Fixing an arbitrary set $\mathcal{F}$ of complex-valued functions over Boolean variables yields a counting problem $\#\mathcal{F}$. Taking only functions from $\mathcal{F}$ to form a tensor network as the problem's input, the counting problem $\#\mathcal{F}$ asks for the value of the tensor network. These dichotomy or quasi-dichotomy theorems form a partial order according to the inclusion relations of the problem subclasses they characterize. As the number of known dichotomy theorems increases, the number of maximal elements in this partially ordered set first grows, and then shrinks when a new dichotomy theorem unifies several previous maximal ones; currently, there are about five or six. More can be artificially defined. However, it might be the timing to directly study the maximum element in the total partial order, namely, the entire class. This paper proposes such a framework, which observes that for the unresolved $\#\mathcal{F}$ problems, the binary functions must be a finite group, formed by 2-by-2 matrices over complex numbers. The framework, divides all unsolved problems according to the group categories, into 9 cases. This paper: introduces this grand framework; discusses the simplification of matrix forms brought by transposition closure property of the group; discusses the barrier reached by the great realnumrizing method, when a quaternion subgroup is involved; advances the order-1 cyclic group case to a position based on a dichotomy theorem conjecture; and resolves the higher-order cyclic group case.

翻译：固定任意一组定义在布尔变量上的复值函数集合 $\mathcal{F}$，可导出一个计数问题 $\#\mathcal{F}$。该问题以仅取自 $\mathcal{F}$ 的函数构成的张量网络作为输入，要求计算该张量网络的值。这些二分或拟二分定理根据其所刻画的问题子类的包含关系，构成一个偏序结构。随着已知二分定理数量的增加，该偏序集中的极大元数量先增长后缩减——当新二分定理统一了若干先前的极大元时，极大元数量减少；目前约有五至六个极大元。此外还可人为定义更多极大元。然而，或许是时候直接研究整个偏序集中的最大元，即全部问题类了。本文提出了这样一个框架，该框架发现：对于尚未解决的 $\#\mathcal{F}$ 问题，其二元函数必须构成一个有限群，该群由复数域上的 2×2 矩阵形成。该框架将所有未解决问题按群类别划分为 9 种情形。本文：介绍这一宏大框架；讨论由群的转置封闭性带来的矩阵形式简化；讨论当涉及四元数子群时，伟大的实归一化方法所遇到的障碍；基于一个二分定理猜想将阶为1的循环群情形推进至明确结论；并解决高阶循环群情形。