This study discusses the computational complexity of the partition function of the symmetric dual-spin system on regular graphs. Based on # exponential time hypothesis (#ETH) and random exponential time hypothesis (rETH), this study develops the classical dichotomies of this problem class into the exponential dichotomies, also known as the fine-grained dichotomies. In other words, this study proves that when the given tractable conditions are satisfied, then the problem is solvable in polynomial time; otherwise, there is no sub-exponential time algorithm when #ETH holds. This study also proposes two solutions to solve the in-effectiveness of existing interpolation methods on building sqrt-sub-exponential time reductions under the restriction of planar graphs. It also utilizes these two solutions to discuss the related fine-grained complexity and dichotomy of this problem under the planar graph restriction.