Ternary Gamma Semirings: From Neural Implementation to Categorical Foundations

This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks (0% accuracy). By introducing a logical constraint -- the Ternary Gamma Semiring -- the same architecture learns a perfectly structured feature space, achieving 100% accuracy on novel combinations. We prove that this learned feature space constitutes a finite commutative ternary $\Gamma$-semiring, whose ternary operation implements the majority vote rule. Comparing with the recently established classification of Gokavarapu et al., we show that this structure corresponds precisely to the Boolean-type ternary $\Gamma$-semiring with $|T|=4$, $|\Gamma|=1$}, which is unique up to isomorphism in their enumeration. Our findings reveal three profound conclusions: (i) the success of neural networks can be understood as an approximation of mathematically ``natural'' structures; (ii) learned representations generalize because they internalize algebraic axioms (symmetry, idempotence, majority property); (iii) logical constraints guide networks to converge to these canonical forms. This work provides a rigorous mathematical framework for understanding neural network generalization and inaugurates the new interdisciplinary direction of Computational $\Gamma$-Algebra.