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用于显式分类与构造性推理的计算与范畴框架。方法：开发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。通过自同构分析、规范标记与剪枝策略确保唯一性与可处理性，同时利用范畴构造形式化代数关系。结果：算法实现了阶数$|T|\!\le\!4$的所有系统分类，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，范畴解释将三元$Γ$-半环与泛代数中的函子模型相关联。结论：本研究为有限三元$Γ$-半环建立了经过验证的计算理论与范畴综合框架，整合了代数结构、算法枚举与符号计算，为未来工业与决策模型应用提供支持。