The $\text{FP}^\text{NP}$ versus \#P dichotomy for \#EO

The complexity classification of the Holant problem has remained unresolved for the past fifteen years. Counting complex-weighted Eulerian orientation problems, denoted as \#EO, is regarded as one of the most significant challenges to the comprehensive complexity classification of the Holant problem. This article presents an $\text{FP}^\text{NP}$ vs. \#P dichotomy for \#EO, demonstrating that \#EO defined by a signature set is either \#P-hard or polynomial-time computable with a specific NP oracle. This result provides a comprehensive complexity classification for \#EO, and potentially leads to a dichotomy for the Holant problem. Furthermore, we derive three additional dichotomies related to the Holant problem from the dichotomy for \#EO.