Researchers have explored various non-systematic satisfiability approaches to enhance the interpretability of Discrete Hopfield Neural Networks. A flexible framework for non-systematic satisfiability has been developed to investigate diverse logical structures across dimensions and has improved the lack of neuron variation. However, the logic phase of this approach tends to overlook the distribution and characteristics of literal states, and the ratio of negative literals has not been mentioned with higher-order clauses. In this paper, we propose a new non-systematic logic named Weighted Random $k$ Satisfiability ( $k=1,3$), which implements the ratio of negative literals in higher-order clauses. The proposed logic, integrated into the Discrete Hopfield Neural Network, established a logical structure by incorporating the ratio of negative literals during the logic phase. This enhancement increased the network's storage capacity, improving its ability to handle complex, high-dimensional problems. The advanced logic was evaluated in the learning phase by various metrics. When the values of the ratio were $r=0.2$, 0.4, 0.6, and 0.8, the logic demonstrated the potential for better performances and smaller errors. Furthermore, the performance of the proposed logical structure demonstrated a positive impact on the management of synaptic weights. The results indicated that the optimal global minimum solutions are achieved when the ratio of negative literals was set to $r=0.8$. Compared to the state-of-the-art logical structures, this novel approach has a more significant impact on achieving global minimum solutions, particularly in terms of the ratio of negative literals.