From Block Orthogonality to Decidability in Complex-Weighted Counting CSP

In a landmark JACM paper recognized with the 2021 G{ö}del Prize, Cai and Chen established a complete complexity dichotomy for counting CSPs over arbitrary finite domains with algebraic complex weights. Its polynomial-time side is characterized by three conditions—Block Orthogonality, Type Partition, and preservation by a common Mal’tsev operation—quantified over the countably infinite family $W_{\mathcal{F}}$ generated from arbitrary $\mathrm{\#}\mathrm{C}\mathrm{S}\mathrm{P}(\mathcal{F})$ instances by partial summation. They asked whether these infinitary conditions are decidable from the finite language $\mathcal{F}$ alone—equivalently, whether the polynomial-time side of this complete fixed-language classification is uniformly recognizable. We settle this problem by giving, for every nonempty finite domain $D$ and every finite exactly encoded algebraic-complex language $\mathcal{F}$, a total exact algorithm that decides all three conditions on the full unbounded family $W_{\mathcal{F}}$. Beyond decidability, we prove that Block Orthogonality alone forces both Type Partition and the existence of a single Mal’tsev operation preserving all generated support and row-equivalence relations. Thus the three-condition characterization collapses to Block Orthogonality, and the finite input $(D,\mathcal{F})$ determines which side of the dichotomy applies. The same framework decides the corresponding conditions in the dichotomy theorem for degree-multiple counting CSP proved by Lin.