq 2,3,5,7.$$

This extends the algorithm beyond characteristic zero; whether these are the only necessary exclusions is part of the characteristic-dependent validity question considered by the paper.