Purpose: This study extends the structural theory of finite commutative ternary $Γ$-semirings into a computational and categorical framework for explicit classification and constructive reasoning. Methods: Constraint-driven enumeration algorithms are developed to generate all non-isomorphic finite ternary $Γ$-semirings satisfying closure, distributivity, and symmetry. Automorphism analysis, canonical labeling, and pruning strategies ensure uniqueness and tractability, while categorical constructs formalize algebraic relationships. \\ \textit{Results:} The implementation classifies all systems of order $|T|\!\le\!4$ and verifies symmetry-based subvarieties. Complexity analysis confirms polynomial-time performance, and categorical interpretation connects ternary $Γ$-semirings with functorial models in universal algebra. \\ Conclusion: The work establishes a verified computational theory and categorical synthesis for finite ternary $Γ$-semirings, integrating algebraic structure, algorithmic enumeration, and symbolic computation to support future industrial and decision-model applications.

翻译：目的：本研究将有限交换三元$Γ$-半环的结构理论扩展为一种用于显式分类与构造性推理的计算与范畴框架。方法：开发了约束驱动的枚举算法，以生成所有满足封闭性、分配律与对称性的非同构有限三元$Γ$-半环。自同构分析、规范标记及剪枝策略确保了唯一性与可处理性，而范畴构造则形式化了代数关系。\\ \textit{结果：} 实现部分对所有阶数$|T|\!\le\!4$的系统进行了分类，并验证了基于对称性的子变种。复杂度分析证实了多项式时间性能，范畴解释则将三元$Γ$-半环与泛代数中的函子模型联系起来。\\ 结论：本工作为有限三元$Γ$-半环建立了一套经过验证的计算理论与范畴综合，整合了代数结构、算法枚举与符号计算，以支持未来的工业与决策模型应用。