From an odd arity signature to a Holant dichotomy

$\mathsf{\text{Holant}}$ is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued $\mathsf{\text{Holant}}$ on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued $\mathsf{\text{Holant}}$ on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for $\mathsf{\text{\#EO}}$, and consequently is an ${\text{FP}}^{\text{NP}}$ vs. #P dichotomy as well, stating that each problem is either in ${\text{FP}}^{\text{NP}}$ or #P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued $\mathsf{\text{Holant}}$ on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in ${\text{FP}}^{\text{NP}}$. We believe that this result is a powerful method for building reductions in complex-valued $\mathsf{\text{Holant}}$, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.